A stochastic compartmental model with continuous infusion

BULLETIN

OF

MATHEMATICAL

BIOLOGY

VOLlYME 38, 1976

A STOCHASTIC COMPARTMENTAL MODEL W I T H CONTINUOUS I N F U S I O N

[] ASHA SETH KAPA])~

School of Public Health, University of Texas, Health Science Center at Houston, Houston, Texas, U.S.A. BAYLISS C. M c I ~ I S College of Engineering, University of Houston, Houston, Texas, U.S.A.

This paper deals with stochastic m-compartmental systems with continuous timedependent infusions into all compartments and reversible time-independent flows between any two compartments. A methodology for the first two moments of the distribution of the number of units in the different compartments at any point in time is outlined without resorting to the usual techniques of generating functions and inverse Laplace transforms. A possible application to a systems analysis of the kidney transplant system is discussed.

Introduction and the Model. Stochastic compartmental models with continuous infusion have been the subject of several recent papers. Purdue (1974, 1975) has studied one and two compartment systems. Cardenas and Matis (1974) have studied n-compartment s y s t e m s with irreversible, time-dependent transition probabilities. Using cummulant generating functions, Cardenas and Matis have obtained the first several moments of the number of units in a particular compartment at time t for some particular cases. Matis and Carter (1970) studied the moment-generating function describing a general two-compartment system in steady state and obtained the first and second stochastic moments. None of the above authors have dealt with the general case discussed in this paper. 695

696

ASI~IA S E T H K A P A D I A AND BAYLISS C. MCINNIS

In this paper we have dealt with an m-compartmental system with continuous time-dependent infusions into all compartments and reversible timeindependent flows between any two compartments. The input processes are assumed Poisson and the transitions between compartments are assumed Markovian in nature. The first two moments of the distribution of the number of units in the different compartments at time t are obtained directly from the differential-difference equation representing the state of a particular compartment. Reference to a parameter-estimation algorithm is given for the case where time series data are available from r out of the m-compartments only. A possible application to the kidney transplant system is described in the last section of the paper. Consider a m-comp~rtment system with m 2 + m parameters described in Figure 1. The bij represent the time-independent flux rates between the compartments, b0~ represents an excretion from compartment i and the A's represent the continuous infusion into the various comp~rtments. The arrival

I X~(t)

Xi (t)

I

i

I

x(t)

I I

lib,.

bo,

I

bji

I I I

b°:

II I I

~'i

, b°~

I

Figure

1.

m-Compartmental

model with infusion

time-dependent

continuous

of units into the different compartments follows the Poisson law. Let ni(t) be a random variable representing the number of units in compartment i(i = 1, .m) at time t. Assume that each of the n~(t) units acts independently, and t h a t the underlying pattern of the movement of units from one compartment into the other is Markovian. In other words, future transitions of a unit are independent of past transitions. Furthermore, given ni(t) -~ R, the probability of exactly one item transfering from compartment i to compartment j in an interval (t, t + At) is t~bjiAt+ 0(At). The probability of exactly one exit from compartment i to the exterior of the system is RboiAt + O(At). The probability that exactly one unit enters compartment i in time (t, t + At) is ~iAt+O(At). •

.

A STOCHASTIC

COMPARTMENTAL

MODEL

WITH

CONTINUOUS

INFUSION

697

Clearly

1 - Z (bj,)At j=0, ]¢i

denotes the probability that an item will remain in compartment i during (t, t + At). Let P~(t) be the probability that there are r units in compartment i at time t. Then, direct enumeration of possible events in (t, t + At) yields the following difference equation (v~riting 2,(t) = 2~).

j#~

]¢i m

+ P~_l(t)2,At + P~-l(t) Z b~jAt,j(t) i=1 ~n

+P~+l(t)(r+l)

~ bjiAt,

r > O, i = 1 . . .

m.

(1)

j=o j¢i

The third term in the above expression is the sum of the following mutually exclusive events (j = 1. . . . m, j # i). There are r - 1 units in compartment i at time t and a unit arrives from compartment j in time At with probability bi~AtIP~(t ) (if there are I units in compartment j). The joint probability of this event is P~r_l(t)b,~AtIP~(t), summed over all values of 1 from 0 to oo gives, P~r-l(t)b~jAt pj(t). Clearly pj(t) represents the mean number of units in compartment j at time t. Equation (1) can next be transformed into the following differential-difference equation.

dP~(t)/dt = -

[ _o°

]

2~ +r ~ bji + ~=1~ bijpj(t) P~(t) j#i

]#i

j¢i 7T~

+P~+l(t)(~+l) 2 b~.~,

~ >= o.

(2)

o'=0 3"¢i

The above equation differs from that of Chiang (1968, p. 173) in that his equation represents the joint probability of there being X1 units in the first compartment, X2 in the s e c o n d . . . , Xs in the sth compartment, given the state of the system at time z. Our equation (2) represents the marginal probability of there being r units in the ith compartment at time t. It is independent of the state of the rest of the system.

698

A S H A S E T H K A P A . D I A A N D B A Y L I S S C. M C I N N I S

Multiplying b o t h sides of (2) b y r and summing over all r from 0 to ~ , we have after a little simplification

r(dP~(t)/dt) = p~(t) = 2~-#~(t) r=O

b:i ]

+ ~ bigpg(t),

i = 1 , . . . m.

:=1

In m a t r i x n o t a t i o n the above system can be represented as ti(t) --- A$(t) + ~(t), where A is the coefficient matrix. The solution to this system is well k n o w n to be it(t) =

eAttt(O)+eAtftoe--A~),(~)d~.

(3)

(See Ogata, 1967, p. 311) where g(0) is the column v e c t o r of initial values and eat = L-I[(sI-A)-I], L -1 represents the Inverse Laplace T r a n s f o r m and s is the p a r a m e t e r of the Laplace Transform. To obtain the variance of the n u m b e r o f units in c o m p a r t m e n t i at time t, we multiply b o t h sides of (2) b y r 2 and sum over all r from 0 to ~ . Denoting ~ r2Pir(t) b y E~(t), after a little simplification we notice t h a t E~(t) satisfies the following first-order differential e q u a t i o n

d/dtE~(t) + 2L~Eqt) = K~(t), where m

j=0 j#i

and m = re(t)

The solution to which is

Ei(t) = Ce-2L't + e-2L,t f e2L,tKi(t)dt,

(4)

where C is a constant to be d e t e r m i n e d from the b o u n d a r y conditions. Now suppose t h a t d a t a is k n o w n from the first r c o m p a r t m e n t s only and t h a t at time t = 0, the n u m b e r of units in c o m p a r t m e n t i follows the n o r m a l distribution with mean N~ and variance a~. W e assume t h a t the c o m p a r t m e n t s

A STOCHASTIC COMPARTMENTAL MODEL WITH CONTINUOUS INFUSION

699

r + 1 . . . . m from which data is not available have zero units in them at t = 0.

Then, E

(0) =

E i ( 0 ) = 0,

1 ....

r,

(5)

i=r+l...m.

(6)

The variance Vi(t) of the number of units in compartment i at time t can be easily obtained using the relationship v

(t) =

i =

1

...

m.

(7)

For the 2-compartment case the above mean and variance are special cases of results obtained b y Purdue (1975). The values of the parameters of such a stochastic model are usually assumed unknown and can be identified b y the generalized least-squares estimation algorithm outlined b y Marls and Hartley (1971). The variance covariance matrix Z in the case where data are available for r compartments only would be the same as that for multivariate stochastic processes (see Goldberger, p. 153).

Possible Application.

Renal transplantation has become a widely available mode of therapy for end-stage kidney disease. Although adequate storage facilities for kidneys of different immuno types are available and sufficient medical knowledge about the various immunosuppressive drugs exists, resource allocation is a major problem in the kidney transplant system. A simplified model of such a system would be a two-compartment model where compartment 1 represents the number of patients on dialysis and compartment 2 the total number of living transplant recipients. B y utilizing time series data on the number of patients in compartments 1 and 2 and assuming the infusion rate into compartment 1 to be known, all the parameters of such a system can be estimated, This information will assist planners for end-stage kidney disease program in allocation of resources for improvement of dialysis units, research on better tissue typing and matching techniques and further development of newer immunosuppressive drugs.

LITERATURE Chiang, C. L. 1968. Introduction to Stochastic Processes in Biostatistics. New York: Wiley. Cardenas, M. and J. It. Matis. 1974. "On the Stochastic Theory of Compartments: Solution for n-Compartment Systems with Irreversible, Time-Dependent Transition Probabilities." Bull. Math. Biol., 361 489-504. Goldberger, A. S. 1964. Econometric Theory. New York: Wiley. Matis, J. H. and H. O. Hartley. 1971. "Stochastic Compartmental Analysis: Model and Least-Squares Estimation for Time Series D a t a . " Biometrics, 27, 77-102.

700

ASHA SETH KAPADIA AND BAYLISS C. MCINNIS

Matis, J. It. and l~I. W. Carter. 1972. "Multicompartmental Analysis in Steady State." Acta. Biotheoretica, 21, 6. 0gata, K. 1967. State Space Analysis of Control Systems. New York: Prentice Hall. Purdue, P. 1974. "Stochastic Theory of Compartments: One- and Two-Compartment Systems." Bull. Math. Biol., 36, 577-587. Purdue, P. 1975. "Stochastic Theory of Compartments." Bull. Math. Biol., 37, 269-275. RECEIVED 7-14-75 REVISED 3-16-76