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Multibreath tracer species dynamics in the lung
Bulletin of Mathematical Biology, Vol. 43, pp. 1-19 Pergamon Press Ltd. 1981. Printed in Great Britain Society for Mathematical Biology
0092-8240/81/0101-0001 $02.00/0
MULTIBREATH TRACER SPECIES DYNAMICS IN THE LUNG
IIGERALD M. SAIDEL and GERALD M. BURMA Dept. of Biomedical Engineering, Case Western Reserve University and Veterans Administration Medical Center, Cleveland, Ohio 44106, U.S.A.
By studying the behavior of various tracer species in the lungs, one can assess many important characteristics which distinguish normal and abnormal function, Quantitative evaluation of function depends on the use of an appropriate model in conjunction with experimental data. A multi-compartment model is derived from mass balances to describe dynamic as well as (breath-averaged) steady-state transport processes between the environment and pulmonary capillary blood_ The breathing cycle is divided into three time periods (inspiration, expiration, and pause) so that the model equations are discrete in time. No other model of tracer species transport in the lungs deals simultaneously with species dynamics, variable breathing pattern, distribution inhomogeneities, and non-equilibrium between alveolar gas and capillary blood_ Models currently in the literature are shown to be special cases of the model presented here.
1. Introduction. F u n c t i o n a l charcteristics of n o r m a l or a b n o r m a l lungs can be o b t a i n e d from experiments using tracer species with different physico-chemical properties such as solubility (Bouhuys, 1977). T h e characteristics of interest include ventilation (distribution a n d mixing of inhaled gas with residual gas), inter-phase diffusion (transport between alveolar gas and capillary blood), perfusion (convective t r a n s p o r t of capillary blood), ventilation/pe~ftt'sion (ventilated alveoli relative to perfused capillaries), and volumes of lung tissue a n d capillary blood. Tracer species m e t h o d s are particularly of interest for non-invasive or minimally invasive e v a l u a t i o n of lung function. Consider, for example, a tracer species that is chemically inert a n d very slightly soluble in blood, such as argon (Ar) or nitrogen (N2). W i t h such a tracer, the dynamics of the gas wash-in or wash-out process can yield i n f o r m a t i o n a b o u t venti-
2
GERALD M. SAIDEL AND GERALD M. BURMA
lation inhomogeneity. Another non-invasive method involves carbon monoxide (CO), which at low concentration diffuses into the blood as if it were an infinite sink. Methods to evaluate alveolar-capillary transport with CO require measurements only at the mouth. With inert soluble gases, lung perfusion and volumes of lung tissue and capillary blood can also be estimated non-invasively under suitable physiological conditions (Stout et al., 1975; Glauser et al., 1974). Invasive techniques with several inert gases of different solubility have been used to evaluate Ventilation/perfusion abnormalities (Hlastala and Robinson, 1978). Many models and experiments have been used to evaluate pulmonary function with tracer species. Ventilation inhomogeneity can be studied by a step change of the inhaled gas for a single breath with a large-volume maneuver or for multiple breaths with spontaneous breathing. A largevolume, single-breath procedure requires subject cooperation; a multibreath experiment does not. In multibreath studies, the dynamics are evaluated from average-expiratory or end-expiratory data on successive breaths. With such limited data, spatially lumped models which are discrete in time are most suitable (Saider et al., 1978). Lumped models are also typically used for the analysis of alveolar-capillary transport from single-breath and multibreath experiments with CO. With few exceptions, these models assume uniform ventilation, i.e. only one alveolar compartment. However, an accurate assessment of this inter-phase diffusion in the presence of ventilation inhomogeneity requires the simultaneous-study of an insoluble tracer in addition to CO (Saidel et al., 1971). In evaluating lung perfusion and tissue volumes, investigators have used various models of the dynamic response of inhaled soluble tracer species (Stout et al., 1975; Glauser et al., 1974). These models are limited, however, by the assumption of uniform alveolar ventilation and perfusion. In contrast, Scrimshire and Tomlin (1973) developed a dynamic model with nine alveolar-Capillary compartments. A fundamental assumption in their model is that N 2 0 equilibration occurs between alveolar gas and capillary blood. This same assumption occurs in models developed to estimate ventilation/perfusion, for example, the model proposed by Farhi (1967) and modified by others (Reid and Hechtman, 1974; Wagner et al., 1974). While these models are based on several inert gases with constant venous infusion, an alternative model by Zwart et al. (1976) deals with a sinusoidally-varying input of a single inhaled tracer species~ Although the input differs in form, this model also suffers from the assumptions of a single alveolar compartment and tracer equilibration between alveolar gas and capillary blood. In this paper, we present a lumped model of the lung suitable for the
MULTIBREATH TRACER SPECIES DYNAMICS IN THE LUNG
3
study of inter-breath dynamics of a variety of tracer species with different input and breathing conditions. The objective is to provide a comprehensive framework for the evaluation o f a b n o r m a l ventilation, inter-phase diffusion, perfusion, ventilation/perfusion, and volumes of lung tissue and capillary blood. Furthermore, we can more clearly appreciate the approximations used in current models from the limiting cases of our more general model. The structure shown in Figure 1 is based on a single dead space (compartment one) and two alveolar spaces (compartments two and three)
2. Model.
INSPIREDGAS-
~
E __
EXPIREDGAS
(i) ALVEOLAR SPACE (2)
I
ALVEOLAR SPACE (3)
I
c'P'LL"YI sP(4, ACE MIXED-VENOUS ~_ BLOOD Figure 1.
CAPILLARYI SPACE (5) ARTERIAL BLOOD
C o m p a r t m e n t a l structure of lung model.
representing two composite groups of alveoli. The parallel compartments are included to represent regional inhomogeneities in concentrations in the lung. With respect to blood-phase transport, the distribution of pulmonary capillary blood flow to each composite group of alveoli is represented as inputs to compartments four and five. In addition, the pulmonary parenchymal tissue volume is divided into compartments six and seven, each associated with alveolar and capillary blood compartments. Transport processes in the system are governed by total and species mass balance equations for all of the system compartments. The general model equations describe the dynamics and distribution of a tracer species after an input change in tracer concentration. Our approach is to obtain
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GERALD M. SAIDEL AND G E R A L D M. BURMA
the total mass balance of each compartment to relate volume rate changes, if any, and volume flow rates. The species mass balances yield dynamic equations for species concentrations. Since the structure of our model is spatially coarse and data over the breathing cycle yields limited information, we choose to analyze dynamics discretely. For any breath k, we consider changes over three discrete time periods (~): inspiration (I), expiration (E), and end-expiratory pause (P). The time periods occur in the sequence {. . . r ( k - l,I), r ( k - l,E), r ( k - l,P), T(k,I), r(k,E),
T(k,P),...}. Gas phase transport.
To keep the equations from becoming unnecessarily complex, several assumptions are made with respect to gas transport. The gas density is assumed constant and equal throughout the lung. In addition, plug transport is assumed through a constant volume dead space, which implies a pure time delay with respect to the transport of gas between the environment and alveolar spaces. Using these assumptions, the total mass balance equations for the dead space reduced t o Ool =Q12+Q13,
"c=I,
Oto=O21+Q3t,
z=E,
where Qu is the flow rate from compartment i to compartment j. These flow rates can be related by gas flow fractions which are assumed constant:
fi=Q1JQol=Qil/Qxo,
i=2, 3,
where f2 +f2 = 1. The total alveolar volume VA(k, ~) at the end of period z of breath k is given by
v~(¢, ~)= v2(k, ~) + G(k, ~), where V~(k,z) is the volume of alveolar compartment ~ at the end of period T(k,'c). We assume that the alveolar volume fractions are independent of breath number;
vi(z)=V~(k,'c)/VA(k,'c), where
i=2, 3,
I)2(T)-~-V3(T)=1. The parameters f / a n d v~(z), which are related by v,(I)VA(k, I) = v,(E)VA(k - 1, E) +~VT(k, I), v~(E) = v,(P),
M U L T I B R E A T H T R A C E R SPECIES D Y N A M I C S IN T H E L U N G
5
where Vr(k,T ) is the volume change over the period T(k,z), characterize the distribution of ventilation. Taking these parameters as constants implies that the mechanical properties of the lungs are time invariant. In deriving the total mass balance equations for the alveolar compartments, the volume of tracer species taken up by p u l m o n a r y tissues and capillary blood is considered small compared to the total alveolar gas volume. This is valid if the fraction of soluble tracer in the alveolar gas is sufficiently small. For the alveolar compartments (i=2,3), the total mass balance equations over inspiration and expiration can be simplified under the conditions given above; consequently,
dV//dt =
f
JiQol,
-f/Q0 t, 0,
r=I,
-c= E , z =P,
(i)
where V~(t) is the volume of alveolar c o m p a r t m e n t i at any time t. In the time interval t k < t < t k + l within breath k, we index the three intervals-inspiration, expiration, and p a u s e - - b y z E {I, E, P}. The volume of compartment i at the end of these intervals within breath k is V/(k,T). Integrating over inspiration T(k, I), we obtain
V~(k,I) = V~(k- 1, P) +f~Vw(k, I) and over expiration
T(k,E) v (k, E) = v (k, i) - f Y r (k,
No volume change occurs during pause, T(k,P):
Vdk, P)= V~(k,E). The species mass balance equation for each well-mixed alveolar compartment on inspiration k is
[d(C,V~)/dt] =
Cl,f/Q01
-
[Ri+2 +R~+4],
i = 2 , 3,
(2)
where C i is the species concentration in alveolar c o m p a r t m e n t i and Cli iS the species concentration entering alveolar c o m p a r t m e n t i from the dead space (compartment 1). The terms R~..2 and Ri--a represent uptake of s o l u b l e g a s by blood and tissue compartments respectively. Dividing equation (2) by equation (1) and integrating over inspiration T(k,I) in a
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GERALD M, SAIDEL AND GERALD M. BURMA
piece-wise fashion (Saidel et al., 1978), we find that C,(k, l)V~(k, I) = C,(k - 1, P)V~ (k - 1, P) + C1 o (k - 1 )fWD + Co
[VT(k, I ) -
- T (k, I ) [ ( R i + 2)k,I
-}- (R~ + 4)k, I1
(3)
where the subscripted bracketed quantity ()k,~ represents an average over the period T(k, ~). On expiration, the species mass balance is d(CiV~)/dt = - C ~ Q l o -
[Ri÷2 +R~+41,
i = 2 , 3.
(4)
Expanding the derivative of equation (4) and combining it with equation (1) yields V/dCJdt = - [R~ + 2 + Ri + 4]. Integrating over expiration T ( k , E ) , we obtain C~(k,E)=C~(k,I)--T(k,E)[(R,+2/Vi)k,E+(Ri+4/V~)k,~].
(5)
During pause, the species mass balance is d ( C y ~ ) / d t = [ R i + 2 +R~+4] ,
i = 2 , 3.
integrating over T ( k , P ) and noting that the alveolar volume is constant, we find C,(k,P)=Ci(k,E)-[T(k,P)/V~(k,E)][(Ri+2)k,p+(Ri+4)k,nl.
(6)
The end-expired gas concentrations from the dead space can be related to end-expired alveolar concentrations if we assume that V T ( k , E ) > VD and that the dead-space time delay toward the end of expiration is sufficiently. small to provide a good estimate of the mixed alveolar concentrations. Thus, a species mass balance over the dead space yields:
Clo(k)Q~lo~-C2(k,E)9~21-t- C3 (k, E ) Q 3 1 ,
MULTIBREATH TRACER SPECIES DYNAMICS IN THE L U N G
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where Clo(k ) is the concentration leaving the dead space and entering the environment at the end of T(k,E). Dividing by Q10, we obtain C1 o (k) =f2C2 (k. E) +f3C3 (k, E).
(7)
Blood phase transport. Blood density, volume, and flow rates are assumed constant for each capillary space. Consequently, the flow rate through c o m p a r t m e n t i is Qz, and the total flow rate is given by Q=Q4+Qs. Capillary blood flow fractions associated with alveolar compartments are defined as
gi=Qi+2/Q
i=2,3,
so that
g2+g3=l.
Gee (1972) studied the relationships between the total pulmonary blood flow rate, p u l m o n a r y blood volume, and extra-vascular fluid volume in anesthetized supine dogs using indicator dilution methods. Pulmonary blood volume, extra-vascular fluid volume, and the rate of pulmonary blood flow were found to be directly proportional to one another. Consequently, we let the blood volume of each c o m p a r t m e n t be proportional to the blood flow to that compartment:
Vi+2/Vb=gi, i=2, 3. The total capillary blood volume is Vb = V4 + V5. The species mass balance equation for each capillary c o m p a r t m e n t is
V~+2(dCi+ 2/dt) = QgiEC~ - Ca, i+2] + Ri+ 2, where C o is the species concentration of mixed-venous blood and C,,~+ 2 is the species concentration leaving capillary c o m p a r t m e n t i + 2 and entering •the systemic arteries. Integrating over T(k, z), we obtain
C,+ 2(k, r)=Ci+ 2(k, r - 1 ) + [ T ( k , z)Q/K][(Co)k,~-(Co, i+ 2)k,~] + [T(k,
(8)
where the argument ( k , r - 1 ) refers to the time period just prior to (k,z) in the cyclical order [_.. (k - 1, I), ( k - 1, E), ( k - 1, P), (k, I), (k, E), (k, P ) . . . }.
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GERALD M. SAIDEL A N D G E R A L D M. BURMA
Tissue phase transport. The tissue spaces are assumed to be of constant volume and density. The total volume is Vp = V6 + V7. The distribution of tissue volume to each alveolar compartment is defined by: hi=Vi+4/Vv,
i=2,3,
so that h 2 + h 3 = 1. The species mass balance equation for well-mixed tissue compartments is V~+4 (dCi/dt ) = Ri +4.
Integrating over time T(k, z), we get Ci+ 4 (k, z) = Ci+ 4 (k, • - 1 ) + I T ( k , "Q/Vph[l(R~+4)k, ~.
(9)
Alveolar-capillary transport. Consider the diffusion of a gas species across a thin homogeneous membrane separating gas and blood. The species flux J across the membrane is given by J = (~rCo - Cb)~/fi.
where C o and Cb are the species concentrations in gas and blood, @ is the diffusivity, 6 is the thickness of the membrane, and the partition coefficient a represents the ratio of the equilibrium concentration of the species in the blood to its concentration in the gas phase. The concentration profile Cb(z) of a gas species in the capillary bed can be described by a one-dimensional, quasi-steady model of a l v e o l a r capillary transport. Consequently, a mass balance equation for a tracer species leads to Q (dCb/dz) = Js = (~C o - Cb)~s/6,
where s is the surface area per unit length along the capillary bed. At z = 0, the species concentration in the b l o o d is the mixed venous concentration entering the capillary bed (Cb(0)=C~). Similarly, at z = L the species concentration becomes the end-capillary concentration (Cb(L) = Ca). Assuming that the alveolar concentration is constant over the blood transit time, this equation can be integrated to obtain C~ = C~ + [ a C o - C,][1 - exp ( - ~sL/DQ)].
MULTIBREATH TRACER SPECIES DYNAMICS IN THE L U N G
9
In terms of the nomenclature used for the multicompartment model, the arterial concentration from capillary compartment i + 2 is related to the gas concentration of alveolar compartment i according to
Ca,i+2-Cv=[ffCi-Cv]~i,
i=2,3,
(10)
where e~= 1 - exp ( - 7/Qg,).
(11)
The relationship described above avoids the usual assumption of equilibrium between alveolar gas and end-capillary blood. To relate a lumped-parameter model of transport to this onedimensional model, we simply integrate the membrane flux over the capillary surface. In particular; the rate of mass transport of species between one alveolar compartment and a blood compartment is given by:
R=
Jdz=(~s/6) o
fL
(aCo--Cb)dZ~--y[aCo--Cb] ,
o
where 7 = ~ s L / 6 is a transport coefficient. The time averaged diffusion rate can be expressed as < R > = 7 = y [ a < C g >
- ].
(12)
Interphase transport.
For alveolar-capillary transport, average diffusion rates between compartments during inspiration and pause (-c--I,P) can be written similar to (12):
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GERALDM. SAIDEL AND GERALD M. BURMA
and for expiration (~=E),
( R~+~/V~)k,~='~h~a[( Ci/V~)k,e - (1/a )( C~+4/V~)k,E], i = 2 , 3 . In these constitutive equations, the b l o o d - g a s and coefficients (a) are assumed to be identical for a assumption is based on the work of Cander (1959) partition coefficients for several inert gases. Also, alveolar-Capillary transport (7) is assumed proportional alveolar-tissue transport (~).
tissue gas partition given species. This who found similar the coefficient of to the coefficient of
3. Model Analysis. Time-averaged approximations. Species concentrations at the end of successive periods {... ( k - 1, I), ( k - 1, E), ( k - 1, P), (k, 1), (k, E), (k, P ) . . . } are more readily related than are concentrations averaged over those periods. The time average of any variable X(t) over the interval T(k,v) during any period "c~{I,E, P} of breath k is defined as
(X)k,~=[1/T(k,r)]
i
T ( k , ~)
X(t)dt.
,3 t - T ( k , 0
Consequently, we express the time-averaged terms, a linear combination of the variables at the start and end of period -c= I or P, as
(Ci)k,~ ---c~Ci(k, ~ - 1 ) + (1 - ~)C~(k, % .and for z = E as
(C,/V~)k, E = ~[C,(k, I)/V~(k, I)] + (1 - ~)[C~(k, E)/V~(k, E)], where the weighting factor e lies in the interval [0, 1). For the analysis of experimental data, it is appropriate to let a take a value such as 0.5. Here, since we are more interested in the characteristics of the model, we let e = 0 to allow a simpler form. That is, the average value is approximated by the end-tidal value. Invariant breathing pattern. Although the model can be applied to spontaneous breathing, we shall restrict our analysis to a time-invariant
M U L T I B R E A T H T R A C E R S P E C I E S D Y N A M I C S IN T H E L U N G
11
breathing pattern, i.e. one independent of breath number k. Thus, for z e{I,E,P} we let
T(k, "c)= T(r),
vr(k, ~)= VT, and for i-- 2, 3
J'Gv~+fYr,
3=I , z=E,P,
V/(z) = (VAVi where v, = vi (E).
Combined model equations. For time-invariant breathing and averaged concentrations equal to end-period concentrations (c~=0), the model equations become A,(~)tp,(k, 3) = O,(k, ~ - 1) + ¢ ) , ( k - 1, ~),
(13)
3
C , o ( k ) = ~ f~C~(k,z),
(14)
/=2
where 0, and qS, are column vectors and A, is a matrix. Specifically, the vectors are
O,(k,z) =
Lci,k,j C,+2(k,r )
qbi(k_l,z) =
C,+4(k,~)
i i,k 1 T(r)QC,,(k,r) v~ 0
and the matrix is
, .
(aT(z)(giy+h,~))
_~T('c)ygi]
_£T(z)~hi. ; -
. (T(z)7 ]
Ai(z) =
(rp
3
To account for c h a n g e s that occur within the breathing cycle, we in-
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GERALD M. SAIDEL AND GERALD M. BURMA
troduced the following functions:
,~i(k-1,z)
= {~ (C1° ( k - 1 ) - C°1 )( V°/VT)+ C° 1]f~VT/VAvi', ~ = I ~:=E, P
#~(z)={jlVr/Vavg+ l, z=I =E,P. In this model, the unknown independent parameters are ]2 (or J3), v2 (or V3), g2 (or g3), h2 (or h3) , Q, Vb, Vp, 7, and ~. Except for the transport
coefficients 7, and }, the other parameters are dependent solely on physiological conditions. Ratios of the distribution parameters are of value in characterizing the effectiveness of the lung in transporting species between the environment and capillary blood. In particular, ventilation inhomogeneity depends on f~/v~ and ventilation/perfusion inhomogeneity depends on fi/gi. Special simplifications of the model occur for species which are: (a) insoluble (0< a <6, where 6 is suitably small) such as N2; (b) slightly soluble in tissue, but highly reactive in blood (viz. CO); and (c) soluble (~ < cr < oo), such as halothane. We shall examine some of the special cases in the following sections. Insoluble species. For this case we set o-=0. Also, if the tracer species enters only with inhaled gas, then the species concentrations in blood (Ci+2) and tissue (Ci+4) are zero. Consequently, the state equation (13) reduces to a scalar form:
C~(k, ~)= [Ci(k, ~ -
1 ) + 2 ~ ( k - 1, r)]/p~(r).
From the definition of 2~ a n d / ~ during expiration and pause, this equation shows species concentration to be constant during expiration and pause:
C,(k, P)-- C,(k, E) = C~(k, I). Consequently, we find
Ci(k, ~) = Ci(k, I)= [C,(k- 1, P) + 4 ( k - 1,/)3/~,(/) = ]-C~(k- 1,E) + 2 ~ ( k - 1, I)]/p~(I). I
(15)
In this case, there are only two unknown independent parameters: f2 (or f3) and v2 (or v3). For uniform ventilation, five=l, ,~ and Pi become
M U L T I B R E A T H T R A C E R SPECIES D Y N A M I C S IN T H E L U N G
13
independent of the index i so that
C~(k,E) = CA(k,E)= [ C l o ( k - 1) + 2 ( k - 1 )]//~, where
CA(k,E ) is the alveolar concentration, Z(k--1)=[ (Clo(k-- a )--Co~ )(VD/VT)+ Col]VT/VA
and
I~= VrlVA + 1. The model output from (14) leads to (71o (k) = C , , ( k E ) = E G o (k - 1 ) + ,~(k - 1 )3/#.
U p o n rearrangement of these equations, we obtain C l o ( k ) - Col = [C~0 ( k - 1 ) - Co~][(VA + VD)/(VA+ Vr)].
The solution is given by
G o ( k ) - c01 = EClo ( 0 ) - Col] r(vA + vD)/(vA + vT)] k
(16)
which is the form of the solution for a one-compartment model (Saidel et al., 1978). Carbon monoxide. If C O is a tracer species in the inhaled gas, its concentrations in blood (Ci+2) and lung tissue (C~+4) will be negligibly small. In the blood, CO reacts with hemoglobin such that its partial pressure in the blood is negligible. In the tissue, the CO solubility is very small. For this case, the state (13) reduces to
[#,(z)+ T(z)g,?a/VAv~]C~(k,z)=Ci(k,~-l)+2~(k-l,z ).
(17)
The u n k n o w n independent parameters in (17,) are f2 (or f3), v2 (ol v3), g 2 (or g3), a n d 7- To reduce the n u m b e r of independent unknowns, one may assume that f~/gi= 1, which means that the relative distribution of ventilation anti perfusion is uniform. While this assumption may not be unreasonable for normal lungs, it is unjustifiable for many if not most abnormal lungs.
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G E R A L D M. SAIDEL AND G E R A L D M. BUR MA
A c o m m o n clinical measure of alveolar-capillary transport is obtained with a (breath-averaged) steady-state analysis of CO in the inhaled and exhaled gas. In particular, a transport coefficient called the diffusing capacity (DLco) is estimated by
DLco = [ Vr/O]EC01 - C~]/P A,
(18)
where 0 is the average breathing period, Pa is the alveolar partial pressure of CO, and CE is the mixed-expired CO concentration. For comparison, a similar expression can be derived for 7- If the ventilation and ventilation/ perfusion distributions are uniform, fjv~ = f/g~ = 1, then our model has only one equivalent transport pathway between the environment and blood. Consequently, (17) becomes
[#(z)+T('c)Ta/VA]CA(k,z)=CA(k,z--1)+2(k--l,v).
(19)
Under steady-state conditions, we let
CA(k,c}=CA('c), 2(k-l,r)=).(r). Note from the output equation (14), Clo=CA(E). To reduce our model to (18) which is based on continuous gas inflow and outflow, we define T(I) =0, the average breathing period, and set T(E)=T(P)=O. These conditions require that
C I(P)=C4(E)=C.,t(I)=C4, /t(r)=/dl)=
1 + I"r l{j
and
;~(v) = , ~ ( / ) = E(cA - C , o ) ( V , , / V r ) + C o d V r / V , Consequently, (19) can be simplified and rearranged as ~;~ = Evr/O]E Co l - CA]E1 - v , , / v d / c ;.
(2o)
For comparison with (20), we express (18) in an equivalent form using the relationships between (a) mixed-expired and alveolar concentration derived from a mass balance, c ~ = C AE1 - V D / V d + Col V,,/VT
MULTIBREATH TRACER SPECIES DYNAMICS IN THE LUNG
15
and (b) alveolar partial pressure and concentration
PA = R TCA, where R is the ideal gas constant and Tis the absolute temperature. When these relationships are substituted into (18), the result is
DLco = [Vr/O][ Col -- CA][ 1 -- VD/Vr]/R T CA.
(21)
Hence, from (20) and (21), the transport coefficients are found to be directly proportional,
7 = (RT/a)DLco. Soluble species. F o r soluble species, we start with the state equations (13) and consider some special limiting cases: (a) high rate of alveolar-capillary transport and (b) uniform distributions. Let the diffusion rate or breathing period be sufficiently large, such that T(z)~/Vp>> 1,
T(r)7/Vb >> 1
and e~= 1 - e x p ( - 7/Qg~)--- 1. Under these conditions, and for 7 =7, the state equations can be written as
[~,,(r) + (a/VAV,)[g,T(z )Q + Vbg , + Vph,]}Ci(k, z) = [ 1 + (Vbg,/VAV,)+ (VvhjVAV,)]C,(k, r - 1) + 2~(k - 1, z) q- (T(z)OgjVav~)Cv(k, ~)
(22)
and a C i ( k , "c)= Ci+ 2(k , "c)= C i + 4 (k, "r).
(23)
This last equation states that equilibrium-exists among alveolar gas, capillary blood, and tissue fluid. Furthermore, if the distributions of perfusion and tissue volume are equal (gi=hi), then we see that Vb and Vp cannot be differentiated and appear only as the sum Vb + Vp = Vbp. For a system with uniform distributions..li=c~=gi=h~, the alveolar, end-
16
GERALD M. SAIDEL AND GERALD M. BURMA
capillary, and output concentrations can be written as
Clo(k)=C (k,E). Thus, with uniform distributions and a relatively large diffusion rate, (22) and (23) reduce to {#(-c) + (a/VA)ET(z)O + V j } C A ( k , v ) = [1 + (aVbp/VA)]CA(k, ~ - 1 ) + 2 ( k - 1, z) + [T(z)QC~(k, r)/VA] , C,(k, ~) = CA(k , "c).
(24) (25)
Constant venous injusion. The model of Farhi (1967) to evaluate a ventilation/perfusion ratio (VA/Q) deals with the experimental situation in which a tracer species is delivered only via a peripheral vein at a constant rate. Hence, inhaled tracer concentration is zero (Co1=0). As a result of the constant venous infusion being balanced by tracer loss by exhalation or tissue uptake, a breath-averaged s t e a d y state is achieved. Farhi models the lungs as having continuous gas inflow and outflow, uniform distributions, and a large diffusion rate. U n d e r these conditions, (24) and (25) lead to equations relating arterial and mixed-venous concentrations only: {(#(~)/a) + (T('c)Q + Vbp)/VA}C,(k , z) = (1 + Vbp/VA)C,(k , "c- 1 ) + 2 ( k - 1, z ) + T(z)QC~(k, z ) / V A.
(26)
To model continuous gas inflow and outflow, we let T(z)=O, the average breathing period, and T ( E ) = T ( P ) = O . For the steady state, the dependence of concentrations on breath number vanishes. With these conditions, we find
C.(k,'c)=Co, CAk,'c)=C and (26) reduces to Farhi's model, co/c, =
+ o],
(27)
where the alveolar ventilation rate is given by l?a = [Vr - VD]/O. Single-breath dynamics. The Cander and Forster (1959) method of estimating pulmonary blood flow and tissue volume is based on the singlebreath dynamics of a soluble tracer species that is inhaled. To compare
MULTIBREATH TRACER SPECIES DYNAMICS IN THE LUNG
17
their model with ours, we assume uniform distributions and a high diffusion rate. The model also assumes that inspired and expired time periods are short compared to a pause at the end of inspiration, T--0. In this single breath-holding maneuver, the alveolar and output concentrations are equal (C A (k, 72) = C A (k, I) = Ca (1, E) = C ~o), the initial condition is CA(k,T,--1)=CA(O)=O and the species concentration in the mixed venous blood is negligible (C~=0). Thus, from (24) we obtain [I + Vr/VA+(Cr/VA)(OQ+ Vbp)]Clo=L=Coa(Vr--VD)/V a.
(28)
If the procedure is repeated for two breath-holding periods (0= T1 and 0 = T2), and all other parameters are held constant, then (28) can be used twice to solve for the blood flow rate: Q = [ ( v a + VT)/a+ V b p l [ C l o ( O 1 ) - C l o ( 0 2 ) l / [ 0 2 C l o ( 0 2 ) - O 1 C l o ( O 1 ) l
(29)
Now, let us compare (29) with the equation derived by Cander and Forster:
Q=E(VA+ VT)/G+ gbp]{lnECxo(01)/Clo(O2)]}/(02 -01).
(30)
F r o m a power series expansion of the logarithmic expression, it can be seen that (29) and (30) are approximately the same provided that Clo(01)/Clo(0z) is not too far from unity. For an insoluble gas species, a = 0 , (28) reduces to
E1 -}- VT/VA]C'~o = C"~1(g T - VD)/VA where C* indicates the concentration of the insoluble gas species. Combining this with (28), we can solve for the pulmonary tissue-blood volume: = [(vA + VT)/aJ[(CToCol/C
oC
I)- 1] - 09_.
As the breath-holding time 0 becomes small,
gbp ~" ~( gA -]- gT)/Cr][ C~oCol/ Cl oC~l ) -
l].
(31)
4. Discussion. We have developed a discrete-time, multicompartment model that describes tracer gas species dynamics for an inhomogeneous
18
GERALD M. SAIDEL AND GERALD M. BURMA
pulmonary system. Although many of the elements of our model have previously been used in other models~ the combination that we deal with is unique. This model is intended gs the basis for analyzing pulmonary function in acutely ill patients. Depending on clinical circumstances, our model can take into account either mechanically-aided or spontaneous breathing. Since data acquisition for assessing acute dysfunction should be as quick as possible, a dynamic rather than steady-state model is preferable. Indeed, what we have in mind is a dynamic experiment in which several tracer gas species are inhaled simultaneously for 10 or 12 breaths. The tracer concentration dynamics at the mouth are not significantly affected by systemic recirculation up to the first 6-10 breaths. At the mouth, the tracer gases are continuously sampled and measured (by a respiratory mass spectrometer). Simultaneously, the flow at the mouth is also measured (with a pneumotachometer). By comparing the measured variables to the simulated model output under corresponding conditions, numerical estimates of model parameters may be obtained. For parameter estimation to be practical with the limited data in this context, the number of model parameters must be minimal. By allowing just two parallel transport pathways between the environment and capillary blood, we have the simplest characterization of pulmonary inhomogeneities associated with species transport. With more parallel compartments (a simple extension of our model), the parameter estimates are likely to be ill-determined. Because of the possible pathological stuations for which the model is to be used, we avoid the usual assumption of chemical equilibrium between alveolar gas and capillary blood. For example, subjects with edematous lungs may have an increased effective diffusion distance between alveolar gas and capillary blood. Also, emphysematous subjects have greatly reduced effective surface area for diffusion and possibly an increased effective diffusion distance due to a thicker concentration boundary layer in the alveolar gas. Consequently, the ratio of alveolar transport to capillary perfusion may be smaller than normal (especially in an inhomogeneously perfused lung) and cause a significant tracer concentration difference between alveolar gas and end-capillary blood. In various ways, models currently used to estimate one or more pulmonary characteristics from species transport studies are limiting cases of our model. Typically, current models require special conditions on the experimental method, e.g. a particular breathing pattern. But for clinical purposes, particular breathing patterns may not be feasible. Commonly, dynamic models do not incorporate pulmonary inhomogeneities, but steady-state models do. In acute situations, however, the time to reach
MULTIBREATH TRACER SPECIES DYNAMICS IN THE LUNG
19
steady state may be inappropriately long. Furthermore, steady-state models cannot yield some important information, namely, characteristic volume parameters. No model except ours deals with tracer species dynamics, variable breathing pattern, pulmonary inhomogeneities, and nonequilibrium between alveolar gas and capillary blood. With appropriate experiments of tracer species dynamics, we believe that our model can be used for a variety of research studies and clinical applications. Supported in part by the Medical Research Service of the Veterans Administration. During some periods of this research, G. M. Burlna xYas a n NIH Pre-doctoral Trainee (GM-01090) and received a Summer Fellowship from the Northern Ohio Lung Association.
LITERATURE Bouhuys, A. 1977. The Physiology of Breathing. New York: Grune & Stratton. Cander, L. 1959. "Solubility of Inert Gases in Human Lung Tissue." J. &ppl. Physiol_, 14,538-540. Cander, L. and R_ Forster. 1959. "Determination of Pulmonary Parenchymal Tissue Volume and Pulmonary Capillary Blood Flow m Man." J. AppI. Physiol., 14, 541 551. Farhi, L. E. 1967. "Elimination of Inert Gas by the Lungs." Resp. Physiol., 3, 1 11. Gee, M. H. 1972. The In l//vo Volume of the Pulmonary Fluid Compartments in Dogs and the Relationship Between These Volumes and the Hemodynamics of the Pulmonary Circulation. Ph.D. Thesis, University of Colorado, Boulder. Glauser, F. L., A. Wilson, M. Hoshiko, M. Watanabe and J. Davis. 1974. "Pulmonary Parenchymal Tissue Changes in Pulmonary Edema." J. Appl. Physiol., 36, 648 652. Hlastala, M. P. and H. T. Robertson_ 1978. ':Inert Gas Elimination Characteristics of the Normal and Abnormal Lung." J. Appl. Physiol. 44, 258-266. Reid, M. H. and H. B. Hechtman_ 1974. "A Multicompartment Analysis of the Lung." Med. Biol. Engng, 12, 405-413. Saidel, G. M., T_ C. Militano and E. H. Chester. 1971. "Pulmonary Gas Transport Characterization by a Dynamic Model." Resp. Physzol., 12, 305 328. - - , J. Saniie and E. H. Chester. 197~."Modeling and Moments of Multibreath Lung Washout." Ann. Biomed. Engng, 6, 126 237. Schrimshire, D. and P. Torhlin. 1973. "Gas Exchange During Initial Stages of N 2 0 Uptake and Elimination in a Lung Model." J. Appl. Physiol., 34, 775 789. Stout, R., H. Wessel and M. Paul. 1975. "Pulmonary Blood Flow Determined by Continuous Analysis of Pulmonary NzO Exchange." J. Appl. Physiol., 38, 913-918. Wagner, P., H. Saltzman and J. B. West. 1974. "Measurement of Continuous Distributions of Ventilation-Perfusion Ratio: Theory." J. Abpl. Physiol., 36, 588-599. Zwart, A., R. C. Seagrave and A. Van Dieren, 1976. "Ventilation Perfusion Ratio Obtained by a Noninvasive Frequency Response Technique." J. Appl. Physiol., 41,419-424. RECEIVED 6 - 1 2 - 7 9 REVISED 1-17-80
0092-8240/81/0101-0001 $02.00/0
MULTIBREATH TRACER SPECIES DYNAMICS IN THE LUNG
IIGERALD M. SAIDEL and GERALD M. BURMA Dept. of Biomedical Engineering, Case Western Reserve University and Veterans Administration Medical Center, Cleveland, Ohio 44106, U.S.A.
By studying the behavior of various tracer species in the lungs, one can assess many important characteristics which distinguish normal and abnormal function, Quantitative evaluation of function depends on the use of an appropriate model in conjunction with experimental data. A multi-compartment model is derived from mass balances to describe dynamic as well as (breath-averaged) steady-state transport processes between the environment and pulmonary capillary blood_ The breathing cycle is divided into three time periods (inspiration, expiration, and pause) so that the model equations are discrete in time. No other model of tracer species transport in the lungs deals simultaneously with species dynamics, variable breathing pattern, distribution inhomogeneities, and non-equilibrium between alveolar gas and capillary blood_ Models currently in the literature are shown to be special cases of the model presented here.
1. Introduction. F u n c t i o n a l charcteristics of n o r m a l or a b n o r m a l lungs can be o b t a i n e d from experiments using tracer species with different physico-chemical properties such as solubility (Bouhuys, 1977). T h e characteristics of interest include ventilation (distribution a n d mixing of inhaled gas with residual gas), inter-phase diffusion (transport between alveolar gas and capillary blood), perfusion (convective t r a n s p o r t of capillary blood), ventilation/pe~ftt'sion (ventilated alveoli relative to perfused capillaries), and volumes of lung tissue a n d capillary blood. Tracer species m e t h o d s are particularly of interest for non-invasive or minimally invasive e v a l u a t i o n of lung function. Consider, for example, a tracer species that is chemically inert a n d very slightly soluble in blood, such as argon (Ar) or nitrogen (N2). W i t h such a tracer, the dynamics of the gas wash-in or wash-out process can yield i n f o r m a t i o n a b o u t venti-
2
GERALD M. SAIDEL AND GERALD M. BURMA
lation inhomogeneity. Another non-invasive method involves carbon monoxide (CO), which at low concentration diffuses into the blood as if it were an infinite sink. Methods to evaluate alveolar-capillary transport with CO require measurements only at the mouth. With inert soluble gases, lung perfusion and volumes of lung tissue and capillary blood can also be estimated non-invasively under suitable physiological conditions (Stout et al., 1975; Glauser et al., 1974). Invasive techniques with several inert gases of different solubility have been used to evaluate Ventilation/perfusion abnormalities (Hlastala and Robinson, 1978). Many models and experiments have been used to evaluate pulmonary function with tracer species. Ventilation inhomogeneity can be studied by a step change of the inhaled gas for a single breath with a large-volume maneuver or for multiple breaths with spontaneous breathing. A largevolume, single-breath procedure requires subject cooperation; a multibreath experiment does not. In multibreath studies, the dynamics are evaluated from average-expiratory or end-expiratory data on successive breaths. With such limited data, spatially lumped models which are discrete in time are most suitable (Saider et al., 1978). Lumped models are also typically used for the analysis of alveolar-capillary transport from single-breath and multibreath experiments with CO. With few exceptions, these models assume uniform ventilation, i.e. only one alveolar compartment. However, an accurate assessment of this inter-phase diffusion in the presence of ventilation inhomogeneity requires the simultaneous-study of an insoluble tracer in addition to CO (Saidel et al., 1971). In evaluating lung perfusion and tissue volumes, investigators have used various models of the dynamic response of inhaled soluble tracer species (Stout et al., 1975; Glauser et al., 1974). These models are limited, however, by the assumption of uniform alveolar ventilation and perfusion. In contrast, Scrimshire and Tomlin (1973) developed a dynamic model with nine alveolar-Capillary compartments. A fundamental assumption in their model is that N 2 0 equilibration occurs between alveolar gas and capillary blood. This same assumption occurs in models developed to estimate ventilation/perfusion, for example, the model proposed by Farhi (1967) and modified by others (Reid and Hechtman, 1974; Wagner et al., 1974). While these models are based on several inert gases with constant venous infusion, an alternative model by Zwart et al. (1976) deals with a sinusoidally-varying input of a single inhaled tracer species~ Although the input differs in form, this model also suffers from the assumptions of a single alveolar compartment and tracer equilibration between alveolar gas and capillary blood. In this paper, we present a lumped model of the lung suitable for the
MULTIBREATH TRACER SPECIES DYNAMICS IN THE LUNG
3
study of inter-breath dynamics of a variety of tracer species with different input and breathing conditions. The objective is to provide a comprehensive framework for the evaluation o f a b n o r m a l ventilation, inter-phase diffusion, perfusion, ventilation/perfusion, and volumes of lung tissue and capillary blood. Furthermore, we can more clearly appreciate the approximations used in current models from the limiting cases of our more general model. The structure shown in Figure 1 is based on a single dead space (compartment one) and two alveolar spaces (compartments two and three)
2. Model.
INSPIREDGAS-
~
E __
EXPIREDGAS
(i) ALVEOLAR SPACE (2)
I
ALVEOLAR SPACE (3)
I
c'P'LL"YI sP(4, ACE MIXED-VENOUS ~_ BLOOD Figure 1.
CAPILLARYI SPACE (5) ARTERIAL BLOOD
C o m p a r t m e n t a l structure of lung model.
representing two composite groups of alveoli. The parallel compartments are included to represent regional inhomogeneities in concentrations in the lung. With respect to blood-phase transport, the distribution of pulmonary capillary blood flow to each composite group of alveoli is represented as inputs to compartments four and five. In addition, the pulmonary parenchymal tissue volume is divided into compartments six and seven, each associated with alveolar and capillary blood compartments. Transport processes in the system are governed by total and species mass balance equations for all of the system compartments. The general model equations describe the dynamics and distribution of a tracer species after an input change in tracer concentration. Our approach is to obtain
4
GERALD M. SAIDEL AND G E R A L D M. BURMA
the total mass balance of each compartment to relate volume rate changes, if any, and volume flow rates. The species mass balances yield dynamic equations for species concentrations. Since the structure of our model is spatially coarse and data over the breathing cycle yields limited information, we choose to analyze dynamics discretely. For any breath k, we consider changes over three discrete time periods (~): inspiration (I), expiration (E), and end-expiratory pause (P). The time periods occur in the sequence {. . . r ( k - l,I), r ( k - l,E), r ( k - l,P), T(k,I), r(k,E),
T(k,P),...}. Gas phase transport.
To keep the equations from becoming unnecessarily complex, several assumptions are made with respect to gas transport. The gas density is assumed constant and equal throughout the lung. In addition, plug transport is assumed through a constant volume dead space, which implies a pure time delay with respect to the transport of gas between the environment and alveolar spaces. Using these assumptions, the total mass balance equations for the dead space reduced t o Ool =Q12+Q13,
"c=I,
Oto=O21+Q3t,
z=E,
where Qu is the flow rate from compartment i to compartment j. These flow rates can be related by gas flow fractions which are assumed constant:
fi=Q1JQol=Qil/Qxo,
i=2, 3,
where f2 +f2 = 1. The total alveolar volume VA(k, ~) at the end of period z of breath k is given by
v~(¢, ~)= v2(k, ~) + G(k, ~), where V~(k,z) is the volume of alveolar compartment ~ at the end of period T(k,'c). We assume that the alveolar volume fractions are independent of breath number;
vi(z)=V~(k,'c)/VA(k,'c), where
i=2, 3,
I)2(T)-~-V3(T)=1. The parameters f / a n d v~(z), which are related by v,(I)VA(k, I) = v,(E)VA(k - 1, E) +~VT(k, I), v~(E) = v,(P),
M U L T I B R E A T H T R A C E R SPECIES D Y N A M I C S IN T H E L U N G
5
where Vr(k,T ) is the volume change over the period T(k,z), characterize the distribution of ventilation. Taking these parameters as constants implies that the mechanical properties of the lungs are time invariant. In deriving the total mass balance equations for the alveolar compartments, the volume of tracer species taken up by p u l m o n a r y tissues and capillary blood is considered small compared to the total alveolar gas volume. This is valid if the fraction of soluble tracer in the alveolar gas is sufficiently small. For the alveolar compartments (i=2,3), the total mass balance equations over inspiration and expiration can be simplified under the conditions given above; consequently,
dV//dt =
f
JiQol,
-f/Q0 t, 0,
r=I,
-c= E , z =P,
(i)
where V~(t) is the volume of alveolar c o m p a r t m e n t i at any time t. In the time interval t k < t < t k + l within breath k, we index the three intervals-inspiration, expiration, and p a u s e - - b y z E {I, E, P}. The volume of compartment i at the end of these intervals within breath k is V/(k,T). Integrating over inspiration T(k, I), we obtain
V~(k,I) = V~(k- 1, P) +f~Vw(k, I) and over expiration
T(k,E) v (k, E) = v (k, i) - f Y r (k,
No volume change occurs during pause, T(k,P):
Vdk, P)= V~(k,E). The species mass balance equation for each well-mixed alveolar compartment on inspiration k is
[d(C,V~)/dt] =
Cl,f/Q01
-
[Ri+2 +R~+4],
i = 2 , 3,
(2)
where C i is the species concentration in alveolar c o m p a r t m e n t i and Cli iS the species concentration entering alveolar c o m p a r t m e n t i from the dead space (compartment 1). The terms R~..2 and Ri--a represent uptake of s o l u b l e g a s by blood and tissue compartments respectively. Dividing equation (2) by equation (1) and integrating over inspiration T(k,I) in a
6
GERALD M, SAIDEL AND GERALD M. BURMA
piece-wise fashion (Saidel et al., 1978), we find that C,(k, l)V~(k, I) = C,(k - 1, P)V~ (k - 1, P) + C1 o (k - 1 )fWD + Co
[VT(k, I ) -
- T (k, I ) [ ( R i + 2)k,I
-}- (R~ + 4)k, I1
(3)
where the subscripted bracketed quantity ()k,~ represents an average over the period T(k, ~). On expiration, the species mass balance is d(CiV~)/dt = - C ~ Q l o -
[Ri÷2 +R~+41,
i = 2 , 3.
(4)
Expanding the derivative of equation (4) and combining it with equation (1) yields V/dCJdt = - [R~ + 2 + Ri + 4]. Integrating over expiration T ( k , E ) , we obtain C~(k,E)=C~(k,I)--T(k,E)[(R,+2/Vi)k,E+(Ri+4/V~)k,~].
(5)
During pause, the species mass balance is d ( C y ~ ) / d t = [ R i + 2 +R~+4] ,
i = 2 , 3.
integrating over T ( k , P ) and noting that the alveolar volume is constant, we find C,(k,P)=Ci(k,E)-[T(k,P)/V~(k,E)][(Ri+2)k,p+(Ri+4)k,nl.
(6)
The end-expired gas concentrations from the dead space can be related to end-expired alveolar concentrations if we assume that V T ( k , E ) > VD and that the dead-space time delay toward the end of expiration is sufficiently. small to provide a good estimate of the mixed alveolar concentrations. Thus, a species mass balance over the dead space yields:
Clo(k)Q~lo~-C2(k,E)9~21-t- C3 (k, E ) Q 3 1 ,
MULTIBREATH TRACER SPECIES DYNAMICS IN THE L U N G
7
where Clo(k ) is the concentration leaving the dead space and entering the environment at the end of T(k,E). Dividing by Q10, we obtain C1 o (k) =f2C2 (k. E) +f3C3 (k, E).
(7)
Blood phase transport. Blood density, volume, and flow rates are assumed constant for each capillary space. Consequently, the flow rate through c o m p a r t m e n t i is Qz, and the total flow rate is given by Q=Q4+Qs. Capillary blood flow fractions associated with alveolar compartments are defined as
gi=Qi+2/Q
i=2,3,
so that
g2+g3=l.
Gee (1972) studied the relationships between the total pulmonary blood flow rate, p u l m o n a r y blood volume, and extra-vascular fluid volume in anesthetized supine dogs using indicator dilution methods. Pulmonary blood volume, extra-vascular fluid volume, and the rate of pulmonary blood flow were found to be directly proportional to one another. Consequently, we let the blood volume of each c o m p a r t m e n t be proportional to the blood flow to that compartment:
Vi+2/Vb=gi, i=2, 3. The total capillary blood volume is Vb = V4 + V5. The species mass balance equation for each capillary c o m p a r t m e n t is
V~+2(dCi+ 2/dt) = QgiEC~ - Ca, i+2] + Ri+ 2, where C o is the species concentration of mixed-venous blood and C,,~+ 2 is the species concentration leaving capillary c o m p a r t m e n t i + 2 and entering •the systemic arteries. Integrating over T(k, z), we obtain
C,+ 2(k, r)=Ci+ 2(k, r - 1 ) + [ T ( k , z)Q/K][(Co)k,~-(Co, i+ 2)k,~] + [T(k,
(8)
where the argument ( k , r - 1 ) refers to the time period just prior to (k,z) in the cyclical order [_.. (k - 1, I), ( k - 1, E), ( k - 1, P), (k, I), (k, E), (k, P ) . . . }.
8
GERALD M. SAIDEL A N D G E R A L D M. BURMA
Tissue phase transport. The tissue spaces are assumed to be of constant volume and density. The total volume is Vp = V6 + V7. The distribution of tissue volume to each alveolar compartment is defined by: hi=Vi+4/Vv,
i=2,3,
so that h 2 + h 3 = 1. The species mass balance equation for well-mixed tissue compartments is V~+4 (dCi/dt ) = Ri +4.
Integrating over time T(k, z), we get Ci+ 4 (k, z) = Ci+ 4 (k, • - 1 ) + I T ( k , "Q/Vph[l(R~+4)k, ~.
(9)
Alveolar-capillary transport. Consider the diffusion of a gas species across a thin homogeneous membrane separating gas and blood. The species flux J across the membrane is given by J = (~rCo - Cb)~/fi.
where C o and Cb are the species concentrations in gas and blood, @ is the diffusivity, 6 is the thickness of the membrane, and the partition coefficient a represents the ratio of the equilibrium concentration of the species in the blood to its concentration in the gas phase. The concentration profile Cb(z) of a gas species in the capillary bed can be described by a one-dimensional, quasi-steady model of a l v e o l a r capillary transport. Consequently, a mass balance equation for a tracer species leads to Q (dCb/dz) = Js = (~C o - Cb)~s/6,
where s is the surface area per unit length along the capillary bed. At z = 0, the species concentration in the b l o o d is the mixed venous concentration entering the capillary bed (Cb(0)=C~). Similarly, at z = L the species concentration becomes the end-capillary concentration (Cb(L) = Ca). Assuming that the alveolar concentration is constant over the blood transit time, this equation can be integrated to obtain C~ = C~ + [ a C o - C,][1 - exp ( - ~sL/DQ)].
MULTIBREATH TRACER SPECIES DYNAMICS IN THE L U N G
9
In terms of the nomenclature used for the multicompartment model, the arterial concentration from capillary compartment i + 2 is related to the gas concentration of alveolar compartment i according to
Ca,i+2-Cv=[ffCi-Cv]~i,
i=2,3,
(10)
where e~= 1 - exp ( - 7/Qg,).
(11)
The relationship described above avoids the usual assumption of equilibrium between alveolar gas and end-capillary blood. To relate a lumped-parameter model of transport to this onedimensional model, we simply integrate the membrane flux over the capillary surface. In particular; the rate of mass transport of species between one alveolar compartment and a blood compartment is given by:
R=
Jdz=(~s/6) o
fL
(aCo--Cb)dZ~--y[aCo--Cb] ,
o
where 7 = ~ s L / 6 is a transport coefficient. The time averaged diffusion rate can be expressed as < R > = 7
-
(12)
Interphase transport.
For alveolar-capillary transport, average diffusion rates between compartments during inspiration and pause (-c--I,P) can be written similar to (12):
l0
GERALDM. SAIDEL AND GERALD M. BURMA
and for expiration (~=E),
( R~+~/V~)k,~='~h~a[( Ci/V~)k,e - (1/a )( C~+4/V~)k,E], i = 2 , 3 . In these constitutive equations, the b l o o d - g a s and coefficients (a) are assumed to be identical for a assumption is based on the work of Cander (1959) partition coefficients for several inert gases. Also, alveolar-Capillary transport (7) is assumed proportional alveolar-tissue transport (~).
tissue gas partition given species. This who found similar the coefficient of to the coefficient of
3. Model Analysis. Time-averaged approximations. Species concentrations at the end of successive periods {... ( k - 1, I), ( k - 1, E), ( k - 1, P), (k, 1), (k, E), (k, P ) . . . } are more readily related than are concentrations averaged over those periods. The time average of any variable X(t) over the interval T(k,v) during any period "c~{I,E, P} of breath k is defined as
(X)k,~=[1/T(k,r)]
i
T ( k , ~)
X(t)dt.
,3 t - T ( k , 0
Consequently, we express the time-averaged terms, a linear combination of the variables at the start and end of period -c= I or P, as
(Ci)k,~ ---c~Ci(k, ~ - 1 ) + (1 - ~)C~(k, % .and for z = E as
(C,/V~)k, E = ~[C,(k, I)/V~(k, I)] + (1 - ~)[C~(k, E)/V~(k, E)], where the weighting factor e lies in the interval [0, 1). For the analysis of experimental data, it is appropriate to let a take a value such as 0.5. Here, since we are more interested in the characteristics of the model, we let e = 0 to allow a simpler form. That is, the average value is approximated by the end-tidal value. Invariant breathing pattern. Although the model can be applied to spontaneous breathing, we shall restrict our analysis to a time-invariant
M U L T I B R E A T H T R A C E R S P E C I E S D Y N A M I C S IN T H E L U N G
11
breathing pattern, i.e. one independent of breath number k. Thus, for z e{I,E,P} we let
T(k, "c)= T(r),
vr(k, ~)= VT, and for i-- 2, 3
J'Gv~+fYr,
3=I , z=E,P,
V/(z) = (VAVi where v, = vi (E).
Combined model equations. For time-invariant breathing and averaged concentrations equal to end-period concentrations (c~=0), the model equations become A,(~)tp,(k, 3) = O,(k, ~ - 1) + ¢ ) , ( k - 1, ~),
(13)
3
C , o ( k ) = ~ f~C~(k,z),
(14)
/=2
where 0, and qS, are column vectors and A, is a matrix. Specifically, the vectors are
O,(k,z) =
Lci,k,j C,+2(k,r )
qbi(k_l,z) =
C,+4(k,~)
i i,k 1 T(r)QC,,(k,r) v~ 0
and the matrix is
, .
(aT(z)(giy+h,~))
_~T('c)ygi]
_£T(z)~hi. ; -
. (T(z)7 ]
Ai(z) =
(rp
3
To account for c h a n g e s that occur within the breathing cycle, we in-
12
GERALD M. SAIDEL AND GERALD M. BURMA
troduced the following functions:
,~i(k-1,z)
= {~ (C1° ( k - 1 ) - C°1 )( V°/VT)+ C° 1]f~VT/VAvi', ~ = I ~:=E, P
#~(z)={jlVr/Vavg+ l, z=I =E,P. In this model, the unknown independent parameters are ]2 (or J3), v2 (or V3), g2 (or g3), h2 (or h3) , Q, Vb, Vp, 7, and ~. Except for the transport
coefficients 7, and }, the other parameters are dependent solely on physiological conditions. Ratios of the distribution parameters are of value in characterizing the effectiveness of the lung in transporting species between the environment and capillary blood. In particular, ventilation inhomogeneity depends on f~/v~ and ventilation/perfusion inhomogeneity depends on fi/gi. Special simplifications of the model occur for species which are: (a) insoluble (0< a <6, where 6 is suitably small) such as N2; (b) slightly soluble in tissue, but highly reactive in blood (viz. CO); and (c) soluble (~ < cr < oo), such as halothane. We shall examine some of the special cases in the following sections. Insoluble species. For this case we set o-=0. Also, if the tracer species enters only with inhaled gas, then the species concentrations in blood (Ci+2) and tissue (Ci+4) are zero. Consequently, the state equation (13) reduces to a scalar form:
C~(k, ~)= [Ci(k, ~ -
1 ) + 2 ~ ( k - 1, r)]/p~(r).
From the definition of 2~ a n d / ~ during expiration and pause, this equation shows species concentration to be constant during expiration and pause:
C,(k, P)-- C,(k, E) = C~(k, I). Consequently, we find
Ci(k, ~) = Ci(k, I)= [C,(k- 1, P) + 4 ( k - 1,/)3/~,(/) = ]-C~(k- 1,E) + 2 ~ ( k - 1, I)]/p~(I). I
(15)
In this case, there are only two unknown independent parameters: f2 (or f3) and v2 (or v3). For uniform ventilation, five=l, ,~ and Pi become
M U L T I B R E A T H T R A C E R SPECIES D Y N A M I C S IN T H E L U N G
13
independent of the index i so that
C~(k,E) = CA(k,E)= [ C l o ( k - 1) + 2 ( k - 1 )]//~, where
CA(k,E ) is the alveolar concentration, Z(k--1)=[ (Clo(k-- a )--Co~ )(VD/VT)+ Col]VT/VA
and
I~= VrlVA + 1. The model output from (14) leads to (71o (k) = C , , ( k E ) = E G o (k - 1 ) + ,~(k - 1 )3/#.
U p o n rearrangement of these equations, we obtain C l o ( k ) - Col = [C~0 ( k - 1 ) - Co~][(VA + VD)/(VA+ Vr)].
The solution is given by
G o ( k ) - c01 = EClo ( 0 ) - Col] r(vA + vD)/(vA + vT)] k
(16)
which is the form of the solution for a one-compartment model (Saidel et al., 1978). Carbon monoxide. If C O is a tracer species in the inhaled gas, its concentrations in blood (Ci+2) and lung tissue (C~+4) will be negligibly small. In the blood, CO reacts with hemoglobin such that its partial pressure in the blood is negligible. In the tissue, the CO solubility is very small. For this case, the state (13) reduces to
[#,(z)+ T(z)g,?a/VAv~]C~(k,z)=Ci(k,~-l)+2~(k-l,z ).
(17)
The u n k n o w n independent parameters in (17,) are f2 (or f3), v2 (ol v3), g 2 (or g3), a n d 7- To reduce the n u m b e r of independent unknowns, one may assume that f~/gi= 1, which means that the relative distribution of ventilation anti perfusion is uniform. While this assumption may not be unreasonable for normal lungs, it is unjustifiable for many if not most abnormal lungs.
14
G E R A L D M. SAIDEL AND G E R A L D M. BUR MA
A c o m m o n clinical measure of alveolar-capillary transport is obtained with a (breath-averaged) steady-state analysis of CO in the inhaled and exhaled gas. In particular, a transport coefficient called the diffusing capacity (DLco) is estimated by
DLco = [ Vr/O]EC01 - C~]/P A,
(18)
where 0 is the average breathing period, Pa is the alveolar partial pressure of CO, and CE is the mixed-expired CO concentration. For comparison, a similar expression can be derived for 7- If the ventilation and ventilation/ perfusion distributions are uniform, fjv~ = f/g~ = 1, then our model has only one equivalent transport pathway between the environment and blood. Consequently, (17) becomes
[#(z)+T('c)Ta/VA]CA(k,z)=CA(k,z--1)+2(k--l,v).
(19)
Under steady-state conditions, we let
CA(k,c}=CA('c), 2(k-l,r)=).(r). Note from the output equation (14), Clo=CA(E). To reduce our model to (18) which is based on continuous gas inflow and outflow, we define T(I) =0, the average breathing period, and set T(E)=T(P)=O. These conditions require that
C I(P)=C4(E)=C.,t(I)=C4, /t(r)=/dl)=
1 + I"r l{j
and
;~(v) = , ~ ( / ) = E(cA - C , o ) ( V , , / V r ) + C o d V r / V , Consequently, (19) can be simplified and rearranged as ~;~ = Evr/O]E Co l - CA]E1 - v , , / v d / c ;.
(2o)
For comparison with (20), we express (18) in an equivalent form using the relationships between (a) mixed-expired and alveolar concentration derived from a mass balance, c ~ = C AE1 - V D / V d + Col V,,/VT
MULTIBREATH TRACER SPECIES DYNAMICS IN THE LUNG
15
and (b) alveolar partial pressure and concentration
PA = R TCA, where R is the ideal gas constant and Tis the absolute temperature. When these relationships are substituted into (18), the result is
DLco = [Vr/O][ Col -- CA][ 1 -- VD/Vr]/R T CA.
(21)
Hence, from (20) and (21), the transport coefficients are found to be directly proportional,
7 = (RT/a)DLco. Soluble species. F o r soluble species, we start with the state equations (13) and consider some special limiting cases: (a) high rate of alveolar-capillary transport and (b) uniform distributions. Let the diffusion rate or breathing period be sufficiently large, such that T(z)~/Vp>> 1,
T(r)7/Vb >> 1
and e~= 1 - e x p ( - 7/Qg~)--- 1. Under these conditions, and for 7 =7, the state equations can be written as
[~,,(r) + (a/VAV,)[g,T(z )Q + Vbg , + Vph,]}Ci(k, z) = [ 1 + (Vbg,/VAV,)+ (VvhjVAV,)]C,(k, r - 1) + 2~(k - 1, z) q- (T(z)OgjVav~)Cv(k, ~)
(22)
and a C i ( k , "c)= Ci+ 2(k , "c)= C i + 4 (k, "r).
(23)
This last equation states that equilibrium-exists among alveolar gas, capillary blood, and tissue fluid. Furthermore, if the distributions of perfusion and tissue volume are equal (gi=hi), then we see that Vb and Vp cannot be differentiated and appear only as the sum Vb + Vp = Vbp. For a system with uniform distributions..li=c~=gi=h~, the alveolar, end-
16
GERALD M. SAIDEL AND GERALD M. BURMA
capillary, and output concentrations can be written as
Clo(k)=C (k,E). Thus, with uniform distributions and a relatively large diffusion rate, (22) and (23) reduce to {#(-c) + (a/VA)ET(z)O + V j } C A ( k , v ) = [1 + (aVbp/VA)]CA(k, ~ - 1 ) + 2 ( k - 1, z) + [T(z)QC~(k, r)/VA] , C,(k, ~) = CA(k , "c).
(24) (25)
Constant venous injusion. The model of Farhi (1967) to evaluate a ventilation/perfusion ratio (VA/Q) deals with the experimental situation in which a tracer species is delivered only via a peripheral vein at a constant rate. Hence, inhaled tracer concentration is zero (Co1=0). As a result of the constant venous infusion being balanced by tracer loss by exhalation or tissue uptake, a breath-averaged s t e a d y state is achieved. Farhi models the lungs as having continuous gas inflow and outflow, uniform distributions, and a large diffusion rate. U n d e r these conditions, (24) and (25) lead to equations relating arterial and mixed-venous concentrations only: {(#(~)/a) + (T('c)Q + Vbp)/VA}C,(k , z) = (1 + Vbp/VA)C,(k , "c- 1 ) + 2 ( k - 1, z ) + T(z)QC~(k, z ) / V A.
(26)
To model continuous gas inflow and outflow, we let T(z)=O, the average breathing period, and T ( E ) = T ( P ) = O . For the steady state, the dependence of concentrations on breath number vanishes. With these conditions, we find
C.(k,'c)=Co, CAk,'c)=C and (26) reduces to Farhi's model, co/c, =
+ o],
(27)
where the alveolar ventilation rate is given by l?a = [Vr - VD]/O. Single-breath dynamics. The Cander and Forster (1959) method of estimating pulmonary blood flow and tissue volume is based on the singlebreath dynamics of a soluble tracer species that is inhaled. To compare
MULTIBREATH TRACER SPECIES DYNAMICS IN THE LUNG
17
their model with ours, we assume uniform distributions and a high diffusion rate. The model also assumes that inspired and expired time periods are short compared to a pause at the end of inspiration, T--0. In this single breath-holding maneuver, the alveolar and output concentrations are equal (C A (k, 72) = C A (k, I) = Ca (1, E) = C ~o), the initial condition is CA(k,T,--1)=CA(O)=O and the species concentration in the mixed venous blood is negligible (C~=0). Thus, from (24) we obtain [I + Vr/VA+(Cr/VA)(OQ+ Vbp)]Clo=L=Coa(Vr--VD)/V a.
(28)
If the procedure is repeated for two breath-holding periods (0= T1 and 0 = T2), and all other parameters are held constant, then (28) can be used twice to solve for the blood flow rate: Q = [ ( v a + VT)/a+ V b p l [ C l o ( O 1 ) - C l o ( 0 2 ) l / [ 0 2 C l o ( 0 2 ) - O 1 C l o ( O 1 ) l
(29)
Now, let us compare (29) with the equation derived by Cander and Forster:
Q=E(VA+ VT)/G+ gbp]{lnECxo(01)/Clo(O2)]}/(02 -01).
(30)
F r o m a power series expansion of the logarithmic expression, it can be seen that (29) and (30) are approximately the same provided that Clo(01)/Clo(0z) is not too far from unity. For an insoluble gas species, a = 0 , (28) reduces to
E1 -}- VT/VA]C'~o = C"~1(g T - VD)/VA where C* indicates the concentration of the insoluble gas species. Combining this with (28), we can solve for the pulmonary tissue-blood volume: = [(vA + VT)/aJ[(CToCol/C
oC
I)- 1] - 09_.
As the breath-holding time 0 becomes small,
gbp ~" ~( gA -]- gT)/Cr][ C~oCol/ Cl oC~l ) -
l].
(31)
4. Discussion. We have developed a discrete-time, multicompartment model that describes tracer gas species dynamics for an inhomogeneous
18
GERALD M. SAIDEL AND GERALD M. BURMA
pulmonary system. Although many of the elements of our model have previously been used in other models~ the combination that we deal with is unique. This model is intended gs the basis for analyzing pulmonary function in acutely ill patients. Depending on clinical circumstances, our model can take into account either mechanically-aided or spontaneous breathing. Since data acquisition for assessing acute dysfunction should be as quick as possible, a dynamic rather than steady-state model is preferable. Indeed, what we have in mind is a dynamic experiment in which several tracer gas species are inhaled simultaneously for 10 or 12 breaths. The tracer concentration dynamics at the mouth are not significantly affected by systemic recirculation up to the first 6-10 breaths. At the mouth, the tracer gases are continuously sampled and measured (by a respiratory mass spectrometer). Simultaneously, the flow at the mouth is also measured (with a pneumotachometer). By comparing the measured variables to the simulated model output under corresponding conditions, numerical estimates of model parameters may be obtained. For parameter estimation to be practical with the limited data in this context, the number of model parameters must be minimal. By allowing just two parallel transport pathways between the environment and capillary blood, we have the simplest characterization of pulmonary inhomogeneities associated with species transport. With more parallel compartments (a simple extension of our model), the parameter estimates are likely to be ill-determined. Because of the possible pathological stuations for which the model is to be used, we avoid the usual assumption of chemical equilibrium between alveolar gas and capillary blood. For example, subjects with edematous lungs may have an increased effective diffusion distance between alveolar gas and capillary blood. Also, emphysematous subjects have greatly reduced effective surface area for diffusion and possibly an increased effective diffusion distance due to a thicker concentration boundary layer in the alveolar gas. Consequently, the ratio of alveolar transport to capillary perfusion may be smaller than normal (especially in an inhomogeneously perfused lung) and cause a significant tracer concentration difference between alveolar gas and end-capillary blood. In various ways, models currently used to estimate one or more pulmonary characteristics from species transport studies are limiting cases of our model. Typically, current models require special conditions on the experimental method, e.g. a particular breathing pattern. But for clinical purposes, particular breathing patterns may not be feasible. Commonly, dynamic models do not incorporate pulmonary inhomogeneities, but steady-state models do. In acute situations, however, the time to reach
MULTIBREATH TRACER SPECIES DYNAMICS IN THE LUNG
19
steady state may be inappropriately long. Furthermore, steady-state models cannot yield some important information, namely, characteristic volume parameters. No model except ours deals with tracer species dynamics, variable breathing pattern, pulmonary inhomogeneities, and nonequilibrium between alveolar gas and capillary blood. With appropriate experiments of tracer species dynamics, we believe that our model can be used for a variety of research studies and clinical applications. Supported in part by the Medical Research Service of the Veterans Administration. During some periods of this research, G. M. Burlna xYas a n NIH Pre-doctoral Trainee (GM-01090) and received a Summer Fellowship from the Northern Ohio Lung Association.
LITERATURE Bouhuys, A. 1977. The Physiology of Breathing. New York: Grune & Stratton. Cander, L. 1959. "Solubility of Inert Gases in Human Lung Tissue." J. &ppl. Physiol_, 14,538-540. Cander, L. and R_ Forster. 1959. "Determination of Pulmonary Parenchymal Tissue Volume and Pulmonary Capillary Blood Flow m Man." J. AppI. Physiol., 14, 541 551. Farhi, L. E. 1967. "Elimination of Inert Gas by the Lungs." Resp. Physiol., 3, 1 11. Gee, M. H. 1972. The In l//vo Volume of the Pulmonary Fluid Compartments in Dogs and the Relationship Between These Volumes and the Hemodynamics of the Pulmonary Circulation. Ph.D. Thesis, University of Colorado, Boulder. Glauser, F. L., A. Wilson, M. Hoshiko, M. Watanabe and J. Davis. 1974. "Pulmonary Parenchymal Tissue Changes in Pulmonary Edema." J. Appl. Physiol., 36, 648 652. Hlastala, M. P. and H. T. Robertson_ 1978. ':Inert Gas Elimination Characteristics of the Normal and Abnormal Lung." J. Appl. Physiol. 44, 258-266. Reid, M. H. and H. B. Hechtman_ 1974. "A Multicompartment Analysis of the Lung." Med. Biol. Engng, 12, 405-413. Saidel, G. M., T_ C. Militano and E. H. Chester. 1971. "Pulmonary Gas Transport Characterization by a Dynamic Model." Resp. Physzol., 12, 305 328. - - , J. Saniie and E. H. Chester. 197~."Modeling and Moments of Multibreath Lung Washout." Ann. Biomed. Engng, 6, 126 237. Schrimshire, D. and P. Torhlin. 1973. "Gas Exchange During Initial Stages of N 2 0 Uptake and Elimination in a Lung Model." J. Appl. Physiol., 34, 775 789. Stout, R., H. Wessel and M. Paul. 1975. "Pulmonary Blood Flow Determined by Continuous Analysis of Pulmonary NzO Exchange." J. Appl. Physiol., 38, 913-918. Wagner, P., H. Saltzman and J. B. West. 1974. "Measurement of Continuous Distributions of Ventilation-Perfusion Ratio: Theory." J. Abpl. Physiol., 36, 588-599. Zwart, A., R. C. Seagrave and A. Van Dieren, 1976. "Ventilation Perfusion Ratio Obtained by a Noninvasive Frequency Response Technique." J. Appl. Physiol., 41,419-424. RECEIVED 6 - 1 2 - 7 9 REVISED 1-17-80