2CompartmentModel

Determination of physiological parameters by breath gas analysis Karl Unterkofler1,2 , Julian King1 , Pawel Mochalski1 , Martin Jandacka1,2 , Helin Koc1 , Susanne Teschl3 , Anton Amann1,4 , and Gerald Teschl5 1 Breath Research Institute, University of Innsbruck, Rathausplatz 4, A-6850 Dornbirn, Austria 2 University of Applied Sciences Vorarlberg, Hochschulstr. 1, A-6850 Dornbirn, Austria 3 University of Applied Sciences Technikum Wien, Höchstädtplatz 6, A-1200 Wien, Austria 4 Univ.-Clinic for Anesthesia and Intensive Care, Innsbruck Medical University, Anichstr. 35, A-6020 Innsbruck, Austria 5 Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria E-mail: anton.amann@i-med.ac.at E-mail: gerald.teschl@univie.ac.at E-mail: karl.unterkofler@fhv.at Version: 24 October 2014 Determination of physiological parameters by breath gas analysis 2 Abstract. In this paper we develop a simple two compartment model which extends the Farhi equation to the case when the inhaled concentration of a volatile organic compound (VOC) is not zero. The model connects the exhaled breath concentration of VOCs with physiological parameters such as endogenous production rates and metabolic rates. Its validity is tested with data obtained for isoprene and inhaled deuterated isoprene-D5. Keywords: Breath gas analysis, Volatile organic compounds, Metabolic rates, Modeling, Isoprene 1. Introduction The progress of different analytical methods in mass spectrometry in the last twenty years has opened the door to breath gas analysis. There is considerable evidence that volatile organic compounds (VOCs) produced in the human body and then partially released by breath have great potential for diagnosis in physiology and medicine [3]. The emission of such compounds may result from normal human metabolism as well as from pathophysiological disorders, bacterial or mycotic processes (see [1] and the references therein), or exposure to environmental contaminants [12, 14]. As human specific chemical fingerprints, VOCs can provide non-invasive and real-time information on infections, metabolic disorders, and the progression of therapeutic intervention. In this paper we develop a simple two compartment model which extends the Farhi equation to the case that the inhaled concentration of a VOC is not zero. The model connects the exhaled breath concentration of VOCs with physiological parameters such as endogenous production rates and metabolic rates. In addition we test the model with data for normal isoprene and inhaled deuterated isoprene-D5. In Section 2.1 we derive the Farhi equation and extend it in Section 2.2 to the case where the inhaled concentration of a VOC is not zero. Section 2.3 answers the question “Is subtracting the inhaled concentration from the exhaled concentration of a VOC the true way to correct breath concentrations for room air concentrations?”. Section 2.4 derives the expressions for determining the total metabolic rate and the total production rate of the body. Section 3 tests the theory with data obtained for isoprene and inhaled deuterated isoprene-D5. 2. A two compartment model 2.1. Derivation of the Farhi equation To derive the classical Farhi equation which relates alveolar concentrations of VOCs with their blood concentrations one uses a simple two compartment model (see Figure 2) which consists of one single lung compartment and one single body compartment. The amount of a VOC transported to and from the lung by blood is given by Q̇c (t)(Cv̄ (t) − Ca (t)), 3 Determination of physiological parameters by breath gas analysis where Q̇c denotes the cardiac output, Cv̄ the averaged mixed venous concentration, and Ca the arterial concentration. On the other hand one exhales the amount V̇A (t)(CI − CA (t)), where V̇A denotes the ventilation, CI denotes the concentration in the inhaled air (normally assumed to be zero), and CA the alveolar air concentration. This leads to the following mass balance equation describing the change in the concentration of a VOC in the lung‡ (see Figure 1) dCA ṼA = V̇A (CI − CA ) + Q̇c (Cv̄ − Ca ), (1) dt where ṼA denotes the volume of the lung. CI V̇A ✲ CA air Cv̄ Q̇c ✲ Ca blood Figure 1. Diagram of gas exchange in an alveoli symbolized by a dashed line. Remark: This model is only valid for VOCs with blood:air partition coefficients λb:air less than 10, where the upper airways have no influence on breath concentrations, that means there exists no wash-in/wash-out behavior such as shown by, e.g., acetone or ethanol [2]. If the system is in an equilibrium state (e.g., stationary at rest) Equation (1) reads 0 = V̇A (CI − CA (CI )) + Q̇c (Cv̄ (CI ) − Ca ) and using Henry’s law Ca = λb:air CA we obtain CA (CI ) = CI Cv̄ (CI ) , + + 1 λb:air + r (2) λb:air r where, r = V̇A /Q̇c is the ventilation-perfusion ratio. Note, however, that CA and Cv̄ depend here on CI ! Assuming that CI = 0 we derive the classical Farhi equation [4] Cv̄ (0) . (3) λb:air + r However, if CI 6= 0, then Equation (2) shows that simply subtracting CI from CA to get CA (0) is not correct since Cv̄ is depending on CI too (more in Subsection 2.3). We summarize the assumptions for the validity of Farhi’s equation: CA (0) = (i) a stationary state is achieved in the lung, i.e., dCA dt = 0, (ii) only alveolar air is sampled which implies CA = Cexhaled (no dead space air contributions, e.g., CO2 controlled sampling), ‡ For notational convenience we have dropped the time variable t, i.e., we write CX instead of CX (t), etc. CX denotes the instant or averaged concentration of X over a small sampling period τ , i.e., R t+τ /2 CX (t) = 1/τ t−τ /2 CX (s)ds. Determination of physiological parameters by breath gas analysis 4 (iii) no influence exists of the upper airways (i.e., low solubility for aqueous solution, or λb:air < 10, see [2]) on the investigated VOC§, k, (iv) the distribution of the blood flow into the different body compartments remains unchanged, (e.g., constant at rest), (v) the inhaled concentration is zero, i.e., CI = 0. Remark: Note that for small values of the blood:gas partition coefficient λb:air small changes of the ventilation-perfusion ratio r (e.g., when hyperventilating) will strongly influence the relationship between CA and Cv̄ . This might lead to false correlations or corrupt existing ones. 2.2. Extension of the Farhi equation To calculate the explicit dependence of CA and Cv̄ on CI we need to consider the mass balance for the body compartment too. The change of the amount of a VOC in the body is given by the amount which enters the body compartment with the arterial blood plus the amount which is produced in the body minus the amount which is metabolized and the amount which leaves it with the venous blood. Thus the change of the amount of a VOC in the body compartment is given by¶,+ dCB ṼB = Q̇c (Ca − Cv̄ ) − λb:B kmet CB + kprod , (4) dt where kmet denotes the metabolic rate, kprod the production rate, ṼB the effective volume of the body∗ , and CB the concentration in the body which is connected to the venous concentration by Henry’s law Cv̄ = λb:B CB . Here λb:B denotes the blood:body partition coefficient. When in an equilibrium state (i.e., dCdtA = 0) we can use Equations (1), (4), and Cv̄ = λb:B CB to eliminate the implicit dependence of CA on CI in Equation (2) CA (CI ) = Cv̄ (CI ) = r+ kprod kmet V̇A + kmet λb:air kprod (r + λb:air ) kmet A r + kV̇met + λb:air + + r+ r+ r+ V̇A kmet V̇A + λb:air kmet λb:air r kmet V̇A + λb:air kmet CI , (5) CI . (6) § When there is an influence of the upper airways then Cexhaled 6= CA . k In that case the lung must be modeled at least by two compartments. ¶ Here we used the usual convention to multiply kmet by λb:B . It would be more natural to use kmet only but this can be incorporated in a redefinition of kmet . + Since the considered inhaled concentrations are low a linear kinetic is sufficient for the description. ∗ The body blood compartment and the body tissue compartment are assumed to be in an equilibrium and therefore can be combined into one single body compartment with an effective volume. For more details about effective volume compare appendix 2 in [8]. 5 Determination of physiological parameters by breath gas analysis ❄ ✲ CA V̇A CI ✲ Q̇c Cv̄ ✻ lung compartment Ca ❄ ✻ Cv̄ ✛ Q̇c Ca body compartment ✲k Cbody ✛ met kprod Figure 2. Two compartment model consisting of a lung compartment (gas exchange) and a body compartment with production and metabolism. Dashed lines indicate equilibrium according to Henry’s law. From Equation (5) and (6) we see that the exhaled concentration CA and the mixed venous concentration Cv̄ solely depend on the inhaled concentration CI and the physiological parameters kprod , kmet , V̇A , Q̇c , λb:air . We now discuss some special cases: (a) For CI = 0 (no trace gas is inspired) this reduces to CA (0) = Cv̄ (0) = r+ kprod kmet V̇A + kmet kprod kmet r+ λb:air (r + λb:air ) V̇A kmet + λb:air , = CA (0) (r + λb:air ). (7) (b) On the other hand, when the production is zero (kprod = 0), this yields CA (CI ) = Cv̄ (CI ) = r+ V̇A kmet r+ V̇A kmet + λb:air r+ λb:air r kmet V̇A + λb:air kmet CI = CI = 1 1+ λb:air CI , CA (CI ) ≤ CI , (8) V̇ r+ k A met λb:air CA (CI ). kmet + Q̇c (9) (c) Assuming CA = CI (zero alveolar gradient) in Equation (5) yields CI = kprod . kmet λb:air (10) 2.3. Is subtracing CI correct? Many breath researchers claim (without evidence) that one must simply subtract the room air concentration (inhaled concentration) to correct the exhaled alveolar concentration, that means they take CA (0) = CA (CI ) − CI . To see if this is correct Determination of physiological parameters by breath gas analysis 6 we consider Equation (5) which we rewrite as 1 CI . CA (CI ) = CA (0) + 1 + λb:air V̇A r+ k Hence CA (0) = CA (CI ) − met 1 1+ λb:air CI . (11) V̇ r+ k A met From this result we conclude that simply subtracting the inhaled concentration is in general false! One has to consider the following factor 1 . (12) 1 + λb:air V̇A r+ k met This factor is approximately 1 for small values of λb:air (e.g., methane λb:air < 0.1) or small values of kmet . But it might be 2/3 if, e.g., λb:air = 1/2 (approximately fulfilled V̇A r+ k for isoprene). met 2.4. Endogenous production and metabolic rates The question remains how to determine the endogenous production rate and the total metabolic rate of the body. When in a stationary state the averaged values of ventilation and perfusion are constant. Thus Equation (5) resembles a straight line of the form CA (CI ) = a CI + b, (13) CI being the variable here. The constants a and b are given by b = CA (0) = r+ kprod kmet V̇A + kmet λb:air (14) and a= V̇A ) kmet (r + r+ V̇A kmet + λb:air . (15) Thus the constants a and b are completely determined by the physiological quantities V̇A , Q̇c , kprod , kmet , and λb:air . The gradient a is independent of kprod , fulfills 0 < a < 1, and is determined by the metabolic rate kmet , the ventilation, and perfusion. The quantity b = CA (0) is proportional to the production rate kprod . Varying CI , one can measure CA (CI ) experimentally and thus determine a and b. Measuring in addition ventilation and perfusion enables to calculate the total production rate and the total metabolic rate of the body from these two equations kprod = kmet = b λ V̇ 1−a b:air A , a λ −r 1−a b:air (16) V̇A , λb:air − r (17) a 1−a Determination of physiological parameters by breath gas analysis 7 or kprod = (V̇A + (r + λb:air ) kmet ) CA (0), (18) when kmet is known. Remark 1: In [13] the effect of inhaled VOCs on exhaled breath concentrations was studied. Unfortunately breath frequency and heart rate were not reported (monitored?). Therefore ventilation and perfusion can not be estimated respectively determined. However this study shows that Equation (5) explains the experimental findings very well. Remark 2: This approach yields total production rates only. However, it will not be able to determine different production rates in different compartments. When more than one production source exists in the body one needs a multi compartment model for the body. Then changes of r, e.g., by exercise will vary the blood distributions into these compartments, which allows to determine the different production rates. Remark 3: Due to the term (−r) in the denominator of Equation (17), errors in measuring a, V̇A , and Q̇c can cause considerable errors in the metabolic rate. 2.5. Changes in production rates. When measuring breath samples or performing ergometer experiments one assumes that the endogenous production rate stays constant during the time frame of these experiments. However, when performing breath analysis during sleep it is possible that the production rate will display, e.g., a circadian rhythm which can be determined by (ventilation and perfusion are considered to be constant, quasi equilibrium) kprod (t) = (V̇A + (r + λb:air ) kmet ) CA,0 (t). (19) 3. Experimental data 3.1. Setup End-tidal isoprene concentration profiles are obtained by means of a real-time setup designed for synchronized measurements of exhaled breath VOCs as well as a number of respiratory and hemodynamic parameters. Our instrumentation has successfully been applied for gathering continuous data streams of these quantities during ergometer challenges [7] as well as in a sleep laboratory setting [6]. These investigations aim at evaluating the impact of breathing patterns, cardiac output or blood pressure on the observed breath concentration and permit a thorough study of characteristic changes in VOCs output following variations in ventilation or perfusion. We refer to [7] for an extensive description of the technical details. In brief, the core of the mentioned setup consists of a head mask spirometer system allowing for the standardized extraction of arbitrary exhalation segments, which subsequently are directed into a Proton-Transfer-Reaction-Time-of-Flight mass spectrometer (PTR-MS-TOF, Ionicon Analytik GmbH, Innsbruck, Austria) for online Determination of physiological parameters by breath gas analysis 8 analysis♯. This analytical technique has proven to be a sensitive method for the quantification of volatile molecular species M down to the ppb (parts per billion) range by taking advantage of the proton transfer H3 O+ + M → M H+ + H2 O from primary hydronium precursor ions [9, 10]. Note that this “soft” chemical ionization scheme is selective to VOCs with proton affinities higher than water (166.5 kcal/mol). Count rates of the resulting product ions M H+ or fragments thereof appearing at specified mass-to-charge ratios m/z can subsequently be converted to absolute concentrations of the compound under scrutiny. Specifically, protonated isoprene is detected in PTR-MS-TOF at m/z = 69, protonated deuterated isoprene-D5 is detected in PTR-MS-TOF at m/z = 74 and can be measured with breath-by-breath resolution. An underlying sampling interval of 4 s is set for each parameter. 3.2. Protocol of the experiments Figure 3. Typical results of an ergometer session with inhalation of deuterated isoprene-D5: rest for 9 min – release of deuterated isoprene-D5 into the sealed laboratory and waiting for 13 min – 75 Watts for 18 min – rest for 6 min – 75 Watts for 12 min – rest for 5 min; deuterated isoprene-D5 green, normal isoprene blue. To test the theory developed we took some existing data from our ergometer experiments with inhaled deuterated isoprene-D5. ♯ The PTR-MS-TOF replaces the formerly used PTR-MS. Determination of physiological parameters by breath gas analysis 9 Deuterated Isoprene-D5 (98%, Campro Scientific GmbH, Germany) was released into the laboratory room with the help of a 0.5-l glass bulb (Supelco, Canada). Prior to the experiment the bulb was evacuated using a vacuum membrane pump and an appropriate volume of liquid isoprene (dependent on the target concentration) was injected through a rubber septum. After the compound complete evaporation both Teflon valves of the bulb were opened and the bulb content was purged with synthetic air at the flow rate of 1 l/min for 3 minutes. Such conditions provided 3 l of the purge gas (six bulb volumes) to be introduced into the bulb and, thereby, completely displaced the original bulb content. During the bulb purging the laboratory air was continuously mixed with the help of a fan to achieve a homogenous isoprene distribution. In contrast to chamber experiments the laboratory serves here as a big reservoir (volume: approx. 60 000 l) with a nearly constant background concentration††. Three of the authors (one female, two males) took part at several ergometer challenges at different room air concentrations ranging from 30 to 1000 ppb. The exact protocol was as follows (see Figure 3): • minutes 0–9: the volunteer rests with mask on the ergometer • minutes 9–12: deuterated isoprene-D5 is released and the room air is mixed by a fan • minutes 12–22: the volunteer rests on the ergometer • minutes 22–40: the volunteer pedals at 75 Watts • minutes 40–46: the volunteer rests on the ergometer • minutes 46–58: the volunteer pedals at 75 Watts • minutes 58–63: the volunteer rests on the ergometer • minutes 63–68: mask is taken off and the room air concentration is measured. As one can see deuterated isoprene-D5 with a partition coefficient of nearly 1 (λb:air = 0.95, [11]) enters the arterial blood stream quickly and it takes only a few minutes untill it appears in breath and an equilibrium is achieved in the room air and the blood of the volunteer. To ensure that a steady state was achieved we waited another ten minutes before starting with exercise. At the onset of exercise normal isoprene shows a peak as is well known. This peak stems from a high concentration in muscle blood caused by the production in this compartment. Deuterated isoprene-D5 is nowhere produced in the body. Hence in every compartment of the body the concentration is the same. However, at the onset of exercise the ventilation perfusion ratio goes up and the deuterated isopren-D5 in exhaled breath declines in accordance with the Farhi equation since the venous blood still has an unaltered isoprene level for 1 to 2 minutes. But then, due to the increased inhalation of deuterated isoprene-D5, the venous blood gains a higher concentration level (compare with Equation (6)) too and the exhaled †† For time frames of a few minutes the room air concentration can considered to be constant; however, over one hour a degrease in the room air concentration is noticeable due to leaks in the sealing of the laboratory. 10 Determination of physiological parameters by breath gas analysis Session 1 Session 2 Session 3 Session 4 Session 5 Mean CI,deuterated [ppb] CI,normal [ppb] CA,deuterated [ppb] CA,normal [ppb] V̇A [l/min] Q̇c [l/min] 86.61 161.88 202.14 447.58 935.78 - 11.46 7.01 5.16 8.81 12.1 8.91±2.93 57.06 131.47 156.16 288.41 390.06 - 159.67 115.12 100.35 137.75 114.79 125.54±23.3 6.40 5.31 9.56 8.74 6.66 7.33±1.76 4.53 4.37 6.38 4.66 4.73 4.93±0.82 Table 1. Volunteer 1 (male, mass: 68 kg, height: 174 cm) normal and deuterated inhaled and exhaled isoprene concentrations with corresponding ventilation and perfusion. Session 6 Session 7 Session 8 Session 9 Session 10 Mean CI,deuterated [ppb] CI,normal [ppb] CA,deuterated [ppb] CA,normal [ppb] V̇A [l/min] Q̇c [l/min] 49.81 104.70 159.72 226.08 515.21 - 5.48 6.18 6.38 5.74 7.12 6.18±0.63 31.77 71.62 106.31 215.86 213.93 - 45.53 50.12 48.73 48.76 36.85 46.0±5.38 5.75 5.67 5.83 8.95 8.28 6.9±1.59 5.69 6.11 5.96 6.88 7.48 6.42±0.74 Table 2. Volunteer 2 (female, mass: 62 kg, height: 168 cm) normal and deuterated inhaled and exhaled isoprene concentrations with corresponding ventilation and perfusion. Session 11 Session 12 Session 13 Session 14 Session 15 Mean CI,deuterated [ppb] CI,normal [ppb] CA,deuterated [ppb] CA,normal [ppb] V̇A [l/min] Q̇c [l/min] 32.09 68.08 127.22 164.33 617.11 - 7.29 6.07 6.37 5.90 7.81 6.69±0.83 22.42 44.91 87.93 137.19 351.69 - 184.59 180.69 190.25 142.16 170.88 173.71±19.0 8.04 8.06 8.65 7.28 8.13 8.03±0.49 4.46 4.79 4.54 4.40 4.09 4.46±0.25 Table 3. Volunteer 3 (male, mass: 90 kg, height: 180 cm) normal and deuterated inhaled and exhaled isoprene concentrations with corresponding ventilation and perfusion. concentration reaches its former level. For the purpose to validate the 2-compartment model we took the average resting values of all measured quantities of the last 3 minutes before starting the ergometer challenge. These values are summarized in Tables 1 – 3. Determination of physiological parameters by breath gas analysis kmet Volunteer 1 16.87 ± 4.8 Volunteer 2 7.95 ± 4.0 11 Volunteer 3 22.63 ± 4.8 Table 4. Metabolic rates (in [l/min]) for deuterated isoprene-D5 for each volunteer. Q̇c [l/min] V̇A [l/min] CA (0) ppb kmet [l/min] ā kprod [nmol/min] 4.93 6.42 4.46 7.33 6.9 8.03 120.6 41.9 169.0 16.87 7.95 22.63 0.669 0.671 0.694 216.4 35.7 439.9 Volunteer 1 Volunteer 2 Volunteer 3 Table 5. Metabolic rates and endogenous production rates for isoprene for each volunteer (with Vmol = 27 l). 3.3. Results From Tables 1 – 3 we can calculate the metabolic rates for deuterated isoprene-D5 for each volunteer. To avoid the problem with small denominators (see Remark 3) we perform a nonlinear least square optimization using Equation (8) !2 A X CA (CI ) (r + kV̇met ) min . (20) − A CI + λb:air r + kV̇met This yields kmet for each volunteer. The results are listed in Table 4. Since normal isoprene and deuterated isoprene-D5 behave similarly from a chemical standpoint, we assume, neglecting isotopic effects, as a first approximation that both have the same metabolic rate. Then we calculate the average rest ventilation V̇A , the average perfusion Q̇c , the gradient ā by Equation (15), and the corrected average exhaled normal isoprene concentration CA (0). Using Equation (18) we can then calculate the corresponding endogenous production rate for normal isoprene. The results are listed in Table 5. Remark: One can also calculate the total production rate kprod and the total metabolic rate kmet from our three compartment model [5] by combining the two body compartments per rpt λb:per Cper λb:rpt Crpt + kmet kmet . Cv̄ Taking the nominal values from Table 2 and Table C1 in [5] yields kmet = 10 l/min and kprod = 125.3 nmol/min. rpt per kprod = kpr + kpr , kmet = 4. Discussion In this paper we developed the simplest possible compartment model which allows to compute metabolic rates and endogenous production rates of VOCs from their breath concentrations. This model also shows how breath concentration should be corrected Determination of physiological parameters by breath gas analysis 12 when the inhaled concentration is not zero. To apply this model in a clinical setting further investigations with a representative number of patients are necessary. In addition one could determine dependences on body mass, sex, etc. To circumvent the intricate measurements of ventilation and perfusion one could use heart frequency and breath frequency. For long time exposure studies one could extend the model by adding a storage compartment which fills up and depletes depending on the partition coefficient. To consider VOCs which interact with the upper airways one could extend the model by at least a two compartment model of the lung, e.g., as developed in [8]. Acknowledgments Acknowledgment J.K., P.M., M.J., and K.U. gratefully acknowledge support from the Austrian Science Fund (FWF) under Grant No. P24736-B23. G.T. also gratefully acknowledges support from the Austrian Science Fund (FWF) under Grant No. Y330. We appreciate funding from the Austrian Federal Ministry for Transport, Innovation, and Technology (BMVIT/BMWA, project 836308, KIRAS). We gratefully appreciate funding from the Oncotyrol-project 2.1.1. The Competence Centre Oncotyrol is funded within the scope of the COMET - Competence Centers for Excellent Technologies through BMVIT, BMWFJ, through the province of Salzburg and the Tiroler Zukunftsstiftung/Standortagentur Tirol. The COMET Program is conducted by the Austrian Research Promotion Agency (FFG). We thank the government of Vorarlberg (Austria) for its generous support. 5. References [1] A. Amann and D. Smith (eds.), Volatile Biomarkers, Elsevier, Boston, 2013. [2] J. C. Anderson, A. L. Babb, and M. P. Hlastala, Modeling soluble gas exchange in the airways and alveoli, Ann. Biomed. Eng. 31 (2003), 1402–22. [3] B. de Lacy Costello, A. Amann, H. Al-Kateb, C. Flynn, W. 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