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Rep Prog Phys. Author manuscript; available in PMC 2015 March 27.
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Rep Prog Phys. 2013 September ; 76(9): 096601. doi:10.1088/0034-4885/76/9/096601.
The physics of functional magnetic resonance imaging (fMRI)
Richard B Buxton
Department of Radiology, University of California, San Diego, USA
Abstract
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Functional magnetic resonance imaging (fMRI) is a methodology for detecting dynamic patterns
of activity in the working human brain. Although the initial discoveries that led to fMRI are only
about 20 years old, this new field has revolutionized the study of brain function. The ability to
detect changes in brain activity has a biophysical basis in the magnetic properties of
deoxyhemoglobin, and a physiological basis in the way blood flow increases more than oxygen
metabolism when local neural activity increases. These effects translate to a subtle increase in the
local magnetic resonance signal, the blood oxygenation level dependent (BOLD) effect, when
neural activity increases. With current techniques, this pattern of activation can be measured with
resolution approaching 1 mm3 spatially and 1 s temporally. This review focuses on the physical
basis of the BOLD effect, the imaging methods used to measure it, the possible origins of the
physiological effects that produce a mismatch of blood flow and oxygen metabolism during neural
activation, and the mathematical models that have been developed to understand the measured
signals. An overarching theme is the growing field of quantitative fMRI, in which other MRI
methods are combined with BOLD methods and analyzed within a theoretical modeling
framework to derive quantitative estimates of oxygen metabolism and other physiological
variables. That goal is the current challenge for fMRI: to move fMRI from a mapping tool to a
quantitative probe of brain physiology.
1. Introduction
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Functional magnetic resonance imaging (fMRI) makes possible an experimental window to
observe the working human brain. Figure 1 shows the measured responses in the motor area
of the human brain based on MR signals sensitive to blood flow and blood oxygenation
when subjects tap their fingers for 2 s. Even a brief stimulus elicits a strong blood flow
change that translates to a weak blood oxygenation level dependent (BOLD) signal change.
Unlike x-ray and nuclear medicine methods for measuring brain function, these fMRI
measurements are completely noninvasive, requiring no injections of contrast agents or
radioactive isotopes. Remarkably, this ability to probe functional changes within the intact
brain is just based on physical principles of nuclear magnetic resonance (NMR) and the
intrinsic effects of blood oxygenation on the MR signal due to the magnetic properties of
deoxyhemoglobin.
© 2013 IOP Publishing Ltd
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The idea that changes in blood oxygenation could drive measurable signal changes in brain
MR images was introduced by Ogawa and colleagues in 1990 and called BOLD contrast [1].
In a rat model they showed that venous vessels, and importantly the tissue near the vessels,
had a low signal with an MRI technique that is sensitive to the local magnetic field
heterogeneity [1]. When the animal breathed a gas mixture containing 10% CO2, there was
much less signal loss near the vessels. The physiological effect in this experiment is that
breathing CO2 dramatically increases brain blood flow, and at high levels reduces oxygen
metabolism. The key effect was that the oxygen extraction fraction (OEF) in the brain—the
fraction of O2 carried by an element of blood that is removed in passing through the
capillary bed—was reduced by breathing CO2. The venous blood was thus more
oxygenated, and the total amount of deoxyhemoglobin was reduced. In this initial
demonstration the change in blood oxygenation was produced by an external agent
(breathing CO2). Ogawa et al [2] and Kwong et al, working independently [3], showed that
intrinsic changes in blood oxygenation happen in normal physiology associated with
changes in neural activity, marking the birth of fMRI as a tool for investigating patterns of
activity in the brain. Interesting perspectives on the early development can be found in a
recent issue of Neuroimage commemorating 20 years of fMRI [4–10].
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The BOLD effect related to neural activity arises because of two distinct phenomena. The
first is that when hemoglobin, the molecule in blood that carries oxygen, loses that oxygen
to become deoxyhemoglobin, the magnetic properties change in a subtle way:
deoxyhemoglobin is paramagnetic, and alters the magnetic susceptibility of blood [11, 12].
The difference in susceptibility between blood vessels and the surrounding tissue creates
local magnetic field distortions that decrease the net MR signal. In the brain a typical OEF is
~40%, and in a 3 T magnetic field this level of deoxyhemoglobin in the veins and capillaries
is sufficient to reduce the MR signal in the brain by ~10% in the baseline state compared
with what it would be if no deoxyhemoglobin was present.
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In short, the MR signal is sensitive to the OEF. This phenomenon alone, while interesting
from a biophysical point of view, would not necessarily produce a useful basis for an
experimental technique, because it is not obvious that the OEF would change with
physiological activity. For example, matched fractional increases in cerebral blood flow
(CBF) and cerebral metabolic rate of oxygen (CMRO2) would leave the OEF unchanged.
This biophysical effect becomes very useful, though, when combined with an unexpected
physiological phenomenon: when an area of brain is activated, the blood flow increases
much more than the oxygen metabolic rate [13]. This leads to a reduction in the OEF, a
seemingly paradoxical scenario in which the venous blood is more oxygenated—despite the
increase in oxygen metabolic rate—because the blood flow has increased more. Taken
together, these two phenomena produce the BOLD effect, a local increase in the MR signal
due to a reduction in the OEF during increased neural activity. Functional MRI based on the
detection of BOLD signal changes has become the leading tool for imaging the working
human brain. Nevertheless, a quantitative physiological interpretation of exactly what is
being measured with BOLD-fMRI is complicated by the complexity of the signal, as
discussed below.
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To put fMRI in perspective in the broader context of other techniques for measuring brain
function, it is useful to think of functional neuroimaging as having two main branches,
depending on whether electromagnetic signals directly related to neural signaling are
measured, or whether associated physiological changes (blood flow or metabolism) are
measured. Although a primary goal of functional neuroimaging is to measure the electrical
activity underlying neuronal signaling, localizing that activity with high spatial resolution is
difficult without placing electrodes directly in the brain. Fluctuating electric potentials at the
scalp and magnetic fields measured near the head provide information on electric currents
within the brain, and from these data the location of sources of activity can be estimated
with electroencephalography (EEG) or magnetoencephalography (MEG) methods. The
alternate approach is to measure the changes in blood flow and metabolic activity that
accompany neural activity changes, and this approach makes a more precise localization of
the activity possible. Positron emission tomography (PET) methods were a critical advance
in spatial localization, using radioactive tracers to measure blood flow, glucose metabolism
and oxygen metabolism. These studies amply demonstrated the localized nature of blood
flow and metabolism changes, suggesting a relatively tight linkage between neural activity,
energy metabolism and blood flow. Functional MRI has better spatial and temporal
resolution than PET methods, avoids the risks of ionizing radiation, and has largely replaced
PET for functional neuroimaging research studies. Nevertheless, PET methods still play an
important role for receptor studies and clinical applications.
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A recurring theme in this review is that the BOLD signal is a complicated function of the
underlying physiological changes, depending on the balance of changes in blood flow and
oxygen metabolism. That is, the physiological changes measured with PET are one step
away from the neuronal activity itself, but at least they clearly measure the defined
physiological variables (e.g. CBF). The BOLD effect is not even a clean reflection of those
physiological variables because it is primarily driven by the change in local
deoxyhemoglobin concentration, which depends on the combined changes of CBF, CMRO2
and the cerebral blood volume (CBV). The problem is that increased neural activity tends to
increase each of these physiological variables, but these changes have conflicting effects on
the BOLD response. Increased CBF tends to wash out the deoxyhemoglobin, while
increased CMRO2 increases local production of deoxyhemoglobin. Increased venous CBV
increases the total deoxyhemoglobin content, partially offsetting the effects of the OEF
change. In contrast, though, increased arterial CBV may increase the measured signal
through a volume exchange effect as increased CBV pushes out extravascular fluid. Arterial
blood typically generates a larger MR signal than tissue, and so this volume exchange leads
to a positive signal change, added to the oxygenation-dependent change associated with
deoxyhemoglobin changes. In short, the BOLD signal is a complex phenomenon, and much
of this review is devoted to understanding that complexity and the difficulties involved in
developing a quantitative interpretation of the BOLD signal. Nevertheless, the importance of
a quantitative understanding is that it offers the potential for interpreting fMRI signals in
terms of quantitative physiological variables, rather than simply a qualitative index of
changing neural activity.
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Functional MRI based on the BOLD effect is the most widely used method, but it is not the
only MRI methodology that has been developed to be sensitive to local brain activation.
Table 1 lists the primary physiological variables that change with neural activation and are
thus potential targets for localizing and measuring activation. Arterial spin labeling (ASL)
methods measure CBF [14], and figure 1(a) is an example of the CBF response to a brief
stimulus. A number of approaches sensitive to CBV have been developed based on injection
of an agent that remains in the blood vessels and alters the local MR signal in proportion to
how much of the agent is present [15, 16]. More recently, a technique called vascular space
occupancy (VASO) has been developed that is sensitive to changes in CBV and does not
require injection of a contrast agent [17]. Magnetic resonance spectroscopy (MRS) methods
have been developed to measure aspects of energy metabolism [18], although these methods
have low spatial and temporal resolution and have not yet been widely adopted for human
studies. Despite the clear importance of measuring CMRO2 for understanding energy
metabolism in the brain, it is a remarkably difficult physiological variable to measure [19].
Steady-state measurements are possible with PET, but require multiple tracers to account for
confounding effects of CBF and CBV [20]. The basic sensitivity of the MR signal to
deoxyhemoglobin clearly suggests the possibility that MR methods have the potential to
provide measurements of CMRO2, and current efforts in this direction are reviewed below.
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This review focuses on the physical and physiological basis of the BOLD effect, with a view
toward both understanding the basic mechanisms and guiding the interpretation of the
BOLD signal in experimental applications. It is important to note that the underlying
physiology is still poorly understood, and part of this review will necessarily be speculative.
An overarching theme is that the complexity of the BOLD effect makes it difficult to
interpret the magnitude of the BOLD response in a quantitative way. That is, a detected
BOLD signal indicates that something is happening in that location, but if the BOLD signal
magnitude differs between two groups (such as a healthy population and a disease
population) it is difficult to interpret this experimental result in a quantitative way related to
the underlying physiology. Nevertheless, a combination of techniques, particularly CBF
measurements with ASL in conjunction with BOLD measurements, provides a much richer
context for a quantitative interpretation of fMRI signals (e.g. see [21]).
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The organization of this review is as follows. Section 2 gives an overview of how fMRI and
related MR methods are currently being used to investigate brain function. Section 3 reviews
the physiological basis of the BOLD effect, including current ideas about the links between
the physiological variables accessible with MRI methods and the underlying neural activity.
Section 4 reviews the physical basis of the BOLD effect, in particular the physics of MR
signal decay in complex biological tissue. Section 5 considers the physiological information
that can be derived potentially with MR methods, including the dynamics of different
physiological variables. Section 6 is a summary and brief overview of the prospects for
quantitative fMRI.
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2. The fMRI experiment
2.1. The NMR signal
NMR is a highly developed field with many sophisticated methods for manipulating the
magnetization associated with nuclear spins to yield informative signals. The basic method
for generating the signal used in fMRI, though, is perhaps the simplest imaginable NMR
signal. The central physical principles underlying NMR are the following:
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1.
Equilibrium magnetization. When placed in a magnetic field B0, the magnetic
moments of nuclei with nonzero spin tend to weakly align with B0, creating a net
macroscopic magnetization M0. Functional MRI manipulates the magnetization due
to hydrogen nuclei (protons), and the hydrogen nuclei in the brain are
overwhelmingly in water molecules (a proton concentration of about 78M,
compared with mM concentrations of most other metabolites).
2.
Precession. If the magnetization M0 is tipped away from alignment with B0, it will
precess around the B0 axis with angular frequency ω0 = γB0, where γ
(gyromagnetic ratio) is a constant for any given nucleus. For protons γ = 2.675 ×
108 rad T−1, and a typical magnetic field for fMRI is 3 Tesla (T), so the precession
frequency ν0 (=ω0/2π) is approximately 128 MHz. After tipping the magnetization
away from B0, the net magnetization vector can be described as two components:
the remaining longitudinal magnetization along the B0 axis, and the rotating
transverse magnetization perpendicular to B0. The rotating component generates an
oscillating magnetic field that induces a current in a nearby coil, creating the basic
measured NMR signal.
3.
Relaxation. Over time, the transverse magnetization decays exponentially to zero
with a time constant T2, and the longitudinal magnetization recovers exponentially
toward its equilibrium value M0 with a time constant T1. In gray matter in the
human brain at a field strength of 3 T, T1 ~1.0 s and T2 ~ 0.1 s.
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The NMR experiment begins with the nuclear magnetization aligned with B0. The central
experimental manipulation of the magnetization is the application of a radio frequency (RF)
transverse magnetic field at the resonant frequency ω0, which has the effect of tipping the
magnetization away from the B0 axis. The degree of tipping depends on the magnitude and
duration of the RF pulse and is described by the flip angle (e.g. a 90° flip angle tips the
magnetization completely from the longitudinal axis into the transverse plane). The newly
created transverse component then precesses at frequency ω0. This precessing magnetization
creates a time-varying magnetic flux that induces a voltage oscillation in a nearby detector
coil. The detected signal is an oscillation at frequency ω0 that decays over time due to
transverse relaxation, called a free induction decay (FID) (figure 2(a)). In practice, the FID
decays faster than would be expected for the T2 of the sample due to magnetic field
inhomogeneities, and the decay constant is described as
(with
). One component
of those additional field offsets is the field distortions around blood vessels containing
deoxyhemoglobin, so the primary NMR effect of blood oxygenation is that it affects
(although there are important subtleties in this idea discussed below).
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The basic distinction between T2 and
is that the added field offsets that contribute to
signal decay are potentially reversible (with important exceptions discussed in section 4). In
general, signal decay is a dephasing process due to the fact that the resolved signal (e.g. the
net signal from a resolved volume element, or voxel, in a macroscopic imaging experiment)
adds signals with different phases. These different phases result from precession in slightly
different fields, and the key difference is whether these field offsets are varying randomly in
time or are fixed in time. The source of T2 decay is randomly varying fields, and for a
hydrogen nucleus in a water molecule there is a significant contribution from the magnetic
field of the other hydrogen nucleus in the molecule as the molecule tumbles. These random
field fluctuations, different for each molecule, lead to a random walk of the phase and a
resulting partial cancellation of the net signal. The effect of these random fluctuations is
irreversible. In contrast, if two nuclei are located in different but constant fields, they will
steadily get out of phase with each other, but this dephasing is reversible. Applying a 180°
RF inversion pulse at a time TE/2 flips the magnetization in the transverse plane like a
pancake, effectively reversing the sign of the accumulated phases. Continued phase
evolution of each spin then unwinds the net phase accumulation acquired before the 180° RF
pulse, so that at time TE (the echo time) the two signals are back in phase creating a spin
echo (SE) (figure 2(b)). After TE the relative phases continue to evolve but another 180°
pulse can bring them back together to form another echo. This SE process reverses the
effects of static field offsets, but not the random field fluctuations, so with each SE the peak
signal is reduced by T2 decay.
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In the field of MRI the usual terminology is that images in which the acquired signal is a
spin echo are called SE images, while images in which the acquired signal is an FID are
called gradient echo (GE or GRE) images. In both cases the delay after the initial excitation
RF pulse is called TE, and TE controls the amount of signal decay that is allowed to happen
before the signal is measured. The term ‘gradient echo’ is somewhat unfortunate here,
because it tends to confuse two concepts. A GE occurs when an external linear gradient of
the magnetic field is turned on as a short pulse, initially creating phase dispersion and signal
loss, and a second gradient pulse with opposite sign is then applied to reverse that signal
loss, creating an echo. Gradient echoes are essentially a technical trick used in imaging pulse
sequences (including SE imaging), and do not have anything to do with intrinsic signal
decay mechanisms. Spin echoes, though, are directly related to these intrinsic decay
mechanisms. So even though a more logical terminology would be SE and FID imaging, we
must work with SE and GE imaging, and for both methods the time delay between
excitation and the center of data collection is called TE.
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2.2. Image acquisition
While the NMR pulse sequence used in fMRI is among the simplest, the imaging methods
are among the most sophisticated. A key technological innovation that made fMRI possible
was the development of single-shot imaging methods that allowed collection of all the data
needed for an image within a few tens of milliseconds after a single RF excitation. Roughly
speaking, an MR image is a snapshot of the transverse magnetization at a particular time TE
after the RF pulse that initially created the transverse magnetization. The essential physical
relationship that makes imaging possible is the enormous range (seven orders of magnitude)
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between the lifetime of the transverse magnetization (
) and the period of the
precession frequency (2π/ω0 ~ 0.008 µs). Because of this, quite small differences in
frequency (less than one part per million) are readily detectable. The approach for imaging is
to measure the NMR signal in the presence of rapidly adjustable linear magnetic field
gradients that create small well-controlled field variations. The basic idea is that when a
linear magnetic field gradient G in the x-direction is applied to a sample, the local resonant
frequency becomes directly proportional to the position of the spins along the x-axis (ω(x) =
ω0 + γGx). At time t, the local phase of the precessing transverse magnetization at x (the
local signal) varies linearly with x. The net measured signal S(t) is the phase-dependent sum
of the signals from all positions. The key is that this process, implemented physically here,
is identical to the mathematical process of calculating the Fourier transform (FT) of a
function: a function f (x) is multiplied by a cosine or sine function in kx and integrated to
find the FT F(k). The spatial frequency k is inversely proportional to the spatial wavelength.
In the imaging process, the result of applying the gradient field is that the signal arising from
each location is multiplied by a cosine or sine in phase ϕ = γGxt, so identifying k(t) = γGt
gives a direct relationship with the FT. In short, if I (x) is the image of the spatial
distribution of the MR signal, in the presence of a linear gradient the net measured signal
over time S(t) is directly proportional to the FT of I (x), and image reconstruction is simply
taking the inverse FT of S(t).
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This remarkable relationship between the net signal over time and the spatial FT of the
distribution of transverse magnetization extends to two and three dimensions, in which field
gradients in multiple directions are varied over time (figure 3). For example, a twodimensional (2D) image I (x, y) has a corresponding 2D k-space in the Fourier domain, and
the net signal traces out a trajectory in k-space as the gradients are modulated. Specifically,
by measuring the net signal over time while manipulating linear field gradients in x and y,
Gx (t) and Gy (t), a range of k-space is mapped, and this range directly determines the
resolution of the image. Higher resolution requires mapping k-space out to higher values of k
(shorter spatial wavelength). Specifically, the spatial resolution of the image is the distance
between two points in space for which the phase difference of their signals is 180° for the
highest value of k measured.
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These ideas illustrate the fundamental flexibility of MRI. Imaging can be based on any
pattern of gradients that causes the net signal to follow a trajectory in k-space that traces out
a sufficiently large block of k-space for the desired image spatial resolution. The most
commonly used single-shot technique is echo planar imaging (EPI), in which the trajectory
in k-space is a back and forth raster pattern (figures 3(a) and (b)). Specifically, the trajectory
in k-space moves along a line in the +kx direction at fixed ky (Gx = constant,Gy = 0), then a
brief pulse of Gy shifts the trajectory to a new ky level and a constant Gx with reversed sign
moves the trajectory in the −kx direction at the new value of ky. A typical acquisition uses 64
such scanning lines in k-space. Note that the amplitude of each point in k-space is the
amplitude of a particular sinusoidal pattern across the image, i.e. one Fourier component
(figures 3(c) and (d)).
This basic imaging scheme also highlights the technical hurdles involved. Time is limited
for scanning through k-space, because for a single-shot image the entire sampling must be
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done under the overarching window of the exponential decay of the MR signal (i.e. an
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interval of duration about ). Thinking of k(t) = γGt for a constant gradient, the ‘speed’ at
which the trajectory moves through k-space (i.e. Δk in a given time interval) is directly
proportional to the magnitude of the gradient G, so stronger gradients mean that k-space can
be sampled faster. But the back and forth EPI trajectory also means that this gradient must
be rapidly reversed, and the time required to do this is determined by the slew rate of the
gradient hardware (how quickly the sign of the gradient can be reversed). These two
hardware characteristics, gradient strength and slew rate, determine the performance of the
gradient system. Interestingly, current limits on MRI systems are not set by hardware
limitations, but by the need to limit slew rates because the associated dB/dt can surpass
nerve stimulation thresholds. An alternative to the back and forth trajectory through k-space
of an EPI acquisition is a spiral acquisition trajectory created by oscillating both x and y
gradients. A spiral trajectory provides an efficient balance of the limiting demands of
gradient strength and slew rate in covering k-space [22].
The development of EPI was crucial for fMRI. Although it originated in the pioneering MRI
work by Peter Mansfield [23], it did not begin to be available on human imaging systems
until the early 1990s, just in time to be exploited for imaging the BOLD effect. The key
advantage of EPI is that it is fast and also has a high signal-to-noise ratio (SNR). The latter
effect may not be obvious, but stems from the low spatial resolution of EPI, which is limited
by the range of k-space that can be covered before the signal decays away. The SNR is
proportional to the volume of a resolved tissue element (voxel), and for EPI the SNR can be
on the order of 100 : 1. This makes this imaging technique well-suited to measure the
dynamic, weak signal changes associated with the BOLD effect.
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2.3. Limits of spatial and temporal resolution for human brain imaging
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The human cerebral cortex is a convoluted sheet with an area of ~2000 cm2, a thickness of
~2.5mm and ~105 neurons mm−2 (i.e. in a column through the cortex with an area of 1 mm2)
[24]. A whole brain image acquisition requires coverage of a three-dimensional (3D)
rectangular volume of about 200 × 180 × 180 mm3. For typical whole brain fMRI studies
the spatial resolution is usually about 3 × 3 × 3 mm3 with a temporal resolution of about 3 s.
(These numbers can be improved if one abandons whole-brain coverage and focuses on a
smaller volume of brain, but for many studies whole brain coverage is the goal.) The
standard acquisition is a series of 2D images, in which the effect of the RF excitation pulse
is limited to one slice, and after collecting the k-space data for that slice an RF excitation
pulse is applied to the next slice, etc. After 3 s, the first slice is acquired again. An
alternative approach is a full 3D acquisition in which gradients in all three spatial directions
are rapidly modulated to create a signal trajectory through a 3D k-space. Currently, most
fMRI studies use the 2D multi-slice approach.
However, the methodology for image acquisition continues to develop in sophistication,
leading to demonstrations of dramatic improvements in spatial and temporal resolution in
recent years. The central challenge is that these two goals conflict with each other: better
spatial resolution means that more of k-space must be sampled, and that takes more time.
The central technological innovation that has driven improvement in spatial and temporal
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resolution is the introduction of arrays of multiple small receive coils (multi-channel coils)
instead of one large RF coil, often described as parallel imaging because the NMR signals
are simultaneously measured with multiple coils and receivers. The key improvement here is
that each of the smaller coils has a limited volume of sensitivity for detecting the NMR
signal, so each has a unique spatial sensitivity pattern. This added spatial information from
the coil locations means that it is not necessary to cover all of k-space to reconstruct an
image with high spatial resolution. By limiting the points in k-space that must be sampled
for each image, the imaging time can be reduced several fold without sacrificing spatial
resolution.
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In addition, with a multi-band approach, several widely separated 2D slices can be acquired
at the same time, and the signals separated using the spatial information from the different
coil sensitivity patterns. By combining the parallel imaging and multi-band approaches,
Moeller et al demonstrated whole brain fMRI acquisitions with spatial resolution of 1 × 1 ×
2 mm3 and temporal resolution of 1.5 s [25]. Feinberg et al introduced a multiplexed method
that adds a third approach to these two in which a few adjacent slices are excited with slight
delays, so that the data from all three slices are measured in a longer data readout window
[26]. With this multiplexed approach they demonstrated acquisition of fMRI data sets with
2mm isotropic resolution in less than 1 s. This method is currently being used in a largescale project to map the patterns of connectivity in the human brain (the Human
Connectome Project) [27].
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The basic goal of increased spatial resolution with decreased acquisition time has a cost in
reduced SNR. For this reason, ultrafast imaging improves at higher magnetic field strengths,
where the intrinsic strength of the NMR signal increases because the equilibrium
magnetization increases in proportion to B0. The current workhorse for human fMRI is the 3
T system, but there are currently > 30 7 T systems around the world [28], and a few 9.4 T
systems. At the higher fields, several studies have pushed the limits of spatial resolution to
resolve functional activity on scales around or below 1mm in humans. Recently, Olman et al
demonstrated changes in the distribution of the BOLD response across the different layers of
the visual cortex with 0.7mm isotropic resolution at 7 T [29]. In addition to layer-specific
activity, the visual cortex is organized in columns on the order of 0.5mm across, with similar
neuronal responses within a column (see [30] for a cautionary discussion about the
heterogeneity of the cortex and the different uses of the term ‘column’ in neuroscience). A
number of studies have used high-field fMRI to probe functional activity at the columnar
level [31, 32].
2.4. fMRI experimental designs
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Just as the technology for fMRI studies has continued to improve, the design of experiments
has evolved in sophistication as well. The classic fMRI experiment is a simple block design,
such as an alternation between 30 s of performing a task and 30 s of baseline, with four
blocks in an experimental run that lasts 4 min. For example, a simple experiment often
repeated in the early days of fMRI was alternation between finger tapping and rest (as in
figure 1). During this time dynamic images are collected, typically with 2D-EPI images
moving sequentially through different slices to cover the brain (figure 4). Current scanners
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can support about 20 EPI acquisitions per second, so that the brain can be covered with 3
mm thick 2D images (a full brain volume acquisition) in about 3 s, so that dynamic images
on any single slice are about 3 s apart. (The newer pulse sequence designs described in the
last section have not yet become standard on 3 T systems used for fMRI.)
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The basic analysis then involves correlating the measured time series for each voxel with a
reference model function defined by the stimulus pattern (e.g. the 30 s on/off blocks) [33].
Voxels that have a significant correlation are identified as activated by the chosen task. The
model function is essentially a guess as to what the fMRI signal from a brain region
responding to the task would look like. Because the hemodynamic response function (HRF)
is slow, taking several seconds to rise to a new level with a block stimulus, the model
function is usually taken as a delayed and rounded version of the stimulus pattern. The
method is referred to as a general linear model (GLM) approach [34, 35], based on the idea
that a voxel’s time course is described as a linear combination of a scaled version of the
model function plus random noise. In addition, other sources of systematic signal variation,
such as scanner drift and physiological noise due to breathing and heart pulsations, can be
included as additional nuisance regressors in the model. This bare-bone description hides
much of the sophistication of the statistical techniques that have been developed for
analyzing fMRI data, and the field of statistical analysis has grown enormously since the
early days of simple block design experiments [36].
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An important innovation was the introduction of event-related designs (for a review, see
[37–40). Rather than grouping stimuli in a block, they are presented randomly, making it
possible to ask questions that cannot be easily answered with block designs. For example, in
a memory task it would be useful to present items that had been previously studied and
novel items to measure differences in the patterns of activation. This is difficult with a block
design because grouping the stimuli prevents analyzing the responses individually. With an
event-related design one can use different model functions for novel and studied stimuli
based on the known pattern of application of these stimuli. In addition, though, by
measuring responses from the subjects on whether they remember each stimulus, one could
also separate the previously studied stimuli into those that were remembered and those that
were not, and develop appropriate model functions for each, and thus identify brain areas
related to successful recall. In addition, with an event-related design it is also possible to
estimate the HRF from the data, provided the data include samples in a range of delays after
each stimulus type was applied [41].
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While block designs and event-related designs are widely used for determining response
patterns to specific stimuli, in recent years a great deal of research has focused on using
fMRI to explore functional connections between brain regions in the absence of any applied
stimuli [42, 43]. For example, an early discovery was that even when the subject was not
performing a motor task, the BOLD signal fluctuations in the motor areas of the left and
right hemispheres were correlated [44]. This approach has expanded enormously, and a
number of recurring resting state networks (RSNs) have been identified as brain regions
with coherent BOLD fluctuations [45, 46]. Newer methods are being developed to look at
the temporal evolution of these connectivity patterns [47–49].
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Brain decoding is another approach looking at patterns of activity across the brain rather
than correlating each voxel with a reference model function. The essential goal is to use a
training set of data showing how an individual subject’s brain responds to particular stimuli
or conditions, and then use multivariate analysis methods to classify the overall pattern of
activity in terms of those stimuli or conditions [50, 51]. These methods have shown a
remarkable ability to identify what a person is seeing or remembering, just from patterns of
fMRI activation, leading to popular accounts describing these methods as ‘mind reading’.
Overall, these studies nicely demonstrate that there is a great deal of information about the
ongoing neural activity in the brain reflected in the patterns of transient blood flow changes
measured with fMRI.
2.5. Diffusion tensor imaging for mapping anatomical connections
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In addition to BOLD-fMRI activation experiments, a number of studies are integrating this
functional information with anatomical connectivity information based on diffusion tensor
imaging (DTI) [52]. Classically, NMR has been one of the most sensitive ways to measure
the diffusion of water [53]. In the brain, water molecules randomly migrate over time due to
thermal motions with a diffusion constant D ~ 1 µm2 ms−1 (10−5 cm2 s−1). In one
dimension, molecules starting at the same location will spread into a Gaussian distribution
of displacements with variance σ2 = 2DT at time T, so that in the brain molecules randomly
move on the order of 12 µm in 100 ms. Although this displacement is much smaller than an
imaging voxel, the effects of diffusion can be encoded in the NMR signal in a detectable
way that makes it possible to measure D.
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In its simplest form, the method involves adding a bipolar gradient pulse to the FID pulse
sequence (figure 5(a)). After excitation, imagine the local magnetization vectors at different
locations in x precessing uniformly. A strong linear x-gradient pulse with amplitude G and
short duration δt is then applied to create a position-dependent phase variation ϕ(x) = γGxδt.
After a delay T the same gradient pulse, except with opposite sign, is applied. If the water
molecules have not moved during the delay T, the phase offsets due to the two lobes of the
bipolar gradient pulse will be precisely opposite in sign. The local precessing magnetization
vectors are then all back in phase, so that the bipolar gradient pulse would have no effect on
the net signal. However, if the water molecules move randomly in x due to diffusion during
the interval T, the phase offsets due to the gradient pulses will not cancel. The net phase
difference after the bipolar gradient pulse is proportional to Δx, the distance moved during T.
For diffusion, Δx has a mean of zero and a variance proportional to D. The net signal is
calculated by multiplying this Gaussian distribution of phases by cosϕ and integrating. The
result is that the net signal is attenuated by the bipolar gradient by a factor exp(−bD), where
b is a lumped factor depending on the amplitude and timing of the bipolar gradient pulse
(figures 5(a) and (b)). Increasing b by increasing the strength or duration of the gradient
pulses improves sensitivity for measuring diffusion effects, and this is often a motivation for
the development of stronger gradient hardware on MRI systems. Figure 5(c) illustrates the
effect of diffusion on the acquired image, along with a map of the estimated value of D,
often referred to as the apparent diffusion coefficient (ADC).
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This basic experiment measures attenuation due to diffusion along a particular spatial axis,
the direction of the bipolar gradient pulse. It can be repeated many times to map out
displacements due to diffusion along many directions. In the Gaussian diffusion model
particles that start at the same location are displaced in an ellipsoidal pattern over time, with
each of the three principal axes of the ellipsoid having a different value of D. The MR
measurement of diffusion, though, is sensitive just to displacements along a single axis, and
in general this axis will not correspond to one of the principal axes. That is, the 3D
ellipsoidal pattern of displacements must be projected onto the single axis defined by the
diffusion gradient direction, and it is the displacements along that axis that determine the
attenuation of the MR signal. The measured value of D along an arbitrary axis is determined
by the diffusion tensor, a 3 × 3 symmetric matrix with six independent components [54, 55].
Physically, there are six independent numbers because a description of the diffusion
ellipsoid requires two angles to specify the orientation of the first principal axis, an
additional angle to specify the orientation of the second principal axis relative to the first,
and three values of D for the three principal axes. By measuring diffusion along at least six
axes the diffusion tensor can be calculated, and from the tensor the principal axes of
diffusion are derived. When the diffusion tensor is expressed in a coordinate system defined
by the principal axes, the diagonal elements are the values of D for the principal axes and
the off-diagonal elements are zero.
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The first principal component is the axis along which diffusion is highest. For gray matter in
the brain, diffusion is reasonably isotropic. Importantly, though, for white matter diffusion is
highly anisotropic. White matter contains nerve bundles of axons, the long range wiring of
the brain connecting neurons in different brain regions. In these white matter fiber tracts
diffusion of water is much higher along the fiber than perpendicular to the fiber, so
identifying the first principal component of the diffusion tensor identifies the local
orientation of the fibers. By connecting these local orientation vectors the fiber tracts can be
traced out, providing measures of anatomical connectivity between one brain region and
another [56, 57]. For a complex structure with multiple crossing fibers, the classic diffusion
tensor cannot provide a complete description of local diffusion. Expanding the number of
directions along which diffusion is measured provides a more complete characterization of
local diffusion [58–60]. The field of fiber tract mapping with DTI has grown enormously
over the last decade, providing highly detailed information on brain anatomical connections,
although more work is still needed to validate these maps [61].
2.6. ASL for measuring CBF
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While the BOLD signal provides a reasonably sensitive measure for detecting changes in
brain activity, it is difficult to interpret quantitatively for two reasons. First, the BOLD
signal is only sensitive to a change between two states during the experiment (e.g. a baseline
state and a task state). For this reason, chronic changes in the baseline state, as might be the
case in disease or in an aging population, cannot be detected unless the baseline state change
alters the acute response in the fMRI experiment [62]. Second, the BOLD effect is a
complex reflection of the underlying physiological change. Because it primarily depends on
the change in deoxyhemoglobin, the blood flow and oxygen metabolism changes with
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activation drive the BOLD signal in opposite directions, so that the BOLD signal magnitude
strongly depends on the exact balance of the two physiological changes.
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An alternative methodology, designed to overcome these limitations by specifically
measuring CBF, is called arterial spin labeling (ASL) [14, 63, 64]. CBF is defined as the
rate of delivery of arterial blood to a tissue element. For the simplest ASL experiment the
goal is to create two images in which the magnetization of the delivered arterial blood has
been manipulated, but the static signal from the rest of the tissue remains the same (figure
6). Subtraction of these images then yields an image just of the delivered arterial blood, and
so is proportional to local CBF. For example, applying a 180° inversion pulse to a band
below the slice of interest will invert the magnetization of blood in the arteries, and after a
delay TI (inversion time) some of the labeled blood is delivered to the imaging slice where it
adds to the net signal with a negative sign due to the inversion. In the second experiment, the
arterial magnetization is not inverted, and after the same delay to allow for delivery to the
imaging slice it adds to the net signal with a positive sign. If this is done carefully so that the
magnetization of the rest of the tissue is precisely the same for the two images, subtraction
will leave just a signal proportional to the delivered arterial blood during the delay TI.
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In practice, ASL techniques have evolved in sophistication to deal with a number of
experimental issues, including relaxation of the labeled magnetization, transit delays from
the tagging region to the imaging slice, and accurate balancing of the labeling and control
pulses to deal with inversion efficiency and off-resonance excitation effects, among others
[65–70]. With current ASL methods it is possible to accurately measure CBF, although the
sensitivity and spatial and temporal resolution are typically worse than what is currently
available with BOLD imaging (compare the noise levels in the two measured responses in
figure 1). For this reason, ASL has not replaced BOLD imaging for activation studies, but
provides additional quantitative physiological information. By providing a quantitative
measurement of a specific, well-defined physiological variable it is more readily
interpretable than the BOLD response, and it also provides a measurement of CBF in the
baseline state, overcoming some of the difficulties of interpreting BOLD responses in
disease. As discussed below, the ability to measure both CBF and BOLD responses opens
possibilities for a much more quantitative assessment of brain function, including estimation
of oxygen metabolism changes.
3. The physiological basis of fMRI
3.1. Hemoglobin and blood oxygenation
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The primary energy source for brain neural activity is the oxidative metabolism of glucose
to carbon dioxide (CO2) and water, with one CO2 molecule produced for each O2 molecule
metabolized. Local brain metabolism requires constant delivery of oxygen and constant
clearance of CO2 by blood flow. The essential problem in transporting O2 through the body
is that it has a low solubility in water. Carbon dioxide, in contrast, readily dissolves by
chemically combining with water to form bicarbonate ions. If O2 and CO2 as gases are
maintained at the same partial pressure above a surface of water, the concentration of
dissolved CO2 in the water is about 30 times higher than that of O2. The clearance of CO2
by blood flow is then relatively simple due to this high CO2-carrying capacity, and the
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delivery of O2 is the difficult task. Evolution has solved this problem with carrier molecules
that readily bind oxygen in the lungs and then release it in the capillary. In mammals, this
molecule is the hemoglobin (Hb) contained in the red blood cells, and it increases the O2carrying capacity of blood by about a factor of 30–50. The O2–Hb binding curve—the
fractional saturation of Hb as a function of the plasma partial pressure of O2—has a
sigmoidal shape, as illustrated in figure 7. As long as pO2 is above about 80 mmHg, the
arterial Hb is nearly fully loaded with oxygen. The point at which the hemoglobin is halfsaturated is called the p50 of the hemoglobin, about 27 Torr for human blood at body
temperature. A number of factors, such as pH or temperature, change the p50 and shift the
dissociation curve to the left or right [71].
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Figure 7 compares typical atmospheric pO2 values for sea level and the top of Mt Everest
with approximate dissolved gas pO2 levels in different vascular compartments and brain
tissue to illustrate the gradients involved in O2 transport. Although the dissolved gas
component is relatively unimportant in terms of carriage of oxygen to the capillary bed, it is
the key component for the actual transfer of O2 from blood to tissue. When the blood
reaches the capillaries (and to some degree the arterioles as well), the dissolved O2 in
plasma diffuses out of the vessel into the tissue. Because the O2 bound to hemoglobin and
the dissolved O2 are in rapid equilibrium, O2 is released from the hemoglobin and partly
replenishes the dissolved gas in the plasma. As more oxygen leaves the blood, the plasma
pO2 largely follows the O2 saturation curve. However, the CO2 increasing in the blood also
leads to a rightward shift of this curve, which complicates modeling of the gas exchange.
3.2. Blood flow and oxygen metabolism changes with brain activation
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CBF is a somewhat subtle concept, in that it is not directly related to the motion of blood
within the capillary network of an element of tissue. Instead, CBF is defined in terms of
delivery of arterial blood to the tissue element, specifically the volume of arterial blood
delivered to the tissue element in a unit time divided by the mass (or volume) of the tissue
element. Classic units of CBF are ml/100 g per min, and a typical value for the human brain
is 50 ml/100 g per min. For imaging experiments a more natural measure of the volume
element is its volume, rather than its weight, and because the density of the brain is close to
1 g ml−1, the CBF value in these units is then about the same. Note that CBF then essentially
has units of inverse time (ml blood per ml tissue per min), and we can think of a basic value
of CBF as 0.5 min−1. In 2 min the volume of arterial blood delivered to a tissue element is
about equal to the volume of the tissue element.
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The CMRO2 of a tissue element is defined as the moles of O2 consumed within the element
per minute divided by the volume of the tissue element, and a typical value in the human
brain is 1.6 µmoles ml−1 min−1. Note that this can be written as a concentration divided by
time, 1.6mM min−1. OEF is the fraction of the O2 molecules delivered to the capillary bed
that are extracted and metabolized. The basic relationship between CBF (F), CMRO2 (JO2)
and OEF (E) is essentially just mass balance:
(1)
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where [O2]a is the concentration of O2 in arterial blood, with a typical value of about 8mM.
The product of CBF and [O2]a is simply the rate of delivery of O2 to the capillary bed, and
the fraction metabolized is then CMRO2. In a healthy subject at sea level the hemoglobin of
arterial blood is nearly saturated (~98%), and most of the arterial O2 is bound to hemoglobin
(only a few percent of the total is present as dissolved gas in the plasma). For this reason,
[O2]a primarily reflects the concentration of hemoglobin in arterial blood (with four O2
molecules bound to each hemoglobin). This means that the hematocrit, the volume of blood
occupied by red cells containing hemoglobin, affects [O2]a.
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The physiological phenomenon at the heart of the BOLD effect is that the fractional increase
in CBF with activation is about twice as large as the fractional increase in CMRO2. From
equation (1), this leads to a decrease in the OEF E, and a resulting increase in the MR
signal. Why this seeming imbalance of CBF and CMRO2 changes occurs is unknown, but a
possible explanation for the function served by decreasing E with activation is that this
preserves the oxygen concentration in the tissue (figure 7(a)) [19, 72]. Returning to the view
of O2 diffusing from plasma to tissue, the diffusion gradient from blood to tissue is
proportional to the difference of the concentrations of dissolved O2 in the blood and tissue
spaces. For CMRO2 to increase, the pO2 gradient between blood and tissue must increase,
which could happen in two ways: opening previously closed capillaries (capillary
recruitment), so that the diffusion distance is reduced, or, increasing the pO2 difference
between capillaries and tissue. Current thinking is that capillary recruitment is a small effect,
if it happens at all [73], so to increase the capillary/tissue pO2 difference, either the tissue
pO2 must drop or the capillary pO2 must rise. The key relationship for the latter possibility
is that for capillary pO2 to rise, the OEF must fall. A simple O2 transport model suggests
that CBF must increase 2–3 times more than CMRO2 in order to preserve the tissue pO2,
consistent with the experimental observations [19] (figure 7(b)). Recent experimental
evidence suggests that it may not be the mean tissue pO2 that is important to preserve, but
rather the pO2 in regions with the lowest value [72]. In short, the evolutionary benefit of the
mismatch between CBF and CMRO2 changes with activation—the physiological effect that
leads to the BOLD effect on the MR signal—may be a homeostatic mechanism to preserve
tissue pO2.
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A possible reason why preserving tissue pO2 is an important physiological function is that
there is very little reserve of O2 in the tissue to act as a buffer [19]. Because O2 is so poorly
soluble in water (solubility 0.0013mM/mmHg at body temperature), a typical tissue pO2 of
25 mmHg corresponds to a concentration of only ~0.03mM. For a typical CMRO2 of
1.6mM min−1, if delivery of O2 stopped the O2 dissolved in tissue would be depleted in ~1
s. The blood within a tissue element provides somewhat more of a buffer. For a blood
volume fraction of 4% and a rough average hemoglobin saturation of 70%, the total O2
concentration in a volume of tissue approaches 0.3mM, and the depletion time rises to ~10 s.
In short, the brain is critically dependent on a continuing supply of O2 delivered by blood
flow.
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3.3. Energy costs of neural activity
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The change in the OEF that drives the BOLD response depends on the balance of changes in
CBF and CMRO2, so it is important to consider two questions. (1) What are the energy costs
of neural activity that would drive CMRO2? (2) What are the mechanisms that drive the
CBF change when neural activity changes? In line with the speculation above that the
biologically useful function of the large CBF change is to maintain tissue pO2, one could
imagine a simple feedback system in which CBF is controlled by an oxygen sensor in tissue,
so that increased energy metabolism associated with neural activity then drives the CBF
change. Interestingly, though, a substantial body of evidence suggests that this is not the
case. An appropriate oxygen sensor has not been found, and instead a number of
mechanisms have been identified by which aspects of neural activity itself drive CBF. The
current picture is that the acute changes in CBF are driven in a feedforward way by the
neural activity, rather than the energy metabolism change [74]. This means that we must
think of CBF and CMRO2 as being driven in parallel by neural activity. While there is an
overall pattern for the CBF change to be about twice as large as the CMRO2 change, this
ratio is not necessarily fixed. In this section and the next the questions of the energy costs
and the CBF drivers are considered in turn.
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The brain represents only about 2% of body weight, yet accounts for about 20% of the
body’s energy metabolism, and most of that energy cost is related to neuronal signaling. All
cellular processes, including neural signaling, are constrained by the physical laws of
thermodynamics: for any transformation of the system energy is conserved, and entropy
increases or at best stays the same. The concept of free energy change (or Gibbs free energy
change) combines the first and second laws of thermodynamics into a single useful
relationship (see [75] for an excellent discussion of free energy in biological systems). For
any transformation, such as a chemical reaction or movement of an ion across a membrane,
there is an associated free energy change ΔG. The free energy is a measure of how far a
system is from equilibrium, with a negative value of ΔG meaning that the transformation
moves the system closer to equilibrium. Cellular work refers to processes that have a
positive ΔG, moving a part of the system away from equilibrium, such as transport of an ion
against its electrochemical gradient. A process with a positive ΔG can only occur if it is
tightly coupled to another process with a more strongly negative ΔG, so that the net ΔG for
the combined transformation is negative. That is, in order to move one system farther from
equilibrium, that transformation must be coupled to another system that is moving closer to
equilibrium.
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For understanding the thermodynamic basis of neuronal signaling, the two key systems that
are far from equilibrium are the adenosine triphosphate (ATP) system (a chemical reaction
out of equilibrium) and the sodium ion (Na+) gradient across the cell membrane (a diffusion
gradient out of equilibrium). These two systems are the primary sources of negative ΔG to
drive thermodynamically uphill reactions. The breakdown of ATP to adenosine diphosphate
(ADP) and inorganic phosphate (Pi) is at equilibrium with a very low ratio of [ATP]/[ADP],
yet in the body the ATP/ADP ratio is much higher, leading to a strong negative ΔG for the
transformation ATP → ADP + Pi [76]. The Na+ distribution across the cell membrane is
also far from equilibrium, with a high extracellular concentration and a low intracellular
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concentration. In addition, the intracellular space has a negative electric potential relative to
the extracellular space, making the Na+ gradient even farther from equilibrium. For this
reason, a sodium ion moving from the extracellular to intracellular space also is associated
with a negative ΔG, although not as strong as the negative ΔG for ATP. The potassium (K+)
distribution also is out of equilibrium, but with a higher concentration inside the cell than
outside the cell it is closer to equilibrium than the Na+ distribution.
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Pumping a Na+ ion out of the cell against its gradient—a strongly uphill thermodynamic
process—can be done by coupling the transport to the more strongly downhill process of
conversion of ATP to ADP. The enzyme that catalyzes this combined process is the sodium/
potassium pump, which couples the uphill processes of moving sodium and potassium
against their gradients to the downhill conversion of ATP to ADP. In this way we can think
of the ATP/ADP system and the Na+ distribution across the cell membrane as two batteries
able to drive cellular work, with the ATP/ADP system able to recharge the Na+ gradient
through the sodium/potassium pump. This analogy with batteries of successively higher
voltage extends to cellular signaling mechanisms as well [77]. Calcium ions (Ca2+) also are
far from equilibrium, with a higher concentration outside than inside the cell. The
intracellular Ca2+ concentration often serves as the primary signal to initiate cellular
activity. For example, the arrival of an action potential at a synapse with another neuron
opens Ca2+ channels on the pre-synaptic side, so that Ca2+ flows into the cell (a downhill
process). The rising intracellular Ca2+ concentration then triggers the release of
neurotransmitter into the synaptic cleft. One way of restoring the initial Ca2+ gradient is by
coupling movement of Ca2+ against its gradient to movement of Na+ down its gradient [78],
recharging the Ca2+ battery.
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In neural signaling, the arrival of an excitatory action potential triggers a downhill cascade
of events, including opening Ca2+ channels so that calcium flows into the presynaptic
terminal, release of neurotransmitter into the synaptic cleft, binding of neurotransmitter to a
receptor on the postsynaptic neuron, and opening of Na+ channels with a rush of Na+ ions
down their gradient. In this way, the Na+ gradient acts like an amplifier, with the opening of
Na+ channels as a switch that connects a strong battery to a circuit. Neural signaling itself is
a downhill process because the system is maintained far from equilibrium, and the energy
cost is in recovery, primarily pumping back the ions against their gradients (figure 8).
Clearing neurotransmitter and resetting the pre-synaptic terminal for the next signal is a
relatively small part of the energy cost compared with pumping Na+, the amplifier signal,
against its gradient. For this reason, the sodium/potassium pump consumes most of the ATP
needed for recovery from excitatory neural signaling [79]. Attwell and co-workers
developed a detailed energy budget for the primate brain based on the associated ATP costs,
and concluded that about 74% of the energy cost was related to synaptic activity, and only
about 20% to the generation of action potentials [74, 80]. The overall energy consumption is
closely related to the spiking rate [81], but it is the integrative activity associated with a
neuron receiving many synaptic inputs that is costly [82].
The energy cost of inhibitory neuronal signaling may be considerably lower. At an
inhibitory cortical synapse the primary action is the opening of chloride (Cl−) or potassium
(K+) channels. The chloride concentration ratio across the membrane is similar to that of
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Na+, but because of its negative charge it is approximately in equilibrium with the resting
membrane potential. The potassium distribution is opposite to that of Na+ (high intracellular
concentration, low extracellular concentration), and would be in equilibrium with a more
negative membrane potential. The membrane potential itself is a balance between the
equilibrium potentials of the different ions weighted with the number of channels open for
each ion (membrane permeability), and if the potential becomes sufficiently depolarized it
fires an action potential. An excitatory signal opens Na+ channels, depolarizing the
membrane potential and moving the cell toward firing. Opening chloride channels tends to
stabilize the resting membrane potential, while opening K+ channels would hyperpolarize
the neuron (i.e. move it farther from firing). Because the ions related to inhibitory signals are
closer to equilibrium than Na+, the associated currents and energy costs of pumping the ions
back should be substantially less than for excitatory activity.
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Neural activity is thus maintained by the consumption of ATP, and the ATP/ADP system is
restored by the oxidative metabolism of glucose [83]. Glucose and O2 are delivered by blood
flow, and in the cytosol the first broad step of metabolism is glycolysis, the conversion of a
glucose molecule to two molecules of pyruvate. This step is coupled to the net conversion of
two ADP molecules to two molecules of ATP. The two pyruvate molecules then diffuse into
the mitochondria, where they are broken down with a net consumption of 6 O2 molecules,
with production of 6 CO2 molecules and ~32 ATP molecules. (Intermediate stores of free
energy in this complex process include both chemical forms, such as the NADH/NAD+
ratio, and an H+ gradient across the inner membrane of the mitochondria.) The full picture of
the dynamics of glucose metabolism in the brain is still somewhat unclear. With neural
activation, glucose metabolism increases more than O2 metabolism [84], suggesting that not
all of the pyruvate produced by glycolysis is going into the mitochondria for oxidative
metabolism. A leading theory is that astrocytes, non-neuronal glial cells that play a key role
in recycling neurotransmitter at a synapse, preferentially use glycolysis to generate ATP
[85]. The excess pyruvate is then converted to lactate and released for uptake by the
neurons, where it is converted back to pyruvate and metabolized in the mitochondria (lactate
shuttle hypothesis). Despite the excess glycolysis, the majority of the energy costs are met
by oxidative metabolism [86].
In summary, the primary energy cost related to neuronal signaling is estimated to be due to
excitatory synaptic activity, and oxidative metabolism of glucose provides most of the
energy. We expect that the change in CMRO2 in a brain region reflects the overall energy
cost of the neural activity.
3.4. Neurovascular coupling
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As noted above, current thinking is that the acute change in CBF associated with neural
activity is driven by aspects of neural activity itself rather than by the energy metabolism
change. There are also feedback mechanisms, undoubtedly, that serve to adjust CBF to the
demands of energy metabolism, but these likely operate on a longer time scale. Numerous
experiments have revealed particular pathways involved in the control of CBF [87–91], but
how these pathways function in a coordinated dynamic network is still largely unknown.
The following section briefly describes some of the current ideas.
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Blood flow increases when the smooth muscle surrounding the arteries and arterioles
relaxes, allowing the vessel to dilate and thus reducing the resistance. Tension created by the
smooth muscle cells is primarily related to the cytosolic Ca2+ concentration, which depends
on exchange between the cytosol and Ca2+ stores within the cell, and on the influx of Ca2+
from the extracellular space through voltage-sensitive Ca2+ channels, which tend to open as
the cellular membrane depolarizes [92]. For this reason, cytosolic Ca2+ tends to follow the
membrane potential, with a graded depolarization producing a graded increase in Ca2+ and a
corresponding contraction of the smooth muscle. Because of this sensitivity to the
membrane potential, a number of agents are thought to exert an effect on the arterial
diameter by opening K+ channels on the smooth muscle cell. Opening K+ channels
hyperpolarizes the cell, reducing cytosolic Ca2+ and relaxing the muscle. Potassium
channels are remarkably diverse, making it possible for a number of vasoactive agents to
modulate CBF by opening or closing potassium channels. Other mechanisms interfere with
the way cytosolic Ca2+ couples to the enzymes that control muscle contraction. For
example, nitric oxide (NO) initiates a chain of events leading to production of the
intracellular messenger cyclic guanine monophosphate (cGMP), and the cGMP is thought to
affect both K+ channels and the sensitivity of the contractile mechanism to Ca2+.
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A wide range of vasoactive agents have been identified, and a partial list suggests the overall
importance of controlling blood flow and the many mechanisms that have evolved to control
it. Cations including H+, K+ and Ca2+ in the fluid space around the arteries dilate the vessels
[93, 94]. The latter two suggest a possible connection with neural activity, while the former
suggests a mechanism linking to energy metabolism through pH, as excess lactic acid dilates
the vessel. Adenosine has a strong vasodilatory effect [87, 95, 96], and is interesting because
of its intertwined roles in both neural activity and energy metabolism. As discussed earlier,
the primary energy storage molecule is ATP. In addition to its role in energy metabolism,
ATP also serves as a neurotransmitter/neuromodulator [97]. In the extracellular space, ATP
is sequentially broken down to produce adenosine, which then has a potent effect by
reducing neuronal excitability. Caffeine competes for adenosine receptors [98], and a
number of studies have shown that caffeine reduces CBF [99], consistent with the idea of a
reduction of the vasodilatory effect of adenosine. A number of studies have shown that CBF
is modulated through an extended metabolic pathway related to arachidonic acid (AA) and
its derivatives [88, 100], including prostaglandins formed from cyclooxygenase (COX)
pathways. Finally, a number of direct neural pathways and associated neurotransmitters have
been found to have vasoactive effects, with norepinephrin and neuropeptide Y producing
vasoconstriction, and vasoactive intestinal peptide (VIP), acetylcholine, NO and other
transmitters, producing vasodilation [89, 101].
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Current thinking is that many of these processes are related to the activity of astrocytes [90,
91, 102–104]. Astrocytes contain receptors for numerous neurotransmitters, including
glutamate, GABA, acetylcholine and adenosine, and activation of these receptors induces
changes in cytosolic Ca2+. With numerous processes contacting neuronal synapses, the
astrocytes are well positioned to play a key role in recycling neurotransmitter, and also to
monitor and integrate local neuronal activity. Additional processes, called end-feet, make
contact with blood vessels, so that astrocytes create a bridge between neuronal activity and
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blood flow. This close anatomical arrangement has long suggested an important functional
arrangement, and in recent years the mechanisms by which changes in neural activity
translate into changes in CBF have become clearer. Because of the close interactions
between neurons, astrocytes and blood vessels, this combination is often referred to as the
neurovascular unit [105, 106].
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Many of these mechanisms are consistent with a basic picture in which excitatory neural
activity increases CBF, consistent with the idea that this is also associated with the primary
energy cost of neural signaling, and that inhibitory neural activity decreases CBF
(vasoconstriction). However, there are a few interesting exceptions. As noted above,
adenosine has an inhibitory effect on neural activity but is also a vasodilator [97]. This is
likely a survival mechanism, in which adenosine overproduction is linked to a lack of
recovery of ATP, possibly from an energy crisis due to a lack of oxygen, and the response
would tend to both increase the delivery of oxygen and reduce the demand for oxygen. A
second example of opposite effects on neural activity and CBF is nitric oxide, a very potent
vasodilator, and yet it is produced by inhibitory interneurons [107]. The presence of these
somewhat counterintuitive mechanisms, which drive neural activity (and presumably
CMRO2) oppositely to CBF suggests that the coupling of CBF and CMRO2 may be
variable.
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These considerations of the many potential mechanisms that could be involved in
neurovascular coupling highlight the complexity of CBF control. For many of these
mechanisms there is a plausible correspondence between flow and energy metabolism
demands, in the sense that the aspects of neural activity that are likely to cost the most
energy also strongly drive CBF changes. This suggests the possibility that the essentially
feedforward mechanisms of CBF control are tuned during development to provide sufficient
O2 to meet the energy demands while maintaining tissue oxygenation, as suggested above.
However, the independence of these mechanisms (i.e. the lack of a rapid feedback signal
from tissue O2 to adjust CBF) means that we should treat the CMRO2/CBF coupling ratio λ
as a physiological quantity that could vary between brain regions, with different types of
stimulus, and with disease. As this parameter strongly affects the BOLD response, the
interpretation of the magnitude of the BOLD response in a physiologically meaningful way
is difficult.
4. The physical basis of fMRI
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This section reviews the physical basis of the BOLD effect, and in particular the quantitative
modeling that can be used to understand and interpret the BOLD response. While the
modeling reveals the complexity of the BOLD effect, it also provides a theoretical
framework for combining different measurements to estimate important physiological
quantities. A major ongoing goal of this work is to develop methods for separating the
effects of CBF and CMRO2 and thus provide tools to make fMRI into a quantitative probe
of brain physiology. These ideas are discussed in more detail in several recent reviews [21,
108–110]. Here the focus is on signal decay, the dominant mechanism exploited in fMRI.
However, the magnetic field offsets produced by magnetized blood vessels also interact with
a rapidly pulsed method called steady-state free precession (SSFP). While the signal
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behavior is more complicated than what is described below, this is nevertheless a sensitive
and promising approach for detecting and mapping brain activity [111, 112].
4.1.
decay
After an excitation RF pulse, the measured MR signal is proportional to the transverse
magnetization at the time of measurement. The central physics for understanding the BOLD
effect is how the decay of that signal is altered by deoxyhemoglobin in the blood vessels. In
a typical fMRI experiment images are repeated, each acquired with the same fixed delay TE
after excitation. If the deoxyhemoglobin content decreases, the signal decays less and is thus
slightly stronger during neural activation. In the following, we consider current physical
models for decay of the transverse magnetization relevant for understanding the BOLD
effect, including the effects of fluctuating fields, static field inhomogeneities, and diffusion
through inhomogeneous fields.
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The decay of the signal, starting at the initial excitation at t = 0, can be described by an
attenuation factor A(t), with A(0) = 1. That is, A(t) represents the fraction of the original
signal remaining at time t. Note that the actual signal is A(t) scaled by the proton density and
potentially by factors related to the repetition time and T1—here the goal is just to model the
decay of the transverse magnetization. In classic NMR theory the attenuation factor for an
SE experiment is a simple exponential decay, A(t) = exp[−R2t], where R2 = 1/T2 is the
transverse relaxation rate. For the GE experiment typical of fMRI (i.e. measuring an FID),
the signal decay is more complicated because of the effects of microscopic magnetic field
distortions around blood vessels containing deoxyhemoglobin. This enhanced decay rate is
described by a relaxation rate
(>R2), although in general
is a function of time. That is,
the attenuation factor is still modeled as nominally an exponential
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with time-dependent
, but
, so the shape of A(t) can depart significantly from a simple
exponential. The inverse of
is the decay time
(<T2).
Now consider an otherwise uniform tissue element containing blood vessels occupying a
volume fraction V (typically ~0.05), and focus on the net signal from the extravascular
space, SE. If the signal from a tissue element with no blood vessels (V = 0) is defined to be 1
at t = 0, the extravascular signal is
(2)
If there is no difference between the magnetic susceptibility of the blood vessels and the
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surrounding tissue, the relaxation rate is assumed to be simply
. Now introducing a
susceptibility difference between the blood vessels and the surrounding space, the form of
can be written as
(3)
The significance of this form is that it breaks the added relaxation into two terms: a part that
can be refocused with an SE experiment (
appears as a change in R2 (ΔR2).
) and a part that cannot be refocused and so
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The physical effect that determines the balance of the two additional terms in equation (3) is
the diffusion of water molecules. If the water molecules were static (no diffusion), then the
local magnetization would simply precess at the local Larmor frequency, which is
proportional to the local field offset. The net signal would then decay more quickly as the
magnetization from microscopic domains precessed at slightly different frequencies
producing a growing phase dispersion. In this static dephasing limit all of the additional
relaxation is in
(ΔR2 = 0), and it depends just on the overall distribution of field offsets
around the vessels. After an evolution time t the local phase of each microscopic domain is
simply proportional to t, and an SE will cleanly reverse the local phase accumulation at each
point.
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With random motions of the water molecules due to diffusion, though, the net phase of each
precessing magnetization at time t reflects the past history of locations of the water
molecules. That is, the local magnetization precesses at different rates as spins undergo a
random walk through the distorted magnetic field. In the fast diffusion limit (or the motional
narrowing regime) water molecules diffuse during the decay time over large regions
compared with the spatial scale of the magnetic field distortions, so that effectively each
spin samples all of the field offsets created by the magnetized vessels. None of this random
phase accumulation can be reversed by an SE, and so in this limit all of the effect of the
local field inhomogeneities is in the ΔR2 term. Note though that because each spin is, on
average, sampling all of the field offsets, there is less dispersion of the accumulated phase
across spins (the origin of the description motional narrowing, as the final distribution of
phases is narrower than it would be without diffusion). For this reason, a given amount of
deoxyhemoglobin in larger blood vessels, where diffusion effects are less important, has a
greater effect on
than when the same amount of deoxyhemoglobin is in capillaries, where
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diffusion considerably diminishes the effect on . Much of the effort in modeling the
BOLD effect has been devoted to exploring the middle ground between the two limits of
static dephasing and fast diffusion.
4.2. Magnetic field distortions around a magnetized cylinder
The field distortion around magnetized blood vessels is usually modeled in terms of the
magnetic field offset at a point near an infinitely long cylinder with radius R oriented at an
angle α to the main magnetic field B0 and with magnetic susceptibility difference Δχ
between the cylinder and the surrounding space:
(4)
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Note that the field offset in equation (4) is the z-component of the added field due to the
magnetized cylinder, because this is the component that adds to the much larger main
magnetic field B0 and so alters the precession frequency. Here (r, θ) is a coordinate system
oriented to the cylinder, so that r is the perpendicular distance from the point to the axis of
the cylinder, and θ is the angle of the point with θ = 0 taken as the angle of projection of the
main magnetic field B0 into the cylinder space (figure 9). The field distortion is maximized
when the cylinder is perpendicular to B0 (α = π/2), and there is no distortion when it is
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parallel to B0 (α = 0). There are also two lines at angles θ = +π/4 and −π/4 along which the
field is not distorted, giving the overall field distribution the characteristic dipole shape
shown in figure 9. Note also that the field falls off with the square of distance scaled by the
radius of the cylinder, but that the field offset at the surface of the cylinder (r = R) is the
same for vessels of all sizes, depending just on geometrical orientation and the susceptibility
difference Δχ.
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The magnetic susceptibility difference is usually taken as Δχ = (1 − Y)HctΔχ0, where Y is the
fractional O2 saturation of hemoglobin, Hct is the hematocrit (the volume fraction of blood
occupied by red blood cells), and Δχ0 is the susceptibility difference that would result if all
of the hemoglobin was deoxygenated (Y = 0) and the hematocrit is one. Recent models [113,
114], though, have modified this in keeping with the experiments of Spees et al [115] who
found that the point of equal susceptibility difference between plasma and red blood cells is
not 100% saturation but rather ~95% saturation due to the different protein contents of
plasma and red cells. Assuming the susceptibility of the extravascular space is the same as
that of the plasma, this effect should be included. For the discussion here we will ignore this
correction, and assume the simpler relation between Y and Δχ. The offset of the angular
frequency of precession is γΔB, and this distribution determines the accumulated phase that
leads to A(t). Taken together, the added angular frequency (rad s−1) distribution around the
vessel is
(5)
with
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(6)
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Note that δω0 is the maximum angular frequency offset at the surface of a cylinder oriented
perpendicular to the magnetic field. For a susceptibility difference of Δχ0 = 0.264 × 10−6
[115], a typical hematocrit of 0.44 (normal range is about 0.42–0.52 for adult males and
0.35–0.47 for adult females), and a baseline venous saturation of Y = 0.6, a typical value of
δω0 is ~230 rad s−1 for B0 = 3 T. In their seminal paper modeling the BOLD effect, Ogawa
and co-workers expressed δω0 as an equivalent frequency ν (in Hz, rather than rad s−1, so ν
= δω0/2π), which would be ~37 Hz for this example. This estimate for a field strength of 3 T
is consistent with assumptions in two recent models [113, 114], but lower values have been
assumed in other models based on either lower assumed values of Hct, Y or Δχ0 [116, 117].
Because of physiological variability in the hematocrit value in the normal healthy
population, this basic parameter could vary in practice by up to 30%, so there is no
definitive value that can be assumed to be accurate for all subjects.
The parameter δω0 (or ν) is the fundamental scaling parameter for magnetic field offsets
around a magnetized cylinder, and the geometry of those offsets is governed by equation (5).
For modeling blood vessels in the brain, the usual assumption is that the vessels are
randomly oriented. This is probably a reasonable assumption for the smallest vessels, but
larger vessels may be more oriented. In the cortex, blood vessels spread over the surface of
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the brain with arterioles diving perpendicular to the surface into the cortex [118]. Detailed
representations of the human brain vascular network are now becoming available as models
or as direct vascular images [119–123], and it will be important to do similar calculations of
effects on these realistic networks to test whether the idealized assumptions of the
numerical models (randomly oriented, infinitely long cylinders) introduce significant
systematic errors.
4.3. Static dephasing
Yablonskiy and Haacke [124] analyzed the extravascular signal in the static dephasing
regime (no diffusion) for randomly oriented magnetized cylinders occupying a volume
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fraction V. In this regime all of the added relaxation from equation (3) is due to . It turns
out to be useful to express the decay relative to an external medium that contains no vessels
(V = 0). Then the attenuation curve, rather than starting at a value of one, starts at a value of
1 − V (as in equation (2), because the vessels themselves are assumed to generate no signal
—intravascular signal changes will be considered later). The signal then gradually decays,
initially equivalent to an
approaches a value of
value that increases with time (figure 10). As time increases
(7)
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This form nicely explained the earlier Monte Carlo simulation results of Ogawa et al [116].
The numerical factor comes from the random orientation of vessels, effectively creating an
average scaling frequency of δωav = 2/3δω0. This monoexponential behavior develops for
time t > 1/δωav. Importantly, the monoexponential portion extrapolates back to a value of 1
at t = 0, so that the ratio of the actual signal at t = 0 to the back-extrapolated value from the
monoexponential regime depends just on the volume fraction.
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We can understand this behavior intuitively by considering two aspects of the field
distortions described by equation (5). First, the field offsets (measured as equivalent angular
frequency offsets) have a finite range δω0; that is, the largest offsets are at the surface of the
cylinder, and depend only on the susceptibility difference and not the cylinder radius.
Second, if we consider a cylinder with radius R, the field distribution for r > R is identical to
the field offsets produced by a thinner cylinder with a larger susceptibility difference,
provided the product R2δω0 remains constant. For the assumed distribution of infinite
cylinders, the blood volume fraction V translates to R2, so this basic combination of physical
variables, R2δω0, is at the heart of equation (7). We can think of the extravascular signal as
arising from a series of thin shells around the magnetized cylinder. As time increases, the
phase variation of the signals within a shell grows, and we can loosely think of the net signal
from a shell as being destroyed once the phase variation reaches a sufficiently high level.
Note, though, that once the signal from a shell near the surface of the cylinder has been
destroyed, information on the separate values of R and δω0 is lost, because the field
distortions in the remaining space generating a signal are the same for any set of cylinders
for which R2δω0 is constant.
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Note that this is not true, though, for the earliest part of the signal decay when the signal
from the shell closest to the cylinder has not been destroyed. Initially, the extravascular
signal is 1 − υ, and as time increases to the point that the signals from the shells closest to
the cylinder are destroyed, the subsequent signal decay is the same as it would be for a set of
very thin cylinders with a large susceptibility difference such that Vδω0 remains the same.
For such cylinders, the signal from the inner shells is destroyed almost immediately, so the
projection back to t = 0 is to a signal of 1 (i.e. it approaches the condition of V = 0). For this
reason, the projection of the decay curve back to zero provides information on the blood
volume fraction V independent of the
on the product of V and δω0.
value observed at later times, which depends only
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Potentially, this provides a way of disentangling the effects of blood volume and blood
susceptibility. This is critical, because δω0 is the parameter that directly relates to blood
oxygenation and to the OEF, and a robust measurement of OEF provides information on
is
oxygen metabolism. Put another way, blood volume is a confounding effect such that
not a clean reflection of OEF alone, and so the ability to separately account for blood
volume effects is critical for quantifying CMRO2 from BOLD measurements. This remains
a challenge, although Yablonskiy et al [110] have developed an approach called qBOLD
based on this key insight related to extravascular static dephasing and the shape of the decay
curve (discussed further below).
4.4. Diffusion effects
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Modeling the effects of diffusion is considerably more complicated, and nearly all studies
are limited by the need to assume relatively simple geometries. Several analytical treatments
have been proposed [125–128], and a number of studies have used Monte Carlo simulations
tracking thousands of random walks through a field of randomly oriented vessels to model
diffusion [114, 116, 129–131]. In thinking about diffusion, the critical characteristic time is
the time required for diffusive motions to be comparable to the spatial scale of the field
distortions, τD ~ R2/D, where R is the radius of the blood vessel creating the field offsets and
D is the diffusion coefficient. If the evolution time of the signal is much longer than τD, then
each diffusing spin will tend to sample all of the field offsets produced by the magnetized
vessel. At this extreme (the fast diffusion limit) we can estimate the expected behavior with
a random walk argument. Let δω2 be the mean squared field offset in the space around the
magnetized cylinder within a radial distance rmax. This maximum radial distance is
determined by the blood volume fraction V (essentially we are averaging out to the point of
overlap of field offsets due to neighboring blood vessels), with V = (R/rmax)2. For parallel
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[127]. For the fast diffusion limit, we
cylinders with low volume fractions,
consider each spin to be randomly sampling field offsets from a Gaussian distribution with
variance δω2, and precessing for a time τD in each field, so that in each step it acquires a
phase (squared) of
. After evolving for a time t (N = t/τD steps), the variance of
the accumulated phase is
. When this Gaussian distribution of phase is
multiplied by cos(ϕ) and integrated to calculate the decay curve, the result is a
monoexponential decay for fast diffusion with ΔR2 of the form
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(8)
with κ a dimensionless numerical constant (for this simplified estimate, κ = 0.25).
Equations (7) and (8) essentially bracket the range of effects on the signal decay curve due
to blood vessels with blood volume fraction V and a susceptibility difference characterized
by δω0. With no diffusion (static dephasing) the added decay is described entirely by
(equation (7)), with a linear dependence on δω0. In the fast diffusion limit, the decay is
described by ΔR2 (equation (8)), with a quadratic dependence on δω0.
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In practice, the decay of the signal in the BOLD effect falls between these two limits for the
smallest blood vessels (e.g. for a capillary with a radius of 2.5 µm and D ~ 1 µm2 ms−1, the
characteristic time is τD ~ 6 ms), and so a number of investigators have approached this
problem with Monte Carlo simulations. Figure 10(b) shows Monte Carlo simulations from
Dickson et al [131] for the change in relaxation rate for a GE experiment with TE = 30 ms
and SE experiment with TE = 60 ms as a function of blood vessel size. (If these simulations
had been performed for the same TE, the SE curve would correspond to ΔR2 in equation (3),
and the difference between the SE and GE curves would correspond to .) Note that from
these calculations we would expect vessels with R > 10 µm to be solidly in the static
dephasing regime, and equation (7) is typically used to model the effects of these larger
vessels. However, the smallest capillaries with R ~ 2.5 µm are part-way toward the fast
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from both ΔR2 and . For an assumed
diffusion limit, with some contribution to
diffusion constant of D = 1 µm ms−1, Ogawa et al [116] found that the effects of the largest
vessels were accurately described by equation (7), while the effect of the smallest vessels (R
= 2.5 µm) could be described by equation (8) with the product κτD = 0.0010 s.
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A key result of the arguments above is that diffusion always tends to reduce the BOLD
effect compared with what it would be in the absence of diffusion. Physically, this is
because any distribution of field offsets produces the widest distribution of phases when
there is no diffusion, because diffusion effectively brings in averaging over the field
distribution (motional narrowing of the phase distribution) as described earlier. The
magnitude of this effect can be estimated from equations (7) and (8) by considering two
blood vessel distributions, one of larger vessels (R > 10 µm) and one of capillaries (R = 2.5
µm), each with the same blood volume fraction and each with the same deoxyhemoglobin
content (equal δω0). Curves showing the signal loss due to each population calculated for
δω0 = 200 rad s−1 at a field of 3 T are plotted in figure 10(c), showing an approximately
three-fold difference for an O2 extraction fraction of 40%. In addition, the hemoglobin
oxygen saturation is expected to be higher on average for capillaries compared with veins, as
it varies down the length of the capillary from a value near the arterial saturation to the
venous value. Taken together, these arguments suggest that the standard BOLD effect
measured in GE imaging is dominated by the veins, with a much smaller contribution from
the capillaries.
However, the physical effects of diffusion also suggest another possible application: using
an SE experiment instead of a GE experiment could isolate signal changes related to the
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smallest vessels. That is, the SE experiment is only sensitive to the ΔR2 effects (the
effects are refocused), and these effects only occur for the smallest vessels (figure 10(b)).
The significance of this idea is related to an issue noted early in the development of fMRI
and referred to as the brain versus vein problem [132, 133]. The largest change in
deoxyhemoglobin is in the veins, but a draining vein may show a significant BOLD effect
that is displaced from the site of the neural activity change. The potential of SE-BOLD
imaging is that such draining veins would contribute less to the net signal than the effects
due to the capillaries, and as a result the signal would be better localized to the true site of
the neural activity change [134]. However, because the SE-BOLD effect is so much weaker
than the GE-BOLD effect, this approach is not practical except at high main magnetic fields
where the SNR is significantly improved [31, 135].
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In addition, the argument for the better selectivity of the SE-BOLD experiment is based on
considerations of the extravascular signal change. In practice, at moderate field strengths
(1.5–3 T), the intravascular signal change is significant, even for an SE experiment [136,
137]. At high fields the intravascular contribution is reduced [138], so again the idea of
using SE-BOLD for better localization is only practical for high fields. Interestingly, a
recent modeling study suggests that the SE-BOLD sensitivity to the microvasculature may
peak at very high fields, and that the SE sensitivity to the vascular signal of capillaries and
arterioles persists even at high field strengths [114].
The different sensitivities of GE and SE imaging to vessel size also suggests the possibility
[139–141]. This ratio steadily
of vessel size imaging, by looking at the ratio of ΔR2 and
diminishes as the vessel size increases, and can provide a way to investigate angiogenesis,
among other applications.
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4.5. Intravascular BOLD effect
The arguments above focused on the signal changes in the extravascular space, and did not
consider changes in the signal generated by blood itself (as noted in the last section). The
intravascular compartment is a small fraction of the total tissue volume (only about 5%), and
so at first glance one might suppose that the intravascular spins would contribute a
comparably small amount to the net BOLD signal change. However, at 3 T and below, the
vascular contribution is comparable to the extravascular contribution [142–144]. Water
molecules in blood are closer to the source of the magnetic susceptibility change
(deoxyhemoglobin), significantly increasing the effect, but this is somewhat moderated by
strong diffusion effects as well [12]. The net result is that the intrinsic signal change in the
blood is more than an order of magnitude larger than the extravascular signal change [142].
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Figure 10(d) shows curves of the
of blood based on empirical data measured at 3 T for
various hematocrit values [145] and fit to a simple quadratic function that includes the
hematocrit as a potential variable [113]. Comparing extravascular and intravascular signal
changes from the curves in figures 10(c) and (d), an activation that reduced the OEF from
40% to 30% would create an extravascular signal change of ~2% and an intravascular signal
change of ~50%. Even with a blood volume fraction of only 0.02, the intravascular signal
change still accounts for about one-third of the total signal change due to its much wider
dynamic range.
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4.6. Modeling the BOLD effect
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The previous sections have described individual components of the BOLD signal, including
static dephasing effects, the role of diffusion and the contribution of intravascular changes.
In practice, the measured signal change also includes a volume exchange effect, as CBV
increases and displaces extravascular water generating a different signal. While this effect is
not ‘blood oxygenation dependent’, it nevertheless contributes to the measured signal and
should be included in the modeling [146]. A full quantitative model of the BOLD effect is
important for understanding the basic mechanisms, for optimizing the image acquisition
technique to maximize sensitivity, and for calibrating the BOLD signal to measure local
CMRO2. The latter calibrated BOLD application (described in more detail in the next
section) is particularly important, because it provides a way to measure the CMRO2
response to a stimulus [108, 109]. The basic idea is that the BOLD response depends on
both the CBF and CMRO2 responses. If the BOLD response is measured in conjunction
with an ASL experiment to measure the CBF response independently, one can in principle
isolate the CMRO2 response. The key to making this approach work is the availability of an
accurate model relating the BOLD signal change to the underlying changes in CBF and
CMRO2. This section reviews modeling efforts in this direction, focusing on the GE-BOLD
signal at 3 T as the most commonly used technique. Uludag et al extended these modeling
ideas in a comprehensive way to include SE-BOLD and a wide range of field strengths
[114].
Historically, the most influential model of the BOLD effect was introduced by Davis et al in
their seminal paper describing the calibrated BOLD approach [130]. In the context of the
terminology used in the current review (equation (3)), they modeled the combined term
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, which can be abbreviated as
(dHb), the additional contribution to the
relaxation rate due to the presence of deoxyhemoglobin. They assumed the form
(9)
where V is the blood volume fraction, [dHb] is the local concentration of deoxyhemoglobin
in blood, and k is a constant that depends on field strength. They suggested a value of β = 1.5
as a compromise between the effects of large and small vessels (as in equations (7) and (8),
where the exponent on δω0 varies from 1 to 2). The fractional BOLD signal change between
a baseline state (denoted by subscript ‘0’) and an activation state (no subscript) is due to the
(dHb) between those two states. In these calculations of a fractional signal
change in
change (ΔS/S0), the effects of R2 cancel out, so this term is dropped from the following
equations. The measured GE signal is
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(10)
where TE is the echo time of the experiment, and the approximation is based on the
assumption that the exponent is small compared with one. The fractional BOLD signal
change between a baseline state and an activated state is then
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(11)
where lowercase symbols are the values in the activated state normalized to the baseline
state values (i.e. υ= V/V0 and c is the ratio of [dHb] in the active state to the baseline state).
The deoxyhemoglobin concentration in blood ([dHb]) was modeled as though it was all in
the venous vasculature, so that the concentration ratio is just the ratio of OEFs (E/E0), and
equation (1) can be used to express this in terms of the CBF normalized to its baseline value
(f) and the CMRO2 normalized to its baseline value (r). Combining the multiplicative
constants into a single scaling factor M, the BOLD signal is
(12)
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Finally, the blood volume change is assumed to be related to the blood flow change with a
power law relationship, υ = fα, with α = 0.38 based on experiments comparing total CBV
with CBF [147], so that the final model is
(13)
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This simple form satisfies the goal of relating the BOLD signal change to the changes in
CBF (f) and CMRO2 (r), and has since been widely used. Note, though, that there are some
issues with this model. The derivation of this expression leaves out intravascular signal
changes and volume exchange effects. In addition, in recent years the question of the right
value of α has been raised, as a number of experiments have found that the volume change
on the venous side is smaller than the overall blood volume change [148–152]. The right
value of β also has been questioned, and as fMRI studies have moved to 3 T fields, an
assumed value of β = 1.3 is becoming more common [153]. A subsequent model from my
group (originally in [154] and then modified in [146, 155]) included intravascular signal
changes and volume exchange effects. This model did not have an exponent β and involved
two constants, rather than the single M in the Davis model.
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Recently, more detailed models have dispensed with the goal of a simple final form and
included all of the effects currently thought to be contributing to the BOLD response [113,
114]. Griffeth and co-workers in my group used such a detailed model to test the accuracy of
the Davis model and found that, despite the limitations of the original derivation, the form of
the Davis model is reasonably accurate [113]. This is because many effects that simply scale
the overall BOLD signal are captured by the scaling constant M. Further analysis with the
detailed model currently underway suggests that an alternative heuristic model is
approximately as accurate as the Davis model:
(14)
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where αV is specifically the exponent for the volume change of the venous compartment
(e.g. 0.2 based on [152]), and λ is the ratio of the fractional CMRO2 change to the fractional
CBF change (e.g. for a 20% CBF change accompanied by a 10% CMRO2 change, λ = 0.5).
The heuristic advantage of equation (14) is that it clearly shows the range of effects that
modulate the BOLD signal. A number of factors primarily just scale the BOLD response,
and so are subsumed in a scaling factor A. A venous volume increase with activation tends
to reduce the BOLD effect by increasing the total deoxyhemoglobin, reflected by the αV
term. The change in blood oxygenation strongly depends on the relative changes in CBF and
CMRO2, reflected in the λ term. Finally, there is an intrinsically nonlinear dependence on
the CBF change, reflected in the last term. This nonlinearity is due to the ceiling effect on
the BOLD signal: the maximum that a large CBF increase can do is to reduce
deoxyhemoglobin to near zero values, and this would correspond to a finite signal change
[156].
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5. Functional MRI as a quantitative probe of brain physiology
5.1. The calibrated BOLD method
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A key feature of the basic BOLD signal models is that they include an overall scaling
parameter (M in equation (13) and A in equation (14), and it will be discussed here as the
more familiar M). This is critical because M captures much of the physiological variability
of the baseline state. That is, M depends on the amount of deoxyhemoglobin in the baseline
state, which is affected by hematocrit, venous blood volume and the baseline OEF. As a
result, M should be assumed to vary across individuals, across brain regions, and even
within the same individual brain region if the baseline state changes. The downside of this is
that M must be measured in each subject, but the upside is that once it is measured it
accounts for much of the variability across subjects. In addition, knowing M makes it
possible to interpret changes in the BOLD response in terms of changes in CMRO2 when
CBF is measured concurrently with an ASL technique. For example, from equation (13), if f
is known from the ASL measurement and M is known in some way, then the measured
BOLD signal change can be used to calculate r, the ratio of the CMRO2 value in the active
state relative to the baseline state. The key requirement then is a method for determining M,
called the BOLD calibration.
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Davis et al [130] proposed the calibrated BOLD approach based on measuring the CBF and
BOLD responses to inhaling a gas containing 5% CO2 (hypercapnia). Based on a number of
studies, mild hypercapnia was thought to produce a large increase in CBF with little or no
change in CMRO2, and so r = 1 is assumed. Then from equation (13), with f known from the
ASL experiment, the value of M can be estimated. This has proven to be a powerful tool for
exploring CMRO2 changes, particularly for exploring how the CMRO2/CBF coupling ratio
λ varies under different conditions (for recent reviews see [108, 109, 157]).
However, in recent years several studies have questioned whether inhaled CO2 really has no
effect on CMRO2 [158–161]. At high levels, CO2 acts as an anesthetic, so we would expect
CMRO2 to decrease for high levels of inhaled CO2. The essential question, which is still
debated [162], is whether modest levels (5% CO2) also have an effect on CMRO2. In part
because of these concerns, there has been an effort to develop alternative methods for
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calibration. One suggestion is to use hyperoxia instead (i.e. breathing a gas mixture with
50% or more O2) [153, 163]. In normal arterial blood the hemoglobin saturation is near
100%, so hyperoxia primarily increases the O2 concentration as dissolved gas in the plasma.
As the hyperoxic blood reaches the capillaries, the excess dissolved O2 diffuses into the
tissue and offsets some of the O2 that previously came from the hemoglobin-bound pool. As
a result, the venous deoxyhemoglobin concentration decreases slightly, creating a BOLD
signal increase. With appropriate assumptions about the physiological variables, an estimate
of M can be derived. Recently, Blockley and co-workers from my group analyzed both the
hyperoxia and hypercapnia methods with our detailed model of the BOLD response, and
found that the hyperoxia method was sensitive to the assumed value of baseline OEF [164].
Given that variability of baseline OEF is one of the key parameters that M is designed to
capture, this suggested that hyperoxia may not be robust for determining M. In addition, a
recent study found evidence that hyperoxia also may alter CMRO2 [165].
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Blockley and co-workers also proposed a different way of estimating M without using
inhaled gases, based on a measurement of
in the baseline state. That is,
can be
measured with appropriate comparisons of GE and SE signals. The assumption is then that
is due just to local deoxyhemoglobin. While this approach does not capture the intrinsic
ΔR2 contributions to the net change in
with activation, those changes are likely to be a
in the baseline state is
minor fraction of the signal change due to . Put another way,
primarily determined by total deoxyhemoglobin, as is M. By modeling the experiment with
the detailed BOLD model one can estimate a scaling factor needed to derive the full value of
M. However, this approach is still speculative. A critical technical problem is that in practice
is contaminated by field inhomogeneities that produce additional signal loss, so these
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field variations must be accurately measured and corrected to derive an estimate of
due to
deoxyhemoglobin [166, 167]. Nevertheless, the promise of this approach is a much simpler
calibration procedure that does not require inhalation of special gases.
In summary, the calibrated BOLD approach is a powerful way to measure fractional changes
in CMRO2, which are very hard to measure with any technique [19]. There are still concerns
about whether hypercapnia is isometabolic as used in the calibration experiment, and
whether the baseline OEF is sufficiently stable across the population (particularly in disease
states) so that the hyperoxia method can be robust with an assumed value of OEF. Simpler
methods of calibration based on
in the baseline state look promising, as they do not
require inhalation of special gas mixtures, but it remains to be demonstrated that the
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confounding effects of field inhomogeneities on
can be controlled sufficiently. Promising
approaches are to use other methods for determining some of the model parameters, such as
the blood volume effects, in combination with BOLD-fMRI and ASL measurements [144,
168–170].
5.2. Interpreting the BOLD response
The complexity of the BOLD effect described in the previous sections leads to a
fundamental problem for interpreting the magnitude of the BOLD response in a quantitative
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way in terms of the underlying physiology. Two recent studies illustrate this problem. In the
first example, Fleisher et al [171] compared the BOLD response in the hippocampus during
a memory task between two groups, one consisting of subjects with specific risk factors for
the development of Alzheimer’s disease, and the other an age-matched control group
without those risk factors. They found a weaker BOLD response in the higher risk group,
suggesting an early effect of the disease. The interesting question, though, is how should this
result be interpreted in terms of the underlying physiological effects? It is tempting to
interpret this as a sign that the neural processing associated with the memory task was
altered. However, the model in equation (14) shows that for the same change in oxygen
metabolism, the BOLD response could differ if the CMRO2/CBF coupling ratio λ changes,
or if the baseline state changes affecting A. In particular, the scaling parameter A essentially
reflects the deoxyhemoglobin content in the baseline state, which can be altered by changes
in baseline O2 extraction fraction or venous blood volume. The latter effect is particularly a
concern in looking at patient populations, where chronic changes in the baseline state due to
the disease or the effects of medications could alter the BOLD response to a standard task,
even though the CBF and CMRO2 responses to the task are unchanged. In the Fleisher study
they also used ASL to measure CBF in both the baseline state and the active state, with the
interesting finding that CBF in the active state was similar in the two groups, but the
baseline state was different. This suggests that the change in the acute BOLD response to the
task may be partly due to a chronic difference in the baseline state.
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A second example used dual measurement of ASL and BOLD responses in a calibrated
BOLD method to untangle the effects of caffeine on both baseline state changes due to the
drug and alterations in the way the brain responds to a visual stimulus with the drug on
board [21]. Interestingly, the BOLD response to the visual stimulus was nearly identical preand post-caffeine. Nevertheless, the underlying physiological changes measured with the
calibrated BOLD method were substantial: baseline CBF decreased while baseline CMRO2
increased, and in response to the visual stimulus the CMRO2 response increased by more
than 60%. These findings are consistent with the known action of caffeine in blocking
adenosine receptors [98], and with other fMRI studies with caffeine [172]. Because
adenosine both increases blood flow and acts as an inhibitor of neural activity, when
adenosine receptors are blocked we expect CBF to go down, but because inhibition of neural
activity is partially lifted we expect CMRO2 to go up. The fact that the BOLD signal failed
to reflect these underlying changes is due to the conflict between two effects: the baseline
state changes should increase the baseline OEF, which would tend to increase the BOLD
response; and the larger ratio of the change in CMRO2 to CBF in response to the visual
stimulus, which tends to decrease the BOLD response. In terms of equation (14) this
amounts to an increase in both A and λ, which have opposing effects on the overall scaling
of the BOLD signal, and in these data the two effects balanced out to produce no change in
the BOLD response. In short, while a physiological interpretation of the BOLD signal alone
is not possible, the addition of other measurements makes a much more quantitative probe of
brain physiology possible.
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5.3. Dynamics of the BOLD response
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The time scale of the BOLD response is much slower than the time scale of neural activity
because the CBF response develops and recovers relatively slowly. Even a brief sub-second
neural stimulus produces a BOLD response that is delayed by a few seconds and may take
about 6 s to evolve [173]. This time scale for the BOLD response may vary across the brain
and across subjects, but a more interesting finding is that it can also vary in the same subject
depending on the physiological baseline state. In particular, altering baseline CBF slows or
speeds up the BOLD response in a counterintuitive way: increased baseline CBF slows the
BOLD response, while decreased baseline CBF leads to a faster BOLD response. For
example, Cohen et al found this pattern when baseline CBF was increased by inhaling a gas
mixture with increased CO2 content (hypercapnia) or reduced by voluntary hyperventilation
(hyocapnia), exploiting the intrinsic sensitivity of CBF to the CO2 content of blood [174].
Other studies with caffeine, which lowers baseline CBF, found faster BOLD response
dynamics [175, 176], consistent with this basic trend.
A possible explanation for this phenomenon was proposed by Behzadi and Liu [177] based
on a biomechanical model in which the net compliance of the artery is due to two factors:
the tension of the smooth muscle, and an elastic component that increases as the vessel is
stretched. The essential physical idea of the model is that when the vessel is constricted, so
that the elastic components are not stretched, the compliance is dominated by the smooth
muscle and responds more quickly to changes in smooth muscle tension. In contrast, when
the artery is dilated the elastic components make a significant contribution to the overall
compliance of the vessel, and the same relaxation of the smooth muscle would not produce
as large a change of the overall compliance.
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The dynamics of the BOLD response also exhibits interesting transient features, in particular
a brief initial dip before the primary positive BOLD response [178] and a much longer poststimulus undershoot after the end of the stimulus [179, 180]. While the initial dip is not
always observed, the post-stimulus undershoot is common. Interestingly, there is still no
consensus on the physiological origin or significance of these features. While it is possible
that these transients reflect underlying transients of the neural response, it is also easy to
imagine that they arise from different time constants for the physiological variables that
contribute to the BOLD signal: CBF, CMRO2 and venous CBV. For example, if the
CMRO2 increases before the CBF begins to change, the BOLD response could show an
initial dip due to the increase in deoxyhemoglobin. A post-stimulus undershoot could occur
if the flow transiently drops below the baseline level, or if the flow returns quickly to the
baseline but the venous blood volume or oxygen metabolism returns more slowly. Because
of this dependence of the BOLD signal on multiple physiological changes, it is not possible
to identify the sources of these transients from BOLD measurements alone.
The initial dip was first detected in animal experiments with optical methods sensitive to the
different reflectance properties of hemoglobin and deoxyhemoglobin [181, 182] and later
found in human fMRI studies [183]. The basic finding was an early increase in
deoxyhemoglobin, prior to the much larger decrease in deoxyhemoglobin that produces the
BOLD effect. The dominant view since the early days has been that the initial dip represents
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a quick increase in CMRO2 before the CBF can increase, so that initially the OEF increases.
However, the interpretation of the initial dip has been controversial because of potential
confounding factors in the optical measurements and because of the possibility of CBV
effects in the BOLD signal. Optical studies in animal models have found evidence for both
an early change in oxygenation before a blood volume change [184] and for blood volume
changes as the earliest effect [185]. An fMRI study showing that caffeine, by quickening the
CBF response, also reduces the initial dip is consistent with the idea of a faster rise in
CMRO2 compared with CBF [176]. However, Uludag et al [114] recently argued that
volume exchange effects with the arterial compartment could lead to an initial dip as well. In
short, the physiological origin of the initial dip is still unsettled [186, 187]. A strong
motivation for understanding the initial dip has been that if it is due to an early CMRO2
increase, then mapping the initial dip may provide a more accurate map of the location of
neural activation, based on the idea that the CMRO2 response is better localized than the
CBF response that dominates the primary positive BOLD response. However, in practice,
the weakness of the initial dip makes it challenging to detect except at higher fields [188],
and even harder to map accurately.
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A post-stimulus undershoot was present in the seminal data from Kwong et al [3]
demonstrating the BOLD effect in humans in response to a visual stimulus, and yet its
interpretation is still controversial (see figure 11 for a recent example). Several mechanisms
could potentially produce this effect, and the current experimental data do not provide a
clear interpretation (see recent reviews [179, 180]). While the effect could be due to an
undershoot of neural activity, the duration of the BOLD post-stimulus undershoot (often >30
s) makes this unlikely as a complete explanation, although there are intriguing suggestions
that the undershoot is modulated by neural activity [189]. An early and still leading
proposed explanation is that it is due to a slow recovery of CMRO2 after CBF has returned
to the baseline, so that the OEF is increased during this period [190, 191]. However, animal
model studies using an agent (MION) that stays in the vasculature and alters the MR signal
in a way that reflects CBV found a slow recovery of CBV during the undershoot period [15].
This motivated the development of two similar models, the balloon model [154] and the
windkessel delayed compliance model [192], as biomechanical explanations in which the
venous CBV recovers slowly, even though CBF and CMRO2 have returned to the baseline
so that OEF is back to the baseline as well.
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Note that these two potential explanations reflect the fundamental ambiguity of BOLD
measurements: both venous CBV and OEF affect the BOLD signal. The evidence favoring
the slow CMRO2 recovery is primarily the absence of evidence for a slow CBV recovery in
other studies using different methods sensitive to CBV. Animal models initially found little
evidence for venous volume changes with short and moderate duration stimuli [149],
although more recent studies found evidence for slowly increasing venous blood volume
with longer stimuli [193, 194]. In human studies using different techniques sensitive to
blood volume most have found no evidence to support a slow CBV recovery in the BOLD
undershoot period [195–199], although one study using a technique sensitive to venous
blood volume did find evidence of a slow recovery [200]. Two recent studies used a breathhold paradigm based on the idea that this would change CBF but not CMRO2, so there
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would be CBV changes but not CMRO2 changes. They found no BOLD post-stimulus
undershoot, consistent with slow CMRO2 recovery as the explanation when the undershoot
is seen [201, 202]. However, a recent animal study found that increased intracranial pressure
reduces the undershoot, consistent with a biomechanical explanation [203]. A third
possibility is that instead of being due to a slow recovery of CBV or CMRO2, the
undershoot is associated with a slight undershoot of CBF [200]. Although ASL studies
sometimes show a small undershoot in the CBF response, they do not reach statistical
significance. However, this may simply be a problem with the sensitivity of the ASL
measurement. Theoretical calculations [180] suggest that the level of CBF undershoot
required to explain the BOLD undershoot is likely to be in the noise of most ASL
measurements. Clearly further work is needed to understand the post-stimulus undershoot,
and it may be that several of these possible explanations contribute.
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5.4. Measuring baseline oxygen metabolism
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Currently, we know much more about the behavior of CBF than CMRO2, because it is much
harder to measure oxygen metabolism [19]. The calibrated BOLD approach described in the
previous section makes it possible to measure the fractional CMRO2 change between a
baseline state and an activated state, but this still does not give an absolute measure of
CMRO2 in either state. In disease the key physiological change may be a chronic shift in
CMRO2, rather than a change in how CMRO2 responds to an acute stimulus [62]. In
contrast to the case for CMRO2, for studying CBF the ASL method does make it possible to
measure absolute CBF in any physiological state. The development of a comparable robust
methodology for measuring absolute CMRO2 in the human brain noninvasively would be an
important advance for physiological studies of brain function. The basic question is how can
we estimate the local OEF? If OEF is known, CMRO2 can be calculated with an additional
ASL measurement of CBF and equation (1).
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One approach to measuring local OEF centers on measuring the venous oxygenation in
specific vessels, based on the effect of deoxyhemoglobin on either the T2 of blood or the
magnetic susceptibility of blood. In the relaxometry approach, the signal of blood in a
venous vessel is isolated with a variation of an ASL method, and a multi-echo acquisition
then provides a measure of the R2 of the blood [204–209]. With a calibration curve
appropriate for the experiment, the measured R2 is converted to venous O2 saturation [210].
The deoxyhemoglobin in venous blood also alters the magnetic susceptibility of the blood,
which alters the magnetic field in and around the vessel. This is essentially the macroscopic
counterpart to the BOLD effect, in that the BOLD effect arises from field distortions that are
too small to measure directly, but their effect is manifested in the signal decay curve. For a
large enough venous vessel the susceptibility-induced phase of the blood can be resolved in
the MR image and measured directly and then related to blood oxygenation [160, 211–213].
While these methods have primarily been used for essentially whole brain measurements,
this work extends these methods to multiple vessels for more localized measurements [214].
In principle, this approach is only limited by the resolution of the images, although the
question of determining the drainage basin for individual vessels for accurate localization of
CMRO2 remains an issue.
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Two alternative approaches for localized CMRO2 measurements are directly related to the
BOLD modeling considerations in the earlier sections. In general, the basic problem in
estimating baseline OEF from BOLD-related measurements is that the relaxation effects
primarily depend on the total deoxyhemoglobin within a voxel, and that depends on both
baseline OEF and venous blood volume. That is, if we could isolate the parameter δω0 it
would provide an estimate of the baseline OEF, but the basic effect is governed by equations
(7) and (8), which depend on venous CBV (V) as well. For these BOLD-based
measurements, the challenge then becomes the question of how to estimate the local venous
CBV.
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The first approach designed to solve this problem is based on a detailed analysis of the full
decay curve of the signal, including the initial rounded portion before monoexponential
decay begins and the decay around an SE (figure 10(a)) [110, 167, 215–217]. As described
in section 4.3, for the extravascular signal the initial value is reduced—in proportion to local
CBV—from the back-extrapolated value from the monoexponential decay portion of the
curve. In short, if the signal decay was purely due to static dephasing in the extravascular
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space, measuring the full decay curve would provide a way to separately measure V and ,
and thus isolate δω0 and estimate OEF. Following the work of Yabonskiy and co-workers,
this general approach is usually described under the label qBOLD. Several authors have
extended the qBOLD method with sophisticated modeling approaches to dealing with the
other factors affecting the signal decay curve (including intravascular effects and diffusion
effects) [110, 117, 167, 218]. The full model for the decay curve has become somewhat
complex, with a number of parameters describing various physiological effects. The central
question with any such model is whether the value of one particular parameter of interest
(baseline OEF in this case) can be robustly estimated in the face of physiological variability
across the population of subjects. Experimental studies in simple model systems that try to
mimic characteristics of a vascular bed found that robust separation of the relevant variables
was difficult [219, 220]. A recent Bayesian analysis suggests that very high SNR is required
to accurately estimate both blood volume and OEF from qBOLD data [221]. An alternative
to trying to derive information on blood volume from the decay curve alone is to combine
the basic ideas of qBOLD with an additional measurement of blood volume [166, 222].
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An alternative approach has evolved from the work of Gauthier and co-workers [223–225]
and Bulte et al [226] related to the measurement of M in the calibrated BOLD experiment.
Essentially, this approach is the positive aspect of the concerns raised in the previous section
about the hyperoxia approach for estimating M due to the strong sensitivity to the assumed
value of baseline OEF. The basic idea is that the combination of hypercapnia and hyperoxia
experiments may be able to provide an estimate of baseline OEF. As originally developed,
the motivation is that both the hypercapnia and hyperoxia experiments should yield the same
value of M, if the correct value of baseline OEF is assumed in calculating M from the
hyperoxia experiment (and the hypercapnia derived M is accurate). Then the assumed value
of OEF in the analysis of the hyperoxia data is adjusted until the M-values agree. Another
recent work from Blockley et al [227] suggests another way of looking at this basic
approach, based on the idea that the primary physiological sensitivity of the hyperoxia
experiment is really to venous blood volume, rather than M itself. Then the value of M
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derived from the hypercapnia experiment depends on both baseline venous CBV and
baseline OEF, while the hyperoxia experiment primarily provides a measure of venous
CBV, and so the two together provide the information needed to estimate baseline OEF. In
short, the basis of the method is that BOLD signal changes with hypercapnia and with
hyperoxia depend on venous CBV and baseline OEF in different ways, so it is possible to
untangle their effects. This approach is promising, although technically involved because of
the need to have subjects breathe special gas mixtures.
6. Summary and future directions
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Functional MRI has grown from the conjunction of two phenomena: the biophysical effect
that deoxyhemoglobin has magnetic properties that affect the MR signal, and the
physiological effect that CBF increases much more than oxygen metabolism as neural
activity changes, so that the deoxyhemoglobin level changes as neural activity changes. The
resulting BOLD effect has been exploited as a sensitive tool for investigating the working
human brain, with the capability of distinguishing subtle differences in patterns of brain
activity. The power of a technique that is noninvasive and can be repeated many times for
longitudinal studies should not be underestimated, because it makes possible extended
studies of the healthy human brain that would not be possible with techniques involving
radioactive tracers. Importantly, this means that fMRI can be used to study the dynamics of
the human brain in a way that other methods cannot. The limits of spatial and temporal
resolution continue to be pushed back, as higher magnetic fields improve the SNR and as
technological innovations such as parallel imaging make it possible to reconstruct accurate
images from less acquired data.
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The challenge for fMRI investigators is to fully understand the depth of information that can
be derived from fMRI signals. The BOLD effect alone has primarily been used as a mapping
tool (‘Where is the activation?’) and as a qualitative comparison tool (‘Is the response to
task A larger than the response to task B in a particular brain region?’). These features alone
have led to innovative studies of the integrated dynamics of the brain, based just on how the
dynamics of the BOLD signal in one area correlate with the dynamics in other areas. Yet
there is still a fundamental problem in the quantitative interpretation of the magnitude of the
BOLD response. If the BOLD response to a standard task is larger in one group than
another, how should this be interpreted? It is tempting to conclude that the neural activity
associated with performance of that task is different, but based on our understanding of the
physical origins of the BOLD effect this conclusion is not justified. Systematic differences
in the baseline state between groups, such as medications, caffeine intake, or even anxiety
level, can alter the scaling of the BOLD response for the same changes in CBF and CMRO2.
In addition, variations in the CMRO2/CBF coupling ratio also strongly affect the magnitude
of the BOLD response. In short, the complexity of the BOLD response means that it is
difficult to attach a quantitative interpretation to BOLD measurements alone. The solution to
this problem is a multi-modal approach combining measurements from other techniques in
addition to BOLD. In particular, ASL measurements of CBF significantly expand the
potential for interpreting the BOLD response.
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In summary, the physics underlying the BOLD effect is reasonably well understood, and
much of the effort now is focused on exploiting this understanding to develop new methods
for quantitative assessment of brain physiology, particularly oxygen metabolism and
cerebral blood volume (especially venous blood volume). If these efforts are successful it
will make possible a much more quantitative assessment of human brain physiology.
Acknowledgments
The author has benefitted from helpful discussions of this material with Nic Blockley, Valerie Griffeth and Aaron
Simon. This work was supported by grants from the National Institutes of Health: NS-36722, NS-081405 and
EB-00790.
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Figure 1.
CBF and BOLD responses in human primary motor cortex to 2 s of finger tapping. (a) The
brief stimulus evokes a strong change in CBF measured with an ASL method. (b) The CBF
change is accompanied by an increase in venous blood oxygenation, giving rise to the
BOLD response measured with fMRI. Figure reproduced from [19] based on data from
[228].
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Figure 2.
NMR signal decay. (a) After an initial RF pulse creates transverse magnetization, the
primary signal measured in fMRI is the FID, called a GE signal in MRI terminology. The
signal decays approximately with a time constant
due to both intrinsic T2 decay plus
dephasing due to magnetic field inhomogeneities within an image voxel. (b) Subsequent RF
pulses create spin echoes of the original signal, reversing the effects of the magnetic field
) on subsequent
inhomogeneities. The signal at each SE decays with time constant T2 (
echoes. The primary origin of the BOLD effect in fMRI is that blood oxygenation affects
. Figure adapted from [77] with permission of the author.
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Figure 3.
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Basic magnetic field gradient pulses for EPI. Applying a pattern of pulsed linear field
gradients along different spatial axes ((a) shows gradient amplitude as a function of time)
produces sinusoidal spatial modulations of the local signals such that the net signal from the
slice traces out a trajectory in the FT space (k-space) of the image (b). The image
reconstruction is then a 2D FT of the acquired data. For example, the spatial contribution to
the image (c) of a single point in the measured data ((d) with the point indicated by the
circle) is a single Fourier component (here emphasized by scaling up the value of that point).
Current techniques of image acquisition are considerably more sophisticated, but are based
on the ideas illustrated here. Adapted from [77] with permission of the author.
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Figure 4.
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Basic data acquisition and analysis for fMRI. (a) Dynamic -weighted images are acquired
with a single-shot technique (typically EPI) while a subject performs a task, here illustrated
with a simple block design alternating 20 s of finger tapping with 20 s of rest. (b) The time
course of the signal for each voxel, illustrated here with a 3 × 3 display of the voxels at the
intersection of the lines in (a), are correlated with the stimulus pattern (shown as the block
pattern in the central voxel). (c) Voxels with a statistically significant correlation with the
stimulus are classed as activated by the stimulus and displayed in color overlay on an
anatomical image. As with the image acquisition methods, current fMRI data analysis
methods have become considerably more sophisticated, but the idea of correlation as a
measure of association remains. In resting state methods there is no external stimulus, and
instead the fluctuations in the BOLD time course for different voxels are correlated with
each other to identify covarying RSNs.
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Figure 5.
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Diffusion-sensitive imaging. The MR signal can be sensitized to the local random diffusion
of water molecules with a bipolar gradient pulse (a) that attenuates the measured signal (b)
by an exponential in bD, where b depends on the gradient strength and timing parameters,
and D is the local diffusion constant. This approach is sensitive to displacements of water
molecules due to diffusion that are on the order of 10 µm, far smaller than the voxel
resolution of the images. (c) Images are shown without the bipolar gradient pulse, with
diffusion weighting, and the calculated ADC. The direction of the applied gradient pulse is
arbitrary, and from measurements of multiple directions the local diffusion pattern can be
determined (in the simplest case, the diffusion tensor). Diffusion in white matter is highly
anisotropic due to the microscopic fiber architecture, and this has led to sophisticated
techniques for mapping white matter fiber tracts to provide measures of anatomical
connectivity between different brain regions. Adapted from [77] with permission of the
author.
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Figure 6.
ASL to measure blood flow. (a) Magnetization of arterial blood is alternately manipulated
by applying an RF inversion pulse (tag image) or leaving it relaxed (control image). (b)
After a sufficient delay to allow the labeled blood to be delivered to a slice of interest, the
signal difference (control–tag) subtracts out the static signal from the slice leaving a signal
proportional to the volume of arterial blood delivered to each voxel during the delay time,
providing a quantitative measurement of CBF.
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Figure 7.
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Possible physiological origin of the mismatch of blood flow and oxygen metabolism
changes as a mechanism to maintain tissue pO2. (a) Physical and physiological variations in
the partial pressure of oxygen (pO2), with the O2 saturation curve of hemoglobin as an inset.
To maintain constant tissue pO2 with increased oxygen metabolism (CMRO2), the capillary
pO2 must increase to increase the diffusion gradient, and this means that the O2 extraction
fraction (E) must decrease. The reduction in E with brain activation is the origin of the
BOLD effect. (Adapted from [77] with permission of the author.) (b) Observed fractional
changes in blood flow (CBF) and CMRO2 from a number of activation studies, with lines of
constant ratio of fractional changes in CMRO2 to CBF (λ, with λ < 1 indicating a decrease
in E). The solid line is a modeling prediction of the CBF/CMRO2 coupling ratio needed to
preserve tissue pO2 (adapted from [19]).
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Figure 8.
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The energy cost of neural activity. (a) Estimates of the ATP consumed for different aspects
of neural activity from [80]. Primates have more synapses per neuron, resulting in the
dominant energy cost being recovery from synaptic signaling. (b) The primary excitatory
synaptic signaling involving pre-synaptic Ca2+ influx, release of neurotransmitter
(glutamate, Glu), opening of post-synaptic Na+ channels, and inward Na+ currents are all
thermodynamically downhill events, and the recovery from these events requires energy
metabolism: clearing neurotransmitter through the astrocytes (1), conversion to glutamine
(Gln) (2), release of glutamine, uptake by the pre-synaptic terminal, conversion back to
glutamate and repackaging the neurotransmitter in vesicles (3), pumping out Ca2+ (4) and
pumping out Na+. The last event consumes the most ATP, consistent with post-synaptic Na+
influx acting like an amplifier of the initial signaling. (Adapted from [77] with permission of
the author.)
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Figure 9.
Magnetic field distortions around a magnetized cylinder. This is the basic physical model for
the extravascular effects of a blood vessel containing deoxyhemoglobin, with the dipole
pattern of distortions on the left and the geometry of equation (4) illustrated in the other
panels.
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Figure 10.
Model curves of signal decay effects due to magnetized venous blood vessels. (a) For static
dephasing (no diffusion) the extravascular signal initially decays slowly but then settles to
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an exponential decay with
given by equation (7). Projecting this portion back to t = 0, the
difference with the actual curve is the blood volume fraction (0.02 in this example). (b)
When effects of diffusion are included, the change in relaxation rate depends on the vessel
size, with motional averaging reducing the net effect for the smallest vessels (data from
Monte Carlo simulations reported in [131] with TE = 30 ms for the GE curve and TE = 60
ms for the SE curve). (c) Extravascular signal attenuation as a function of the O2 extraction
fraction for a population of large vessels with radius > 10 µm and a population of the
smallest vessels with radius = 2.5 µm, each with the same total volume of deoxyhemoglobin,
calculated from equations (7) and (8). (d) Intravascular signal as a function of O2 extraction
fraction based on experimental curves measured at a field strength of 3 T [145]. Curves in
(a), (c) and (d) were calculated to be consistent with the GE curve in (b) with B0 = 3T, TE =
30 ms, OEF = 0.4, V = 0.02, and δω0 = 200 rad s−1 (ν = 32 s−1).
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Figure 11.
The BOLD post-stimulus undershoot in human visual cortex. The CBF response (a) and the
BOLD response (b) to a 24s visual stimulus show a strong positive response to the stimulus
plus a prominent post-stimulus undershoot of the BOLD signal. Data are from a study
comparing luminance and color stimuli which found no difference in the responses [229],
and the data for the two types of stimulus were combined for these curves. The origin of the
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BOLD undershoot is still debated, and it could potentially be due to vascular or metabolic
effects.
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Table 1
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Physiological variables related to neural activation. The standard abbreviations used in physiology, and the
symbols used in the equations in this paper are given in the first column. Typical baseline values are for
healthy human adults, but may vary significantly. The fractional changes with activation vary widely with
different stimuli; the values given are meant to suggest typical relative values and not standard absolute
values. The units chosen are convenient for the later modeling discussion, but not necessarily the standard
units used in the physiology literature.
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Variable
Description
Typical
baseline value
Example change
with activation
CBF (F)
Cerebral blood flow (ml arterial blood per ml tissue per min)
0.5 min−1
0.65 min−1 (+30%)
OEF (E)
Oxygen extraction fraction (dimensionless)
0.4
0.34 (−15%)
CMRO2 (R)
Cerebral metabolic rate of oxygen (micromoles O2 per ml tissue per min, or mM
min−1)
1.6mM
min−1
1.8mM min−1 (+12%)
CMRGlc
Cerebral metabolic rate of glucose (micromoles glucose per ml tissue per min, or
mM min−1)
0.3mM min−1
0.4mM min−1 (+30%)
CBV
Cerebral blood volume (dimensionless fraction of tissue volume)
0.05
0.055 (+10%)
[O2]a
Arterial oxygen concentration (micromoles O2 per ml blood)
8mM
8mM (―)
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