Modeling-based determination of physiological parameters of systemic VOCs by breath gas analysis: a pilot study

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 systemic 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.


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.

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 where Qc denotes the cardiac output, C v the averaged mixed venous concentration, and C a the arterial concentration.
On the other hand one exhales the amount where VA denotes the ventilation, C I denotes the concentration in the inhaled air (normally assumed to be zero), and C A 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) where ṼA denotes the volume of the lung.air blood Remark: This model is only valid for VOCs with blood:air partition coefficients λ b: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 = VA (C I − C A (C I )) + Qc (C v(C I ) − C a ) and using Henry's law C a = λ b:air C A we obtain where, r = VA / Qc is the ventilation-perfusion ratio.Note, however, that C A and C v depend here on C I !Assuming that C I = 0 we derive the classical Farhi equation [4] However, if C I = 0, then Equation (2) shows that simply subtracting C I from C A to get C A (0) is not correct since C v is depending on C I too (more in Subsection 2.3).We summarize the assumptions for the validity of Farhi's equation: (i) a stationary state is achieved in the lung, i.e., dC A dt = 0, ‡ For notational convenience we have dropped the time variable t, i.e., we write C X instead of C X (t), etc. C X denotes the instant or averaged concentration of X over a small sampling period τ , i.e., C X (t) = 1/τ t+τ /2 t−τ /2 C X (s)ds.
(ii) only alveolar air is sampled which implies C A = C exhaled (no dead space air contributions, e.g., CO 2 controlled sampling), (iii) no influence exists of the upper airways (i.e., low solubility for aqueous solution, or λ b:air < 10, see [2]) on the investigated VOC § , , (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., C I = 0.
Remark: Note that for small values of the blood:gas partition coefficient λ b:air small changes of the ventilation-perfusion ratio r (e.g., when hyperventilating) will strongly influence the relationship between C A and C v. This might lead to false correlations or corrupt existing ones.

Extension of the Farhi equation
To calculate the explicit dependence of C A and C v on C I 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 ¶ where k met denotes the metabolic rate, k prod the production rate, ṼB the effective volume of the body * , and C B the concentration in the body which is connected to the venous concentration by Henry's law Here λ b:B denotes the blood:body partition coefficient.
When in an equilibrium state (i.e., dC A dt = 0) we can use Equations ( 1), ( 4), and § When there is an influence of the upper airways then C exhaled = C A .In that case the lung must be modeled at least by two compartments.¶ Here we used the usual convention to multiply k met by λ b:B .It would be more natural to use k met only but this can be incorporated in a redefinition of k met .+ 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].
lung compartment body compartment 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 C A and the mixed venous concentration C v solely depend on the inhaled concentration C I and the physiological parameters k prod , k met , VA , Qc , λ b:air .We now discuss some special cases: (a) For C I = 0 (no trace gas is inspired) this reduces to (b) On the other hand, when the production is zero (k prod = 0), this yields (c) Assuming C A = C I (zero alveolar gradient) in Equation ( 5) yields

Is subtracing C I 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 To see if this is correct we consider Equation ( 5) which we rewrite as From this result we conclude that simply subtracting the inhaled concentration is in general false!One has to consider the following factor 1 1 + λ b:air This factor is approximately 1 for small values of λ b:air (e.g., methane λ b:air < 0.1) or small values of k met .But it might be 2/3 if, e.g., λ b:air for isoprene).

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 C I being the variable here.The constants a and b are given by and Thus the constants a and b are completely determined by the physiological quantities VA , Qc , k prod , k met , and λ b:air .The gradient a is independent of k prod , fulfills 0 < a < 1, and is determined by the metabolic rate k met , the ventilation, and perfusion.The quantity b = C A (0) is proportional to the production rate k prod .
Varying C I , one can measure C A (C I ) 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 or when k met 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, VA , and Qc can cause considerable errors in the metabolic rate.

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)

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 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 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.To test the theory developed we took some existing data from our ergometer experiments with inhaled deuterated isoprene-D5.

Protocol of the experiments
The PTR-MS-TOF replaces the formerly used PTR-MS.
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 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 (λ b: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.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.

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) This yields k met for each volunteer.The results are listed in Table 5. Metabolic rates and endogenous production rates for isoprene for each volunteer (with V mol = 27 l).
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 VA , the average perfusion Qc , the gradient ā by Equation (15), and the corrected average exhaled normal isoprene concentration C A (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 k prod and the total metabolic rate k met from our three compartment model [5]  Taking the nominal values from Table 2 and Table C1 in [5] yields k met = 10 l/min and k prod = 125.3nmol/min.

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 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].

Figure 1 .
Figure 1.Diagram of gas exchange in an alveoli symbolized by a dashed line.

Figure 3 .
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.
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 by combining the two body compartmentsk prod = k rpt pr + k per pr , k met = k rpt met λ b:rpt C rpt + k per met λ b:per C per C v .

Table 1 .
C I,deuterated C I,normal C A,deuteratedC Volunteer 1 (male, mass: 68 kg, height: 174 cm) normal and deuterated inhaled and exhaled isoprene concentrations with corresponding ventilation and perfusion.

Table 4 .
C I,deuterated C I,normal C A,deuterated C