Modeling-based determination of physiological parameters of systemic VOCs by breath gas analysis, part 2

In a recent paper (Unterkofler et al 2015 J. Breath Res. 9 036002) we presented a simple two compartment model which describes the influence of inhaled concentrations on exhaled breath concentrations for volatile organic compounds (VOCs) with small Henry constants. In this paper we extend this investigation concerning the influence of inhaled concentrations on exhaled breath concentrations for VOCs with higher Henry constants. To this end we extend our model with an additional compartment which takes into account the influence of the upper airways on exhaled breath VOC concentrations.


Introduction
In their paper [11] Španěl et al. investigated the short-term effect of inhaled volatile organic compounds (VOCs) on exhaled breath concentrations.They showed for seven different VOCs with very different Henry constants (blood:air partition coefficients) that the exhaled breath concentration closely resembles an affine function (straight line) of the inhaled concentration.
This motivated our theoretical investigation [15] regarding the impact of inhaled concentrations for VOCs with low blood:air partition coefficients, i.e., compounds with exhalation kinetics that are described by the Farhi equation [3].For these VOCs the exhaled end-tidal breath concentration resembles the alveolar concentration.
Here we extend this investigation to VOCs with higher blood:air partition coefficients where the influence of the upper airways cannot be neglected.For such VOCs the exhaled end-tidal breath concentration does not equal the alveolar concentration but the bronchial concentration.
Consider for example acetone with typical concentrations of 1 [µg/l] in breath.Assuming that the exhaled end-tidal breath concentration equals the alveolar concentration and using the Farhi equation ‡ the blood:air partition coefficient of acetone λ b:air = 340 would lead to a concentration of 0.341 mg/l in blood which differs considerably from typically measured values in blood of 1 [mg/l].
Hence one can not neglect the influence of the upper airways when investigating VOCs with higher partition coefficients, see e.g., [1].

A three compartment model
To incorporate the influence of the upper airways on exhaled VOC concentrations we choose the simplest possible model.It consists of three compartments as sketched in Figure 1: a two compartment lung (bronchioles and alveoli) as used in [7] and one body compartment.
We consider the bronchial compartment separated into a gas phase and a mucus membrane, which is assumed to inherit the physical properties of water and acts as a reservoir.The part of a VOC dissolved in this layer is transferred to the bronchial circulation, whereby the major fraction of the associated venous drainage is postulated to join the pulmonary veins via the post capillary anastomoses [8].
The amount of a VOC transported at time t via exhalation and inhalation to the bronchial compartment equals therefore where VA denotes the ventilation, C I denotes the concentration in the inhaled air (normally assumed to be zero), and C bro the bronchial air concentration §.Moreover, we state that the measured (exhaled) end-tidal breath concentration equals the ‡ The Farhi equation [3] relates the mixed venous concentration Cv with the alveolar concentration C A by Here λ b:air is the blood:air partition coefficient and r is the ventilation-perfusion ratio which is approximately 1 at rest.§ Note: we have suppressed the time variable t, i.e., we write VA instead of VA (t), and so on.
. Sketch of the model structure.The body is divided into three distinct functional units: bronchial/mucosal compartment (gas exchange), alveolar/endcapillary compartment (gas exchange) and body compartment (metabolism and production).Dashed boundaries indicate a diffusion equilibrium.Thus in each case two compartments can be combined into one compartment with an effective volume Ṽ , e.g., 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, ṼB = V body blood +λ B:b V body tissue .For more details about effective volumes compare appendix 2 in [7].The conductance parameter D has units of volume divided by time and quantifies an effective diffusion barrier between the bronchial and the alveolar tract.
bronchial level, i.e., The contribution of the blood flow through the pulmonary veins via the post capillary anastomoses is where q denotes the fractional blood flow through the bronchioles, Qc the cardiac output, C a the arterial blood concentration, λ muc:b the mucus:blood partition coefficient, and λ muc:air the temperature dependent mucus:air partition coefficient (see Appendix B for details).
Then the arterial blood concentration C a is given by with λ b:air denoting the blood:air partition coefficient and C A the alveolar concentration.
The exchange between the bronchial compartment and the alveolar compartment is modeled as a diffusion process with a diffusion constant D which takes values between zero and infinity.
Thus the total mass balance for the bronchial compartment reads Analogously we derive the mass balance equations from Figure 1 for the alveolar compartment and the body compartment where k met denotes the total metabolic rate of the body and k pr the production rate.Ṽbro , ṼA , and ṼB denote the effective volume of the bronchiols, alveoli, and the body, respectively.C B is the concentration in the body which is connected to the mixed venous concentration C v by Henry's law C v = λ b:B C B where λ b:B denotes the blood:body tissue partition coefficient.
Remark: A single body compartment can be derived from the combination of the liver and tissue compartment of the model in [7].
Thus the three compartment model for VOCs with higher Henry constant consists of the system of the three linear differential equations ( 1) -( 3) Remarks: (i) Summing up these three linear differential equations yields the total change of mass m tot of a VOC, i.e.,

Ṽbro dC
Equation (5) shows that the total change of mass of a VOC is given by what is inhaled minus what is exhaled plus what is produced by the body minus what is eliminated by metabolism (metabolism includes all losses, e.g., by liver, urine, skin, etc.), so that the total mass balance is fulfilled.
(ii) In general, ventilation VA and cardiac output Qc are non-constant functions of time.Nevertheless one can show that all solutions of the system (4) starting in R 3 >0 remain bounded (see appendix B, proposition 1 in [7]).
(iii) Rearranging Equation (4) yields a system of the form We assume that the ambient air is not severely contaminated and hence metabolism can be modeled with a linear kinetics.
for the vector c of the three concentrations If ventilation VA and cardiac output Qc are kept constant and assuming that the production k pr is constant, too, the solution of this system can be given explicitly (see, e.g., chapter 3.2 in [14] ¶).All eigenvalues of the constant matrix N are negative and the concentrations approach exponentially (the eigenvalues of N are the exponential constants) the equilibrium state c(∞) = −N −1 h.
When in a stationary state, namely where all quantities and concentrations are constant, the left hand sides of the system (4) are zero and the system of differential equations reduces to a linear algebraic system of the form where the matrix M and the vector b are given by Trivial linear algebra lets us write the solution of the system (7) with the help of Cramer's rule where M j denotes the matrix M where the j-th column, j = 1, 2, 3, is replaced by the vector b and det(M ) denotes the determinant of a matrix M .From equation (9) we conclude that all concentrations are indeed affine functions (straight lines) of the inhaled concentration C I .C I appears in the first component of the vector b only.Hence det(M ) is independent of C I .The multilinearity of the determinant of the matrix M j implies the affine dependence on C I , i.e., where a j and b j , j = 1, 2, 3 are dependent on D, VA , etc.For the special case D = 0 (this is the case for very high partition coefficients λ b:air > 100) + we get C A = 1/λ b:air C v and furthermore 11) ¶ A pdf version of this book is available from http://www.mat.univie.ac.at/ ~gerald/ftp/book-ode/index.html+ The decoupled case D = q = 0 will be excluded from now on as it lacks physiological relevance. with Furthermore, the connection between the mixed venous blood concentration and the measured exhaled concentration is given by For exogenous VOCs (i.e., Since the fractional blood flow of the bronchial circulation q is very small (q ≈ 0.01 [8]) we have q(1 − q) ≈ q and the following approximations are valid Further simplifications are possible under further assumptions, e.g., k met → 0 leads to Remarks: (i) Looking at the equation C bro (C I ) = a 1 C I + b 1 we see that b 1 is the contribution to the exhaled breath by the endogenous production when no room concentration is present and (1−a 1 ) is the proportion of the room concentration which is taken up by the body.
(ii) For D = 0 the calculation is straight forward but the expressions are quite lengthy.However, these calculation can be easily done with a computer algebra system, e.g., using Mathematica.The results are supplied in Appendix E.

Correction method in order to account for inhaled VOC concentrations
From Equation (10) we conclude that to correct the measured exhaled concentration for the inhaled one, one has simply to subtract the inhaled concentration multiplied by the gradient a 1 , i.e., Example 1: With the data from Section 2.3 we therefore get for acetone Example 2: To estimate a 1 for ethanol we use the following nominal values: q = 0.01, VA = 5.2 [l/min], Qc = 6 [l/min] (from table 1 and 2 in [7]), k met = 0.15 [l/min] (= 7 [g/h] from [2]), λ b:air = 1756 (from [6]), λ muc:air = 2876.7 at 32 • C, λ muc:b = 1.17 (from [12]).This yields This shows that in contrast to methane [13] where one must subtract the total inhaled concentration, for ethanol the inhaled concentration is nearly neglectable.

Endogenous production and metabolic rates
The question remains how to determine the endogenous production rate and the total metabolic rate of the body using the theoretical framework introduced above?When in a stationary state, the averaged values of ventilation and perfusion are constant, then Equation ( 10) resembles an affine function (straight line) of the form C I being the variable here.The constants a 1 and b 1 are given for D = 0 by Equation (12).However, for all cases of D the constants a j and b j , j = 1, 2, 3 are completely determined by the physiological quantities VA , Qc , k pr , k met , q, and partition coefficients.The gradient a 1 is independent of k pr , fulfills 0 < a 1 ≤ 1, and depends on the metabolic rate k met but not the production rate k pr .The quantity b 1 = C bro (0) is proportional to the production rate k pr .
Varying C I , one can measure C bro (C I ) experimentally and thus determine a 1 and b 1 .Measuring in addition ventilation and perfusion allows for calculating the total production rate and the total metabolic rate of the body from these two equations.For D = 0 this yields or if k met is already known.Remarks: (i) Note that the numerators in Equations ( 20) and ( 21) are small which will cause large errors when there are no good data available.
(ii) For D = 0 the calculation is straightforward, too, but the expressions are also quite lengthy.The results are supplied in Appendix E.

Test of the theory with data available from literature
Since Španěl et al. did not provide any data for blood flow (cardiac output Qc ) and breath flow (alveolar ventilation VA ) we took the data for acetone provided by Wigaeus [16] (i.e., series 1).This data which we have already used in [7] are listed in Table 1.Note that D equals zero at rest for acetone.This data determine a 1 = 0.384 and b 1 = 0.0016 in Equation (12).Then the following values can be calculated from Equation (12).They are listed in Table 2.These values are in good agreement with the values from the more detailed model developed in [7].

Discussion
In this paper we extended our investigation of the short-term effect * of inhaled volatile organic compounds (VOCs) on exhaled breath concentrations to VOCs with higher Henry constants.For such VOCs the exhaled end-tidal breath concentration does not equal the alveolar concentration but equals the bronchial concentration and hence it is essential to take the influence of the upper airways into account.
In particular, a special focus is given to the case when the inhaled (e.g., ambient air) concentration is significantly different from zero.The model elucidates a novel approach for computing metabolic/production rates of systemic VOCs with high blood:air partition coefficients from the respective breath concentrations.Moreover, it clarifies how breath concentration of such VOCS should be corrected when the inhaled concentration cannot be neglected.The model predicts an affine relationship (straight line) between exhaled breath concentrations and inhaled concentrations as shown by measurements by Spanel et al. [11] and are in good agreement with data available from Wigaeus [16].
The gradient of this line is completely determined by the physiological quantities VA , Qc , k pr , k met , q, and partition coefficients.However, for practical use it might be easier to determine this gradient directly by experiments for the VOC one is interested in.Even labeled inhaled VOCs might be used to exclude effects from endogenous production.
Nevertheless, a number of limitations should be mentioned here.Firstly, in order to apply this model for the estimation of metabolic/production rates, further studies with a representative number of patients will be necessary.In particular, the individual and population ranges of these quantities will have to be determined.In addition, it should be investigated how these parameters vary with age, body mass, sex, etc.. To circumvent the intricate measurements of ventilation and perfusion, one could measure heart frequency and breath frequency and deduce ventilation and perfusion from these parameters.
In order to account for long-term exposure, the model should be extended to incorporate a storage compartment which fills up and depletes according to its partition coefficient.This yields then at least a 4-compartment model.However, for short-term exposure experiments the influence of such a storage compartment will merely be reflected by a slightly different metabolic rate.There is strong experimental evidence that airway temperature constitutes a major determinant for the pulmonary exchange of highly soluble VOCs, cf.[5].How this influences the λ muc:air (T ) partition coefficient was described in detail for acetone in [7].However, this can immediately be adapted to other highly soluble VOCs.The decrease of solubility in the mucosa -expressed as the water:air partition coefficient λ muc:air -with increasing temperature can be described in the ambient temperature range by a van't Hoff-type equation [12]  as λ muc:air is monotonically decreasing with increasing temperature.In a typical situation the absolute sample humidity at the mouth is 4.7% (corresponding to a temperature of T ≈ 32 • C and ambient pressure at sea level, cf.[9,4]).Thus the local solubility of a VOC in the mucus layer increases considerably from the lower respiratory tract up to the mouth, thereby predicting a drastic reduction of air stream VOC concentrations along the airways.

Appendix A. List of symbols
The vapor equilibrium pressure of water is the pressure at which water vapor is in thermodynamic equilibrium with its condensed state. .
When we breathe air into the lungs it is warmed up to body temperature t body = 37 [C] and moisturized to 100% humidity.However, the pressure is immediately balanced.Using the ideal gas equation for constant pressure we arrive at C lung,dry (t body ) = C room,dry (t) T T body .
In addition when we take 100% humidity into account we end up with Hence a temperature difference between body or lung compartment and the bronchial compartment can safely be ignored since there is no measurable effect on concentrations.
Appendix E. The general case where D = 0.
Here we present the general form of the coefficients a j , b j , j = 1, 2, 3 where the diffusion constant D is not zero, i.e., and a j , b j , j = 1, 2, 3 are dependent on D, VA , etc.In addition we did not introduce dimensionless quantities (e.g., r := VA Qc , etc.) to get a more compact form for these coefficients since we did not want to introduce a batch of new symbols.However, we rearranged the coefficients in such a way that the limit D → 0 (λ b:air > 100 large enough) or D → ∞ (upper airways have no influence) can be read off directly.Taking the limit D → 0 we immediately recover the results in Equation (12).Taking the limit D → ∞ and q → 0 we recover the results of the 2-compartment model of [15].
For the metabolic rate and the production rate we get in the general case where D is not zero And taking the limit D → ∞ and q → 0 we recover the results of the 2compartment model of [15].

log 10 λ
muc:air (T ) = −A + B T + 273.15 , (B.1) where A and B (in Kelvin) are proportional to the entropy and enthalpy of volatilization, respectively.λ b:air will always refer to 37 • C. Similarly, the partition coefficient between mucosa and blood λ muc:b is treated as a constant defined by λ muc:b := λ muc:air (37 • C)/λ b:air .(B.2) Note, that if the airway temperature is below 37 • C we always have that λ muc:air /λ muc:b ≥ λ b:air .(B.3)

Table 2 .
List of calculated values

Table A1 .
Abbreviations It depends solely on the temperature t and can be computed accurately enough by the Buck equation § What we denote by C I is hence C lung (t body ), which is C room (t) converted to body conditions.