sensors
Article
Real-Time Electrical Bioimpedance Characterization
of Neointimal Tissue for Stent Applications
David Rivas-Marchena 1,2 ID , Alberto Olmo 3,4 , José A. Miguel 2 , Mar Martínez 2 ,
Gloria Huertas 1,3 and Alberto Yúfera 3,4, * ID
1
2
3
4
*
Department of Electronics and Electromagnetism, Faculty of Physics, University of Seville,
Av. Reina Mercedes sn, Seville 41012, Spain; david.rivas-marchena@ieee.org (D.R.-M.);
gloria@imse-cnm.csic.es (G.H.)
Group of Microelectronics Engineering, Department of Electronics Technology,
Systems Engineering and Automation, University of Cantabria, Santander 39005, Spain;
jamd@teisa.unican.es (J.A.M.); martinez@teisa.unican.es (M.M.)
Seville Institute of Microelectronics, Microelectronics National Center, Consejo Superior de Investigaciones
Científicas (IMSE-CNM-CSIC), Av. Americo Vespuccio, sn, Seville 41092, Spain; aolmo@dte.us.es
Computer Engineering School (ETSII), University of Seville, Av. Reina Mercedes sn, Seville 41012, Spain
Correspondence: yufera@imse-cnm.csic.es; Tel.: +34-954-466-666
Academic Editor: Nicole Jaffrezic-Renault
Received: 19 May 2017; Accepted: 25 July 2017; Published: 28 July 2017
Abstract: To follow up the restenosis in arteries stented during an angioplasty is an important current
clinical problem. A new approach to monitor the growth of neointimal tissue inside the stent is
proposed on the basis of electrical impedance spectroscopy (EIS) sensors and the oscillation-based test
(OBT) circuit technique. A mathematical model was developed to analytically describe the histological
composition of the neointima, employing its conductivity and permittivity data. The bioimpedance
model was validated against a finite element analysis (FEA) using COMSOL Multiphysics software.
A satisfactory correlation between the analytical model and FEA simulation was achieved in most
cases, detecting some deviations introduced by the thin “double layer” that separates the neointima
and the blood. It is hereby shown how to apply conformal transformations to obtain bioimpedance
electrical models for stack-layered tissues over coplanar electrodes. Particularly, this can be applied
to characterize the neointima in real-time. This technique is either suitable as a main mechanism
for restenosis follow-up or it can be combined with proposed intelligent stents for blood pressure
measurements to auto-calibrate the sensibility loss caused by the adherence of the tissue on the
micro-electro-mechanical sensors (MEMSs).
Keywords: bioimpedance; atherosclerosis; cardiology; oscillation-based test; stent
1. Introduction
Ischemic heart disease is the leading cause of morbidity and mortality in the world [1,2].
The interior of the elastic and muscular arteries shrinks (stenosis), mainly by the growth of a lipid
layer (atherosclerosis), causing angina chest pain and acute myocardial infarction. Among the existing
treatments, the most popular employs a flexible metal mesh (stent) to mechanically support the
opening of stenotic segments and ensure permeability [3,4]. However, despite improvements to the
surgery techniques and the stent design [5–7], there is a high probability of medium-term restenosis [8].
In [9], the use of micro-electro-mechanical pressure sensors (MEMSs) to monitor restenosis with
intelligent stents is proposed. The sensors are implemented as parallel-plate capacitors: one is a fixed
plate, whereas the other bends with increasing pressure. The distance between the plates determines
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a capacitance that is read out with an electronic circuit [10]. However, the growth of the neointimal
tissue on top of the flexible plate compromises its flexibility and degrades the sensor sensitivity.
Characterizing the neointima in real-time would ease the integration of an auto-calibration block
in the readout circuit to compensate for the sensitivity loss of the MEMSs [11]. Different publications
have recently emerged that use bioimpedance techniques to follow up the restenosis by studying
the growth of new tissue in the artery (neointimal layer). Some solutions provide the catheter with
a matrix of microelectrodes [12,13]. Other works place the electrodes directly on the stent [14]. It is also
proposed to use the stent itself as a large electrode [15]. These alternatives still need catheterization to
connect external measurement equipment, however.
A recently published method monitors cell-cultures in vivo based on their bioimpedance [16].
It consists of converting the cell-culture under test (CCUT) into a “biological oscillator”, whose
characteristic parameters (i.e., frequency, amplitude, phase, etc.) are related with the cell-culture
evolution and can be easily determined. This approach, known as an oscillation-based test (OBT), is
also valid for monitoring the neointimal tissue growth. A system topology, such as in Figure 1, amplifies
the bioimpedance and limits the outcome signal amplitude with a non-linear comparator. Closing the
loop, a filter imposes a quasi-sinusoidal signal to stimulate the tissue. The input resistor keeps the
μ
current injected to the blood vessel within safe margins, typically below 20 µA, for cell protection [16].
At the node Vout2 , the oscillation is analyzed, and all the relevant parameters are extracted.
Figure 1. Bioimpedance self-stimulated monitoring system based on an oscillation-based test (OBT) [16].
HBP is the Band-Pass filter, and Zbio is the biological impedance.
A solid understanding of the biological model is essential to link the experimental results with
the neointima variations. This paper is primarily focused on the development of such an analytical
model for the neointima, along with providing early proof-of-concept simulations for the applicability
of OBTs to the real-time characterization of neointimal tissue variations. In the Materials and Methods
section, a mathematical model of the neointima bioimpedance is presented from the different tissues
comprising it. The model is compared to finite element analysis (FEA) simulations, and the validity
of the model is shown in Section 3, along with a comparison between the analytical and simulated
data. The results are further discussed in the final section, for generalizing the model and proposing
improvements to it.
2. Materials and Methods
2.1. Analytical Model
The neointima bioimpedance depends on the thickness of the tissue layers that comprise it—the
lipid core, smooth muscle, fiber cap and endothelium. These are assumed to be stacked in parallel and
immersed in blood, as in Figure 2a. Selecting a four-electrode setup [17], the bioimpedance value of
the neointima layer is incorporated directly into the proposed system performance, without the need
to consider the electrode impedance contributions in a first analysis of the system, as happens when
using two-electrode setups [16]. The performance limitations imposed by the influence of electrodes in
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tetrapolar topologies should be considered for a correct circuit design. These electrodes are placed, in
our design, around the MEMS, as shown in Figure 2b, to maximize the transference impedance.
Figure 2. (a) Stack of neointimal tissue layers and parallel partial capacitances. (b) Layout of
current-carrying (CC) and pick-up (PU) electrodes.
2.1.1. Neointima–Blood Interface (“Double Layer”)
Blood is an electrolyte—it has free charge carriers—but the cellular structure immobilizes charge
in the neointima. The difference between the media produces a charge zone in which the blood
ions that are closer to the interface are fixed by the outermost tissue layer [18]. This charge zone
extends through the electrolyte, creating a capacitor that is included in the analytical model. The use
ε of εr = 17.8
of a triple-layer model [19] to describe the charge zone results in a relative permittivity
and a Debye length of 0.64 nm (4 to 5 orders of magnitude smaller than the usual tissue thicknesses).
Further influence of the intra-arterial blood pressure (depending on volume and flow) on electrical
impedance is studied in [20], although an analytical model explaining this relation was not presented
therein and should be object of future work.
2.1.2. Dielectric Interface
From the set of Maxwell equations, it is derived that the electric field angle (α)
α in a dielectric
interface changes by
E /E
E
D /ε
ε
tan α2
= 2,t 2,n = 1,n = 1,n ε 1 = 2
(1)
tan α1
E1,t /E1,n
E2,n
D2,n /ε 2
ε1
where Ei,t and Ei,n are the tangential and normal components of the electric field and Di,n are the
normal components of the electric displacement for each dielectric. For interfaces at which the media
permittivity varies largely, two different boundary conditions may apply:
Dirichlet Boundary Condition (DBC), for ε2 ≫ ε1 —constant electric potential at the interface
(Φ = 0).
ε ≫ε
Φ
•
Neumann Boundary Condition (NBC), for ε2 ≪ ε1 —no potential gradient normal to the interface
ε ≪ε
(∂Φ⁄∂n = 0).
∂Φ ∂
2.1.3. Partial Capacitance Method
•
Ghione and Goano [21] presented that the total equivalent capacitance between coplanar metal
tracks in printed circuit boards (PCB) can be derived from the partial contribution of each layer of
material deposited in parallel with them.
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The electrodes of the intelligent stent are so close, in comparison with the stent diameter, that they
can also be assumed to be coplanar; the neointima stratified in parallel layers of tissue are stacked on
top of them (Figure 2a).
If the permittivity decreases with the distance to the electrode plane, all the dielectric interfaces
can be modeled with NBCs. Then, the partial capacitances are effectively connected in parallel (parallel
partial capacitance—PPC), so that the equivalent capacitance is
Ceq =
i = n −1
∑
i =1
[ε r,i − ε r,i+1 ] CPPC (hi ) + ε r,n C (∞)
(2)
where n is the number of layers, hi is the height of each layer measured from the electrode plane, and
CPPC (hi ) is the void capacitance of each layer limited by a NBC.
For the opposite, if the permittivity increases, the partial capacitances are assumed to be in series
(series partial capacitance—SPC) with DBCs at the interfaces. In this case, the equivalent capacitance is
i = n −1
1
1
1
1
1
1
= ∑
−
+
Ceq
ε r,i
ε r,i+1 CSPC (hi ) ε r,n C (∞)
i =1
(3)
Because the plates of these capacitors—the electrodes—are coplanar, there is no simple formula to
compute capacitances. To cope with this, Igreja and Dias [22] apply conformal mapping techniques to
transform the disposition of electrodes into a parallel-plate configuration, whose capacitance is
C partial = ε r ε 0 L
K (k)
K (k′ )
(4)
and which depends on the tissue relative permittivity εr , the electrode depth L and the equivalent
width, derived from the mathematical transformation and described as a ratio of complete elliptic
√
integrals of the first kind (K) with modulus k (see Table 1) and its complementary k' = (1 − k2 ).
Table 1. Conformal mapping parameters [21].
Series Partial Capacitance (SPC)
q
t3
k SPC = tt44−
−1
π (1− η )
t3 = cosh
4
r
π (1+ η )
t4 = cosh
4r
Parallel Partial Capacitance (PPC)
r
t2 − t2
1
k PPC = t3 t42 −13
4
π (1− η )
t3 = cosh
8
r
π (1+ η )
t4 = cosh
8r
η = w/(w + g)
r = 2 h/(w + g)
In the equations, w and g are the width of the electrodes and the gap between them, respectively, and h is the height
of a given tissue layer.
2.1.4. Iterative Complex Method
The proposed PC method is only valid for stacks of material whose permittivities vary
monotonically. Blume [23] shows an iterative method compatible with any kind of stack that groups
layers in pairs and computes equivalent permittivities:
SPC : ε eq,i =
ε eq,i−1 ε i+1 κ (hi )
κ (hi+1 ) ε i+1 − ε eq,i−1 + ε eq,i−1 κ (hi )
SPC : ε eq,i = ε eq,i−1 − ε i+1
κ ( hi )
+ ε i +1
κ ( h i +1 )
where κ = K(k)/K(k’) is used to simplify the expressions.
(5)
(6)
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By replacing the permittivity with the complex permittivity (ε* = ε − jσ/ω), the iterative method
can calculate the equivalent impedance of any stack of parallel tissue
layers as
ω
εε εε −− σσ ω
h
i
Z = j ω L κ (∞) ε∗eq,n−1
(7)
κκ
A diagram of the iterative method is shown in Figure 3. The number of iterations that the
method needs to complete is equal to the number of stacked tissues minus 1. This is implemented
in MATLAB [24] and uses mainly the permittivity and conductivity data compiled by Gabriel and
Gabriel [25].
Figure 3. Iterative method to derive the analytical bioimpedance model.
The comparison between permittivities (Figure 4) determines the frequency range for which the
iterative method should apply SPC or PPC. Therefore, the method is predicted to be precise where
εε
≫
≪
≈ εεε1 (nearly no dielectric
ε≫
εε ε1 or
εε ε≪
εε ε1 (clearly defined boundary conditions), as well as where εε2 ≈≈
2 ≫
2 ≪
interface) [13].
Figure 4. Relative permittivity versus frequency for the blood and neointimal tissues.
The increment in total thickness after adding the “double layer” is quite small. This leads to
similar elliptic integrals whose difference is comparable to the methodical error introduced by their
analytical calculation. Hence, the results need an experimental adjustment to improve the accuracy.
2.2. Finite Element Analysis Simulations
The analytical bioimpedance model described previously was verified against FEA with COMSOL
Multiphysics software [26]. Several sample configurations aimed to emulate the growth of neointimal
tissue. In Table 2 is a list of tissue thicknesses for the different configurations.
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Table 2. Configurations of thicknesses for neointimal tissue layers.
Config.
1
2
3
4
5
6
Lipid Core
μ
0 µm
μ0 µm
25
μ µm
50
μ µm
100 µm
μ µm
200
μ
Muscle
μ
Fibrous Cap
Endothelium
25 µm
50μµm
50μµm
100μµm
100 µm
100μµm
μ
0 µm
0μ
µm
5μ
µm
10 μ
µm
20 µm
40 μ
µm
μ
10 µm
10μµm
10μµm
10μµm
10 µm
10μµm
μ
μ
μ
The permittivity of the endothelium was much larger than that derived for the double
ε layer
(εr = 17.8). Hence, this charge-zone effect could be modeled in the FEA simulation as an impedance
contact condition between the endothelium and the blood [27].
The simplest model of the system is a 2D cross-section, as shown in Figure 5. This replicates the
geometry of the mathematical transformations. The same assumptions made during the theoretical
analysis are repeated here (i.e., no border effects, same electric fields and current in all the cross-sections,
etc.). Hence, the validity of the bioimpedance can be easily proved against a simulation with the
same limitations.
Figure 5. Two-dimensional finite element analysis (FEA) simulation model for the configuration #5 in
Table 2. Electrodes were placed on the bottom left and right corners of the image (modeled without
thicknesses). The color plane represents the normalized current density, whereas the grey lines are
equipotentials (V). Higher current densities near the two electrodes can be observed.
Only two electrodes were modeled, with no thickness, to dismiss the perturbation that their
interface with the neointima would introduce. We note that this perturbation was minimized with
the four-electrode configuration chosen. These electrodes were used for both injecting AC voltage to
stimulate the tissues and to pick up the bioimpedance—derived by integrating along the current lines
between electrodes.
The mesh was automatically created by the software, following a sequence controlled by the
physics with a fine element size. This resulted in around 8000 triangles for the 2D model and
40,000 tetrahedrons in the 3D model. Their size shrunk around the boundaries.
3. Results
3.1. Analytical Model and 2D Finite Element Analysis Simulation Results
The histological configurations of Table 2 were sequentially fed into the iterative analytical
method to derive the equivalent permittivity and conductivity of the neointima shown in Figure 6a.
It was observed that the relative error for the six configurations was small and remained inside
the expected accuracy, given the error introduced by mathematical transformations and the ideal
boundary conditions.
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The final equivalent bioimpedance was derived by also computing the effect of the blood. At this
stage, an experimental adjustment was required to overcome the limitations of handling thin layers
such as the double layer, as mentioned in the Analytical Model section.
The comparison between the analytical model and the 2D FEA simulation results is depicted
in Figure 6b. The accuracy of the impedances is consistent with the relative errors computed for the
equivalent permittivities and conductivities, related by Equation (7). We can observe a larger relative
difference in phase than in magnitude, but it is a manageable difference for practical applications, such
as the analysis of cell growth, by tuning the frequency of oscillations appropriately [16].
Figure 6. Comparison of the analytical model (circles) and 2D FEA simulation (solid). (a) Relative
permittivity and conductivity. (b) Bioimpedance. In (a) and (b), dashed lines represent the relative error.
3.2. Generalization of the Bioimpedance Model
So far, the analytical model requires executing the iterative method for each histological
configuration. However, based on several samples, a general expression for the bioimpedance electrical
model in terms of its frequency dependence can be derived. It is assumed to have a gain, three poles
and three zeroes, such that
Zeq (s = jω ) = A
(s + z1 )(s + z2 )(s + z3 )
(s + p1 )(s + p2 )(s + p3 )
(8)
By running least-squares fitting on the data samples, the third zero is obtained to be fixed at
20 GHz, whereas the gain, the poles and the remaining zeroes depend linearly on the thickness of each
layer of neointimal tissue:
x = αα1 dlip + α
α2 dmus +αα3 d f ib +
(9)
α α4
where x is the parameter (gain, pole or zero), αi are
α the linear coefficients and dj are the thicknesses of the
lipid, muscle and fiber layers. Different linear coefficients are computed for histological configurations
without neo-atherosclerosis (i.e., no lipid core or fibrous cap), shown in Table 3 and, with it, Table 4.
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Table 3. Coefficients for Equation (9) in configurations without neo-atherosclerosis.
Param.
Lipid Core (α1 )
10−1
—
—
—
—
—
—
A
p1
p2
p3
z1
z2
Muscle (α2 )
4.60 ×
−1.17 × 10
2.45 × 103
2.56 × 106
2.30 × 10–1
−9.72 × 103
Fibrous Cap (α3 )
Independent (α4 )
—
—
—
—
—
—
1.72 × 102
4.52 × 102
1.82 × 106
8.59 × 108
6.28 × 102
4.24 × 106
Table 4. Coefficients for Equation (9) in configurations with neo-atherosclerosis.
Param.
Lipid Core (α1 )
Muscle (α2 )
Fibrous Cap (α3 )
Independent (α4 )
A
p1
p2
p3
z1
z2
1.17
−1.70 × 10−1
3.02 × 104
4.40 × 104
−2.10 × 10−1
−4.79 × 104
4.41
−9.50 × 10−1
1.80 × 105
2.99 × 105
−1.23
2.86 × 105
2.30 × 10−1
−3.00 × 10−2
6.03 × 103
8.80 × 103
−4.00 × 10−2
9.57 × 103
4.14 × 102
2.24 × 102
1.36 × 107
5.46 × 108
2.80 × 102
1.97 × 107
3.3. Oscillation-Based Test Auto-Calibration Circuit
The set of histological configurations can be enlarged to that contained in Table 5. This recreates
the usual case in which the patient firstly develops a neointima without atherosclerosis. Two subsets
of configurations with neo-atherosclerosis show the growth of a fibrous cap over thin and thick
muscle layers.
Table 5. Extended configurations of thicknesses of neointimal tissue layers.
Config.
Lipid Core
Muscle
Fibrous Cap
Endothelium
1
2
3
4
5
6
7
8
9
10
0 µm
0 µm
0 µm
0 µm
25 µm
100 µm
300 µm
50 µm
100 µm
200 µm
25 µm
50 µm
100 µm
200 µm
50 µm
50 µm
50 µm
100 µm
100 µm
100 µm
0 µm
0 µm
0 µm
0 µm
5 µm
20 µm
50 µm
10 µm
20 µm
40 µm
10 µm
10 µm
10 µm
10 µm
10 µm
10 µm
10 µm
10 µm
10 µm
10 µm
An interesting frequency to implement the OBT at is around 70 Hz, as there are large phase
variations in this area, as depicted in Figure 7. The circuit loop depicted in Figure 1 was implemented
in MATLAB for this frequency, as in [8]. The changes in neointimal histology caused the variations in
oscillation amplitude and frequency shown in Figure 8.
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Figure 7. Bioimpedance of neointima for the histological configurations in Table 5: (a) large Bode plot,
and (b) detailed plot at around 70 Hz.
Figure 8. OBT auto-calibration circuit amplitude and frequency for the histological configurations in
Table 5, measured at node Vout2 of Figure 1.
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These results prove the detection of neo-atherosclerosis by comparing the oscillation frequency
with its nominal value. Additionally, changes in the oscillation amplitude help to distinguish the
growth of different neointimal tissue layers. To simplify the measurements, the range of the oscillation
frequency variation can be enlarged by changing the quality factor of the band-pass filter.
4. Discussion
An analytical method has been presented to obtain the bioimpedance of a stack of parallel tissues
over coplanar electrodes, depending on the thickness of each layer and their electromagnetic properties.
It consists of an iterative approach based on conformal spatial transformations. The correlation
obtained between the analytical model and 2D FEA simulations proves to be enough for predicting
the evolution of the bioimpedance with a reasonable accuracy and minimal computational cost.
Based upon these results, Sections 3.2 and 3.3 delved into the application of the analytical model to
monitor the restenosis in vivo.
The analytical bioimpedance model gives a good prediction for the incremental changes in
the neointima layer because the output circuit response obtained (frequency and amplitude) is
monotonically dependent on neointima impedance changes. A calibration technique, for which some
initial OBT reference values are known, allows for the identification of variations in the neointimal
histology by observing increments in the amplitude and/or frequency of the oscillation, which could
be used in the real-time monitoring of neointimal tissue in stent applications.
Further work is required to improve the overall system accuracy, including for border effects,
anisotropy and other 3D non-idealities. Finally, a workaround to incorporate very thin layers, such as
the double layer, or to incorporate effects of blood volume and flow to the analytical model is needed.
Acknowledgments: This work was carried out in collaboration with the Cardiology Department of the UHMV
Hospital, Santander (Spain) and was funded by the Spanish Government’s “Ministerio de Economía, Industria y
Competitividad” under the joint projects TEC2013-46242-C3-1-P and TEC2013-46242-C3, co-financed with FEDER.
Author Contributions: David Rivas-Marchena developed the analytical model, and performed simulations with
MATLAB and COMSOL. Mar Martínez and José Ángel Miguel generated the basic idea and its application to
stents. Gloria Huertas, Alberto Olmo and Alberto Yúfera supervised the simulation work and its derivation to
oscillation-based solutions. All authors collaborated actively in the writing process.
Conflicts of Interest: The authors declare no conflict of interest.
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