Stochastic compartmental models with Prendville growth mechanisms

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ELSEVIER

Stochastic Compartmental Models with Prendville Growth Mechanisms P. R. PARTHASARATHY ANDB. KRISHNA KUMAR Department of Mathematics, Indian Institute of Technology, Madras--600 036, India Received 29 June 1993; revised 4 April 1994

ABSTRACT We consider a stochastic compartmental model in which cells reproduce in accordance with a regulated birth and death process. We find expressions for the mean vector and covariance matrix for the number of cells in these compartments. We obtain the asymptotic behavior of the mean vector in the general case and explicit expressions for two compartmental and mammillary systems.

1.

INTRODUCTION

Compartmental analysis is a phenomenological and macroscopic approach for modeling physico-chemical processes. In their excellent review, Jacquez and Simon [1] present a unified and simplified theory for such systems. Stochastic modeling of and statistical inference in compartmental models have also witnessed a remarkable development in the recent past [2]. The problems of growth and division of cells is of great importance in biological studies ranging from the chemistry of D N A or proteins to the complex growth of large cellular systems. The cells of the bone marrow, the intestinal crypts, and the skin epithelium provide some intricate examples. Many cell populations, e.g., embryos, organs, and tumors, show limitations in growth due to several factors. Certain tissues show a sudden burst of growth in early life; in later life the growth rate greatly decreases as in lymphoid tissue or virtually there is no cell division as in neural tissue. The density dependent logistic growth law describing such population growth in a limited environment has been the subject of much theoretical and experimental investigation [3]. In view of the complexity of such biological systems, several stochastic analogues of the logistic law have also been developed; see, e.g., [4-8].

MATHEMATICAL BIO SCIENCES 125:51-60 (1995)

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52

P.R. PARTHASARATHY AND B. KRISHNA KUMAR

An adequate stochastic model of hematopoiesis is built by associating a multidimensional birth and death process with transitions to account for cell differentiation. Renshaw [9] has given a comprehensive treatment of stochastic multipopulation birth, death, and migration models. There are cell populations which reach completion and will not grow further. Pharr et al. [10] have developed a discrete time model for cell proliferation of mast cells in which a mast cell may die or divide at the end of a cell cycle into two proliferating or nonproliferating cells. Macken and Perelson [11] observe that in CFU-S and C F U - G M cells, stem cells differentiate into progenitor cells which have less proliferative capacity leading to "completion." In view of the above discussion, it is appropriate to study a compartmental model wherein populations reproduce according to a logistic process. In this paper, we consider the most familiar and simple Prendville [12] model of a birth and death process in which the state space is confined to a preassigned strip 0 ~
M O D E L D E S C R I P T I O N AND ANALYSIS

We consider a system which has N states denoted by C i, i = 1, 2 ..... N. Let X i ( t ) denote the number of cells in state C~ at time t. Let f ( s l , s2,..., s N) = E(l-[l u s xi~°)) be the probability generating function of the initial number of particles with k i = E(Xi(O)), i = 1 , 2 ..... N. We assume that for Li ~< n ~
P{a el-cell dies in ( t, t + A t ) lX i ( t ) = n} = ( n - Li ) ~ i A t + o ( A t ) P{a ci-cell transfers to state C~ in ( t , t + A t ) J X i ( t ) = n} = nyijAt + o(At) P{a ci-cell undergoes no transition in (t, t + At)IXi(t ) = n} = 1 - {(U/- n)A i + (n - Li)tx i + nyq}At + o(At)

P{a c~-cell undergoes more than =o(At).

one transition in (t, t + A t ) l X i ( t ) = n}

PRENDVILLE GROWTH COMPARTMENTAL MODEL

53

All these events that happen to a cell in (t, t + At) are independent of the event happening to the other cells and of the event that happened to this cell in the past. With these assumptions, the vector process ( X I ( t ) , X 2 ( t ) , . . . , X N ( t ) ) is a regulated Markov process. Here the transition rates )ii, /xi, and %j are nonnegative and are assumed to be independent of t. Let P(sl, s 2 ..... SN; t) = E(FIN= 1 SiXi(t)) be the probability generating function of (Xl(t), Xz(t),..., X u ( t ) ) and m i ( t ) = E ( X i ( t ) ) . This probability generating function satisfies the Kolmogorov forward differential equation -'~-=

U/Ai+ (si-1)P i=1 N N + Y'~ • yij(sj - si) OP

i=1 j=l

+

i =1

(AiSi+ l a , i ) ( 1 - s i ) O t °~si

(2.1)

t~Si

with P(sl, s z ..... SN; O) = f ( s 1, S 2..... SN). This equation is intractable; we solve this in a special case in the next section. However, it can be used in certain special cases to obtain the moments of the process ( X l ( t ) , Xz(t),..., X u ( t ) ) . The mean vector M r ( t ) = (ml(t), mz(t) ..... m s ( t ) ) satisfies dM(t) dt = AM(t) + B,

M r ( 0 ) = ( k l , k z ..... k u ) ,

(2.2)

where

A=

-da )t12

"Y21 -d2

"'" "'"

TN 1 ] TN2

d i = A i + [dbi -4- E "Yij

J

and

B r = [Ol)i 1 + L 1 ~£1, U2)i2 +

L 2 ~LI, 2 ..... ON)t N

+ LNtzu].

The solution of this nonhomogeneous differential equation is M ( t ) = foteAUBdu + e m M ( O).

54

P . R . P A R T H A S A R A T H Y AND B. KRISHNA K U M A R

C l e a r l y (Xl(t),X2(t) ..... X N ( t ) ) is b o u n d e d by t h e d e f i n i t i o n o f t h e m o d e l . F r o m t h e P e r r o n - F r o b e n i u s t h e o r y f o r M L - m a t r i c e s [13],

lim M( t ) = -A t ---,oo

(2.3)

- lB.

W e i l l u s t r a t e this in F i g u r e 1 f o r a 3 - c o m p a r t m e n t a l s y s t e m w i t h p a r a m e t e r v a l u e s A~ = 0.5, A2 = 0.7, A3 = 0.8, /x I = 0.2, /z z = 0.5, /~3 = 0.6, L 1 = L 2 = L 3 = 100, U 1 = U 2 = U 3 = 500, k 1 = 200, k 2 = 300, k 3 = 400, T12 = 0.05, 3/21 = 0.01, ')/23 = 0.02, ')/32 = 0.03, "Y13= 0.04, T31 = 0.06. W e d e t e r m i n e M(t) explicitly in c e r t a i n s p e c i a l c a s e s in t h e n e x t section. L e t us n o w f i n d t h e c o v a r i a n c e m a t r i x . W e n e e d s o m e n o t a t i o n : F o r i = 1,2 . . . . . N , let

Cij(t ) =

E( Xi( t ) Xj( t ) ), E(Xi(t)(Xi(t ) -

j --/:i 1)),

j = 1'

C(t) =(cij(t)),

~oo~ 3so -6

Ml(t) M2(t) M3(t)

3°°I/

E "6 IE

0

5

Time FIG. 1.

I0

15

PRENDVILLE GROWTH COMPARTMENTAL MODEL

55

b i = UiAi + Z i izi, d~ ) =

li i ( bi - )ti)

j = i or k = i , j = k = i, j~iandk~i,

o(i':((d)ik')),

E = diag( L 1/Zl, L 2 ~2 ..... LN ttN) and

d i = Ai -F [zi -'}- E ~/ij" J Note that, only the ith row and ith column in D (° can have nonzero elements. Differentiating (2.1) with respect to s i and sj along s i = 1, i = 1, 2 ..... N, we get

dCij (t) d-------7---= ~-" "AiCkj + ~" 7gJCik - ( di + dj)Ci: k

k

+ bjmi(t ) + bimj(t ),

i~ j

acii( ~kiCki - 2diCii + 2 ( b i - Ai)mi( t ) - 2 A i / z i. dt t ) = 2 ~'~ k These equations can be put in the matrix form N

dC(t) = A C ( t ) + C ( t ) A ' + dt

~,mi(t)D(i)-2E. i=1

The unique solution of this matrix differential equation is

C ( t ) = e A r t c ( O ) e A t + [teAr(t-T) ( ~_,mi(z)D(i)-2E)eAtt-~)dz, ~0

\ i

where C(0)=(32f//OsiOsy) at s i 1 for every i. The covariance matrix is then given by =

C( t ) +diag(ml( t ),m2( t) ..... mN( t ) ) - M( t ) M r ( t).

(2.4)

56

P . R . PARTHASARATHY AND B. KRISHNA KUMAR

From (2.3), (2.4), and the Perron-Frobenius theory for ML matrices it follows that

c ( t ) ~c,

t--,~

for a suitable c > 0. 3.

SPECIAL CASES

3.1. TWO COMPARTMENTAL REVERSIBLE SYSTEM In this case A takes a simple form:

A =

-d,

'~21 ]

"}/12

-d2

where d l = •1 +/-L1 + Y12 and d 2 = A2 +/~2 + Y2,The eigenvalues /31 and /32 are negative and they are given by

-(d,

+

d2) __+~ ( d , -

/3'' /32 =

d2) 2 + 4')',2'Y21

2

The corresponding matrix T of eigenvectors is

T=

[, 1] /3, -~- d, 'Y2,

/32 + d2

.

'Y2,

From (2.3), after considerable simplification, we get 1

Ml(t) = /31-/32 [ ×

I [ ~1/31( /3, + d2) + r21~2 /31 + bl /31 + ~'2162 +

e Bl t

b,d2]

/31

- [1,1/32(/32 + ,t2) + ~'211'1/32 + b,/32 + ~2,t'2 + t',a2] /32 j + "Yzlb2 + b,d 2

/31/32

PRENDVILLE GROWTH COMPARTMENTALMODEL

57

We get M2(t) using symmetry, by interchanging the subscripts 1 and 2. Further, lim Ml(t ) =

t-, ~

Y21b2 + bid 2 /31 &

and lim Mz(t )

t--~oo

y12bl + b2d 1 ~1 ~2

3.2. N-COMPARTMENTAL MAMMILLARY SYSTEM This system has one central compartment and ( N - l ) peripheral compartments; cells in the central compartment can move to the peripheral compartments but not vice versa and there is no movement of cells between peripheral compartments. In other words Yo >~0, j = 2, 3,..., N and Yi~ = 0 otherwise. This system is used in hematopoiesis where the precursor population generates several populations such as erythrocytes, neutrophils, eosinophils, basophils, and thrombocytes which are well differentiated from morphological and physiological point of view and have their own evolution, from the young cell to the mature cell entering the blood stream. From (2.2) m'l(t ) = - d l m l ( t ) + b 1 m'i(t)=Yliml(t)-(Ai+

tzi)mi(t)+bi,

i>~2.

Solving these, ml(t ) = - ~ ( 1 -

e -dlt) + kl e-d''

and

m i ( t ) = Yli

((hi

-~a + kl + bi

)[1

Ai + t-ti

+ k2e-(a~+~i) t. Further, as t ~ 0% ml( t ) ~ bl dl

l

b1

e-d1 t _

e-(;t~+~0t

J

P. R. PARTHASARATHY AND B. KRISHNA KUMAR

58 and

)

i~>2.

mi(t)--+Tli -d-71+ kl + bi / ( Ai + lzi),

3.3. N-COMPARTMENTAL IRREVERSIBLE SYSTEM In this case, cells can move only to any one of the succeeding compartments, i.e., Yij >I 0, for j = i + 1,i = 2 .... , N and Yi~ = 0 otherwise. Such systems occur often in biological modeling (e.g., lipoprotein synthesis). F r o m (2.2),

[ edltml( t ) ] ' = bl edit

These equations can be solved recursively. 3.4. TWO COMPARTMENTAL IRREVERSIBLE SYSTEM T h e probability generating function of (2.1), P(s 1, $2, t) can be determined explicitly when )t 1 = Y21 = 0. For simplicity, let us assume that L 2 = L 1 and Ua = Ur In this case (2.2) takes the form

OP ( LlP, I + ( U 1 A 2 + L I I ~ 2 ) ( s 2 _ I ) } p Ot = ( S l - 1 ) S1 S2 + [/-'1( 1 - Sl) + y,2(s2 - s1) ] ~OP + ( AzSz

+/z2)(l_ s 2 )

OP

t9S2

which is a first-order partial differential equation. Using the m e t h o d of characteristics, after a fairly large a m o u n t of calculations, we get 4

P( sl,s2,t ) = l-I Ai( sl,s2;t), i=1

PRENDVILLE GROWTH COMPARTMENTAL MODEL

59

where

kl

Al(Sl,Sz;t)=(sle-(~'+~n)' + f/[ Izl +Y12g(t-u)]e-(Ul+v'2)Udu

,

Az(Sl,Sz;t) = e x p { L 1/xl

× for[ 1-

sle-~"l+~2)'+£[

Z3(Sl,S2;t ) =

~le-( ~l +"n)u

+rlzg(t--Y)]e-(~'+'12)YdY

du/

SlL, ( }[2 "~-Ida2) UI- L, x{(x s

+

- 1)e

Z4( Sl,S2;t) = {h2s 2 + t~2 - h2( s2 --1)e-(AE+~2)t} U'-k2 and

g(t) =

( '~2s2 '{- ]'~2) "+- ]-£2($2 -

1) e-('~2+''~)t

()[2S2 "q'-]'/"2) -- ~2($2 --1)

e-('~:+ ~:)t "

Remark. The above analysis can be extended to open compartmen* tal systems which account for immigration or emigration of cells. The authors are indebted to the referees for their valuable suggestions which considerably improved the presentation. One of the authors (B. Krishna Kumar) thanks the Council of Scientific Industrial Research in India for their financial support during this research. REFERENCES 1 J.A. Jacquez and C. P. Simon, Qualitative theory of compartmental systems, SIAM Rev. 35:43-79 (1993). 2 J. H. Matis, B. C. Patten, and G. C. White, eds., Compartmental Anab/sis of Ecosystem Models, Int. Coop. Publ. House, Fairland, 1979. 3 T.G. Hallam, Population dynamics in a homogeneous environment, in Mathematical Ecology, An Introduction, T. G. Hallam and S. A. Levin, eds., Springer-Verlag, Heidelberg, 1986, pp. 61-94.

60

P. R. PARTHASARATHY AND B. KRISHNA KUMAR

4 C. E. Smith and H. C. Tuckwell, Some stochastic growth processes, in Mathematical Problems in Biology, Van der Driessche, ed., Springer-Verlag, New York, 1974, pp. 211-224. 5 W.Y. Tan, Logistic stochastic growth models and applications, in Handbook of the Logistic Distribution, N. Balakrishnan, ed., Marcel Dekker, 1992, pp. 397-425. 6 K.P. Tognetti and G. K. Winley, Stochastic growth models with logistic mean population, J. Theor. Biol. 82:167-169 (1980). 7 P. R. Parthasarathy and B. Krishna Kumar, A birth and death process with logistic mean population, Commun. Statist. Theory Methods 20:621-629 (1991). 8 P. R. Parthasarathy and B. Krishna Kumar, Some logistic growth models in ecology, in Handbook of the Logistic Distribution, N. Balakrishnan, ed., Marcel Dekker, 1992, pp. 540-551. 9 E. Renshaw, A survey of stepping-stone models in population dynamics, Adv. in Appl. Probab. 18:581-627 (1986). 10 P.N. Pharr, J. Nedelman, H. P. Downs, M. Ogawa, and A. J. Gross, A Stochastic model for mast cell proliferation in culture, J. Cell. Physiol. 125:379-386 (1985). 11 C.A. Macken and A. S. Perelson, Stem cell proliferation and differentiation, in A Multitype Branching Process Model, Lecture Notes in Biomath. 76, SpringerVerlag, Berlin, 1988. 12 B.J. Prendville, Discussion: Symposium on stochastic processes, J. Roy. Statist. Soc. Bl1:273 (1949). 13 E. Seneta, Nonnegative Matrices, George Allen and Unwin, London, 1973.