Cancer Letters, 3 (1977) 203--208
203
© Elsevier/North-Holland Scientific Publishers, Ltd.
MICROMETASTASES FORMATION: A PROBABILISTIC MODEL
LANCE A. LIOTTA*, CHARLES DELISI**, GERALD SAIDEL~" and JEROME KLEINERMANtt *Laboratory o f Pathology and **Laboratory o f Theoretical Biology, DCBD, NCI, National Institutes of Health, Bethesda, Maryland 20014; fDepartment of Biomedical Engineering, Case Western Reserve, Cleveland, Ohio 44106; and f f Department of Pathology, St. Lukes Hospital, Cleveland, Ohio 44104 (U.S.A.)
(Received 18 March 1977) (Revised version received 2 May 1977) (Accepted 20 May 1977)
SUMMARY A mathematical model of the process of metastases is formulated in which the hematogenous metastatic process from a solid t u m o r is considered to consist of a series of stages. A mathematical expression is obtained for the probability that no metastases will have been established by a characteristic time interval after t u m o r initiation. The murine T241 fibrosarcoma that rapidly and reproduceably produces pulmonary metastases was studied. Estimates of parameters required for the expression of probability of metastases formation were derived experimentally. The probability remains close to one for a characteristic time at which point it drops to zero. This indicates that at least in this experimental system there is a predictable critical time period beyond which micrometastases are virtually certain to have been formed.
INTRODUCTION The importance of stochastic (i.e. probabilistic) phenomena in biological and physical systems [1], especially those in which the events of interest are rare, is well known. In the area of cancer biology, neoplasm initiation [2] as well as the seeding of distant organs by t u m o r cells released from the primary t u m o r are inherently stochastic processes. The a t t e m p t to model and explore these processes therefore requires coming to grips with their probabilistic nature. Hematogenous metastasis formation is, of course, a very complex phenomenon involving a variety o f poorly understood processes including the interaction of t u m o r cells with vascular endothelium, vascular basement membrane [7], hematogenous components such as platelets, immune cells and numerous other factors [14]. Consequently, the task of developing a viable dynamic
204
model for metastatic spread might seem exceedingly difficult. P~ecent w o r k suggests, however, that the process can be compartmentalized in a conceptually simple manner thus opening the possibility of a quantitative approach to the analysis and interpretation of data in this area. The purpose of this communication is to develop one o f the more surprising mathematical consequences of the model and present its experimental verification in a particular experimental system. Specifically, we show h o w the theory can be used to predict the probability of no micrometastases as a function o f time after t u m o r initiation. We s h o w experimentally and theoretically that this quantity remains close to one for some characteristic time, at which point it drops precipitously to zero. The parameters which control this switch are identifiable from the t h e o r y and were determined by an independent series of experiments. MODEL FORMULATION
The model u p o n which the predictions are based considers the hematogenous metastatic process from a solid t u m o r to consist of a series of £tages: (a) Vascularization of the neoplasm provides a circulatory entrance route for t u m o r cells [3,6]. (b) T u m o r cells in the t u m o r mass penetrate the walls of t u m o r vessels and are dislodged in clumps of various sizes into the t u m o r venous drainage [3]. (c) T u m o r cells are carried by the circulation to the target organ where they arrest in the small b l o o d vessels [ 3 ] . :, (d) A small fraction o f the arrested t u m o r cells survive to traverse the vascular wall and initiate metastases [4]. The mathematical model summarized here describes the dynamics of the t u m o r cell arrival in a target organ with the subsequent initiation and growth o f metastatic foci. Since we are interested in ~he formation of micrometastases when their n u m b e r is small, the process to be considered is inherently probabilistic; and
TumorCells Arrestedin the Vessels of the TargetOrgan
Metastatic FOCI
X(t) = ~ , 3 m A t
"
~
~
Death or
Dislodgement
Fig. 1. The f o r m a t i o n o f metastatic l o c i can be considered as a stochastic p o p u l a t i o n balance. T u m o r cells disseminate f r o m the p r i m a r y t u m o r and arrest in the target organ at a rate k(t). The c h a n c e of one death in t h e p o p u l a t i o n o f arrested cells re(t) in time A t is ~2mA t. The chance t h a t one metastatic focus will be initiated f r o m the p o p u l a t i o n rn(t) in t i m e A t is ~3mA t. The p o p u l a t i o n o f m e t a s t a t i c foci is d e n o t e d by M(t). See ref. 1 for a c o m p r e h e n s i v e p r e s e n t a t i o n of stochastic (probabilistic) processes and their applicat i o n to natural sciences.
205
the q u a n t i t y of particular interest is Po(t), the probability t h a t no metastases will have been initiated by time t after t u m o r initiation. In order to obtain an expression for this probability, we suppose, as discussed previously [8], t h a t t u m o r cell dissemination and metastasis formation can be represented b~ the c o m p a r t m e n t a l model shown in Fig. 1. The three parmeters entering this model, and u p o n which P0 (t) will depend, are: (a) the ra~e h(t) at which t u m o r cells arrest in the target organ; (b) the probability ~2rnA t t h a t an arrested t u m o r cell will die (or dislodge) in a short time interval, where rn is the n u m b e r of arrested t u m o r cells; (c) the probability/33mA t t h a t an arrested t u m o r cell will initiate a metastatic focus in a short time interval. The death rate for m t u m o r cells, which follows first-order kinetics, includes destruction by mechanical factors, host defenses or therapy. First order kinetics in this model means t h a t the probability t h a t a death will occur or a metastatic focus will be formed from the population of arrested t u m o r cells is proportional to the population size. Justification for this form can be f o u n d in the studies of Phillips et al. [12] on cellular immune cell killing of t u m o r cells, and of Schabel [13] with regard to chemotherapy. The rate of initiation of metastases by rn t u m o r cells, having the form ~3m, can be justified from numerous studies of metastasis formation following intraveonous injection of graded doses of t u m o r cells [4,5,14]. The probability t h a t no metastases will have f o r m e d by time t can now be obtained as follows. Let M(t) be the n u m b e r of metastatic loci at time t. Let N(t) = rn(t) + M(t), and let y(t) be the probability that a r a n d o m l y selected cell has been incorporated into a metastatic focus (entered compartm e n t M(t)); i.e.
y = M/N
(1)
where the argument t has been omitted for notational simplicity. Assuming t h a t the chance of forming a metastatic focus is independent of previously f o r m e d loci, the probability of exactly x loci is Px(t) = (x N) yx(1 - y)N -- x and the probability of no loci is P0 (t) = (1- y ) N = (1 - M/N) N. Metastasis formation is a relatively rare event among all the t u m o r cells arrested in the target organ [4,10,14]. This means m is large and y is small. Then P0 (t) ---- exp(- M(t)).
(2)
Equation (2) is the result one expects of a Poisson process. That the initial stages of metastasis f o r m a t i o n can, in fact, be modeled by such a process has been discussed previously and is in accord with a variety of evidence [10]. The mean n u m b e r of metastases M(t) can for linear Markov processes be f o u n d by solving the deterministic differential equation corresponding to the model in Fig. 1. The result is ~3 M(t) = - - f 0 t k(r) (1 - exp[ - (~2+/~3) ( t - z)] )d r. (3)
206 Equation (3) together with Eqn. (2) furnishes the probability that no micrometastases will have been formed b y time t. MATERIALS AND METHODS The test Eqn. (2) we applied it to a particular experimental system with a transplantable, poorly immunogenic murine fibrosarcoma that metastasizes to the lungs [8]. In this system t u m o r cells are released from the primary t u m o r at an exponentially increasing rate. Estimates of the parameters ~2 and ~3 can be obtained experimentally b y following the kinetics with which intravenously injected t u m o r cells colonize the lung, as described in detail elsewhere [ 1 0 ] . The function X(t) has been experimentally determined by measuring t h e rate at which t u m o r cells enter the primary t u m o r venous drainage [8]. T u m o r cells entering the venous drainage were tagged with Cr sl and injected intravenously into tumor-bearing mice. Virtually all the cells arrested in the lung within 5 min. Thus it can be assumed in this t u m o r host system that the rate at which t u m o r cells enter the t u m o r venous drainage is equivalent to their arrest rate in the lungs. Based on experimental data [8] the best functional form is X(t) = cek(t--
o ), t > 0
=0
,t
8 : P0 = e x p ( -
(~3/~:) c [ 1 / k ( e k ( t - o) -- 1) + 1 / ( k + ~ )
×(ek(t-O)-e-~
(t - o))]).
(5a)
207
When t > > O, the s o l u t i o n is a p p r o x i m a t e l y
(5b)
P0 ~ exp( -- (/~3/~) c e k t [ 1 / k + 1/(k + f12)] )EFFECT OF INCREASING DEATH RATE (,G2) .
1.0
__.=~2=i~4
/'62=33"6 .5,2=134.4
a
c..f;
~< .90 c,O
~< .80 LJJ
\
\ k
\
%k
~) .70
¢2=4".2 \
cc
-~ .60
#3 =.0005
o .50
', \
K = 0.48
Z
, \
"~'%
,, ",•, "\
"~. ~ \\\
\
C = 1440
~.4o
% \
%\\
\\
\\
"\\ \\
\\,, ,,,
>-
.30 =n
< .20 (23 or,,- .10 o_ 0
\"x ~
I
1,
I
l
2
4
6
8
\\\
%Nx-,
~
"-2_
10
\\\
\% m~ '
12
\x ~J
b ....
14
16
l
•
18
20
18
20
DAYS POSTIMPLANT
EFFECT OF DECREASING RATE OF CELL ARREST (K) 1,0 ~ CO
b
.90
F.80 CO o
Z
,,%
.60
\
'~ \ " "\ ,.. x . ,
,~ .30 L
K='24//~
c~
0
X,',,
~' x ",
.50 .40
i 2
r 4
6
", "'.
/K=.015
". "''.
l 8 10 12 DAYS POSTIMPLANT
14
16
Fig. 2. Comparison of model prediction with experimental data for the probability of no micrometastases existing at the time of t u m o r excision at a sequence o f times following transplantation. The data (m) shown above is the proportion of 10 mice with no metastases. The solid curve in (a) and (b) is the solution of Eqn. (5) for the parameter values listed. The parameter values were determined experimentally. The dashed curves are solutions of Eqn. (5) when a specified parameter is varied.
208
The value P0 is the probability no micrometastases have been formed b y time t. That is, this is the probability o f surgical cure b y primary t u m o r excision. In F i g 2 the value of P0 is compared with experimental data for which ~2 = 4.2 d a y s - 1, ~3/~2 = 1.19 • 10 -4 , k = 0.48 days -1 , c = 1440 cells/day and 0 = 4.0 (days). Also illustrated are some aspects of the expected sensitivity of the results to system dependent parameters. It is apparent that the probability of no micrometastases remains close to one for some characteristic time, after which it drops precipitously to zero, a pattern which has also been observed in other experimental tumors such as the Lewis lung carcino m a [6]. The theory described here has been applied to one thoroughly studied murine system. Further investigations into the extent o f application to other t u m o r types are currently under investigation. An understanding of the factors which determine the onset and duration of the critical period when micrometastases are first formed m a y provide insights into the timing of therapy [13] as well as into the examination frequency required for a successful cancer screening protocol. Some o f these implications will be developed in detail elsewhere. The main point here is that the quantitative agreement between theory and experiment presented in this report opens the possibility for a relatively simple, though systematic, rational and quantitative approach to an exceedingly complex and important field. REFERENCES 1 Bailey, T.J. (1964) The Elements of Stochastic Processes with Applications to the Natural Sciences. John Wiley, New York. 2 Bell, G.I. (1976) Model of carcinogenesis as an escape from mitotic inhibitors. Science, 192,569--572. 3 Coman, D.R. (1953) Mechanisms responsible for the origin and distribution of bloodborne tumor metastases: a review. Cancer Res., 13, 397--404. 4 Fidler, I.J. (1970) Metastases: quantitative analysis of distribution and fate of tumor emboli labeled with IUDR-125. J. Natl. Cancer Inst., 45, 775--782. 5 Fisher, B. and Fisher, E.R. (1967) Metastases of cancer cells. In: Methods in Cancer Research, Vol. 1, pp. 243--286. Editor: H. Busch. Academic Press, New York-London. 6 James, S.E. and Salsbury, A.J. (1974) Effect of ICRF 159 on tumor blood vessels and its relationship to the antimetastatic effect in the Lewis lung carcinoma. Cancer Res., 34,839--842. 7 Liotta, L., Kleinerman, J., Catanzaro, P. and Rynbrandt, D. (1977) Degradation of basement membrane by murine tumor cells. J. Natl. Cancer Inst., in press. 8 Liotta, L.A., Kleinerman, J. and Saidel, G.M. (1974) Quantitative relationships of intravascular tumor cells, tumor vessels, and pulmonary metastases following tumor implantation. Cancer Res., 34, 997--1004. 9 Liotta, L.A., Kleinerman, J. and Saidel, G. (1976) Significance of hematogenous tumor cell clumps in the metastatic process. Cancer Res., 36,889--894. 10 Liotta, L.A., Saidel, G.M. and Kleinerman, J. ( 1 9 7 6 ) A stochastic model of metastases formation. Biometrics, 32, 535--550. 11 Parzen, E. (1962) Stochastic Processes. Holden Day, San Francisco. 12 Phillips, R.A., Clark, D.A., Schilling, R.M. and Miller, R.G. (1974) Quantitation and characterization of effector cells and their progenitors. In : Cell Biology and Tumor Immunology, Proceedings XI International Cancer Congress, Vol. 1, pp. 264--274. Editors: Bucalossi, Veronesi and Cascinelli. American Elsevier. 13 Schabel, F.M., Jr. (1975) Concepts for systemic treatment of micrometastasis. Cancer, 35, 15--24. 14 Weiss, L. (1976) Fundamental Aspects of Metastasis. North-Holland, Amsterdam.
203
© Elsevier/North-Holland Scientific Publishers, Ltd.
MICROMETASTASES FORMATION: A PROBABILISTIC MODEL
LANCE A. LIOTTA*, CHARLES DELISI**, GERALD SAIDEL~" and JEROME KLEINERMANtt *Laboratory o f Pathology and **Laboratory o f Theoretical Biology, DCBD, NCI, National Institutes of Health, Bethesda, Maryland 20014; fDepartment of Biomedical Engineering, Case Western Reserve, Cleveland, Ohio 44106; and f f Department of Pathology, St. Lukes Hospital, Cleveland, Ohio 44104 (U.S.A.)
(Received 18 March 1977) (Revised version received 2 May 1977) (Accepted 20 May 1977)
SUMMARY A mathematical model of the process of metastases is formulated in which the hematogenous metastatic process from a solid t u m o r is considered to consist of a series of stages. A mathematical expression is obtained for the probability that no metastases will have been established by a characteristic time interval after t u m o r initiation. The murine T241 fibrosarcoma that rapidly and reproduceably produces pulmonary metastases was studied. Estimates of parameters required for the expression of probability of metastases formation were derived experimentally. The probability remains close to one for a characteristic time at which point it drops to zero. This indicates that at least in this experimental system there is a predictable critical time period beyond which micrometastases are virtually certain to have been formed.
INTRODUCTION The importance of stochastic (i.e. probabilistic) phenomena in biological and physical systems [1], especially those in which the events of interest are rare, is well known. In the area of cancer biology, neoplasm initiation [2] as well as the seeding of distant organs by t u m o r cells released from the primary t u m o r are inherently stochastic processes. The a t t e m p t to model and explore these processes therefore requires coming to grips with their probabilistic nature. Hematogenous metastasis formation is, of course, a very complex phenomenon involving a variety o f poorly understood processes including the interaction of t u m o r cells with vascular endothelium, vascular basement membrane [7], hematogenous components such as platelets, immune cells and numerous other factors [14]. Consequently, the task of developing a viable dynamic
204
model for metastatic spread might seem exceedingly difficult. P~ecent w o r k suggests, however, that the process can be compartmentalized in a conceptually simple manner thus opening the possibility of a quantitative approach to the analysis and interpretation of data in this area. The purpose of this communication is to develop one o f the more surprising mathematical consequences of the model and present its experimental verification in a particular experimental system. Specifically, we show h o w the theory can be used to predict the probability of no micrometastases as a function o f time after t u m o r initiation. We s h o w experimentally and theoretically that this quantity remains close to one for some characteristic time, at which point it drops precipitously to zero. The parameters which control this switch are identifiable from the t h e o r y and were determined by an independent series of experiments. MODEL FORMULATION
The model u p o n which the predictions are based considers the hematogenous metastatic process from a solid t u m o r to consist of a series of £tages: (a) Vascularization of the neoplasm provides a circulatory entrance route for t u m o r cells [3,6]. (b) T u m o r cells in the t u m o r mass penetrate the walls of t u m o r vessels and are dislodged in clumps of various sizes into the t u m o r venous drainage [3]. (c) T u m o r cells are carried by the circulation to the target organ where they arrest in the small b l o o d vessels [ 3 ] . :, (d) A small fraction o f the arrested t u m o r cells survive to traverse the vascular wall and initiate metastases [4]. The mathematical model summarized here describes the dynamics of the t u m o r cell arrival in a target organ with the subsequent initiation and growth o f metastatic foci. Since we are interested in ~he formation of micrometastases when their n u m b e r is small, the process to be considered is inherently probabilistic; and
TumorCells Arrestedin the Vessels of the TargetOrgan
Metastatic FOCI
X(t) = ~ , 3 m A t
"
~
~
Death or
Dislodgement
Fig. 1. The f o r m a t i o n o f metastatic l o c i can be considered as a stochastic p o p u l a t i o n balance. T u m o r cells disseminate f r o m the p r i m a r y t u m o r and arrest in the target organ at a rate k(t). The c h a n c e of one death in t h e p o p u l a t i o n o f arrested cells re(t) in time A t is ~2mA t. The chance t h a t one metastatic focus will be initiated f r o m the p o p u l a t i o n rn(t) in t i m e A t is ~3mA t. The p o p u l a t i o n o f m e t a s t a t i c foci is d e n o t e d by M(t). See ref. 1 for a c o m p r e h e n s i v e p r e s e n t a t i o n of stochastic (probabilistic) processes and their applicat i o n to natural sciences.
205
the q u a n t i t y of particular interest is Po(t), the probability t h a t no metastases will have been initiated by time t after t u m o r initiation. In order to obtain an expression for this probability, we suppose, as discussed previously [8], t h a t t u m o r cell dissemination and metastasis formation can be represented b~ the c o m p a r t m e n t a l model shown in Fig. 1. The three parmeters entering this model, and u p o n which P0 (t) will depend, are: (a) the ra~e h(t) at which t u m o r cells arrest in the target organ; (b) the probability ~2rnA t t h a t an arrested t u m o r cell will die (or dislodge) in a short time interval, where rn is the n u m b e r of arrested t u m o r cells; (c) the probability/33mA t t h a t an arrested t u m o r cell will initiate a metastatic focus in a short time interval. The death rate for m t u m o r cells, which follows first-order kinetics, includes destruction by mechanical factors, host defenses or therapy. First order kinetics in this model means t h a t the probability t h a t a death will occur or a metastatic focus will be formed from the population of arrested t u m o r cells is proportional to the population size. Justification for this form can be f o u n d in the studies of Phillips et al. [12] on cellular immune cell killing of t u m o r cells, and of Schabel [13] with regard to chemotherapy. The rate of initiation of metastases by rn t u m o r cells, having the form ~3m, can be justified from numerous studies of metastasis formation following intraveonous injection of graded doses of t u m o r cells [4,5,14]. The probability t h a t no metastases will have f o r m e d by time t can now be obtained as follows. Let M(t) be the n u m b e r of metastatic loci at time t. Let N(t) = rn(t) + M(t), and let y(t) be the probability that a r a n d o m l y selected cell has been incorporated into a metastatic focus (entered compartm e n t M(t)); i.e.
y = M/N
(1)
where the argument t has been omitted for notational simplicity. Assuming t h a t the chance of forming a metastatic focus is independent of previously f o r m e d loci, the probability of exactly x loci is Px(t) = (x N) yx(1 - y)N -- x and the probability of no loci is P0 (t) = (1- y ) N = (1 - M/N) N. Metastasis formation is a relatively rare event among all the t u m o r cells arrested in the target organ [4,10,14]. This means m is large and y is small. Then P0 (t) ---- exp(- M(t)).
(2)
Equation (2) is the result one expects of a Poisson process. That the initial stages of metastasis f o r m a t i o n can, in fact, be modeled by such a process has been discussed previously and is in accord with a variety of evidence [10]. The mean n u m b e r of metastases M(t) can for linear Markov processes be f o u n d by solving the deterministic differential equation corresponding to the model in Fig. 1. The result is ~3 M(t) = - - f 0 t k(r) (1 - exp[ - (~2+/~3) ( t - z)] )d r. (3)
206 Equation (3) together with Eqn. (2) furnishes the probability that no micrometastases will have been formed b y time t. MATERIALS AND METHODS The test Eqn. (2) we applied it to a particular experimental system with a transplantable, poorly immunogenic murine fibrosarcoma that metastasizes to the lungs [8]. In this system t u m o r cells are released from the primary t u m o r at an exponentially increasing rate. Estimates of the parameters ~2 and ~3 can be obtained experimentally b y following the kinetics with which intravenously injected t u m o r cells colonize the lung, as described in detail elsewhere [ 1 0 ] . The function X(t) has been experimentally determined by measuring t h e rate at which t u m o r cells enter the primary t u m o r venous drainage [8]. T u m o r cells entering the venous drainage were tagged with Cr sl and injected intravenously into tumor-bearing mice. Virtually all the cells arrested in the lung within 5 min. Thus it can be assumed in this t u m o r host system that the rate at which t u m o r cells enter the t u m o r venous drainage is equivalent to their arrest rate in the lungs. Based on experimental data [8] the best functional form is X(t) = cek(t--
o ), t > 0
=0
,t
8 : P0 = e x p ( -
(~3/~:) c [ 1 / k ( e k ( t - o) -- 1) + 1 / ( k + ~ )
×(ek(t-O)-e-~
(t - o))]).
(5a)
207
When t > > O, the s o l u t i o n is a p p r o x i m a t e l y
(5b)
P0 ~ exp( -- (/~3/~) c e k t [ 1 / k + 1/(k + f12)] )EFFECT OF INCREASING DEATH RATE (,G2) .
1.0
__.=~2=i~4
/'62=33"6 .5,2=134.4
a
c..f;
~< .90 c,O
~< .80 LJJ
\
\ k
\
%k
~) .70
¢2=4".2 \
cc
-~ .60
#3 =.0005
o .50
', \
K = 0.48
Z
, \
"~'%
,, ",•, "\
"~. ~ \\\
\
C = 1440
~.4o
% \
%\\
\\
\\
"\\ \\
\\,, ,,,
>-
.30 =n
< .20 (23 or,,- .10 o_ 0
\"x ~
I
1,
I
l
2
4
6
8
\\\
%Nx-,
~
"-2_
10
\\\
\% m~ '
12
\x ~J
b ....
14
16
l
•
18
20
18
20
DAYS POSTIMPLANT
EFFECT OF DECREASING RATE OF CELL ARREST (K) 1,0 ~ CO
b
.90
F.80 CO o
Z
,,%
.60
\
'~ \ " "\ ,.. x . ,
,~ .30 L
K='24//~
c~
0
X,',,
~' x ",
.50 .40
i 2
r 4
6
", "'.
/K=.015
". "''.
l 8 10 12 DAYS POSTIMPLANT
14
16
Fig. 2. Comparison of model prediction with experimental data for the probability of no micrometastases existing at the time of t u m o r excision at a sequence o f times following transplantation. The data (m) shown above is the proportion of 10 mice with no metastases. The solid curve in (a) and (b) is the solution of Eqn. (5) for the parameter values listed. The parameter values were determined experimentally. The dashed curves are solutions of Eqn. (5) when a specified parameter is varied.
208
The value P0 is the probability no micrometastases have been formed b y time t. That is, this is the probability o f surgical cure b y primary t u m o r excision. In F i g 2 the value of P0 is compared with experimental data for which ~2 = 4.2 d a y s - 1, ~3/~2 = 1.19 • 10 -4 , k = 0.48 days -1 , c = 1440 cells/day and 0 = 4.0 (days). Also illustrated are some aspects of the expected sensitivity of the results to system dependent parameters. It is apparent that the probability of no micrometastases remains close to one for some characteristic time, after which it drops precipitously to zero, a pattern which has also been observed in other experimental tumors such as the Lewis lung carcino m a [6]. The theory described here has been applied to one thoroughly studied murine system. Further investigations into the extent o f application to other t u m o r types are currently under investigation. An understanding of the factors which determine the onset and duration of the critical period when micrometastases are first formed m a y provide insights into the timing of therapy [13] as well as into the examination frequency required for a successful cancer screening protocol. Some o f these implications will be developed in detail elsewhere. The main point here is that the quantitative agreement between theory and experiment presented in this report opens the possibility for a relatively simple, though systematic, rational and quantitative approach to an exceedingly complex and important field. REFERENCES 1 Bailey, T.J. (1964) The Elements of Stochastic Processes with Applications to the Natural Sciences. John Wiley, New York. 2 Bell, G.I. (1976) Model of carcinogenesis as an escape from mitotic inhibitors. Science, 192,569--572. 3 Coman, D.R. (1953) Mechanisms responsible for the origin and distribution of bloodborne tumor metastases: a review. Cancer Res., 13, 397--404. 4 Fidler, I.J. (1970) Metastases: quantitative analysis of distribution and fate of tumor emboli labeled with IUDR-125. J. Natl. Cancer Inst., 45, 775--782. 5 Fisher, B. and Fisher, E.R. (1967) Metastases of cancer cells. In: Methods in Cancer Research, Vol. 1, pp. 243--286. Editor: H. Busch. Academic Press, New York-London. 6 James, S.E. and Salsbury, A.J. (1974) Effect of ICRF 159 on tumor blood vessels and its relationship to the antimetastatic effect in the Lewis lung carcinoma. Cancer Res., 34,839--842. 7 Liotta, L., Kleinerman, J., Catanzaro, P. and Rynbrandt, D. (1977) Degradation of basement membrane by murine tumor cells. J. Natl. Cancer Inst., in press. 8 Liotta, L.A., Kleinerman, J. and Saidel, G.M. (1974) Quantitative relationships of intravascular tumor cells, tumor vessels, and pulmonary metastases following tumor implantation. Cancer Res., 34, 997--1004. 9 Liotta, L.A., Kleinerman, J. and Saidel, G. (1976) Significance of hematogenous tumor cell clumps in the metastatic process. Cancer Res., 36,889--894. 10 Liotta, L.A., Saidel, G.M. and Kleinerman, J. ( 1 9 7 6 ) A stochastic model of metastases formation. Biometrics, 32, 535--550. 11 Parzen, E. (1962) Stochastic Processes. Holden Day, San Francisco. 12 Phillips, R.A., Clark, D.A., Schilling, R.M. and Miller, R.G. (1974) Quantitation and characterization of effector cells and their progenitors. In : Cell Biology and Tumor Immunology, Proceedings XI International Cancer Congress, Vol. 1, pp. 264--274. Editors: Bucalossi, Veronesi and Cascinelli. American Elsevier. 13 Schabel, F.M., Jr. (1975) Concepts for systemic treatment of micrometastasis. Cancer, 35, 15--24. 14 Weiss, L. (1976) Fundamental Aspects of Metastasis. North-Holland, Amsterdam.
