A modeling-based evaluation of isothermal rebreathing for breath gas analyses of highly soluble volatile organic compounds Julian Kinga,b , Karl Unterkoflera,c, Gerald Teschlb , Susanne Teschld , Paweł Mochalskia,e, Helin Koçc , Hartmann Hinterhuberf, Anton Amanna,g,∗ a Breath Research Institute, Austrian Academy of Sciences, Rathausplatz 4, A-6850 Dornbirn, Austria of Vienna, Faculty of Mathematics, Nordbergstr. 15, A-1090 Wien, Austria c Vorarlberg University of Applied Sciences, Hochschulstr. 1, A-6850 Dornbirn, Austria d University of Applied Sciences Technikum Wien, Höchstädtplatz 5, A-1200 Wien, Austria e Institute of Nuclear Physics PAN, Radzikowskiego 152, PL-31342 Kraków, Poland f Innsbruck Medical University, Department of Psychiatry, Anichstr. 35, A-6020 Innsbruck, Austria g Innsbruck Medical University, Univ.-Clinic for Anesthesia, Anichstr. 35, A-6020 Innsbruck, Austria arXiv:1105.0864v1 [q-bio.QM] 4 May 2011 b University Abstract Isothermal rebreathing has been proposed as an experimental technique for estimating the alveolar levels of hydrophilic volatile organic compounds (VOCs) in exhaled breath. Using the prototypic test compound acetone we demonstrate that the end-tidal breath profiles of such substances during isothermal rebreathing show characteristics that contradict the conventional pulmonary inert gas elimination theory due to Farhi. On the other hand, these profiles can reliably be captured by virtue of a previously developed mathematical model for the general exhalation kinetics of highly soluble, blood-borne VOCs, which explicitly takes into account airway gas exchange as major determinant of the observable breath output. This model allows for a mechanistic analysis of various rebreathing protocols suggested in the literature. In particular, it clarifies the discrepancies between in vitro and in vivo blood-breath ratios of hydrophilic VOCs and yields further quantitative insights into the physiological components of isothermal rebreathing. Keywords: breath gas analysis, volatile organic compounds, highly soluble gas exchange, rebreathing, acetone, modeling 1. Introduction Over the past decade, much advance has been made in the attempt to extract the diagnostic and metabolic information encapsulated in a number of endogenous volatile organic compounds (VOCs) appearing in human exhaled breath (Amann et al., 2007; Amann and Smith, 2005; Miekisch et al., 2004). However, the quantitative understanding of the relationships between breath concentrations of such trace gases and their underlying systemic levels clearly lags behind the enormous analytical progress in breath gas analysis. In particular, formal means for evaluating the predictive power of various breath sampling regimes are still lacking. Recently, several efforts have been undertaken to complement VOC measurements with adequate physical models mapping substance-specific distribution mechanisms in the pulmonary tract as well as in the body tissues. Some major breath VOCs have already been investigated in this form, e.g., during rest and exercise conditions or exposure scenarios (Tsoukias and George, 1998; Kumagai and Matsunaga, 2000; Anderson et al., 2003; Mörk and Johanson, 2006; King et al., 2010a, 2011; Koc et al., 2011). Such mechanistic descriptions of the observable exhalation kinetics give valuable ∗ Corresponding author. Tel.: +43 676 5608520; fax: +43 512 504 6724636. Email address: anton.amann@oeaw.ac.at (Anton Amann) Preprint submitted to Respiratory Physiology & Neurobiology insights into the relevance of the measured breath concentrations with respect to the endogenous situation and hence are mandatory to fully exploit the diagnostic potential of breath VOCs. Within this context, the main focus of this article will be on a modeling-based review of isothermal rebreathing, which has been proposed as an experimental technique for estimating the alveolar levels of hydrophilic exhaled trace gases (Jones, 1983; Ohlsson et al., 1990; O’Hara et al., 2008). This class of compounds has been demonstrated to significantly interact with the water-like mucus membrane lining the conductive airways, an effect which has become known as wash-in/wash-out behavior. For further details we refer to Anderson et al. (2003); Anderson and Hlastala (2007). As a phenomenological consequence, exhaled breath concentrations of such highly water soluble substances tend to be diminished on their way up from the deeper respiratory tract to the airway opening. The resulting discrepancies between the “true” alveolar and the measured breath concentration can be substantial (even if breath samples are drawn in a strictly standardized manner employing, e.g., CO2 - or flow-controlled sampling) and will depend on a variety of factors, including airway temperature profiles and airway perfusion as well as breathing patterns (Tsu et al., 1991). In particular, the above-mentioned effect considerably departs from the classical Farhi description of pulmonary May 5, 2011 inert gas exchange (Farhi, 1967), on the basis of which the operational dogma has been established that end-tidal air will reflect the alveolar level and that arterial concentrations can be assessed by simply multiplying this value with the blood:gas partition coefficient λb:air at body temperature. This “common knowledge” has first been put into question in the field of breath alcohol testing, revealing observable blood-breath concentration ratios of ethanol during tidal breathing that are unexpectedly high compared to the partition coefficient derived in vitro (Jones, 1983). Similarly, excretion data (i.e., the ratios between steady state partial pressures in expired air and mixed venous blood) of highly water soluble compounds (including the MIGET test gas acetone) have been shown to underestimate the values anticipated by treating the airways as an inert tube (Zwart et al., 1986; Anderson and Hlastala, 2010). at m/z = 59 (dwell time 200 ms). Additionally, we routinely measure the mass-to-charge ratios m/z = 21 (isotopologue of the primary hydronium ions used for normalization; dwell time 500 ms), m/z = 37 (first monohydrate cluster H3 O+ (H2 O) for estimating sample humidity; dwell time 2 ms), m/z = 69 (protonated isoprene; dwell time 200 ms), m/z = 33 (protonated methanol; dwell time 200 ms) as well as the parasitic precursor ions NH+4 and O+2 at m/z = 18 and m/z = 32, respectively, with dwell times of 10 ms each. In particular, following O’Hara et al. (2008), the pseudo concentrations associated with m/z = 32 determined according to standard PTR-MS practice (see, e.g., (Schwarz et al., 2009a, Eq. (1))) will be used as a surrogate for the end-tidal oxygen partial pressure PO2 (relative to an assumed nominal steady state level at rest of 100 mmHg). Similarly, calibrated pseudo concentrations corresponding to m/z = 37 are considered as an indicator for absolute sample humidity Cwater , see (King et al., 2011) for further details. Partial pressures PCO2 of carbon dioxide are obtained via a separate sensor. Table 1 summarizes the measured quantities used in this paper. In general, breath concentrations will always refer to end-tidal levels. In a previously published mathematical model for the breath gas dynamics of highly soluble blood-borne trace gases, airway gas exchange is taken into account by separating the lungs into a bronchial and alveolar compartment, interacting via a diffusion barrier mimicking pre- and post-alveolar uptake (King et al., 2011). This formulation has proven its ability to reliably capture the end-tidal breath profiles as well as the systemic dynamics of acetone in a variety of experimental situations and will be used here for illuminating the physiological processes underlying isothermal rebreathing tests as carried out in the literature. For comparative reasons, the illustration in the sequel will mainly be limited to acetone, with possible extrapolations to other highly soluble VOCs indicated where appropriate. Variable Symbol Nominal value (units) Acetone concentration Cmeasured 1 (µg/l) (Schwarz et al., 2009b) CO2 partial pressure PCO2 40 (mmHg) (Lumb, 2005) Water content Cwater 4.7 (%) (Hanna and Scherer, 1986) O2 partial pressure P O2 100 (mmHg) (Lumb, 2005) Table 1: Summary of measured breath parameters together with some nominal values in end-exhaled air during tidal breathing at rest. 2. Methods 2.1. Experiments 2.2. Physiological model Extensive details regarding our experimental setup are given elsewhere (King et al., 2009, 2010b, 2011). Here, we will only briefly discuss the parts relevant in the context of isothermal rebreathing. All phenomenological results were obtained in conformity with the Declaration of Helsinki and with approval by the Ethics Commission of Innsbruck Medical University. A schematic sketch of the model structure is presented in Fig. 1. For the associated compartmental mass balance equations we refer to Appendix A as well as to the original publication (King et al., 2011). Briefly, the body is divided into four distinct functional units, for which the underlying concentration dynamics of the VOC under scrutiny will be taken into account: bronchial/mucosal compartment (Cbro ; gas exchange), alveolar/end-capillary compartment (CA ; gas exchange), liver (Cliv ; production and metabolism) and tissue (Ctis ; storage). The nomenclature is detailed in Table A.1. The rebreathing system itself consists of a Tedlar bag (SKC, Dorset, UK) with a volume of Ṽbag = 3 l, that can directly be connected to a spirometer face mask covering mouth and nose. From the latter, end-tidal exhalation segments are drawn into a Proton Transfer Reaction Mass Spectrometer (PTR-MS; Ionicon Analytik GmbH, Innsbruck, Austria), which allows for VOC detection and quantification on a breath-by-breath resolution as described in (King et al., 2009). The rebreathing bag is warmed to 37 ± 1◦ C by means of a specially designed outer heating bag (Infroheat Ltd., Wolverhampton, UK), cf. (O’Hara et al., 2008). Heating is intended to assist the thermal equilibration between the alveolar tract and the upper airways and prevents condensation and subsequent losses of hydrophilic VOCs depositing onto water droplets forming on the surface wall of the bag. End-tidal acetone concentrations are determined by monitoring the protonated compound The measurement process is described by Cmeasured = Cbro , (1) i.e., we assume that the measured (end-tidal) VOC concentration reflects the bronchial level. Dashed boundaries indicate a diffusion equilibrium, described by the appropriate partition coefficients λ, e.g., λb:air . The nomenclature is detailed in Table A.1. 2 the model. More specifically, the factor z captures the change of VOC solubility in the airway mucosa and bronchial blood in response to fluctuations of the characteristic mean airway and bronchial blood temperature T̄ (in ◦ C). In King et al. (2011, Eqs. (3)-(5)) the latter has been shown to be estimable by virtue of the measured sample humidity Cwater and typically ranges around 34◦ C during tidal breathing of room air at rest. It is assumed that the temperature dependence of λb:air is proportional to the temperature change of the mucosa:air partition coefficient λmuc:air over the temperature range considered, i.e. z is given by z(T̄ ) := λmuc:air (T̄ )/λmuc:air (37◦ C). (4) Remark 1. Note that z decreases with increasing temperature and takes values greater than one for T̄ below body temperature. A particular convenience of the above representation stems from the fact that, as a first approximation, the mucosa layer can be assumed to inherit the physico-chemical properties of water. Correspondingly, λmuc:air can be estimated via the respective water:air partition coefficient, which is usually available from the literature (see, e.g., the extensive compendium in (Staudinger and Roberts, 2001)). A central role is played by the gas exchange location parameter D, mimicking pre- and post-alveolar uptake in the mucosa. As has been discussed in (King et al., 2011), this quantity is a re-interpretation of stratified conductance according to the series inhomogeneity model by Scheid et al. (1981) (see also (Hlastala et al., 1981)) and will be close to zero for highly water soluble substances during tidal breathing at rest. Moreover, an increase of D with ventilatory flow can be expected, which will be modeled here as Figure 1: Sketch of the model structure used for capturing dynamic VOC concentrations (C). Subscripts connote as follows: bag–rebreathing bag; I–inhaled; bro–bronchial; muc–mucosal; A–alveolar; c′ –end-capillary; liv– liver; tis–tissue; b–blood. In particular, for highly water- and blood-soluble compounds such as acetone the present model replaces the familiar Farhi equation describing the steady state relationship between inhaled (ambient) concentration CI , measured breath concentraFarhi tions Cmeasured , mixed venous concentrations Cv̄ and arterial concentrations Ca during tidal breathing (Farhi, 1967), Farhi Cmeasured = CA = V̇A C Q̇c I + Cv̄ λb:air + V̇A Q̇c = Ca , λb:air D := kdiff,1 max{0, VT − VTrest } + kdiff,2 max{0, V̇A − V̇Arest }, (5) where V̇A and VT denote alveolar ventilation and tidal volume, respectively. The positive coefficients kdiff,i can be estimated from experimental data, cf. (King et al., 2011) and Table A.1. (2) 3. Results and discussion with the expression Cmeasured = Cbro = rbro CI + Ca rbro CI + (1 − qbro )Cv̄ = . (1 − qbro )z(T̄ )λb:air + rbro z(T̄ )λb:air + rbro 3.1. Heuristic considerations As has already been indicated in the introduction, Equation (2) is inappropriate for capturing experimentally obtained arterial blood-breath concentration ratios (BBR) of highly water soluble trace gases during free breathing at rest (i.e., assuming CI = 0). For instance, in the specific case of acetone, multiplying the proposed end-tidal population mean of approximately 1 µg/l (Schwarz et al., 2009b) with a blood:gas partition coefficient of λb:air = 340 (Anderson et al., 2006) at body temperature appears to grossly underestimate arterial blood levels spreading around 1 mg/l (Kalapos, 2003; Wigaeus et al., 1981). In contrast, Equation (3) asserts that the observable arterial blood-breath ratio in this case is (3) Here, qbro ≈ 0.01 is an estimate of the effective fractional bronchial perfusion (which might be substance-specific), while rbro = V̇A qbro Q̇c denotes the associated bronchial ventilation-perfusion ratio. The term z(T̄ )λb:air reflects an explicit temperature dependence of airway gas exchange that has been incorporated into BBR = 3 Ca Cmeasured = z(T̄ )λb:air + rbro , (6) and will thus depend on airway temperature and airway blood flow as mentioned in the previous section. In particular, note that the BBR will be greater than the in vitro blood:gas partition coefficient λb:air for T̄ below body temperature, cf. Remark 1. Formally, a model capturing the experimental situation during isothermal rebreathing can simply be derived by augmenting the model equations in Appendix A with an additional compartment representing the rebreathing receptacle, i.e., Remark 2. From Equation (6) it is clear that the more soluble a VOC under scrutiny, the more drastically its observable BBR will be affected by the current airway temperature, with an inverse relation between these two quantities. This deduction is consistent with measurements by Ohlsson et al. (1990) conducted in the field of alcohol breath tests, reporting a monotonous decrease of ethanol BBR values with increasing exhaled breath temperature. In the same contribution, BBRs of ethanol during normal tidal breathing were shown to typically exceed a value of 2500, which differs from the expected value λb:air = 1756 by more than 40%. For comparison, based on a mean characteristic airway temperature of T̄ = 32◦ C and the water:air partition coefficient λmuc:air provided by Staudinger and Roberts (2001, p. 568), by substituting the parameter values in Table A.1 we find that Equation (6) predicts an experimentally observable blood-breath ratio of ethanol in the range of 2560. Equation (6) also suggests that in the case of VOCs with an extremely high affinity for both blood and water (e.g., ethanol, methanol), the discrepancies between in vivo and in vitro BBR values might largely be removed by warming the airways to body temperature before the breath sampling procedure. Contrarily, blood-breath ratios of less soluble VOCs, e.g., acetone, will additionally be affected by the comparatively large value of rbro , which stems from the diffusion disequilibrium between the alveolar and bronchial space. dCbag Ṽbag = V̇A (Cbro − Cbag ) dt (7) and by setting CI = Cbag in Equation (A.1). Following the line of argument in (King et al., 2011, Appendix B), it can easily be checked that all fundamental system properties discussed there remain valid. In particular, if we assume that the conducting airways are warmed to body temperature, i.e., if z(T̄ ) in Equation (4) approaches unity as temperature increases, we may conclude that the compartmental concentrations will tend to a globally asymptotically stable steady state obeying rebr rebr rebr Cmeasured = Cbro = Cbag = CArebr = Cv̄rebr Carebr = . λb:air λb:air (8) This is a simple consequence of substituting CI with Cbro in Equation (3) according to the steady state relation associated with Equation (7). It is important to realize that if steady state conditions hold they will depend solely on the blood:air partition coefficient λb:air at 37◦ C, thus rendering isothermal rebreathing as an extremely stable technique for providing a reproducible coupling between VOC levels in breath and blood. Particularly, it theoretically avoids the additional measurement of ventilation- and perfusion-related variables that would otherwise affect this relationship, thereby significantly simplifying the required technical setup for breath sampling. However, as will be illustrated in the following, the major practical obstacle is to guarantee that a steady state as in Equation (8) is effectively attained. Apart from providing some experimental evidence for the validity of Equation (6), these ad hoc calculations suggest that the common practice of multiplying the measured breath concentration Cmeasured with λb:air to obtain arterial concentrations for highly soluble trace gases will result in an estimation that might drastically differ from the true blood level. Isothermal rebreathing has been put forward as a valuable method for removing the above-mentioned discrepancies. The heuristic intention leading to isothermal rebreathing is to create an experimental situation where the alveolar levels of highly soluble VOCs are not altered during exhalation due to loss of such substances to the cooler mucus layer of the airways. This can be accomplished by “closing the respiratory loop”, i.e., by continuous re-inspiration and -expiration of a fixed mass of air from a rebreathing receptacle (e.g., a Tedlar bag), causing the airstream to equilibrate with the mucosa linings over the entire respiratory cycle (Ohlsson et al., 1990; Anderson et al., 2006; O’Hara et al., 2008). Additionally, warming the rebreathing volume to body temperature will ensure a similar solubility of these VOCs in both regions, alveoli and airways. For the purpose of comparing the qualitative implications of Equation (2) and Equation (3), assume that Cv̄rebr ≈ Cv̄ , i.e., the mixed venous concentrations stay constant during rebreathing (which – at least in the first phase of rebreathing – can be justified to some extent by reference to tissue-to-lung transport delays of the systemic circulation (Grodins et al., 1967)). Then the ratios between measured rebreathing concentrations and end-exhaled concentrations during free tidal breathing at rest predicted by Equation (2) and Equation (3) are found to follow an entirely different trend. To this end, note that while in the first case we find that Farhi,rebr Cmeasured Farhi Cmeasured λb:air + → λb:air V̇A Q̇c , (9) the present model yields rebr Cmeasured (1 − qbro )z(T̄ )λb:air + rbro → , Cmeasured (1 − qbro )λb:air Remark 3. For the sake of completeness, it should be noted that a small part of the air within the rebreathing circuit outlined in Section 2.1 is actually consumed by the PTR-MS device (roughly 40 ml/min), which, however, can safely be negelected in the present setting. (10) where z(T̄ ) ≥ 1, cf. Remark 1. For highly soluble trace gases, this observation constitutes a simple test for assessing the adequacy of the Farhi formulation regarding its ability to describe 4 the corresponding exhalation kinetics. Indeed, for sufficiently large λb:air , the right-hand side of Equation (9) will be close to one, while the right-hand side of Equation (10) suggests that rebreathing will increase the associated end-tidal breath concentrations. In other words, for this class of compounds a markedly non-constant behavior during the initial isothermal rebreathing period indicates that Equation (2) will fail to capture some fundamental characteristics of pulmonary excretion. Such tests are of particular importance in the context of endogenous MIGET methodology (Multiple Inert Gas Elimination Technique, based on endogenous rather than externally administered VOCs (Anderson and Hlastala, 2010)), as they might be used for detecting deviations of the employed gases from the underlying Farhi description (see also (King et al., 2010b)). during free breathing, cf. Fig. 2. Here, for simplicity it is assumed that end-tidal PCO2 levels reflect those of the peripheral and central chemoreceptor environment. The alveolar ventilation follows from V̇A = f (VT − VD ), (12) using a deadspace volume of VD = 0.1 l. Since cardiac output can be expected to stay relatively constant during the rebreathing phase (O’Hara et al., 2008), we fix its value at a nominal level of 6 l/min. Using the parameter values in Table A.1 for a male of height 180 cm and weight 70 kg this completes the necessary data for simulating the above-mentioned rebreathing experiment. More specifically, the model response in the first panel of Fig. 2 is obtained by integrating the differential equations (A.1)–(A.4) and (7), setting CI = Cbag for t ∈ [tstart , tend ] and CI ≡ 0 otherwise. The initial concentrations for each compartment as well as the underlying endogenous acetone production rate kpr = 0.055 mg/min for this specific experiment are derived from the algebraic steady state conditions at t = 0. 3.2. Single cycle rebreathing In this section we will discuss some simulations and preliminary experiments conducted in order to study the predictive value of isothermal rebreathing within a realistic setting. For this purpose we will first mimic isothermal rebreathing as it has been carried out by various investigators (Jones, 1983; Ohlsson et al., 1990). The volume of the rebreathing bag is Ṽbag = 3 l according to Section 2.1. [µg/l] 1 Typical profiles of end-tidal acetone, water, CO2 and oxygen content during normal breathing and isothermal rebreathing at rest are shown in Fig. 2. These representative data correspond to one single normal healthy male volunteer from the study cohort in (King et al., 2009). As has been explained in Section 2.1, PO2 is derived by scaling the end-tidal steady state of the PTR-MS pseudo concentration signal at m/z = 32 to a basal value of 100 mmHg during free tidal breathing. Rebreathing was instituted at tstart = 2 min by inhaling to total lung capacity and exhaling until the bag was filled, thereby providing an initial bag concentration which can be assumed to resemble the normal end-exhaled steady state, i.e., 0.6 0 1 2 [µg/l] C [µg/l] 3 4 5 6 0.3 1 A C bag [µg/l] Cliv [mg/l] C tis [mg/l] 0.8 0.2 0.6 0.4 0 1 2 3 4 5 6 100 (11) 6 4 50 P CO2 Rebreathing was then continued until either the individual breathing limit was reached or the CO2 partial pressure increased above 55 mmHg (corresponding to tend ≈ 4.6 min for the volunteer in Fig. 2). Breathing frequency f and tidal volume VT were simulated on the basis of the monitored values for PCO2 and PO2 . Under iso-oxic conditions, after a certain threshold value is exceeded, both frequency and tidal volume are known to increase linearly with alveolar PCO2 . Moreover, the corresponding slopes (reflecting the chemoreflex sensitivity of breathing) are dependent on the current alveolar oxygen partial pressure (Mohan and Duffin, 1997; Lumb, 2005). More specifically, hypoxia during rebreathing enhances chemoreflex sensitivity, yielding a hyperbolic relation between the mentioned slopes and PO2 . These findings result in a simple model capturing the chemoreflex control of breathing in humans (Duffin et al., 2000; Duffin, 2010), which has been reimplemented in order to compute f and VT from basal values 0 0.1 0 1 [mmHg] 2 P [mmHg] H O content [%] O2 3 2 2 4 5 6 3 150 tidal volume [l] alveolar ventilation [l/min] 100 [l] 2 1 0 [%] [µg/l] bro [mg/l] C 50 0 1 2 3 [min] 4 5 6 [l/min] 0.4 [mmHg] Cbag (0) = Cbro (0). breath acetone breath methanol [a.u.] present model Farhi model 0.8 0 Figure 2: Representative outcome of an isothermal rebreathing experiment during rest. Data correspond to one single normal healthy male volunteer from the study cohort in (King et al., 2009). Isothermal rebreathing starts at tstart = 2 min and ends at approximately tend = 4.6 min. Measured or derived quantities according to the experimental setup are shown in the first and third panel (red tracings), while data in the second and fourth panel correspond to simulated variables as described in the text (black tracings). From Fig. 2, the proposed model is found to faithfully reproduce the observed data, which extends the range of exper5 {kpr , kmet , qbro , kdiff,1 , kdiff,2 }. Among these, the decisive quantities are the blood:gas partition coefficient (negative influence; ς(λb:air ) = 2.07) as well as the initial concentrations and partition coefficients for the liver and tissue compartment (positive influence; ς(Cliv (0)) = 0.06; ς(λb:liv ) = 0.06; ς(Ctis (0)) = 1.39; ς(λb:tis ) = 1.39). In particular, the latter essentially define the mixed venous blood acetone concentration Cv̄ according to Equation (A.5). As this concentration stays almost constant throughout the entire experiment, the above sensitivity analysis results are in direct agreement with Equation (8), predicting a rebreathing steady state that exclusively depends on λb:air and Cv̄ . All other sensitivity indices take values below 0.015. imental regimes for which the underlying formalism has been validated. Further improvement of the goodness-of-fit might be achieved by employing parameter estimation techniques as described, e.g., in King et al. (2010a, 2011) which, however, would be beyond the scope of this paper. In contrast, as can be anticipated from Equation (9), the classical Farhi model fails to capture the given breath profile1 . In particular, the presented data appear to consolidate the heuristic considerations in Section 3.1 and confirm that the alveolar concentration of acetone during free tidal breathing can differ from the associated bronchial (i.e., measured endexhaled) level by a factor of more than 1.5. This is due to an effective diffusion disequilibrium between the conducting airways and the alveolar space. During isothermal rebreathing, the diffusion barrier slowly vanishes and causes the measured breath concentration to approach the underlying alveolar concentration (which itself stays relatively constant during the entire experiment). We stress the fact that in order to simulate a similar response by using the conventional Farhi model, one essentially would have to postulate a temporarily increased endogenous acetone production during rebreathing, which, however, lacks physiological plausibility. 3.3. Sequential rebreathing On the basis of the results from the previous subsection, in the following we will briefly discuss a sequential rebreathing protocol developed by O’Hara et al. (2008). This regime aims at improving the patient compliance of conventional rebreathing by repeatedly providing cycles of five rebreaths (postulated to last approximately 0.5 min) with intermediate periods of free tidal breathing lasting approximately 10 min. According to above-mentioned protocol, isothermal rebreathing is again instituted by inhaling to total lung capacity and exhaling to residual volume into a Tedlar bag with a volume of Ṽbag = 3 l. After each rebreathing cycle, the bag is closed, a small amount of bag air (< 100 ml) is measured and the volunteer starts the next cycle by exhaling to residual volume and inhaling from the bag. For comparative reasons, in Fig. 2 we also display the simultaneously obtained PTR-MS concentration profile of breath methanol, scaled to match the initial level of breath acetone. Taking into account a methanol blood:gas partition coefficient of λb:air = 2590 at body temperature (Kumagai and Matsunaga, 2000), from Equation (10) it can be deduced that for this compound the differences between concentrations extracted during free breathing and rebreathing primarily stem from the thermal equilibration between airways and alveolar tract. The corresponding rise in temperature is mirrored by a steady increase of sample water vapor Cwater , approaching an alveolar level of about 6.2%. In particular, the presented profile for methanol shows the necessity of including an explicit temperature dependence in models describing the rebreathing behavior of highly soluble VOCs. From the data in the last two panels of Fig. 2 it can be inferred that all physiological input variables will have returned to pre-rebreathing values within the 10 min breaks. For simulation purposes, it will hence be assumed that their behavior during repeated rebreathing segments is identical to the profiles within the first 0.5 min of single cycle rebreathing. Values for the initial compartment concentrations as well as for the additional parameters are adopted from the previous subsection. These premises allow for simulating the evolution of breath acetone within a repeated rebreathing regime as displayed in Fig. 3. Here, the initial bag concentration at the onset of each individual rebreathing cycle is determined by the final bag concentration after the preceding rebreathing cycle, i.e., no fresh room air enters the bag. The bag concentration profile in Fig. 3 qualitatively resembles the data presented by O’Hara et al. (2008). However, what emerges from this modeling-based analysis is that in spite of steadily increasing bag concentrations (finally reaching a plateau level), the latter might not necessarily approach the underlying alveolar concentration as in the case of single cycle rebreathing. The major reason for this is a lack of complete thermal and diffusional equilibration between the airways and the alveolar region within the individual rebreathing segments. One potential way to circumvent this issue would be to reduce the desaturation and cooling of the airway tissues between consecutive rebreathing segments by keeping the intermediate time interval of free tidal breathing as short as possible (while si- Remark 4. A ranking of specific model parameters and initial conditions pi with respect to their impact on the observable breath acetone concentration during the isothermal rebreathing period [tstart , tend ] can be obtained by numerically approximating the squared L2 -norm of the corresponding normalized sensitivities, viz., ς(pi ) := Ztend tstart pi ∂Cmeasured (t) ∂pi max s |y(s)| !2 dt. (13) Adopting the nomenclature in Appendix A, these indices were calculated for all effective compartment volumes, partition coefficients, initial concentrations, and for pi ∈ 1 Considering the fact that the Farhi formulation is included in the present model as a limiting case for qbro = 0 and D → ∞ (King et al., 2011, Fig. 4), its associated output can be computed in a similar manner as described above. 6 multaneously maintaining a regime allowing for comfortable breathing). The second panel in Fig. 3 displays the evolution of the predicted blood-bag concentration ratios during the course of experimentation. Note that the in vitro blood:gas partition coefficient λb:air = 340 is never attained. This observation can offer some explanation for the discrepancies that continue to exist with regard to theoretical and experimentally measured ratios between blood and (rebreathed) breath levels (O’Hara et al., 2009). Furthermore, the final plateau value and the observable BBR of acetone will vary with temperature (cf. Equation (6)), which is consistent with similar observations made in the case of breath ethanol measurements (Ohlsson et al., 1990). quantitative insights into the discrepancies between in vitro and measured blood-breath ratios of such VOCs during steady state. Dynamic data are presented for one single representative subject only, inasmuch as our main emphasis was on describing and clarifying some fundamental features and physiological mechanisms related to the observable VOC behavior during the above-mentioned experimental regime. Several practical implications emerge from this modelingbased analysis. Firstly, it is demonstrated that the classical Farhi setting will fail to reproduce the experimentally measured acetone exhalation data during isothermal rebreathing if a constant endogenous production and metabolism rate is postulated. This is due to the fact that airway gas exchange, being a major determinant affecting highly soluble gas exchange, is not taken into account within this formalism. Furthermore, multiplying end-tidal breath concentrations during free tidal breathing with the substance-specific blood:gas partition coefficient λb:air will generally underestimate the true arterial blood concentration for highly blood and water soluble VOCs. Excessive hypoxia and hypercapnia are the main factors limiting the duration of the rebreathing maneuver and preventing a complete equilibration of VOC partial pressures in the alveoli and the conducting airways. From an operational point of view, our data indicate that even if isothermal rebreathing is continued until the individual breathing limit is reached, a steady state according to Equation (8) might not necessarily be attained. As can be deduced from Fig. 2 in the case of acetone, end-exhaled breath (or bag) concentrations extracted after about 0.5 min of rebreathing (corresponding to the common protocol of providing around five consecutive rebreaths) are still likely to underestimate the underlying alveolar level CA . Analogous conclusions can be drawn for sequential rebreathing protocols designed to allow for a recovery of the volunteer during the individual rebreathing segments. A reliable extraction of meaningful breath levels for highly soluble VOCs by virtue of isothermal rebreathing hence appears to require more sophisticated setups incorporating the continuous removal of CO2 and replacement of metabolically consumed oxygen. Provided that the influence of chemical CO2 -absorption on the measured breath and bag concentrations is negligible, such setups might for instance be adapted from closed chamber techniques (Filser, 1992) or general closedcircuit anesthesia systems. Alternatively, alveolar concentrations might be extrapolated to some extent from partially equilibrated rebreathing samples, using numerical parameter estimation schemes for reconstructing VOC exhalation kinetics according to a given model structure. While further validation and data gathering needs to be carried out before such estimates can become practically relevant, the mechanistic descriptions discussed in this paper are intended as a first step towards achieving this goal. Figure 3: Simulation of a sequential rebreathing protocol according to O’Hara et al. (2008) with intermediate pauses of 10 min characterized by free tidal breathing. The underlying model parameters correspond to the individual fit in Fig. 2. Dash-dotted lines represent upper and lower bounds for the bag concentration as well as for the observable blood-bag ratio with respect to changes in airway temperature T̄ . These bounds were obtained by assuming that the airway temperature either instantaneously rises to body core temperature during the individual rebreathing periods or remains constant (i.e., at its initial level T̄ 0 ) throughout the entire experiment. 4. Conclusions Here we have successfully applied a previously published compartment model for the exhalation kinetics of highly soluble, blood-borne VOCs to the experimental framework of isothermal rebreathing. The proposed model has proven sufficiently flexible for capturing the associated end-tidal breath dynamics of acetone, which can be viewed as a prototypical test compound within this context. Moreover, it gives new Acknowledgements The research leading to these results has received funding from the European Communitys Seventh Framework Pro7 respectively. Here, gramme (FP7/2007-13) under grant agreement No. 217967. We appreciate funding from the Austrian Federal Ministry for Transport, Innovation and Technology (BMVIT/BMWA, Project 818803, KIRAS). Gerald Teschl and Julian King acknowledge support from the Austrian Science Fund (FWF) under Grant No. Y330. We greatly appreciate the generous support of the government of Vorarlberg and its governor Landeshauptmann Dr. Herbert Sausgruber. Cv̄ := qliv λb:livCliv + (1 − qliv )λb:tisCtis (A.5) Ca := (1 − qbro )λb:airCA + qbro z(T̄ )λb:airCbro (A.6) and are the associated concentrations in mixed venous and arterial blood, respectively. Appendix A. Model equations and nomenclature Parameter This appendix serves to give a roughly self-contained outline of the model structure sketched in Fig. 1 (King et al., 2011). The time evolution of VOC concentrations is captured by taking into account standard conservation of mass laws for the individual compartments. Local diffusion equilibria are assumed to hold at the airtissue, tissue-blood and air-blood interfaces, the ratio of the corresponding concentrations being described by the appropriate partition coefficients λ, e.g., λb:air . Unlike for low blood soluble compounds, the amount of highly soluble gas dissolved in the local blood volume of perfused compartments cannot generally be neglected, as it might significantly increase the corresponding compartmental capacities. This is particularly true for the airspace compartments. We hence use the effective compartment volumes Ṽbro := Vbro + Vmuc λmuc:air , ṼA := VA + Vc′ λb:air , Ṽliv := Vliv + Vliv,b λb:liv as well as Ṽtis := Vtis and neglect blood volumes only for the mucosal and tissue compartment. Values for the individual compartment volumes, temperature-dependent partition coefficients, and physiological variables such as cardiac output (Q̇c ) and alveolar ventilation (V̇A ) are given in Table A.1. Fractional blood flows to the bronchial tract (qbro ) and the liver (qliv ) as well as the endogenous production (kpr ) and metabolic elimination rates (kmet ) are treated as constants in the context of isothermal rebreathing. According to Fig. 1, the mass balance equation for the bronchial compartment reads dCbro Ṽbro = V̇A (CI − Cbro ) + D(CA − Cbro ) dt
+ qbro Q̇c Ca − z(T̄ )λb:air Cbro , Bronchioles Vbro 0.1 (l)a Mucosa Vmuc 0.005 (l)a Alveoli VA 4.1 (l)a End-capillary Vc′ 0.15 (l)b Liver Vliv 0.0285 LBV (l)a Blood liver Vliv,b 1.1 (l)c Tissue Vtis 0.7036 LBV (l)a Rebreathing bag Ṽbag 3 (l)a Respiratory parameters Breathing frequency f 12.5 (tides/min)d Tidal volume VT 0.593 (l)d Alveolar ventilation V̇A 6.2 (l/min)e Cardiac output Q̇c 6 (l/min) f Fractional flow bronchioles qbro 0.01g Fractional flow liver qliv 0.32a λb:air 340h Mucosa:air λmuc:air 392i, j Blood:liver λb:liv 1.73 j Blood:tissue λb:tis 1.38h Linear metabolic rate kmet 0.0074 (l/kg0.75 /min)l Endogenous production kpr 0.19 (mg/min)l Stratified conductance D 0 (l/min)l Blood flows Partition coefficients (37◦ C) Blood:air Metabolic and diffusion constants (A.1) Constant Eq. (5) kdiff,1 14.9 (min−1 )l Constant Eq. (5) kdiff,2 0.76l Table A.1: Basic model parameters and nominal values during rest. LBV denotes the lean body volume in liters calculated according to LBV = −16.24 + 0.22 bh + 0.42 bw, with body height (bh) and weight (bw) given in cm and kg, respectively (Mörk and Johanson, 2006); a (Mörk and Johanson, 2006); b (Hughes and Morell, 2001); c (Ottesen et al., 2004); d (Duffin et al., 2000); e cf. Eq. (12); f (Mohrman and Heller, 2006); g (Lumb, 2005); h (Anderson et al., 2006); i (Staudinger and Roberts, 2001); j (Kumagai and Matsunaga, 1995); l (King et al., 2011). dCA
ṼA = D(Cbro − CA ) + (1 − qbro )Q̇c Cv̄ − λb:air CA , (A.2) dt and (A.3) References and
dCtis Ṽtis = (1 − qliv )(1 − qbro )Q̇c Ca − λb:tisCtis , dt Nominal value (units) Compartment volumes where CI denotes the inhaled (ambient/bag) gas concentration. Similarly, for the alveolar, liver and tissue compartment we find that dCliv Ṽliv = kpr − kmet λb:livCliv dt
+ qliv (1 − qbro )Q̇c Ca − λb:livCliv , Symbol Amann, A., Smith, D. (Eds.), 2005. Breath Analysis for Clinical Diagnosis and Therapeutic Monitoring. World Scientific, Singapore. (A.4) 8 Amann, A., Spanel, P., Smith, D., 2007. Breath analysis: the approach towards clinical applications. Mini reviews in Medicinal Chemistry 7, 115–129. Anderson, J. C., Babb, A. L., Hlastala, M. P., 2003. Modeling soluble gas exchange in the airways and alveoli. Ann. Biomed. Eng. 31, 1402–22. Anderson, J. C., Hlastala, M. P., 2007. Breath tests and airway gas exchange. Pulm. Pharmacol. Ther. 20, 112–117. Anderson, J. C., Hlastala, M. P., 2010. Impact of Airway Gas Exchange on the Multiple Inert Gas Elimination Technique: Theory. Ann. Biomed. Eng. 38, 1017–1030. Anderson, J. C., Lamm, W. J., Hlastala, M. P., 2006. Measuring airway exchange of endogenous acetone using a single-exhalation breathing maneuver. J. Appl. Physiol. 100, 880–9. Duffin, J., 2010. The role of the central chemoreceptors: a modeling perspective. Respir. Physiol. Neurobiol. 173, 230–243. Duffin, J., Mohan, R. M., Vasiliou, P., Stephenson, R., Mahamed, S., 2000. A model of the chemoreflex control of breathing in humans: model parameters measurement. Respir. Physiol. 120, 13–26. Farhi, L. E., 1967. Elimination of inert gas by the lung. Respir. Physiol. 3, 1–11. Filser, J. G., 1992. The closed chamber technique – uptake, endogenous production, excretion, steady-state kinetics and rates of metabolism of gases and vapors. Arch. Toxicol. 66, 1–10. Grodins, F. S., Buell, J., Bart, A. J., 1967. Mathematical analysis and digital simulation of the respiratory control system. J. Appl. Physiol. 22, 260–76. Hanna, L. M., Scherer, P. W., 1986. A theoretical model of localized heat and water vapor transport in the human respiratory tract. J. Biomech. Eng. 108, 19–27. Hlastala, M. P., Scheid, P., Piiper, J., 1981. Interpretation of inert gas retention and excretion in the presence of stratified inhomogeneity. Respir. Physiol. 46, 247–259. Hughes, J. M. B., Morell, N. W., 2001. Pulmonary Circulation. From basic mechanisms to clinical practice. Imperial College Press, London. Jones, A. W., 1983. Role of rebreathing in determination of the blood-breath ratio of expired ethanol. J. Appl. Physiol. 55, 1237–1241. Kalapos, M. P., 2003. On the mammalian acetone metabolism: from chemistry to clinical implications. Biochim. Biophys. Acta 1621, 122–39. King, J., Koc, H., Unterkofler, K., Mochalski, P., Kupferthaler, A., Teschl, G., Teschl, S., Hinterhuber, H., Amann, A., 2010a. Physiological modeling of isoprene dynamics in exhaled breath. J. Theor. Biol. 267, 626–637. King, J., Kupferthaler, A., Unterkofler, K., Koc, H., Teschl, S., Teschl, G., Miekisch, W., Schubert, J., Hinterhuber, H., Amann, A., 2009. Isoprene and acetone concentration profiles during exercise on an ergometer. J. Breath Res. 3, 027006 (16pp). King, J., Mochalski, P., Kupferthaler, A., Unterkofler, K., Koc, H., Filipiak, W., Teschl, S., Hinterhuber, H., Amann, A., 2010b. Dynamic profiles of volatile organic compounds in exhaled breath as determined by a coupled PTR-MS/GC-MS study. Physiol. Meas. 31, 1169–1184. King, J., Unterkofler, K., Teschl, G., Teschl, S., Koc, H., Hinterhuber, H., Amann, A., 2011. A mathematical model for breath gas analysis of volatile organic compounds with special emphasis on acetone. J. Math. Biol., DOI 10.1007/s00285–010–0398–9. Koc, H., King, J., Teschl, G., Unterkofler, K., Teschl, S., Mochalski, P., Hinterhuber, H., Amann, A., 2011. The role of mathematical modeling in voc analysis using isoprene as a prototypic example. J. Breath Res., accepted. Kumagai, S., Matsunaga, I., 1995. Physiologically based pharmacokinetic model for acetone. Occup. Environ. Med. 52, 344–352. Kumagai, S., Matsunaga, I., 2000. A lung model describing uptake of organic solvents and roles of mucosal blood flow and metabolism in the bronchioles. Inhal. Toxicol. 12, 491–510. Lumb, A. B., 2005. Nunn’s Applied Respiratory Physiology, 6th Edition. Butterworth-Heinemann, Oxford. Miekisch, W., Schubert, J. K., Noeldge-Schomburg, G. F. E., 2004. Diagnostic potential of breath analysis–focus on volatile organic compounds. Clin. Chim. Acta 347, 25–39. Mohan, R., Duffin, J., 1997. The effect of hypoxia on the ventilatory response to carbon dioxide in man. Respir. Physiol. 108, 101–115. Mohrman, D. E., Heller, L. J., 2006. Cardiovascular Physiology, 6th Edition. Lange Medical Books/McGraw-Hill, New York. Mörk, A. K., Johanson, G., 2006. A human physiological model describing acetone kinetics in blood and breath during various levels of physical exercise. Toxicol. Lett. 164, 6–15. O’Hara, M. E., Clutton-Brock, T. H., Green, S., Mayhew, C. A., 2009. Endoge- nous volatile organic compounds in breath and blood of healthy volunteers: examining breath analysis as a surrogate for blood measurements. J. Breath Res. 3, 027005 (10pp). O’Hara, M. E., O’Hehir, S., Green, S., Mayhew, C. A., 2008. Development of a protocol to measure volatile organic compounds in human breath: a comparison of rebreathing and on-line single exhalations using proton transfer reaction mass spectrometry. Physiol. Meas. 29, 309–30. Ohlsson, J., Ralph, D. D., Mandelkorn, M. A., Babb, A. L., Hlastala, M. P., 1990. Accurate measurement of blood alcohol concentration with isothermal rebreathing. J. Stud. Alcohol 51, 6–13. Ottesen, J. T., Olufsen, M. S., Larsen, J. K., 2004. Applied Mathematical Models in Human Physiology. SIAM, Philadelphia. Scheid, P., Hlastala, M. P., Piiper, J., 1981. Inert gas elimination from lungs with stratified inhomogeneity: Theory. Respir. Physiol. 44, 299–309. Schwarz, K., Filipiak, W., Amann, A., 2009a. Determining concentration patterns of volatile compounds in exhaled breath by PTR-MS. J. Breath Res. 3, 027002 (15pp). Schwarz, K., Pizzini, A., Arendacka, B., Zerlauth, K., Filipiak, W., Schmid, A., Dzien, A., Neuner, S., Lechleitner, M., Scholl-Burgi, S., Miekisch, W., Schubert, J., Unterkofler, K., Witkovsky, V., Gastl, G., Amann, A., 2009b. Breath acetone – aspects of normal physiology related to age and gender as determined in a PTR-MS study. J. Breath Res. 3, 027003 (9pp). Staudinger, J., Roberts, P. V., 2001. A critical compilation of Henry’s law constant temperature dependence relations for organic compounds in dilute aqueous solutions. Chemosphere 44, 561–576. Tsoukias, N. M., George, S. C., 1998. A two-compartment model of pulmonary nitric oxide exchange dynamics. J. Appl. Physiol. 85, 653–666. Tsu, M. E., Babb, A. L., Sugiyama, E. M., Hlastala, M. P., 1991. Dynamics of soluble gas exchange in the airways: 2. Effects of breathing conditions. Respir. Physiol. 83, 261–276. Wigaeus, E., Holm, S., Astrand, I., 1981. Exposure to acetone. Uptake and elimination in man. Scand. J. Work Environ. Health 7, 84–94. Zwart, A., Luijendijk, S. C., de Vries, W. R., 1986. Excretion-retention data of steady state gas exchange in tidal breathing. I. Dependency on the blood-gas partition coefficient. Pflügers Arch. 407, 204–210. 9