On steady state inert gas exchange

On Steady State Inert Gas Exchange* JOHN W. EVANS Department of Mathematics,

Unicersi~

of Cak$orniG San Diego, La JoIIa, Calijonia

92093

Received IO October 1978; revised 9 April 1979

ABSTRACT Under the usual conditions in steady state retention-solubility studies a variety of compartmental lung models are equivalent to models with parallel ventilation and perfusion. This is seen through considerations of the generalized eigenvalue problem for A +hB, where A is a matrix describing the airflow and B is a matrix describing the bloodflow. Symmetric interchange between compartments assures the existence of a parallel ventilation and perfusion model with equivalent steady state inert gas exchange. These considerations are applied to the pulmonary dead space and to series ventilation.

1.

INTRODUCTION

In this paper the steady state exchange of soluble inert gases is examined for a class of compartmental models of the lungs. Particular attention is given to models which are equivalent in their inert gas retention to those models given by Farhi [4] and Farhi and Yokoyama [S] which consist of a number of compartments ventilated and perfused in parallel. These parallel models, with up to 50 compartments, are widely used in interpreting inert gas retention studies [3, 8, 10, 13-15, 17-191. A related paper has treated two compartment models [ 161. It is seen that equivalence to a parallel ventilation and perfusion model is related to a generalized eigenvalue problem and that the eigenvalues correspond to ventilation-perfusion ratios in parallel models. The paper is organized as follows: Section 2 describes the inert gas retention test and gives some physiological motivation for the present investigation. Section 3 gives the class of compartmental models of the lung studied here. Section 4 is devoted to definitions which allow a concise description of gas exchange in later parts of the paper. *This work was supported by Public Health Service Grant HL-1773 1. MATHEMATICAL

BIOSCIENCES

cQElsevierNorth Holland, Inc., 1979

46~209-222

209

(1979)

002555&t/79/080209

+ 14.902.25

JOHN W. EVANS

210

Section 5 gives the equations of steady state exchange in matrix form. It is here that the matrix A + ?d, where A describes air flow, B describes blood flow and h is the blood gas partition coefficient, is given. Section 6 presents the dependence of inert gas retention on solubility (the retention-solubility relation) using the matrix formalism. Section 7 is a brief technical digression in which fundamental results from general transport theory are related to the invertibility of A +XB. Section 8 presents the usual parallel ventilation and perfusion models for later comparison. Section 9 introduces the main topic in treating the generalized eigenvalue problem for A +hB and relating this to the retention-solubility relations. Section 10 treats the special case in which there is symmetric interchange between compartments. It is here that the equivalence to parallel ventilation and perfusion models is seen to be complete. A knowledge of the general properties of symmetric matrices and orthogonal subspaces is assumed. Section 11 is devoted to two examples satisfying the symmetry conditions of the previous section. The first concerns pulmonary dead space, and the second deals with purely series ventilation. Section 12 concludes the paper with three counterexamples selected to show that equivalence to parallel models is not always present. 2.

PHYSIOLOGICAL

MOTIVATION

An important test of lung function is based on the steady state exchange of inert gases of varying solubility. The gases are administered in dissolved form by intravenous infusion, and partial pressures are determined for the mixed venous and arterial blood and for mixed expired air [18]. The test is generally interpreted as though the lung were composed of a system of compartments ventilated and perfused in parallel. A description is sought of the distribution of perfusion over the range of all possible ventilation-perfusion ratios. With this interpretation the steady state exchange in certain asthmatic subjects appears to arise from a bimodal distribution of perfusion. It has been suggested by Wagner et al. [15] that this experimental finding is best explained by assuming that there is collateral ventilation to lung units whose direct broncheolar inflow and outflow is obstructed. For this and related reasons, and because of rebreathing from conducting airways, it appears to be worthwhile to investigate nonparallel ventilation and perfusion lung models. In this paper we show among other things that if perfusion is parallel and if the air exchange between compartments is symmetric, then there is a parallel ventilation and perfusion model with the same inert gas retention. We take this as implying that when the blood supply is parallel and the airflow of expiration is the reverse of inspiration, there is the (approximate) equivalence as stated, since the corresponding continuous ventilation and perfusion model would give

211

STEADY STATE GAS EXCHANGE

symmetric mixing. In our estimation this goes a long way why the parallel ventilation and perfusion description is the presence of (conducting airway) dead space. This is earlier work [l] on insoluble gas washout studies. We begin by giving the class of models treated in this

toward explaining successful despite closely related to

3.

LUNG

CONSTANT

VENTILATION

AND PERFUSION

paper. MODELS

In the lung models treated here it is assumed that there is an inert soluble test gas with blood-gas partition coefficient h which is present in trace amounts. The blood-gas partition coefficient h is a measure of the linear solubility of an inert gas and, at equilibrium conditions at a fixed partial pressure of the gas, is given by the gas content of a volume of blood divided by the gas content of an equal volume of air. The fluctuations due to cyclical breathing [9] and pulsitile blood flow are taken as having been averaged out, and the following assumptions are made: (i) There are n compartments with n > 1. A compartment may be thought of loosely as a region of the lungs with homogeneous gas transfer properties. (ii) There is a blood transport system. A flow rate Qti > 0 gives the blood flow from the jth compartment to the ith compartment for i#j with 1 0 gives the input flow (from mixed venous blood) to the ith compartment, and a flow rate & > 0 gives the output flow (to mixed arterial blood), for 1 < i
for 1
JOHN W. EVANS

212 We assume leaving mixture 4.

state the above in matrix form after giving some definitions. We in the sequel that arterial blood is solely the mixture of the blood the n compartments, and similarly that expired air is solely the of the air leaving these compartments.

PRELIMINARIES

AND DEFINITIONS

Let pr, pE, pV and pa give the respective inspiratory, expiratory, mixed venous and mixed arterial partial pressures of the test gas, and let the n-column-vector p have as its ith entry the equilibrium partial pressure of the test gas in the ith compartment. Let A be the n by n matrix with entries A,, where

so that for the ith row of A the off diagonal entries give the air flows into the ith compartment from other compartments with negative sign, while the diagonal entry gives the total air flow out of the ith compartment. Similarly let B be n by n with entries

def

Bii s

-

Qg

for

i#j,

kto Qki

for

i-j.

k#i

In addition let e be the n-column-vector with all entries identically 1, and let a superscript r denote the transpose, so that er is an n-row-vector. Now let a be the vector of inspiratory air flows whose ith entry is cc,, and let b be the corresponding vector for blood flow whose ith entry is &,, for i=l,...,n. Finally, note that the ith entry of eTA (of eTB) is voi (is ~oi) for j = 1, . . . , n, and that by assumption (vi) we have that the ith entry of Be is

kgo Q/ci- j$,QU=&I k#i

jzi

for i=l,..., n, so that b= Be. We are now ready to state the basic equations values of p, pE and p. given pr and pV.

which determine

the

STEADY STATE GAS EXCHANGE

5.

THE BALANCE

213

EQUATIONS

IN MATRIX

FORM

With the definitions of the preceding sections, the mass balance tion (v) at equilibrium (iv) is given simply as (A +XB)p=pla+@p,b,

assump-

(1)

since the amount of test substance per unit time carried in an air channel is equal to the partial pressure times the flow rate, and the amount in a blood channel is X times the partial pressure times the flow rate. Assuming for the moment that the matrix A +hB has an inverse, we then have p=PI(A+AB)-‘a+&,(A+Td?)-‘b.

(2)

Now the arterial partial pressure p, satisfies

so i

&Pi

which is e =Bp/e ‘Be. In the same way the expiratory eTAp/eTAe. The results are the expressions p,eTB(A

+XB)-‘a+&eTB(A

PO’

partial pressure pE is

+AB)-‘b (3)

e TBe

and pIeTB(A+XB)-‘a+XpveTB(A+XB)-Lb PE=

In the next section the fractional assumption that pI = 0 is given. 6.

THE RETENTION-SOLUBILITY

(4)

eTAe retention

of test substance

under the

RELATION

When p, - 0 the expressions (3) and (4) simplify. The fraction of test gas retained in the blood (with pI =0) after a pass through the lungs as a function of solubility h is called the retention R(h). This is equal top,/py,

JOHN W. EVANS

214 and we have R(A)=AeTB(A

+hB)-‘Be/e=Be,

(5)

where we have used the fact that b = Be. It is the retention that is measured experimentally and is the main object of investigation in this paper. In the next section we discuss the invertibility of A +u. Following that, the retention of models with zero off diagonal terms in A and B is given and the equivalence of more general models is investigated. 7.

CONNECTIVITY

AND INVERTIBILITY

The matrices A, B and A + a variety of transport systems and nonpositive off diagonal with respect to columns. That

Ml for X > 0 are typical of matrices describing in that they have nonnegative diagonal terms terms and have a dominant main diagonal is,

IAii(-

ff lAjil = lioi > 0 j= 1 jzi

for i=l , . . . ,n, with similar inequalities for B and for A +m with h >O. Such matrices have been widely studied (see [7] and references therein). It is a fact that A is singular if and only if there is a collection of compartments which are “dead end” compartments in the sense that no path of air flow leads from any of these compartments directly or indirectly (through any number of other compartments) to the outside. Of course a similar statement holds for B with respect to bloodflow. When these considerations are applied to A + ?tB for X > 0, we see that this matrix is singular if and only if there are dead end compartments with respect to flow which may proceed by either air or blood at any stage. In such a model a test gas would be trapped for all time in the dead end compartments. This may be made precise by stating that A + AB for h > 0 is singular if and only if there is a nonempty subset J of 1,. . . , n such that Zy, ,Aji + Bji = ~~,+(j,=0fora11iinJandAi,=Bi,=0fora11iinJand1~j~nnotinJ. We exclude such singular A + AB from consideration in this paper. An additional fact of interest about A +hB is that all entries of (A + AB)-’ are positive (see Ortega and Rheinboldt on M-matrices [ll]). 8.

PARALLEL

VENTILATION

AND PERFUSION

MODELS

The n compartments are said to be ventilated (perfused) in parallel if the off diagonal terms of A (of B) are zero. When both ventilation and perfusion are parallel we have A = diag( pai,. . . , po6,)and B = diag( Qi,. . . , &),

STEADY STATE GAS EXCHANGE

215

where def Qj

SE

pie=

for

Qi

i=l,...,n.

Also b = (a,. . . , &)’ and a = ( &, . . . , r’,,)’ as usual. Equation (5) then becomes [4,5]

i-1

(W where & = ~i/Z~_, 0;. and xi = pai/ Qi for i = 1,. . . , n. In writing R(X) in this form, a compartment with oi =0 has 6 -0, and xi = cc and is taken as contributing nothing to the sum. The quantity Fi is called the fractional perfusion to the ith compartment. Compartments with identical ventilation-perfusion ratios xi are generally lumped together. In the next section the study of models in which R(A) has the form given by (511)is begun. 9.

DIAGONABLE

MODELS

In this section we examine the retention-solubility relation R(A) under the assumption that A +XB is diagonable in the sense given below. Say as usual [20] that g is an eigenvector of A +hB if either (i) there is a complex number (ii) Bg=O.

p such that (A - @)g

= 0, or

In case (i) [case (ii)] we say that t_r+[m] is an eigenvalue of A + AB associated with gi. Call the system A + U? diagonable with extended real nonnegative eigenvalues if there are linearly independent eigenvectors g,, . . . ,g, with associated eigenvalues p,, . . . , p,, which are nonnegative or infinite. We now state and prove a theorem. THEOREM

I

Suppose that A +kB is diagonable with eigenvectors g,, . . . ,gl having eigenvalues CL,,. . . , p, > 0 and eigenvectors g,, ,, . . . ,g,, with eigenvakes equal to

216

JOHN W. EVANS

infini@. Then

where F,+-.*

+F,=l.

Proof. We have (A +xB)gi=(h++X)Bgi or (A +hB)-‘Bgi=[l/(A+ h)]gi for i-I,...,/. We may now express e as y, g, + ** * + yng, for real numbers y,, . . . , yn, so that

where F, = yte TBgi/(Z:_, yie *B@ and consequently completes the proof.

F, + * * * + Fl = 1. This

We do not yet have complete equivalence to the parallel ventilation and perfusion models, since diagonability alone does not insure that 4 > 0 for i=l , . . . , [. We point out here that formally each eigenvector with eigenvalue f.+> 0 contributes to the above expression for R(A), just as a compartment with ventilation perfusion ratio b contributes to the retention-solubility relation for a parallel ventilation-perfusion lung model. An eigenvector with eigenvalue cc does not contribute to the expression and thus acts in a similar fashion to dead space (with infinite ventilation-perfusion ratio) in a parallel model. In the next section we see that a symmetry assumption on A +AB results in a complete identification with a parallel model. In the section on counterexamples it is shown that it is possible to have some Fi
STEADY STATE GAS EXCHANGE

10.

SYMMETRIC

217

MODELS

We say that A +AB is symmetric if A, = Aii and Bu = Bji for 1 < ij
2

If A +M is symmetric, then the system is diagonable with extended nonnegative real eigenvalues, and E. > 0 for i = 1,. . . , I, where I and F,, . . . , F, are as in the previous section. Proof: The eigenvalue problem for A +M has been thoroughly studied [6, 12, 201. Suppose first that B is nonsingular. Because B is symmetric, it has n real nonzero eigenvalues. Because B is diagonally dominant, it can have no negative eigenvalues by the Gershgorin circle theorem [ 1 I]. Thus B is positive definite and has a unique positive definite square root B ‘1’ which has an inverse B --1/2. Because A is diagonally dominant and symmetric, the matrix B -‘/‘AB -1/2 is positive semidefinite and has an orthonormal set of eigenvectors h,, . . . , h,, with eigenvalues CL,,. . . , p,, > 0. Thus B-‘/2AB-‘/2hi=~hi, or

where gi = B - ‘/2hi for i = 1,. . . , n. Moreover

for 1
The case in which B is singular can be handled by a conjugacy given in [12]. We outline the main features without proof. Let %+ l, . ..,g, be a basis for the null space of B, and let

operation

def u2

Let

u ,,

A&+,,...,

=

(g/+r~.**x”).

. . .,u, be a basis for the orthogonal Ag,,, and set

compliment

def UI

s

(u ,,...,

ur)

and

U~(lJ,,U2).

of the span

of

JOHN W. EVANS

218 Now consider the system

where the By the such that It then

O’s represent blocks of zero matrices of appropriate dimension. previous development there are vectors h,,. . .,h, and p1,..., p/ > 0 A,hi=bB,hi and hiTB,h=Si for 1
U,hi=U

we have Ag, = bBgi and gTBgi = 8: for 1 < ij 4 I. The vectors g, , . . . , g, are eigenvectors of A + hB with eigenvalues p,, . . . , I_LI >O, and g,,, ,..., g, are eigenvectors with eigenvalues equal to infinity. For details the reader is referred to [12]. We have used the fact that since A + XB is nonsingular for h > 0, the null spaces of A and B have zero intersection. The previous construction depends on the fact that A U, has rank n - 1, that B, has rank I and is thus symmetric and positive definite, and that U is nonsingular. To complete the proof we must show that I$ > 0 for i = 1,. . . ,1. For e=27_,yigi we now have eTB(A+XB)-‘Be=

i$l yig? (

B(A+XB)-‘B( )

i i==l

and

($,YigiT)B( i, Yigi)

eTBe=

TigT)

219

STEADY STATE GAS EXCHANGE Thus

with

for i= 1, . . . ,I, and the proof is complete. A related result for symmetric airflow in insoluble derived in our earlier work [l]. In the next section examples are given. 11.

gas washout studies is

EXAMPLES

In this section we briefly present some examples and counterexamples. In this and the following section any entry not indicated in a matrix is equal to zero. Example I (Dead space). r

c&” 0

A=

c

Let 0 C&r

0

-ori

0

-CXtin

0 _&i

...

0

...

-LTV”

F,

r’,

B=diag(O,O,& ,..., $) and a=Ae, where QT= vi + . . . + v,,. In this model the first two compartments together constitute the dead space. A fraction (r of the total inspiratory and expiratory volume VT goes to and from the first compartment, with no communication between this and other compartments, and the same fraction (Yof the total ventilation c to and from the and (2 + i)th compartment for i = 1, . . . , n goes to the second compartment back. The first two compartments are unperfused, and the remaining compartments are perfused in parallel. Since this is a symmetric model, there is an equivalent parallel ventilation and perfusion model. This will be treated in detail in a later paper along with Example 2.

220

JOHN W. EVANS

A related treatment is given in [2].

of the dead space for insoluble

gas washout

studies

Let

Example 2 (Series ventilation).

A=

B=diag(o

,,..., 8)

and

a=Ae,

where fl,>ir,, pi= (ii_,+ oi+, for i=2,...,n-1, and pn= &_,. Here the n compartments are symmetrically ventilated in series. Again there is an equivalent parallel ventilation and perfusion model. 12.

COUNTEREXAMPLES Counterexample I.

A=(

Let

-d

_;

8),

B=(;

-b

-y

and a=Ae. Then erBe=l and eTB(A+XB)-‘Be=(l,O,O)(A +AB)-‘(O,O, the upper right hand entry of (A +AB)-‘. This is

i’ ‘_‘: ( det(A + hB)

1) is equal to

det

= 1 +x+:2+x3 1

=&-

=

(A+

I)(h:

i)(h-

i)

$(i- 1) $(i+ 1) ~ ~ + X-i A+i

In this model blood flows in compartment 3, through compartment 2 and out compartment 1 while air flows in the opposite direction. There are complex eigenvalues i and - i of A + ?d, where of course i2 = - 1, and R(h)=h/(l +X+hZ+X3) has poles at - 1, i and -i.

STEADY

STATE

GAS EXCHANGE

Counterexanple

2.

where I is the identity

221

Let

matrix. Then

eTB(_4 +AB-‘Be=(I,O

,...,

-

O)T(A

+AB)-'(0 ,..., 0,l)

A”--’ (1 +A>”

and R(A) =A”/( 1 +A)“. Here 1 is the only eigenvalue of A + AB (LO,..., 0) is the only eigenvector, so the system is not diagonable. Counterexample 3.

IS-(:

Let

i),

A=(:,

1+‘,)

and

a=Ae.

Then R(A) =

A( 1+ &)/2& _ A( I- &)/2& 2+2&+h ’ 2+A

so that in the parallel equivalent sions is negative if 0
interpretation

one of the fractional

perfu-

The author would like to thank James R. Bunch, William B. Gragg and Peter D. Wagner for many helpful discussions.

REFERENCES John W. Evans, The gas washout determination under a symmetry assumption, Bull Math. Biophys. 32:59-63 (1970). John W. Evans, David G. Cantor and Joe R. Norman, The dead space in a compartmental lung model, BulI. Math. Biophys. ?9:71.1-718 (1967). John W. Evans and Peter D. Wagner, Limits on V,/Q distributions from analysis of experimental inert gas elimination, J. AppI. Physiol. 42:889-898 (1977). L. E. Farhi, Elimination of inert gas by the lung, Respirat. Physiol. 3: l-l 1 (1967). L. E. Farhi and T. Yokoyama, Effects of ventilation-perfusion inequality on elimination of inert gases, Respirat. Physiol. 3: 12-20 (1967).

222

JOHN W. EVANS

6 F. R. Gantmacher, The Theory of Matrices, Chelsea, New York, 1959. I John Z. Hearon, Theorems on linear systems, Ann N. Y. Acud Sci. 108:36-68 (1963). a S. A. Jaliwala, R. E. Mates and F. J. Rlocke, An efficient optimization technique for recovering ventilation-perfusion distributions from inert gas data. Effects of random experimental error, J. Chin Znwst. 55: 188-192 (1975). 9 R. E. Nye, Influence of cyclical pattern of ventilatory flow on pulmonary gas exchange, Respirut. Physiol. lo:321 -337 (1970). 10 A. J. Olsxowka, Can V,/Q distributions in the lung be recovered from inert gas retention data, Respirat. Physiol. 25: 191-198 (1975). 11 J. M. Ortega and W. C. Rheinboldt, Zteratiw Solution of Nonlinear Equations in Several Variables, Academic, New York, 1970. 12 G. W. Stewart, On the sensitivity of the eigenvalue problem Ax=USx, SIAM J. Numer. Anal. 9:669-686

(1972).

13 P. D. Wagner, D. R. Dantxker, R. Dueck, J. L. Clausen and J. B. West, Ventilationperfusion inequality in chronic obstructive pulmonary disease, J. Clin. Znwst. 59:203-216

(1977).

14 P. D. Wagner, D. R. Dantaker, V. E. Iacovoni, R. F. Schillaci and J. B. West, Distributions of ventilation-perfusion ratios in asthma (Abstract), Amer. Reu. Respirut. Discuses 111:940 (1975). 15 P. D. Wagner, D. R. Dantzker, V. E. Iacovoni, W. C. Tomkin and J. B. West, Ventilation-perfusion inequality in asymptomatic asthma, Amer. Reu. Resp. Dis. 118:511-524 (1978). 16 P. D. Wagner and J. W. Evans, Conditions for equivalence of gas exchange in series and parallel models of the lungs, Req. Physiol. 31: 117-138 (1977). 17 P. D. Wagner, R. B. Laravuso, R. R. Uhl and J. B. West, Continuous distributions of ventilation-perfusion ratios in normal subjects breathing air and 100% 4, J. Chin Znwst. 54:54-68 (1974). 18 P. D. Wagner, P. F. Naumann and R. B. Laravuso, Simultaneous measurement of eight foreign gases in blood by gas chromatography, J. Appl. Physiol. 36:600-605 (1974).

19 P. D. Wagner, H. A. Saltxman and J. B. West, Measurement of continuous distributions of ventilation-perfusion ratios: theory, J. Appl. Physiol. 36:588-599 (1974). 20 Wilkinson, The Algebraic Eigenwlue Problem, Oxford U. P., Oxford, 1965.