BULLETIN OF MATHEMATICAL BIOLOGY
VOLVME 37, 1975
M A T R I X P R O O F OF F L O W , V O L U M E A N D M E A N TRANSIT TIME THEOREMS FOR REGIONAL AND COMPARTMENTAL SYSTEMS
W. PERL Departments of Medicine and Physiology, New Jersey Medical School, Newark, New Jersey 07103 N. A. LASSEN Department of Clinical Physiology Bispebjerg Hospital 2400 Copenhagen NV, Denmark and R. M. EFFROS Division of Respiratory Physiology and Medicine Harbor General Hospital Section University of California at Los Angeles Torrance, California 90509
The relations (inflow) -- (dose)/(area under indicator curve), and (volume of distribution) -- (throughflow) • (mean transit time) are derived by a matrix method for a system of interconnected subsystems, within which spatial indicator activity gradients may exist, and for compartments, within which the indicator activity is spatially uniform. The inflow theorem is different from the outflow theorem. Equivalent labeling of multi-input systems reduces them formally to single input systems. Foreign indicator flow-volume kinetics are more general than, and include as a special case, tracer flux-mass (metabolic) kinetics. Volume of distribution in the indicator steady state may be different from the equilibrium volume of distribution. T h e well-known relations for a single inflow, single outflow s y s t e m : v o l u m e = flow • m e a n t r a n s i t t i m e (Stephenson, 1948, 1958; Meier a n d Zierler, 1954) a n d 573
574
W. P E R L , N. A. LASSEN AND R. IVL E F F R O S
flow = dose/area under emergent indicator curve, have been derived for a compartmental system by matrix methods (Bright, 1973). In this derivation it was assumed that the dose of indicator was injected into the same compartment in which subsequent measurement of the concentration-time curve was made. ~n view of previous derivations using stochastic (Bergner, 1964a, b) or convolution integral methods (Perl et al., 1969c; Perl, 1971) this assumption can be weakened in that the inflow compartment into which indicator is injected can be different from the outflow compartment in which measurements are made. Bright's final result in his equation (5), combining the flow and volume theorems in terms of exponential parameters, was given essentially by Nosslin (1964, equation 44) who also used a matrix approach, although expressed in 'metabolic' terminology. The pioneering work of Nosslin and Bergner showed that the flow theorem is more properly an inflow theorem which is in general quite different from the Stewart-Henriques-Hamilton outflow theorem (Hamilton, 1962). It is of interest to retrace Nosslin's proof by a simplified matrix method. The matrix method shows directly that the often cited correspondence between the 'foreign indicator' case and the 'native tracer' case is incomplete. The former includes the latter as a special case. A limitation in Nosslin's proof will be removed, namely, the assumption that the system itself, before indicator or tracer is introduced, is described by a set of linear equations. Also, some generalizations over previous matrix treatments axe included: compartment is replaced by 'compartment-region'; multi-inflow multi-outflow systems are considered; the equivalent labeling assumption (Bergner, 1964a) which was weakened by Roberts et aI. (1973) is weakened still further in certain cases. All theorems are derivable from a single underlying relation, the indicator influx theorem, which corresponds in the present matrix approach to the stimulus-response theorem in the convolution approach (Perl,
1971). Foreign Indicator and Native Tracer. A foreign indicator, denoted simply indicator, is a detectable substance introduced into a physiological system, which substance is chemically different from any relevant systemic substance and yet which yields physiological information about the system. For example, antipyrine may be used to measure total body water, raffinose may be used to measure a permeability of a blood-tissue barrier, etc. The indicator may be measured as the amount (mg, cps, etc.) in a sample of systemic substance and expressed as a concentration = (amount in sample)/(cm3 of systemic water or other systemic substance in sample). Deduced information might be a flux of indicator (amount see-1), a volumetric flow (cm3 sec- 1) of a systemic substance
REGIONAL AND COMPARTMENTAL SYSTEMS
575
such as blood water or a volume (era 3) of a systemic substance such as tissue water. A native tracer, denoted simply tracer, is a substance introduced into a physiological system, which substance is chemically identical to a systemic substance of interest (tracee, mother substance) but separately detectable from the systemic substance. For example, 131I-albumin may usually be considered a tracer for systemic albumin. The tracer may be measured as the amount (rag, cps, etc.) in a systemic sample and expressed as a specific activity = (amount of tracer in sample)/(amount of tracer + amount of mother substance in sample). Deduced information might be a flux of tracer (amount sec -1) a flux of mother substance (amount see-l), or a systemic mass (g) of mother substance. The indicator case and its terminology will be used. As will be shown, the indicator case is of wider scope than the tracer case, namely, where the system handles the indicator differently from all other systemic substances of interest. If the system cannot distinguish between the indicator and a systemic substance of interest, then the indicator may be regarded as a tracer of that systemic substance (Bergner, 1967) and the two cases become formally identical, with the one-to-one correspondence: concentration--> specific activity, flow--~ flux, volume --~ mass.
Indicator Influx Theorem.
The system consists of a number, assumed three for illustration, of 'compartment-regions' which communicate with each other and with the outside b y means of 'clearances' (Figure 1). A compartment-region consists of a 'regional portion', tissue for example, within which the concentration of indicator can vary both spatially and temporally and a 'compartmental portion', the venous blood volume contained within the tissue for example, within which the concentration of indicator can vary only temporally. Indicator can exchange between a regional portion and its corresponding compartmental portion. The unidirectional indicator flux j~s(t) (aInount/sec) is defined as to t h e / t h compartment-region from t h e j t h compartment portion, in which the indicator concentration is cj(t). The crossing of the boundary surfaces b y the arrows in Figure 1 is intended to represent that indicator can enter anywhere within a regional portion or a compartmental portion of a compartment-region. The clearance F~j is defined as the ratio of indicator flux to concentration or
jtj(t) = F~jcj(t).
(1)
As with clearance defined elsewhere in physiology, F~j can be an actual systemic convective flow (glomerular filtration rate) or a parameter characterizing indicator transport across a barrier (permeability • surface area) or through a
576 distance
W. PERL, N. A. LASSEN AND R. M. EFFROS (diffusion coefficient x area/distance),
etc.
The indicator
effiux
joj(t) to t h e outside from the j t h c o m p a r t m e n t portion is defined similarly in t e r m s o f an o u t p u t clearance/v0j and an indicator concentration cj(t) in the j t h c o m p a r t m e n t a l portion b y (1) with i = O. The indicator influx ijo(t), if a n y , into t h e j t h c o m p a r t m e n t - r e g i o n from the outside can likewise be defined in t e r m s o f a n i n p u t clearance Fjo a n d a c o n c e n t r a t i o n cjo(t ) a t which the indicator is delivered, b y an e q u a t i o n o f the form o f (1).
i,o(t)
~3
%1
Fibre 1. System of eompartmerit-regions interconnected by clearances lv~i (cma see-1). Compartment portion is within square. Regional portion is between circle and square. Amount m~(t) of indicator is in the jth compartment-region at time t (see) after start of indicator influx ilo(t ) (amount see- 1) into F10 systemic input. Indicator exchange occurs between regional portion and compartmental portion. Indicator concentration cj(t) is spatially uniform in compartmental portion
Afinite dose experiment is p e r f o r m e d in which a finite a m o u n t ml0 (amount) of indicator is injected from outside into the regional portion of c o m p a r t m e n t region 1 a t r a t e ilo(t), a n d into no other compartment-regions. Thus i~o(t) = i3o(t ) = 0 and mio =
Jil0(t) J
dt
(2)
REGIONAL AND COMPARTMENTAL SYSTEMS
577
where the integration, here and later, is from time zero to infinity. At time t, measured from the start of injection, the amount mj(t) of indicator is in the j t h compartment-region and the concentration cj(t) is in the compartmental portion of, hence also in all the exit chao_nels from, the j t h compartment-region. Conservation of indicator in each compartment-region gives the customary set of differential equations (dot denotes time derivative)
~hl = Fllcl + F12c2 + F18c8 + ilo m2 = F21Cl + F22C2 + F23c3 + 0
(3)
rh 3 = .F31c 1 + F32c 2 + F33c z + 0
where - F ~ I - Fol + F~I + F81
-F22 = Fo2 + F12 + F32 -F83 -= F03 + Fz3 + F23.
(4)
In (3) the indicator quantities are time dependent. In (4), - Fjj represents the total clearance out of the j t h compartment-region. The clearances F~j are assumed constant (systemic steady state) and independent of the indicator quantities (small perturbation of the system). The assumption of constancy of the systemic parameters F~j. = (indicator flux jtj(t))/(indicator concentration cj(t)) does not imply constancy of any (systemic flux)/(systemie concentration). The systemic fluxes may be nonlinear functions of the systemic concentrations. The present results require only the linear relations (3) between indicator amounts and indicator concentrations. That such relations are possible, for a sufficiently small perturbation of the system by the indicator, despite nonlinear relationships of systemic quantities in the absence of indicator, is wellknown (see, for example, Bergner, 1961). The reason for the introduction of compartment-regions is that this seems to be the most general structure permitted by equations (3) and (4) which yield the indicator influx theorem. The compartment is the limiting case of the compartment-region when the regional portion goes to zero. Integrating (3) from time zero to infinity with the initial condition mj(0) = 0 (system devoid of indicator before injection) and the final condition mj(~) = 0 (all compartment portions connected directly or indirectly to the outside) and using (2) gives
Fll f cl dt +
f c2dt +
F18 f c 3 dt = - m l o
F21f c~ dt + F22 f c2 dt + F 2 3 f c 3dt = 0
F31 fc~dt + F32fc2dt
+ F 3 a f c 3dr = O.
(5)
578
W. P E R L , N. A. LASSEN AND R. M. E F F R O S
An indicator steady state experiment is either performed or considered to be performed, in which the constant infusion rate i~0(~ ) of indicator, into the same inflow as in the finite dose experiment, is maintained constant (or approaches constancy sufficiently closely) long enough for the time derivatives on the left in (3) to have vanished which then gives Fllc~(oo) + F12c2(oo) + F~3c3(oo) = -ilo(OO)
F21c1(o0) + F29.c2(oo) + F23c3(oo) -- 0
(6)
F3~c~(oo) + F82c~(~) + F33%(oo) = O. Comparison of (6) with (5) shows that the simultaneous equations for determining ~ cj dt/mlo are the same as those for determining cj(oo)/ilo(OO). Hence these two sets of unknowns are respectively equal and f cs dt = cj(oo) mlo i~o(OO)
(7)
Equation (7) is the indicator influx theorem, a special case of the stimulus-response theorem (Perl, 1971). It applies to a multi-systemic input, multi-systemic output system. I t states that if a finite amount ( = dose) of indicator is injected over a finite time into any one input channel then the area under the concentration per unit dose curve measured in any compartmental portion, or exit therefrom, of the system equals the concentration per unit infusion rate for the same two locations in the indicator steady state.
Systemic Inflow Theorems. I f the indicator steady state influx in (7) is expressed as a systemic input clearance Flo (em 3 sec -1) times a reference input concentration Clo(OO) (amount em-8), or ilo(~) = Floclo(OO)
(8)
Flo = ~c1--~] ~f cj dt]
(9)
then (7) yields
Equation (9) is the systemic input theorem for multi-input, multi-output systems in which only a single input is labeled. The second parenthetical factor on the right in (9) is obtained from a finite dose experiment. The first parenthetical factor is obtained from an indicator steady state experiment. This 'dilution' factor would generally be less than unity (when expressed as an activity ratio) as it represents a dilution of the indicator input by the non-labeled systemic inputs. In the important special ease of a single inflow, multioutflow system in which all clearances are actual convective flows of the same
R E G I O N A L AND COMPARTMENTAL SYSTEMS
579
substance (such as water, or blood of a constant hematocrit) the dilution factor in (9) is unity. This result follows from the systemic steady state condition of constancy of each regional volume which requires that Flz + F12 + Fz3 = -Fie F2z + F2s + Fs3 = 0
(10)
Fsl + F 3 2 + F88= 0 comparison of (6) and (10) shows that the solution of (6) for c1(oo)/ilo(OO), c2(oo)/izo(oo), Cs(OO)/ilo(oo)is the same as the solution l/Fie, 1/Fie, 1/Fzo which follows from (10). Hence c1(oo) = c2(oo) = c 3 ( ~ ) = ilo(Oo)/F~o = C~o(OO) (II) in which (8) has been used. A more formal proof of the equal concentration theorem (11) has recently been given for the metabolic case (Hearon, 1974). In the metabolic case F~j is an actual exchange rate (g/see)of systemic substance and cj is the specificactivity in the compaxtmental portion j. In this case if F~j existsin (6) it also existsin (10). If, therefore, the simultaneous equations (6) possess a 'non-pathologic' solution, then (11) expresses equality of specific activity of tracer in every compartmental portion of the system. In the present foreign indicator situation, however, an F~j can existin (6) but not in (I0). Thus Fij might represent the product of the permeability P (era/see)and the surface area S (cm 2) of the boundary surface (blood-tissue barrier) between the compartmental portion j and the regional portion i and Fj~ (r F~j) might represent an active transport mechanism of indicator. Such F~/s would contribute to indicator mass balance in (6) but not to systemic volume balance in (I0). Consequently (ll) does not hold, although the solutionfor the cj(oo)can be obtained in this case by solving (6). I n the extreme case where all the Ftj in (6) represent permeability mechanisms, or in the limit of infinitesimal regions, diffusional mechanisms, it is evident that a general gradient in the indicator steady state concentrations cj(oo) could exist from the input to the outputs, the free energy for which comes from the outside mechanism that maintains a higher concentration in the input channel than in all the output channels. In general, the equal concentration condition in the indicator steady state need not hold if 'demixing' mechanisms which can distinguish between the introduced indicator and the systemic substances exist, such as selective permeability barriers and chemical (active) transport and if the outlets are suitably situated (Roberts et al., 1973). With such mechanisms the indicator steady state conservation equations are different from the systemic steady state conservation equations. In this respect the foreign indicator case is not equivalent to, but is of wider scope than, the native tracer case.
580
W. F E R L , N. A. LASSEN AND R. M. E F F R O S
In the all-convective (equivalent metabolic), single input case, (9) and (11) give the systemic inflow theorem for single inflow, multi-outflow systems (Noss]in, 1964; Bergner, 1964a, b).
Fie=
role~f c, dt.
(12)
Equation (12) states that the inflow of a single inflow, multi-outflow system equals the dose over the area under the concentration curve measured in the outflow from any compartmental portion in the system. Equation (12) implies the equal area rule, that the area under the concentration curve measured in any channel of an all-convective single inflow system is the same, although the curve shape is in general different. Conversely, the measurement of equal areas under concentration curves in all systemic channels would imply a single inflow system.
Systemic Outflow Theorem.
The Nosslin-Bergner inflow theorem (12) is quite different from the Stewart-Henriques-Hamilton outflow theorem (Hamilton, 1962). The outflow theorem expresses conservation of indicator in a convective channel. If the dose me of indicator is injected anywhere into a system and the amount m~j of this indicator is known to be transported convectively from time zero to infinity through the channel from compartment portion j to compartment-region i then the flow Ftj in the channel is given by F~j=
m,j/f c~dt.
(13)
The utility of (13) lies in those cases for which the recovery m J m 0 is known. In the general case mtj. could be less than me (bypass channels around region j ) or greater than m 0 (feedback or recirculation channels from outputs of region j). Equation (13) is usually applied to the outflow channel of a multi-inflow, single outflow system in which case m0j = m0 (recovery is unity) and F0~.=
mo/fC s dr.
(14)
Equations (13) and (14) express conservation of indicator but do not require linearity with respect to indicator or even stationarity of systemic parameters (if F~j is time dependent (13) gives a particular time average of Ftj(t))*. The inflow theorem (12), on the other hand, does not require conservation but does require linearity and stationarity and applies to a single inflow, multi-outflow system. For a single inflow, single outflow, indicator linear, systemic stationary system, (12) and (14) become identical which is probably the reason that the * Although (13) will not generally give the arithmetic average of F~j(t), if ~o is time-dependent, it will give a value which must lie between the minimum and maximum values of F~j(t).
REGIONAL AND COMPARTMENTAL SYSTEMS
581
inflow theorem has not generally been recognized as distinct from the outflow theorem. A verification of the inflow theorem (12) has been indicated elsewhere (Perl, 1971) for the kidney, considered as a 1 inflow (renal artery), 2 outflow (renal vein, urine) system (this example was suggested by F. P. Chinard).
Equivalent Single Inflow Labeling of Multi-inflow Systems. Suppose the total dose mso in a finite dose experiment is apportioned between the j = 1 and 2 inputs of the three input system in the same proportion as the total infusion rate iso(~) in the indicator steady state. Thus (subscripts so denote to system from outside) ml--------~~ m2------~~= A = ms-------~~ (15) ilo(~) i2o(~) iso(~) where mso = mlo + m20 (16) iso(~) = i l o ( ~ ) + i2o(~). (17) Equations (15)-(17) define equivalent labeling of a number of systemic inflows. Equations (5), (6) are modified by replacing 0 in the respective second equations by -m2o, -i~o(~). Substitution of (15) into the modified equation (5) and comparison with the modified equation (6) shows that the simultaneous equations for determining f cj dt/A are the same as those for determining cs(~ ). Hence these two sets of quantities are respectively equal and by (15) the indicator influx theorem becomes fcj dt = .... cj(~). (18) mso iso(~ ) Equation (18) is analogous to (7) and applies to a multi-systemic inflow system in which some but not all of the inflows are equivalently labeled (partially equivalently labeled systems). For convective entry an equation analogous to (9) holds. Again, the important special ease of this equation is for an allconvective system with all systemic inflows equivalently labeled. The systemic steady state condition is then the set of equations (10), modified by replacing the 0 in the second equation by -F2o. The additional equivalent labeling condition is imposed that the indicator steady state inputs are in proportion to the convective inflows or i~o(aC) = i2o(~) . . . where
Ylo
Y2o
iso(~) .
Yso
cso(~)
Fso = Flo + F~o iso = ilo(OO) + i2o(OO) -~ Floclo(OO) + F2oc2o(~).
(19)
(20) (21) (22)
582
W. P E R L , 1~. A. LASSEI~I AND R. M. E F F R O S
Substitution of (19) into the modified equations (6) and comparison with the modified equations (10) gives as before the equal concentration or, in metabolic terminology, the equal specific activity theorem c1(
) = c (oo) = c3( o) = c,o(oo).
(23)
Equations (18) and (23) give the inflow theorem for equivalently labeled allconvective (equivalent to metabolic) systems
Fso= m,o/fcj dr.
(24)
Comparison of (18), (23) and (24) with (7), (1t) and (12) shows that equivalent labeling effectively reduces a multi-input system to a single input system, as though all the inputs were tied together on a parallel arrangement to produce additively a single total input. The original demonstration of equivalent labeling by Bergner (1964a) for the metabolic case assumed that the individual input rates il0(t), i2o(t)in the finite dose experiment were respectively proportional to the systemic input rates F~o, F2o. Roberts et al. (1973) pointed out that this condition could be weakened by requiring only that the total individual doses role, m20 be proportional to Fie, F20 as expressed by combining (15) and (19). This condition can evidently be weakened still further. To derive the indicator influx theorem (18), only (15) is required which makes no reference to systemic inflows and does not require an all-convective system. A pharmacological application of (18) has been indicated elsewhere (Perl, 1971). The equivalent labeled inflow theorem (24) explains why, after injection of a dose of vascular indicator into the vena cava or upper aorta, the indicator curve measured at various downstream locations and in various arterial branches has the same area, although a different curve shape (Bassingthwaighte, 1966). Either the primary injection or the subsequent passage of indicator through the heart produces sufficient mixing so that the laminar flow proceeding downstream from the mixing region automatically labels itself in proportion to the local flow in each streamtube. The subsequent time dependent diffusional and dilutional effects on the indicator, although complicated, are linear and therefore do not disturb the applicability of (24). The outflow theorem (13) cannot explain the equal area but different shape of the various measured indicator curves. All real convective channels are finite in cross section and constitute multiple channels in parallel. The apparent success of the outflow theorem in deducing total flow at a location downstream of a labeled location must therefore actually be ascribed to equivalent labeling, absence of indicator steady state demixing and hence the equal area rule based on the inflow theorem.
REGIONAL AND COMPARTMENTALSYSTEMS
583
Compartmental Volume of Distribution.
The compartment-regions are now specified to be compartments. Thus, compartment-region j is defined as comp a y m e n t j ff the amount mj(t) of indicator within the region at time t is related to the concentration cj(t) by mj(t) = Vajcj(t) (25) where Vaj (eroS), denoted the equilibrium volume of distribution of compartment j for the paxticular indicator, is independent of time. An alternate expression of (25) is in terms of the equilibrium partition coefficient )t~q defined by d )
mj(t) = 6 Crj(t)dVj = ~qVfj(t) dvt
(26)
where the integration is spatial, over the geometric volume Vj (era 3) of region j, and crj(t ) = dmj(t)/dV s is the concentration of indicator in the regional volume element dVj at time t. Equations (25) and (26) yield
where, by (26)
Vaj = a~qVj ;~q = ~,j(t)/cj(t)
rj(t) = fo,,(t) dV,/V,.
(27)
(28)
(29)
The constancy of )t~q in (27) implies instantaneous thermodynamic equilibration of indicator at any time t, between the concentration crj(t) at any interior location and the concentration cj(t) in the outflow. That is, the activity of solute is spatially constant throughout the compartment and moreover equal to the activity of solute at exit [flow limit condition (Kety, 1951)]. I f the activity coefficient of the solute is constant throughout the compartment, as when the compartmental contents are homogeneous, then the solute concentration itself is constant throughout the compartment (this is the usual definition of a compartment).
Indicator Mass Theorem. Multiplying both sides of (18) by Vaj and summing over all compartments gives the indicator mass theorem for partially equivalently labeled systems (Meier and Zierler, 1954; Stephenson, 1958, 1960; Zierler, 1965)
f m(t) dt m(oo) mso iso(oo) where, defining ~ as summation over j,
m(t) = ~ Va/%(t)
(30)
(31)
584
W. P E R L , N. A. LASSEN AND R. M. E F F R O S
is the total amount of indicator within the system at time t in the finite dose experiment and m ( ~ ) = ~ Vdjcj(~ ) (32) is the total amount of indicator within the system in the indicator steady state with the partially equivalent labeling in the denominators specified by (15)(17). As with (7), equation (30) is derivable directly from the stimulusresponse theorem (Perl, 1971). This derivation, or direct inspection of (30), shows that (30) is independent of the number of compartments. Therefore, the previous specification of compartment-regions as compartments is no real restriction in deriving (30) because the system could be considered as composed of a sufficiently large number of sufficiently small compartment-regions. A sufficiently small compartment-region can be considered as always instantly equilibrated spatially, hence describable by (25) which suffices to yield (30) from (18). Equation (30) requires only stationarity of the system (systemic steady state) and linearity of the system with respect to indicator.
Indicator Mean Transit Time Theorem. The left-hand side of (30) is well-known to give a definition of the mean transit time t of the system (Stephenson, 1958; Bergner, 1964b; Zierler, 1965; Perl et al., 1969a, c; Roberts et al., 1973). Thus, integration by parts and the assumption tm(t) -+ 0 as t --> ~ give
fmdt= -ftdm.
(33)
Conservation of indicator for the whole system is expressed by
dm/dt = iso(t ) - jos(t)
(34)
where iso(t ) is the total (equivalently labeled) influx of indicator to the system and jos(t) is the total effiux of indicator from the system. Integration of (34) gives
fisodt=fjo~dt=m~o.
(35)
The mean transit time ~ is defined as
i = ] tj~
]jo at
ftisodt
f i.
(36)
which requires measurements in all inflow and outflow channels (Perl et al., 1969c). Equations (33)-(36) enable expression of (30) as = m(~)/~so(~)
(37)
REGIONAL AND COMPARTMENTALSYSTEMS
585
which is the indicator mean transit time theorem for multi-input, multioutput systems.
Systemic Volume of Distribution.
The volume of distribution Va(~) of the system in the indicator steady state is defined by m(~) = V~(~)%(~)
where %(oo) is a reference (often plasma) concentration. gives an alternate definition of Va(~) as
(38)
Comparison with (32)
Va(~) = ~= t~ Vaj
(39)
:k? = cj(~))/%(~)
(40)
where is the (compartment j)/(refercnce fluid) distribution coefficient of indicator in the indicator steady state. This distribution coefficient is not, in the general nonconvective case, an equilibrium partition coefficient but may reduce to one in special cases (Roberts et al., 1973). The most important special case is the equivalently labeled all-convective case previously discussed, in which the equal concentration condition (23) obtains. I f %(oo) is taken as one of the concentrations then )~ = 1 and the systemic volume of distribution V~(~) in the indicator steady state, equation (39), reduces to the usual systemic volume of distribution Va, that would be measured by equilibrium dilution, and is given by =
= y.
Systemic Mean Transit Time Theorem. expressed by
qvj.
(41)
The right-hand side of (30) can be
(32) and (19) as m(oo)
V~(oo) % ( ~ )
r~(oo) = - -
(42)
if the equivalent inflow concentration is taken as the reference concentration. Equations (37) and (42) yield the systemic mean transit time theorem in the equivalently labeled non-convective case as Va(~) =
F,ot
(43)
and in the equivalently labeled all-convective case
Va -- Fso~.
(44)
In metabolic terminology (43) is written as M = x~.
(45)
586
w. PERT,, 2. A. LASSEN AND R. M. ~PFROS
where M is the mass of systemic substance, I is the throughput flux of systemic substance and t is the mean transit time of tracer measured in a finite dose experiment. Equation (44) is most often applied to the single-inflow, single-outflow system in which casejo 8in (36) can be replaced by the outflow concentration and iso can be replaced by the inflow concentration. The formula of Nosslin (1964), Perl and Samuel (1969) and Bright (1973) in terms of exponential parameters corresponds to assuming the plasma volume to be both the inflow and the outflow compartment of the system, and fitting the plasma indicator concentration by a sum of exponentials
%(t)/mso = ~ a, e -~,t where ~ denotes summation over i.
(46)
Substitution of (46) into (24) gives
.F,o = (~, atilt,) -1
(47)
Substitution of (46) into (36) with j(t) = Fo~%(t ) gives, if the input mean time is negligible, = ft% dt _ 5 a,/k~ (48) f % dt 5 a,[k, If the mean time of input is not negligible, it must be subtracted from the right-hand side of (48). Equations (44), (47), (48) give the desired volume of distribution formula 5 a, lk~ Va = (5 ai/]fi) 2" (49) Equation (49) holds also if the single systemic inflow into which indicator is injected, enters at a different location of the system from that of the single systemic outflow in which the indicator concentration is measured, and if the system is noncompartmental (Perl, 1971). If, however, in a compartmental system the indicator is injected into the outflow compartment, in which concentration is measured, but the single inflow is into another compartment, then (49) is no longer valid (nonequivalent labeling). An example of this case for a two-compartment system is explicitly calculated elsewhere (Appendix of Samuel et al., 1968). In conclusion, the present matrix method corresponds to a system of compartment-regions connected by clearances. The indicator influx theorem (7), relating a finite dose experiment to the indicator steady state, was derived for a multi-input, multi-output system. This single underlying relation by the matrix method corresponds to the single underlying relation by the convolution method, the stimulus-response theorem (Perl, 1971). From the indicator influx
REGIONAL AND COMPARTMENTALSYSTEMS
587
theorem was derived the systemic inflow theorem (9) for multi-inflow, multioutflow systems which involves both the finite dose experiment and the indicator steady state. The special case for all convective single inflow, multioutflow systems (12) involves only the finite dose experiment and yields the equal concentration rule in the indicator steady state and the equal area rule in the finite dose experiment. The systemic inflow theorem is in general quite different from the Stewart-Henriques-Hamilton outflow theorem. Several types of equivalent labeling were discussed, which have the effect of combining multi-inputs into equivalent single inputs. The compartment-regions were then specialized to compartments, the compartmental volume of distribution was defined, and the indicator mass theorem was derived for multi-input, multi-output systems and converted by integration by parts to an indicator mean transit time theorem. The systemic volume of distribution in the indicator steady state (38), (39) was defined and distinguished from the systemic volume of distribution as measured by equilibrium dilution (41). For each of these volumes the indicator mean transit time theorem yields a systemic mean transit time theorem (43), (44). This theorem was finally expressed in terms of exponential parameters. This investigation was supported in part by U.S. Public Health Service Grants HE 12974 and H E 12879. Dr. Effros is a Career Development Awardee of the National Heart and Lung Institute (5 K04 H L 70649 and 7 K04 H L 00132). LITERATURE Bassingthwaighte, J.B. 1966. "Plasma Indicator Dispersion in Arteries of the Human Leg." Circ. Res., 19, 332-346. Bergner, P. -E.E. 1961. "Tracer Dynamics: I. A Tentative Approach and Definition of Ftmdamental Concepts." J. Theor. Biol., 2, 120-140. 1964a. "Tracer Dynamics and the Determination of Pool-sizes and Turnover Factors in Metabolic Systems." Ibid., 6, 137-158. 1964b. "Kinetic Theory: Some Aspects on the Study of Metabolic Processes." In Dynamic Clinical Studies with Radioisotopes, eds. R. M. Knisely and W. N. Tauxe. TID-7678, Clearinghouse, U.S. Department of Commerce, Springfield, Virginia 22151, 1-18. 1967. "The Concepts of Mass, Volume and Concentration." In Compartments, Pools and Spaces in Medical Physiology, eds. P. -E.E. Berger and C. C. Lushbaugh. CONF--661010, Clearinghouse, U.S. Department of Commerce, Springfield, Virginia 22151, 21-37. Bright, P.B. 1973. "The Volumes of Some Compartment Systems with Sampling and Loss from One Compartment." Bull. Math. Biol., 35, 69-79. Hamilton, W. F. 1962. "Measurement of the Cardiac Output." In Handbook of Physiology, Sec. 2, Vol. 1, pp. 551-584, eds. W. F. Hamilton and P. Dew, Washington, D.C. : American Physiological Society. Hearon, J.Z. 1974. "A Note on Open Linear Systems." Bull. Math. Biol., 36, 97-99.
588
W. PERL, N. A. LASSEN AND R. M. EFFROS
Kety, S . S . 1951. "The Theory and Applications of the Exchange of I n e r t Gas at the Lungs and Tissues. Pharrnacol. Rev., 3, 1-41. Meier, P. and K. L. Zierler. 1954. "On the theory of the Indicator-Dilution Method for Measurement of Blood Flow and Volume." J. Appl. Physiol., 6, 731-744. Nosslin, B. 1964. "Mathematical Model of Plasma Protein Turnover Determined with 131I-Labelled Protein." I n Metabolism of Human Gamma Globulin (yss-Globulin), pp. 115-120, ed. by S. B. Andersen. Philadelphia: F. A. Davis. Perl, W. 1971. "Stimulus-Response Method for Flows and Volumes in Slightly Perturbed Constant Parameter Systems." Bull. Math. Biophys., 33, 225-233. - and F. P. Chinard. 1969a. "Occupancy Principle: Nonidentity with Mean Transit Time." Science 166, 260. - and P. Samuel. 1969b. " I n p u t - O u t p u t Analysis for Total I n p u t Rate and Total Traced Mass of Body Cholesterol in Man." Circ. Res. 25, 191-199. - - , R. M. Effros and F. P. Chinard. 1969c. "The Indicator Equivalence Theorem for I n p u t Rates and Regional Masses in Multi-inlet Steady State Systems with Partially Labeled I n p u t . " J. Theor. Biol., 25, 297-316. Roberts, G. W., K. B. Larson and E. E. Spaeth. 1973. "The Interpretation of Mean Transit Time Measurements for Multiphase Tissue Systems." J. Theor. Biol., 39, 447-475. Samuel, P., C. M. Holtzman, E. Meilman and W. Perl. 1968. "Effect of Neomycin on Exchangeable Pools of Cholesterol in the Steady State." J. Clin. Invest., 47, 18061818. Stephenson, J . L . 1948. "Theory of the Measurement of Blood Flow by the Dilution of a n Indicator." Bull. Math. Biophys., 10, 117-121. 1958. "Theory of Measurement of Blood Flow by Dye Dilution Technique." I R E Trans Med. Electron., PGME-12, 82-88. 1960. "Theory of Transport in Linear Biological Systems: I. F u n d a m e n t a l Integral Equation." Bull. Math. Biophys., 22, 1-17. Zierler, K . L . 1965. "Equations for Measuring Blood Flow by External Monitoring of Radioisotopes." Circ. Res., 16, 309-321. RECEIVED 9-3-74 I~WWSED 4-15-75
VOLVME 37, 1975
M A T R I X P R O O F OF F L O W , V O L U M E A N D M E A N TRANSIT TIME THEOREMS FOR REGIONAL AND COMPARTMENTAL SYSTEMS
W. PERL Departments of Medicine and Physiology, New Jersey Medical School, Newark, New Jersey 07103 N. A. LASSEN Department of Clinical Physiology Bispebjerg Hospital 2400 Copenhagen NV, Denmark and R. M. EFFROS Division of Respiratory Physiology and Medicine Harbor General Hospital Section University of California at Los Angeles Torrance, California 90509
The relations (inflow) -- (dose)/(area under indicator curve), and (volume of distribution) -- (throughflow) • (mean transit time) are derived by a matrix method for a system of interconnected subsystems, within which spatial indicator activity gradients may exist, and for compartments, within which the indicator activity is spatially uniform. The inflow theorem is different from the outflow theorem. Equivalent labeling of multi-input systems reduces them formally to single input systems. Foreign indicator flow-volume kinetics are more general than, and include as a special case, tracer flux-mass (metabolic) kinetics. Volume of distribution in the indicator steady state may be different from the equilibrium volume of distribution. T h e well-known relations for a single inflow, single outflow s y s t e m : v o l u m e = flow • m e a n t r a n s i t t i m e (Stephenson, 1948, 1958; Meier a n d Zierler, 1954) a n d 573
574
W. P E R L , N. A. LASSEN AND R. IVL E F F R O S
flow = dose/area under emergent indicator curve, have been derived for a compartmental system by matrix methods (Bright, 1973). In this derivation it was assumed that the dose of indicator was injected into the same compartment in which subsequent measurement of the concentration-time curve was made. ~n view of previous derivations using stochastic (Bergner, 1964a, b) or convolution integral methods (Perl et al., 1969c; Perl, 1971) this assumption can be weakened in that the inflow compartment into which indicator is injected can be different from the outflow compartment in which measurements are made. Bright's final result in his equation (5), combining the flow and volume theorems in terms of exponential parameters, was given essentially by Nosslin (1964, equation 44) who also used a matrix approach, although expressed in 'metabolic' terminology. The pioneering work of Nosslin and Bergner showed that the flow theorem is more properly an inflow theorem which is in general quite different from the Stewart-Henriques-Hamilton outflow theorem (Hamilton, 1962). It is of interest to retrace Nosslin's proof by a simplified matrix method. The matrix method shows directly that the often cited correspondence between the 'foreign indicator' case and the 'native tracer' case is incomplete. The former includes the latter as a special case. A limitation in Nosslin's proof will be removed, namely, the assumption that the system itself, before indicator or tracer is introduced, is described by a set of linear equations. Also, some generalizations over previous matrix treatments axe included: compartment is replaced by 'compartment-region'; multi-inflow multi-outflow systems are considered; the equivalent labeling assumption (Bergner, 1964a) which was weakened by Roberts et aI. (1973) is weakened still further in certain cases. All theorems are derivable from a single underlying relation, the indicator influx theorem, which corresponds in the present matrix approach to the stimulus-response theorem in the convolution approach (Perl,
1971). Foreign Indicator and Native Tracer. A foreign indicator, denoted simply indicator, is a detectable substance introduced into a physiological system, which substance is chemically different from any relevant systemic substance and yet which yields physiological information about the system. For example, antipyrine may be used to measure total body water, raffinose may be used to measure a permeability of a blood-tissue barrier, etc. The indicator may be measured as the amount (mg, cps, etc.) in a sample of systemic substance and expressed as a concentration = (amount in sample)/(cm3 of systemic water or other systemic substance in sample). Deduced information might be a flux of indicator (amount see-1), a volumetric flow (cm3 sec- 1) of a systemic substance
REGIONAL AND COMPARTMENTAL SYSTEMS
575
such as blood water or a volume (era 3) of a systemic substance such as tissue water. A native tracer, denoted simply tracer, is a substance introduced into a physiological system, which substance is chemically identical to a systemic substance of interest (tracee, mother substance) but separately detectable from the systemic substance. For example, 131I-albumin may usually be considered a tracer for systemic albumin. The tracer may be measured as the amount (rag, cps, etc.) in a systemic sample and expressed as a specific activity = (amount of tracer in sample)/(amount of tracer + amount of mother substance in sample). Deduced information might be a flux of tracer (amount sec -1) a flux of mother substance (amount see-l), or a systemic mass (g) of mother substance. The indicator case and its terminology will be used. As will be shown, the indicator case is of wider scope than the tracer case, namely, where the system handles the indicator differently from all other systemic substances of interest. If the system cannot distinguish between the indicator and a systemic substance of interest, then the indicator may be regarded as a tracer of that systemic substance (Bergner, 1967) and the two cases become formally identical, with the one-to-one correspondence: concentration--> specific activity, flow--~ flux, volume --~ mass.
Indicator Influx Theorem.
The system consists of a number, assumed three for illustration, of 'compartment-regions' which communicate with each other and with the outside b y means of 'clearances' (Figure 1). A compartment-region consists of a 'regional portion', tissue for example, within which the concentration of indicator can vary both spatially and temporally and a 'compartmental portion', the venous blood volume contained within the tissue for example, within which the concentration of indicator can vary only temporally. Indicator can exchange between a regional portion and its corresponding compartmental portion. The unidirectional indicator flux j~s(t) (aInount/sec) is defined as to t h e / t h compartment-region from t h e j t h compartment portion, in which the indicator concentration is cj(t). The crossing of the boundary surfaces b y the arrows in Figure 1 is intended to represent that indicator can enter anywhere within a regional portion or a compartmental portion of a compartment-region. The clearance F~j is defined as the ratio of indicator flux to concentration or
jtj(t) = F~jcj(t).
(1)
As with clearance defined elsewhere in physiology, F~j can be an actual systemic convective flow (glomerular filtration rate) or a parameter characterizing indicator transport across a barrier (permeability • surface area) or through a
576 distance
W. PERL, N. A. LASSEN AND R. M. EFFROS (diffusion coefficient x area/distance),
etc.
The indicator
effiux
joj(t) to t h e outside from the j t h c o m p a r t m e n t portion is defined similarly in t e r m s o f an o u t p u t clearance/v0j and an indicator concentration cj(t) in the j t h c o m p a r t m e n t a l portion b y (1) with i = O. The indicator influx ijo(t), if a n y , into t h e j t h c o m p a r t m e n t - r e g i o n from the outside can likewise be defined in t e r m s o f a n i n p u t clearance Fjo a n d a c o n c e n t r a t i o n cjo(t ) a t which the indicator is delivered, b y an e q u a t i o n o f the form o f (1).
i,o(t)
~3
%1
Fibre 1. System of eompartmerit-regions interconnected by clearances lv~i (cma see-1). Compartment portion is within square. Regional portion is between circle and square. Amount m~(t) of indicator is in the jth compartment-region at time t (see) after start of indicator influx ilo(t ) (amount see- 1) into F10 systemic input. Indicator exchange occurs between regional portion and compartmental portion. Indicator concentration cj(t) is spatially uniform in compartmental portion
Afinite dose experiment is p e r f o r m e d in which a finite a m o u n t ml0 (amount) of indicator is injected from outside into the regional portion of c o m p a r t m e n t region 1 a t r a t e ilo(t), a n d into no other compartment-regions. Thus i~o(t) = i3o(t ) = 0 and mio =
Jil0(t) J
dt
(2)
REGIONAL AND COMPARTMENTAL SYSTEMS
577
where the integration, here and later, is from time zero to infinity. At time t, measured from the start of injection, the amount mj(t) of indicator is in the j t h compartment-region and the concentration cj(t) is in the compartmental portion of, hence also in all the exit chao_nels from, the j t h compartment-region. Conservation of indicator in each compartment-region gives the customary set of differential equations (dot denotes time derivative)
~hl = Fllcl + F12c2 + F18c8 + ilo m2 = F21Cl + F22C2 + F23c3 + 0
(3)
rh 3 = .F31c 1 + F32c 2 + F33c z + 0
where - F ~ I - Fol + F~I + F81
-F22 = Fo2 + F12 + F32 -F83 -= F03 + Fz3 + F23.
(4)
In (3) the indicator quantities are time dependent. In (4), - Fjj represents the total clearance out of the j t h compartment-region. The clearances F~j are assumed constant (systemic steady state) and independent of the indicator quantities (small perturbation of the system). The assumption of constancy of the systemic parameters F~j. = (indicator flux jtj(t))/(indicator concentration cj(t)) does not imply constancy of any (systemic flux)/(systemie concentration). The systemic fluxes may be nonlinear functions of the systemic concentrations. The present results require only the linear relations (3) between indicator amounts and indicator concentrations. That such relations are possible, for a sufficiently small perturbation of the system by the indicator, despite nonlinear relationships of systemic quantities in the absence of indicator, is wellknown (see, for example, Bergner, 1961). The reason for the introduction of compartment-regions is that this seems to be the most general structure permitted by equations (3) and (4) which yield the indicator influx theorem. The compartment is the limiting case of the compartment-region when the regional portion goes to zero. Integrating (3) from time zero to infinity with the initial condition mj(0) = 0 (system devoid of indicator before injection) and the final condition mj(~) = 0 (all compartment portions connected directly or indirectly to the outside) and using (2) gives
Fll f cl dt +
f c2dt +
F18 f c 3 dt = - m l o
F21f c~ dt + F22 f c2 dt + F 2 3 f c 3dt = 0
F31 fc~dt + F32fc2dt
+ F 3 a f c 3dr = O.
(5)
578
W. P E R L , N. A. LASSEN AND R. M. E F F R O S
An indicator steady state experiment is either performed or considered to be performed, in which the constant infusion rate i~0(~ ) of indicator, into the same inflow as in the finite dose experiment, is maintained constant (or approaches constancy sufficiently closely) long enough for the time derivatives on the left in (3) to have vanished which then gives Fllc~(oo) + F12c2(oo) + F~3c3(oo) = -ilo(OO)
F21c1(o0) + F29.c2(oo) + F23c3(oo) -- 0
(6)
F3~c~(oo) + F82c~(~) + F33%(oo) = O. Comparison of (6) with (5) shows that the simultaneous equations for determining ~ cj dt/mlo are the same as those for determining cj(oo)/ilo(OO). Hence these two sets of unknowns are respectively equal and f cs dt = cj(oo) mlo i~o(OO)
(7)
Equation (7) is the indicator influx theorem, a special case of the stimulus-response theorem (Perl, 1971). It applies to a multi-systemic input, multi-systemic output system. I t states that if a finite amount ( = dose) of indicator is injected over a finite time into any one input channel then the area under the concentration per unit dose curve measured in any compartmental portion, or exit therefrom, of the system equals the concentration per unit infusion rate for the same two locations in the indicator steady state.
Systemic Inflow Theorems. I f the indicator steady state influx in (7) is expressed as a systemic input clearance Flo (em 3 sec -1) times a reference input concentration Clo(OO) (amount em-8), or ilo(~) = Floclo(OO)
(8)
Flo = ~c1--~] ~f cj dt]
(9)
then (7) yields
Equation (9) is the systemic input theorem for multi-input, multi-output systems in which only a single input is labeled. The second parenthetical factor on the right in (9) is obtained from a finite dose experiment. The first parenthetical factor is obtained from an indicator steady state experiment. This 'dilution' factor would generally be less than unity (when expressed as an activity ratio) as it represents a dilution of the indicator input by the non-labeled systemic inputs. In the important special ease of a single inflow, multioutflow system in which all clearances are actual convective flows of the same
R E G I O N A L AND COMPARTMENTAL SYSTEMS
579
substance (such as water, or blood of a constant hematocrit) the dilution factor in (9) is unity. This result follows from the systemic steady state condition of constancy of each regional volume which requires that Flz + F12 + Fz3 = -Fie F2z + F2s + Fs3 = 0
(10)
Fsl + F 3 2 + F88= 0 comparison of (6) and (10) shows that the solution of (6) for c1(oo)/ilo(OO), c2(oo)/izo(oo), Cs(OO)/ilo(oo)is the same as the solution l/Fie, 1/Fie, 1/Fzo which follows from (10). Hence c1(oo) = c2(oo) = c 3 ( ~ ) = ilo(Oo)/F~o = C~o(OO) (II) in which (8) has been used. A more formal proof of the equal concentration theorem (11) has recently been given for the metabolic case (Hearon, 1974). In the metabolic case F~j is an actual exchange rate (g/see)of systemic substance and cj is the specificactivity in the compaxtmental portion j. In this case if F~j existsin (6) it also existsin (10). If, therefore, the simultaneous equations (6) possess a 'non-pathologic' solution, then (11) expresses equality of specific activity of tracer in every compartmental portion of the system. In the present foreign indicator situation, however, an F~j can existin (6) but not in (I0). Thus Fij might represent the product of the permeability P (era/see)and the surface area S (cm 2) of the boundary surface (blood-tissue barrier) between the compartmental portion j and the regional portion i and Fj~ (r F~j) might represent an active transport mechanism of indicator. Such F~/s would contribute to indicator mass balance in (6) but not to systemic volume balance in (I0). Consequently (ll) does not hold, although the solutionfor the cj(oo)can be obtained in this case by solving (6). I n the extreme case where all the Ftj in (6) represent permeability mechanisms, or in the limit of infinitesimal regions, diffusional mechanisms, it is evident that a general gradient in the indicator steady state concentrations cj(oo) could exist from the input to the outputs, the free energy for which comes from the outside mechanism that maintains a higher concentration in the input channel than in all the output channels. In general, the equal concentration condition in the indicator steady state need not hold if 'demixing' mechanisms which can distinguish between the introduced indicator and the systemic substances exist, such as selective permeability barriers and chemical (active) transport and if the outlets are suitably situated (Roberts et al., 1973). With such mechanisms the indicator steady state conservation equations are different from the systemic steady state conservation equations. In this respect the foreign indicator case is not equivalent to, but is of wider scope than, the native tracer case.
580
W. F E R L , N. A. LASSEN AND R. M. E F F R O S
In the all-convective (equivalent metabolic), single input case, (9) and (11) give the systemic inflow theorem for single inflow, multi-outflow systems (Noss]in, 1964; Bergner, 1964a, b).
Fie=
role~f c, dt.
(12)
Equation (12) states that the inflow of a single inflow, multi-outflow system equals the dose over the area under the concentration curve measured in the outflow from any compartmental portion in the system. Equation (12) implies the equal area rule, that the area under the concentration curve measured in any channel of an all-convective single inflow system is the same, although the curve shape is in general different. Conversely, the measurement of equal areas under concentration curves in all systemic channels would imply a single inflow system.
Systemic Outflow Theorem.
The Nosslin-Bergner inflow theorem (12) is quite different from the Stewart-Henriques-Hamilton outflow theorem (Hamilton, 1962). The outflow theorem expresses conservation of indicator in a convective channel. If the dose me of indicator is injected anywhere into a system and the amount m~j of this indicator is known to be transported convectively from time zero to infinity through the channel from compartment portion j to compartment-region i then the flow Ftj in the channel is given by F~j=
m,j/f c~dt.
(13)
The utility of (13) lies in those cases for which the recovery m J m 0 is known. In the general case mtj. could be less than me (bypass channels around region j ) or greater than m 0 (feedback or recirculation channels from outputs of region j). Equation (13) is usually applied to the outflow channel of a multi-inflow, single outflow system in which case m0j = m0 (recovery is unity) and F0~.=
mo/fC s dr.
(14)
Equations (13) and (14) express conservation of indicator but do not require linearity with respect to indicator or even stationarity of systemic parameters (if F~j is time dependent (13) gives a particular time average of Ftj(t))*. The inflow theorem (12), on the other hand, does not require conservation but does require linearity and stationarity and applies to a single inflow, multi-outflow system. For a single inflow, single outflow, indicator linear, systemic stationary system, (12) and (14) become identical which is probably the reason that the * Although (13) will not generally give the arithmetic average of F~j(t), if ~o is time-dependent, it will give a value which must lie between the minimum and maximum values of F~j(t).
REGIONAL AND COMPARTMENTAL SYSTEMS
581
inflow theorem has not generally been recognized as distinct from the outflow theorem. A verification of the inflow theorem (12) has been indicated elsewhere (Perl, 1971) for the kidney, considered as a 1 inflow (renal artery), 2 outflow (renal vein, urine) system (this example was suggested by F. P. Chinard).
Equivalent Single Inflow Labeling of Multi-inflow Systems. Suppose the total dose mso in a finite dose experiment is apportioned between the j = 1 and 2 inputs of the three input system in the same proportion as the total infusion rate iso(~) in the indicator steady state. Thus (subscripts so denote to system from outside) ml--------~~ m2------~~= A = ms-------~~ (15) ilo(~) i2o(~) iso(~) where mso = mlo + m20 (16) iso(~) = i l o ( ~ ) + i2o(~). (17) Equations (15)-(17) define equivalent labeling of a number of systemic inflows. Equations (5), (6) are modified by replacing 0 in the respective second equations by -m2o, -i~o(~). Substitution of (15) into the modified equation (5) and comparison with the modified equation (6) shows that the simultaneous equations for determining f cj dt/A are the same as those for determining cs(~ ). Hence these two sets of quantities are respectively equal and by (15) the indicator influx theorem becomes fcj dt = .... cj(~). (18) mso iso(~ ) Equation (18) is analogous to (7) and applies to a multi-systemic inflow system in which some but not all of the inflows are equivalently labeled (partially equivalently labeled systems). For convective entry an equation analogous to (9) holds. Again, the important special ease of this equation is for an allconvective system with all systemic inflows equivalently labeled. The systemic steady state condition is then the set of equations (10), modified by replacing the 0 in the second equation by -F2o. The additional equivalent labeling condition is imposed that the indicator steady state inputs are in proportion to the convective inflows or i~o(aC) = i2o(~) . . . where
Ylo
Y2o
iso(~) .
Yso
cso(~)
Fso = Flo + F~o iso = ilo(OO) + i2o(OO) -~ Floclo(OO) + F2oc2o(~).
(19)
(20) (21) (22)
582
W. P E R L , 1~. A. LASSEI~I AND R. M. E F F R O S
Substitution of (19) into the modified equations (6) and comparison with the modified equations (10) gives as before the equal concentration or, in metabolic terminology, the equal specific activity theorem c1(
) = c (oo) = c3( o) = c,o(oo).
(23)
Equations (18) and (23) give the inflow theorem for equivalently labeled allconvective (equivalent to metabolic) systems
Fso= m,o/fcj dr.
(24)
Comparison of (18), (23) and (24) with (7), (1t) and (12) shows that equivalent labeling effectively reduces a multi-input system to a single input system, as though all the inputs were tied together on a parallel arrangement to produce additively a single total input. The original demonstration of equivalent labeling by Bergner (1964a) for the metabolic case assumed that the individual input rates il0(t), i2o(t)in the finite dose experiment were respectively proportional to the systemic input rates F~o, F2o. Roberts et al. (1973) pointed out that this condition could be weakened by requiring only that the total individual doses role, m20 be proportional to Fie, F20 as expressed by combining (15) and (19). This condition can evidently be weakened still further. To derive the indicator influx theorem (18), only (15) is required which makes no reference to systemic inflows and does not require an all-convective system. A pharmacological application of (18) has been indicated elsewhere (Perl, 1971). The equivalent labeled inflow theorem (24) explains why, after injection of a dose of vascular indicator into the vena cava or upper aorta, the indicator curve measured at various downstream locations and in various arterial branches has the same area, although a different curve shape (Bassingthwaighte, 1966). Either the primary injection or the subsequent passage of indicator through the heart produces sufficient mixing so that the laminar flow proceeding downstream from the mixing region automatically labels itself in proportion to the local flow in each streamtube. The subsequent time dependent diffusional and dilutional effects on the indicator, although complicated, are linear and therefore do not disturb the applicability of (24). The outflow theorem (13) cannot explain the equal area but different shape of the various measured indicator curves. All real convective channels are finite in cross section and constitute multiple channels in parallel. The apparent success of the outflow theorem in deducing total flow at a location downstream of a labeled location must therefore actually be ascribed to equivalent labeling, absence of indicator steady state demixing and hence the equal area rule based on the inflow theorem.
REGIONAL AND COMPARTMENTALSYSTEMS
583
Compartmental Volume of Distribution.
The compartment-regions are now specified to be compartments. Thus, compartment-region j is defined as comp a y m e n t j ff the amount mj(t) of indicator within the region at time t is related to the concentration cj(t) by mj(t) = Vajcj(t) (25) where Vaj (eroS), denoted the equilibrium volume of distribution of compartment j for the paxticular indicator, is independent of time. An alternate expression of (25) is in terms of the equilibrium partition coefficient )t~q defined by d )
mj(t) = 6 Crj(t)dVj = ~qVfj(t) dvt
(26)
where the integration is spatial, over the geometric volume Vj (era 3) of region j, and crj(t ) = dmj(t)/dV s is the concentration of indicator in the regional volume element dVj at time t. Equations (25) and (26) yield
where, by (26)
Vaj = a~qVj ;~q = ~,j(t)/cj(t)
rj(t) = fo,,(t) dV,/V,.
(27)
(28)
(29)
The constancy of )t~q in (27) implies instantaneous thermodynamic equilibration of indicator at any time t, between the concentration crj(t) at any interior location and the concentration cj(t) in the outflow. That is, the activity of solute is spatially constant throughout the compartment and moreover equal to the activity of solute at exit [flow limit condition (Kety, 1951)]. I f the activity coefficient of the solute is constant throughout the compartment, as when the compartmental contents are homogeneous, then the solute concentration itself is constant throughout the compartment (this is the usual definition of a compartment).
Indicator Mass Theorem. Multiplying both sides of (18) by Vaj and summing over all compartments gives the indicator mass theorem for partially equivalently labeled systems (Meier and Zierler, 1954; Stephenson, 1958, 1960; Zierler, 1965)
f m(t) dt m(oo) mso iso(oo) where, defining ~ as summation over j,
m(t) = ~ Va/%(t)
(30)
(31)
584
W. P E R L , N. A. LASSEN AND R. M. E F F R O S
is the total amount of indicator within the system at time t in the finite dose experiment and m ( ~ ) = ~ Vdjcj(~ ) (32) is the total amount of indicator within the system in the indicator steady state with the partially equivalent labeling in the denominators specified by (15)(17). As with (7), equation (30) is derivable directly from the stimulusresponse theorem (Perl, 1971). This derivation, or direct inspection of (30), shows that (30) is independent of the number of compartments. Therefore, the previous specification of compartment-regions as compartments is no real restriction in deriving (30) because the system could be considered as composed of a sufficiently large number of sufficiently small compartment-regions. A sufficiently small compartment-region can be considered as always instantly equilibrated spatially, hence describable by (25) which suffices to yield (30) from (18). Equation (30) requires only stationarity of the system (systemic steady state) and linearity of the system with respect to indicator.
Indicator Mean Transit Time Theorem. The left-hand side of (30) is well-known to give a definition of the mean transit time t of the system (Stephenson, 1958; Bergner, 1964b; Zierler, 1965; Perl et al., 1969a, c; Roberts et al., 1973). Thus, integration by parts and the assumption tm(t) -+ 0 as t --> ~ give
fmdt= -ftdm.
(33)
Conservation of indicator for the whole system is expressed by
dm/dt = iso(t ) - jos(t)
(34)
where iso(t ) is the total (equivalently labeled) influx of indicator to the system and jos(t) is the total effiux of indicator from the system. Integration of (34) gives
fisodt=fjo~dt=m~o.
(35)
The mean transit time ~ is defined as
i = ] tj~
]jo at
ftisodt
f i.
(36)
which requires measurements in all inflow and outflow channels (Perl et al., 1969c). Equations (33)-(36) enable expression of (30) as = m(~)/~so(~)
(37)
REGIONAL AND COMPARTMENTALSYSTEMS
585
which is the indicator mean transit time theorem for multi-input, multioutput systems.
Systemic Volume of Distribution.
The volume of distribution Va(~) of the system in the indicator steady state is defined by m(~) = V~(~)%(~)
where %(oo) is a reference (often plasma) concentration. gives an alternate definition of Va(~) as
(38)
Comparison with (32)
Va(~) = ~= t~ Vaj
(39)
:k? = cj(~))/%(~)
(40)
where is the (compartment j)/(refercnce fluid) distribution coefficient of indicator in the indicator steady state. This distribution coefficient is not, in the general nonconvective case, an equilibrium partition coefficient but may reduce to one in special cases (Roberts et al., 1973). The most important special case is the equivalently labeled all-convective case previously discussed, in which the equal concentration condition (23) obtains. I f %(oo) is taken as one of the concentrations then )~ = 1 and the systemic volume of distribution V~(~) in the indicator steady state, equation (39), reduces to the usual systemic volume of distribution Va, that would be measured by equilibrium dilution, and is given by =
= y.
Systemic Mean Transit Time Theorem. expressed by
qvj.
(41)
The right-hand side of (30) can be
(32) and (19) as m(oo)
V~(oo) % ( ~ )
r~(oo) = - -
(42)
if the equivalent inflow concentration is taken as the reference concentration. Equations (37) and (42) yield the systemic mean transit time theorem in the equivalently labeled non-convective case as Va(~) =
F,ot
(43)
and in the equivalently labeled all-convective case
Va -- Fso~.
(44)
In metabolic terminology (43) is written as M = x~.
(45)
586
w. PERT,, 2. A. LASSEN AND R. M. ~PFROS
where M is the mass of systemic substance, I is the throughput flux of systemic substance and t is the mean transit time of tracer measured in a finite dose experiment. Equation (44) is most often applied to the single-inflow, single-outflow system in which casejo 8in (36) can be replaced by the outflow concentration and iso can be replaced by the inflow concentration. The formula of Nosslin (1964), Perl and Samuel (1969) and Bright (1973) in terms of exponential parameters corresponds to assuming the plasma volume to be both the inflow and the outflow compartment of the system, and fitting the plasma indicator concentration by a sum of exponentials
%(t)/mso = ~ a, e -~,t where ~ denotes summation over i.
(46)
Substitution of (46) into (24) gives
.F,o = (~, atilt,) -1
(47)
Substitution of (46) into (36) with j(t) = Fo~%(t ) gives, if the input mean time is negligible, = ft% dt _ 5 a,/k~ (48) f % dt 5 a,[k, If the mean time of input is not negligible, it must be subtracted from the right-hand side of (48). Equations (44), (47), (48) give the desired volume of distribution formula 5 a, lk~ Va = (5 ai/]fi) 2" (49) Equation (49) holds also if the single systemic inflow into which indicator is injected, enters at a different location of the system from that of the single systemic outflow in which the indicator concentration is measured, and if the system is noncompartmental (Perl, 1971). If, however, in a compartmental system the indicator is injected into the outflow compartment, in which concentration is measured, but the single inflow is into another compartment, then (49) is no longer valid (nonequivalent labeling). An example of this case for a two-compartment system is explicitly calculated elsewhere (Appendix of Samuel et al., 1968). In conclusion, the present matrix method corresponds to a system of compartment-regions connected by clearances. The indicator influx theorem (7), relating a finite dose experiment to the indicator steady state, was derived for a multi-input, multi-output system. This single underlying relation by the matrix method corresponds to the single underlying relation by the convolution method, the stimulus-response theorem (Perl, 1971). From the indicator influx
REGIONAL AND COMPARTMENTALSYSTEMS
587
theorem was derived the systemic inflow theorem (9) for multi-inflow, multioutflow systems which involves both the finite dose experiment and the indicator steady state. The special case for all convective single inflow, multioutflow systems (12) involves only the finite dose experiment and yields the equal concentration rule in the indicator steady state and the equal area rule in the finite dose experiment. The systemic inflow theorem is in general quite different from the Stewart-Henriques-Hamilton outflow theorem. Several types of equivalent labeling were discussed, which have the effect of combining multi-inputs into equivalent single inputs. The compartment-regions were then specialized to compartments, the compartmental volume of distribution was defined, and the indicator mass theorem was derived for multi-input, multi-output systems and converted by integration by parts to an indicator mean transit time theorem. The systemic volume of distribution in the indicator steady state (38), (39) was defined and distinguished from the systemic volume of distribution as measured by equilibrium dilution (41). For each of these volumes the indicator mean transit time theorem yields a systemic mean transit time theorem (43), (44). This theorem was finally expressed in terms of exponential parameters. This investigation was supported in part by U.S. Public Health Service Grants HE 12974 and H E 12879. Dr. Effros is a Career Development Awardee of the National Heart and Lung Institute (5 K04 H L 70649 and 7 K04 H L 00132). LITERATURE Bassingthwaighte, J.B. 1966. "Plasma Indicator Dispersion in Arteries of the Human Leg." Circ. Res., 19, 332-346. Bergner, P. -E.E. 1961. "Tracer Dynamics: I. A Tentative Approach and Definition of Ftmdamental Concepts." J. Theor. Biol., 2, 120-140. 1964a. "Tracer Dynamics and the Determination of Pool-sizes and Turnover Factors in Metabolic Systems." Ibid., 6, 137-158. 1964b. "Kinetic Theory: Some Aspects on the Study of Metabolic Processes." In Dynamic Clinical Studies with Radioisotopes, eds. R. M. Knisely and W. N. Tauxe. TID-7678, Clearinghouse, U.S. Department of Commerce, Springfield, Virginia 22151, 1-18. 1967. "The Concepts of Mass, Volume and Concentration." In Compartments, Pools and Spaces in Medical Physiology, eds. P. -E.E. Berger and C. C. Lushbaugh. CONF--661010, Clearinghouse, U.S. Department of Commerce, Springfield, Virginia 22151, 21-37. Bright, P.B. 1973. "The Volumes of Some Compartment Systems with Sampling and Loss from One Compartment." Bull. Math. Biol., 35, 69-79. Hamilton, W. F. 1962. "Measurement of the Cardiac Output." In Handbook of Physiology, Sec. 2, Vol. 1, pp. 551-584, eds. W. F. Hamilton and P. Dew, Washington, D.C. : American Physiological Society. Hearon, J.Z. 1974. "A Note on Open Linear Systems." Bull. Math. Biol., 36, 97-99.
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W. PERL, N. A. LASSEN AND R. M. EFFROS
Kety, S . S . 1951. "The Theory and Applications of the Exchange of I n e r t Gas at the Lungs and Tissues. Pharrnacol. Rev., 3, 1-41. Meier, P. and K. L. Zierler. 1954. "On the theory of the Indicator-Dilution Method for Measurement of Blood Flow and Volume." J. Appl. Physiol., 6, 731-744. Nosslin, B. 1964. "Mathematical Model of Plasma Protein Turnover Determined with 131I-Labelled Protein." I n Metabolism of Human Gamma Globulin (yss-Globulin), pp. 115-120, ed. by S. B. Andersen. Philadelphia: F. A. Davis. Perl, W. 1971. "Stimulus-Response Method for Flows and Volumes in Slightly Perturbed Constant Parameter Systems." Bull. Math. Biophys., 33, 225-233. - and F. P. Chinard. 1969a. "Occupancy Principle: Nonidentity with Mean Transit Time." Science 166, 260. - and P. Samuel. 1969b. " I n p u t - O u t p u t Analysis for Total I n p u t Rate and Total Traced Mass of Body Cholesterol in Man." Circ. Res. 25, 191-199. - - , R. M. Effros and F. P. Chinard. 1969c. "The Indicator Equivalence Theorem for I n p u t Rates and Regional Masses in Multi-inlet Steady State Systems with Partially Labeled I n p u t . " J. Theor. Biol., 25, 297-316. Roberts, G. W., K. B. Larson and E. E. Spaeth. 1973. "The Interpretation of Mean Transit Time Measurements for Multiphase Tissue Systems." J. Theor. Biol., 39, 447-475. Samuel, P., C. M. Holtzman, E. Meilman and W. Perl. 1968. "Effect of Neomycin on Exchangeable Pools of Cholesterol in the Steady State." J. Clin. Invest., 47, 18061818. Stephenson, J . L . 1948. "Theory of the Measurement of Blood Flow by the Dilution of a n Indicator." Bull. Math. Biophys., 10, 117-121. 1958. "Theory of Measurement of Blood Flow by Dye Dilution Technique." I R E Trans Med. Electron., PGME-12, 82-88. 1960. "Theory of Transport in Linear Biological Systems: I. F u n d a m e n t a l Integral Equation." Bull. Math. Biophys., 22, 1-17. Zierler, K . L . 1965. "Equations for Measuring Blood Flow by External Monitoring of Radioisotopes." Circ. Res., 16, 309-321. RECEIVED 9-3-74 I~WWSED 4-15-75
