elimination by cytotoxic cells. Stephen J. Merrill. Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee,. W153233, USA.
Journal of J. Math. Biology (1984) 20:305-320
Mathematical
Biology
(~) Springer-Verlag1984
Stochastic models of tumor growth and the probability of elimination by cytotoxic cells Stephen J. Merrill Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, W153233, USA
Abstract. The probability of tumor extinction due to the action of cytotoxic cell populations is investigated by several one dimensional stochastic models of the population growth and elimination processes of a tumor. The several models are made necessary by the nonlinearity of the processes and the different parameter ranges explored. The deterministic form of the model is T'= 70T- k'6T/(K[ + T) where 70, k~ and K~ are positive constants. The parameter of most import is )to = 70-k'6/K[ which determines the stability of the T = 0 equilibrium. With an initial tumor size of one, a (linear) branching process is used to estimate the extinction probability. However, in the case A = 0 when the linearization of the deterministic model gives no information ( T = 0 is actually unstable) the branching model is unsatisfactory. This makes necessary the utilization of a density-dependent branching process to approximate the population. Through scaling a diffusion limit is reached which enables one to again compute the probability of extinction. For populations away from one a sequence of density-dependent jump Markov processes are approximated by a sequence of diffusion processes. In limiting cases, the estimates of extinction correspond to that computed from the original branching process. Table 1 summarizes the results. Key words: Cytotoxic cells - - stochastic models - - elimination probability - tumor growth O. Introduction The early stages of the growth of a tumor and its interaction with the immune defense play critical roles in determining whether the tumor will escape the defense or be eliminated. It is a reasonable assumption that the probability of elimination is some function of the parameters of the host defense and the properties of the particular tumor. The question examined here involves an attempt to describe that functional dependence for a simple model based on the interaction of a tumor with natural killer (NK) cells, a population of naturally cytotoxic leukocytes. The parameters and the functions chosen use the notation developed in [ 15, 16] where a deterministic system consisting of four ordinary differential equations
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s.J. Merrill
was used to describe the surveillance of the immune defense against tumors by NK cells. Under certain natural hypotheses, solutions of this system behave like solutions of the one-dimensional equation dT - y( T) - k6( T) dt
(1)
where T(0)=k6(0)=0, 3'(T) and k6(T ) are nonnegative, d k 6 / d T > O , T'(0)>0 and T ( T ) ~< ToT for some To. Here we specify the functions 3' and k6 as k'6 T 3,(T)= ToT and k 6 ( T ) - - KI+T where 3'0, k~ and K1 are positive. The move to specific forms for the functions makes it possible to explore the possible roles of the parameters in tumor escape. Also on the technical level, it makes it possible to deal with the discrete stochastic processes and their approximations by continuous (diffusion) processes. The function k6 represents the "saturation" of the cytolytic ability of the cytotoxic population. This form of the function is based on an enzyme-kinetic analogy for cellular cytotoxicity [17] and has the advantage of containing parameters which can be experimentally measured. It has the further advantage of being always positive and bounded. The use of exponential growth for the tumor in the early stages, at least before crowding problems arise is well accepted. It is possible to use a more general function y to represent the growth but it seems to only add to the technical complexity without a corresponding benefit in the (biological) result. Related problems concern calculation of the probability of extinction for a growing tumor [10] and other stochastic (non-linear) models describing the interaction of the tumor with the immune defense [3]. The advantage of the present approach is in the functional forms used. Hopefully this yields tractible problems involving measurable parameters while maintaining a degree of biologic integrity, at least during the early growth stages. There are several deterministic models of the escape from the immune defense (e.g. [2, 5, 6, 12, 19]) but they generally ignore the central importance of the stochastic nature of the interactions when the tumor is very small. It is true that when the tumor is large, the stochastic and deterministic models provide the same information. It is not to say the stochastic nature has been ignored [5]. There are also several models of the spread of a tumor which are based on branching processes and the spacial distribution (e.g. [7, 21]) and others, which wish to dealwith optimal treatment, which compartmentalize the tumor population by a cell's position in the cell cycle as well as considering the special aspects (nicely introduced in [20]). Without exception, these models are much more detailed and complicated and generally deal with asymptotic ( t ~ oo) results. As the goals of this paper differ somewhat, the model has been simplified accordingly. I. Construction of the general model
The population of the tumor at time t, T(t), is assumed to be a birth-death process, that is, a homogeneous Markov process with state space S = {0, 1,...}
Stochastic models of tumor growth
307
and such that
Pr{AhT(t) = T(t + h ) - r(t) = liT(0 ) = To} = y( To)h +o(h), Pr{Ahr(t) = - l I T ( 0 ) = To} = k6(To)h +o(h) and Pr{AhT(t) = 01T(0) = To} = 1 - (y(To) + k6( To))h + o(h). The functions y ( T ) and k6(T) are smooth nonnegative functions of T. These functions are assumed to satisfy y(0) = k6(0) --=0,
a ldY[
G--~ T=0----"yo> 0 ,
T(T) ~ ToT and
dk6
0.
The assumption of homogeneity involves assuming a constancy in the state of the immune defense. This assumption is better for naturally cytotoxic populations than with T-lymphocytes. This assumption will be relaxed at a later time. The specific functions which will be used throughout this paper are
y( T) = yoT and
k'6T KI+T
k6(T) - - -
where Y0, k~ and K1 are positive. These functions satisfy the hypotheses in [16] which give the behavior of the four-dimensional model the " s a m e " behavior as solutions of (1). The hypotheses are: (H1)' k6(T) is a bounded function of T, d2y(T) (H2)'
dT 2
d2k6(T) dT 2 > 0
for T > 0 ,
and
(H3)' there exists a T* > 0 such that y ( T * ) - k6(T*) > 0. To be able to compare the stochastic with deterministic model we present Theorem 1. For the deterministic equation (l ) with initial condition T(O)= To > 0 and Ao = Yo- U6/ Kl: (a) I f ho < 0, the equilibrium solution T = 0 is asymptotically stable and there exists another equilibrium solution 7"> 0 which is unstable. I f To> T, the solution of (1), T(t)-~oo as t~oo. If To< T, limt_~oo T ( t ) - - 0 . (b) I f h o > O, T=-0 is unstable and any solution with To> 0 satisfies T ( t ) ~ oo
as t ~ . (c) I f A o = 0 , (1) becomes
T'
Y~ KI + T
( y~ ~ \ K I + T] T"
For (2) solutions with To> 0 satisfy T( t) >i exp( yoTo/ ( K1 + To) )t.
(2)
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s.J. Merrill
Our examination of the problem of calculating the probability of tumor extinction if To = 1 will attempt to make more realistic the model based on (1) by using stochastic models where there is some reason to believe the deterministic model is lacking. For instance, from Theorem 1, all small tumors would be eliminated if Ao < 0 and all tumors would escape if ;to> 0. However if To = 1, and Ao>0, there would be a reasonable probability (~-k'6/Kj/(yo+k'6/K~)) that it would be killed before it could divide the first time. We begin by a description of T(t) as a continuous time branching process [8]. The advisability of using a branching process to approximate the early history of the growth process has been recognized by Barbour [1], who also pointed out several limitations in using diffusion approximations to growth processes.
2. Small tumor approximation--a continuous branching process For this approximation to T(t), it is assumed that each tumor cell acts independently of the others. Each cell living a random length of time (exponentially distributed with parameter A) after which the cell divides into two cells or is destroyed by the cytotoxic defense. The assumption of an exponentially distributed time to the next event being independent of the population size and the time since the start of the process is a major assumption. Recall that this is to describe the early history of the tumor (so the defense is relatively unchanging) and that there is some evidence that the growth portion of the process may not be strongly dependent on population or time (at least until later in the process) [4]. For u small, we assume
k~u k6(u) =
K1 + u
k'~ ~--u.
K~
The branching process U(t) is specified by the infinitesimal probabilities
~lk+akh+o(h)
for k = 0 , 1 , 2 , . . . .
The k refers to the number of offspring after a cell's lifetime is over and 6~ is the Dirac delta function. Here 9
r
k6 -K1
if k = 0
Yo
if k = 2
ak= -
+Yo
0
if k = 1 if k > 2 .
The mean time to an event is 1/A where
A = yo+k'JK~. Letting u(s)=Y~k~ akSk be the probability generating function, it is wellknown that the probability of extinction (for U(O) = 1), q, is the smallest nonnega-
Stochastic models of tumor growth tive root of
u(s)= 0
309
[8]. As
k~
I
'
.
k'
)K 13/o if 3'o> ~ , q =] (1
k' if 9 3'o ~ 2.
The mean number of offspring M, produced by an individual in a population of size i satisfies
Mn
23'0 3'o+a"(i)
(23'0 ) 1+ 3'o+~-(i)- 1
rll
3"oiln \ 1 1 r
iln
~]
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S . J . Merrill
= l+l(y~ +o(1/n) n \ AK1 ] = 1+lfl(i/n)+o(1/n) n
where fl(x) = yox/AKl. The variance in the number of offspring, V, satisfies V, =(2) 2
To
(
yo+Ct'(i)
To
~2
2yo+a"(i)]
so that lim,_~oo V, = 4yo/A ( 1 - To/A)= 1 (when Ao = 0, 2To = A). Moreover lim ~ j3p;"/n = lim 8 ( To,, ~ = 0 " - ~ j=o ,-.oon\yo+a (i)] uniformly in i. Define X , ( t ) = Z,(nt)/n. Then X, converges weakly to a diffusion X(t) [22] with infinitesimal operator A given by
af(x) = Ax{fl (x)f'(x) +l(1)f,,(x)} "Y0 X 2
r X
Define Timex(c) = rain{t*/> to such that X(t*) = clX(to) = x} (the hitting time). Let v(x) -- Pr{Timex(a) > Timex(b)la < x < b}. v(x) is the probability that X(t) reaches b before a. Then v(x) satisfies
Av(x) = 0 with boundary conditions v(a)=O,
v(b)= 1 [9].
(5)
The probability of extinction, q(x), can be calculated by setting b = +oe and a = 0 where q(x) = 1 - v(x). Then Aq(x) = 0 and 9
q(x)
= 1 --
e -(1/2K0~2 dE.
(6)
q = q(1), the probability of extinction with a unit initial population size is q = 1-
e -(1/2K0e2 dE
,/2
l - ~lcrK~ \
6KI]"
(7)
As K~ get large, the process is approximated by the linear branching process and q o 1. Although q does not depend on 3'0, the time to extinction will be dependent on To.
Stochastic models of tumor growth
311
For if wb(x) is the E[min{Timex(a), Timex(b)}], then A(wb(k)) = - 1 [9]. Set W(X)=wb(x). N o w as A(w(x))=((yo/KOx2)w'(x)+(yoX)W"(x)=-I where w(a)=0 and w ( b ) = 0 , one has w"(x)+(x/K~)w'=-l/yoX, so that e~2/2K,w'(x) = w'(a) e a2/2K'- 1/yo J~ 1/s e sV2K' ds. Integrating and utilizing the boundary conditions one finds
w(x)=w'(a) f f e-(~-~)/2n, d s - l f f e-S~/2K,(f~ le~2/2~ de) ds' '
W(X) =
l{o
e--S2/2Kl
Yo
a 7 eZ2/2KI
dz ds
ix
a e--S2/2Kl
ds
/;;
e -~/2K~ ds
- f f e-~'/2~,fflez~/Zr, dzds} 1
=-- F(gl, a, b).
(8)
Y0
If the interval [a, b] is viewed as those population sizes when the outcome is still in doubt, then as w(x) gives an estimate of the time one expects to stay within the interval, slowly growing tumors will spend more time within the interval without having any change in the extinction probability (holding)to = 0 and A constant). In the previous construction some care was taken to make the assumption that the mean lifetime is a constant for all n reasonable. One wonders if the result would be changed if this hypothesis is abandoned. Let Xn(t) be a birthdeath process with
f yo/h + o( h )
ifa=l
_~a (i). ih +o(h)
ifa=-I
P(Xn(t+h)-Xn(t)=alX"(t)=i)=ll-(yoi+a
(i). ih)+o(h)
[ o (h) Then if
ifa=O otherwise.
zn(t) = Xn(nt)/n,
E( Ah.Zn( t) = Z~( t + h") - Z~( t)lZ"( t) = i/ n) = l [ ( y o i - a ~(i)i)nh ~ + o(nhn)] n
= h"[yo- an(i)]i +o(h ~) = , ~ [ l f yo i'~.
LKl\n] Thus
+o
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S. J. Merrill
Also,
So that
2!m0~nl~( (AhnZn(t))2lZn(t)~i) ~1~(~)"~0(~). Similarly
E[(Ah"Z~(t)4lZ~(t)=i/n]=-~[h~{A
t , / n ) ~-Kik'6i/n2+i/n2]] ..... +o(h")].
We expect Z " ( t ) to converge in distribution to a diffusion process with infinitesimal operator A defined by
As Ao = 0 implies that h = 27o, this is the same diffusion limit as in (4). 3. Tumor approximation away from 0 - - a density-dependent jump Markov process As many immune processes involve a period of recognition and mobilization before the tumor is recognized and since several experimental systems involve the transplantation of "small" tumors we now examine the growth process when the initial population is no longer small (near 1) but not so large that the bounded nature of k 6 is being felt. This corresponds to critical tumor sizes for rejection of transplanted tumors (104-105 typically in mice) as well as when Ks and the population are the same order of magnitude. As we will eventually scale the state space by dividing by n, let
k'6i k~(i) = KI +i/---~" Then
k~( i) = nk6( i/ n ). Also, with 7n(i) = 7o" (i),
y~(i)=ny(i/n). The birth-death process Tn(t) defined with infinitesimal probability of birth 7"(i) and death k~(i) at population i are density-dependent jump Markov processes [11]. Concerning these processes, let
F(x) = 7oX
k'6x K l +x
Stochastic models of tumor growth and let
y(t)
313
be the solution of the deterministic equation
y'(t) = F(y(t))
with y(0) = Xo.
(10)
Let X,(t) be the process defined by scaling the T" process accomplished by dividing by n. That is
t k'6X"(s) ds)} X,(t)= Xo+l{ y~(n lo YoX,(s) ds)- Y-l(n fo Kl +X,(s) '
(11)
where the Y~(t) are independent Poisson processes with intensity 1 [11]. Then as n~m, X,(t)~y(t) in the following sense (Theorem 8.1 of [11]), lira sup n~cX ~
s~_t
[X,(s)-y(s)l=O
a.s. forall t > 0 .
Moreover (Theorem 8.2 of [11]),
x/n(X,(t) -y(t))
converges weakly to
V(t)
where V(t) is a Gaussian process. Now Xn(t) ~y(t) +(1/~/n) V(t) =.~,(t) for large n, and for large n, the infinitesimal drift of Z,(t) is approximately y(t) (for DF(x) bounded) while the infinitesimal variance is
K, +y(t)J" Since X.(t) ~ y(t). it is reasonable to approximate Zn(t) by Z.(t) where Zn(t) is the diffusion process which is the solution of the stochastic differential equation
~-t6Zn~ dt+ dZn(t)= ( ToZn Kl+Zn]
~t6Zn'~dB(t) 3'Ozn+ K,+Z,]
(12)n
with Z,(0)--x0. This equation is to be interpreted in Ito sense and riB(t) is the increment due to Brownian motion. (12)~ was studied with general functions y and k 6 satisfying (HI)', (H2)' and (H3)' in [18]. The diffusion processes Z,(t) have infinitesimal operators A" defined by '
. . . . .
voX * r , +xJ ] tx).
(13)
The probability of extinction q,,(x) for Z, satisfies A"q,,(x)= 0 with boundary conditions lim q,,(a)= 1 and lira q,,(b)=O. a~O
b~oo
As
K~ + x] q~(x) -- O, ffe-2ns(1 4
q,(x) =
1
Yo ~ 4nk;/'y~ 7oK, + ki s } ds f?e_2,,(1.+ yoK17o+ k{ s ) 4"gU'~ ds
(14)
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S.J. Merrill
This expression (14) has no restrictions on the relative sizes of the parameters and is well defined for all parameter values [18].
9
ho>O
If we restrict our attention to the case when ho = 3/o-k'6/K~ > O, then the change of variable z = ns gives
Io
e -2z
1~
(
4"k;/T~
To
yoK1 + k'6
dz
ToKl + k~ Applying the dominated convergence theorem to the denominator and taking the limit as n-~ co gives
IOn( ~ina
~/O~)4nk~/3,OIoX~ dz
e -2z 1-~ y o K l + k'6
=
r,,
e-2Zex / l
\4nk6/To"9]
PL~,~og~3'0+ g J~
zj dz
h
=2Ao
(16)
where ho = y o - k r 6 / K 1 and h = y o + k ' 6 / K l . Using (16), we find for n large,
Io
q . ( x ) -~ 1 -
e -2z(x~
dz
/
A .
2Ao
Thus
z h
2Ao\
[o
-- 1]
= 1 + [ e -2x~
e--2Aonx/A. Of interest here is when the population is of size 1 so that Z.(O) = 1/n, then qn ( 1/ n ) ~ e -2~~ This should compare in some way to the linear branching model result for ho>O, q=
k~ Kjyo"
Note that as ho = T o - k'6/Kl, I
Ao k~ To KI yo"
Stochastic models of tumor growth
315
Also h = 2 y o - A o and ho/A < 1. Thus
(
ho/2yo e x p \ - 2 1 - ho/2yo]"
e--2ho/A
Keeping only linear part we find e -2~o/~ ~ 1 - h o To The prediction from the branching model. For ho/h small, the two will give very similar results, with the nonlinear effects entering as the ratio increases towards 1. ho
