Bulletin of Mathematical Biology Vol. 53, No. 4, pp. 639 656, 199
Printed in Great Britain.
0092 8240/91S3.00+ 0.00 Pergamon Press plc © 1991 Society for Mathematical Biology
A U T O C R I N E AND PARACRINE G R O W T H FACTORS IN TUMOR GROWTH: A MATHEMATICAL MODEL •
SETH
MICHELSON*
Institute for Research Data Management, Syntex Corporation, Palo Alto, CA 94303, U.S.A. •
JOHN
LEITH~
Brown University, Radiation Research Laboratories, Providence, RI 02912, U.S.A. A mathematical model of tumor growth including autocrine and paracrine control has been developed. The model starts with the logistic equation of Verhulst: dV/dt=rV(1-V/K). Autocrine controls are described as modifiers of the Malthusian growth rate (r), while paracrine controls modify the carrying capacity (K) of the system. The control mechanisms are expressed in terms of "candidate" functions, which are based upon the dynamic distribution of TGF-alpha and TGF-beta in the local tumor environment. Three paradigms of tissue growth have been modeled: normal tissue wound repair, unrestricted, unperturbed tumor growth, and tumor growth in a (radiation) damaged environment (the Tumor Bed Effect, TBE). These scenarios were used to test the dynamics of the system against known phenomena. Computer simulations are presented for each case. The model is being extended to include the description of heterogeneous tumors, within which subpopulations can express differential degrees of growth activity. Heterogeneous tumor models, with and without emergent subpopulations, and models of terminal differentiation are also discussed.
1. Introduction. There is overwhelming evidence showing that tumor growth depends not only upon the tumor cells and on their environment, but also upon the reciprocal interactions of each with the other (Keski-Oja et al., 1988; Goustin et al., 1986). Mechanisms by which a tumor can manipulate its environment have been identified previously (Ignotz and Massague, 1986; Keski-Oja et al., 1988). Endogenous compounds which promote the growth of cell populations have been identified for both normal and malignantly transformed cells. Of particular interest in this regard are the Transforming Growth Factors (TGFs), alpha and beta (Tucker et al., 1983; Shipley et al., 1984; Coffey et al., 1986, 1987). The TGFs are augmented by other growth factors, e.g. Epidermal Growth Factor (EGF), Tumor Necrosis Factor (TNF), Platelet Derived Growth Factor (PDGF), and the interleukins, to provide a complex system of regulatory controls. * T o w h o m c o r r e s p o n d e n c e s h o u l d be a d d r e s s e d . t S u p p o r t e d in p a r t b y N I H G r a n t N o . C A 5 0 3 5 0 . 639
640
SETH M I C H E L S O N A N D J O H N L E I T H
Two complementary pathways for the involvement of growth factors in the control of cell growth have been proposed (Sporn and Todaro, 1980; Sporn and Roberts, 1985; Loef et al., 1986). The classical pathway states that if a cell secreting a particular growth factor also has a surface receptor for that factor, a positive feed-forward (autocrine) control loop may be established. Therefore, any growth advantage conferred by the growth factor upon a target cell would also act upon the producer cell itself. Recently, this concept has been modified to include negative or inhibitory control mechanisms (Roberts et al., 1985). The evidence for negative autocrine response suggests that TGF-beta actually slows and stops the growth of cells of epithelial origin (King and Sartorelli, 1989; Howe et al., 1989; Sonnenschein and Soto, 1989). The paracrine pathway describes a control mechanism in which local release of growth factors affects other types of cells in the local milieu. The primary targets for the paracrine factors are the cells of the stroma, and the primary result of paracrine stimulation is an increase in the stroma's ability to support the tumor. It is through this pathway that a tumor manipulates its environment to its own advantage (Goustin et al., 1986). In this paper we modify a basic equation of logistic growth to include these adaptive strategies. We believe that these modifications are necessary for a number of reasons. Michelson et al. (1987a) showed that even with liberal measures of statistical goodness-of-fit, the simple logistic equations of Verhulst and Gompertz do not provide sufficient modeling flexibility to precisely fit t u m o r growth in vivo. Additionally, a simple logistic model ignores any dynamic evolution in the tumor population, and tries to describe adaptive tumor growth through a phenomenological function with constant parameters. To overcome these limitations we have modeled the actions exerted by the TGFs on the Malthusian growth rate (the autocrine pathway) and the competition rate (the paracrine pathway) of the Verhulst model as functions of TGF concentration and receptor activity. To test the theory, we have modeled three scenarios of tissue growth: normal wound healing, unrestricted, unperturbed tumor growth, and the Tumor Bed Effect (TBE). 2. The Basic Model. A variety of mathematical models have been used to describe logistic growth (e.g. historically, Mayneord, 1932; Delthlefsen et al., 1968; Steel and Lamerton, 1966). The simplest are model variations on a theme first presented by Gompertz and Verhulst. Additionally, various modifications of each have been developed. Each model attempts to ascribe the deceleration of population growth rate to an ultimate environmental carrying capacity. The Gompertz equation describes this deceleration phenomenologically. On the other hand, the Verhulst model, described as follows: dV/dt=rV(1-
V/K)
(1)
GROWTH FACTORS IN TUMOR GROWTH
641
includes a specific competition term to account for environmental limits. We term this equation the Basic Equation. The model depends upon three parameters: (1) r, the Malthusian growth rate of the population; (2) K, the carrying capacity of the environment (i.e. the size of the population at which the growth and death rates equilibrate due to intraspecific competition for resources, represented by the t e r m -rV2/K); and (3) Vo, the size of the population at an initial observation time. The model is intuitively more descriptive than the Gompertz equation because both r and K can be associated with explicit biological dynamics (cell cycle and competition for resource). That these mechanisms can be controlled by the autocrine and paracrine pathways make the Basic Equation a natural starting point for this research. For most tumor volume curves, the Basic Equation with constant parameters r and K does not explain the observed growth phenomena well (Michelson et al., 1987a). Adaptive strategies, such as angiogenesis, are displayed across a wide range of mammalian tumors and these mechanisms suggest that both autocrine and paracrine systems work together to alter the growth dynamics of the tumor. One can consider r to be some average measure of cell division time. As r increases, the average intermitotic period decreases. In this very simple model, if r remains positive, the average cell continues to divide, and logistic growth across the population is achieved by competition induced by limited resources. Therefore, we have modified the Basic Equation as follows: we assume that the autocrine and paracrine responses can be represented strictly by r and K in the equation. We derive a series of functions, termed "candidates", for r and K which represent autocrine and paracrine dynamics in terms of TGFalpha/EGF and TGF-beta concentrations at the tumor site. The functions depend explicitly upon the cell type and type of growth being modeled. Three typical growth situations are described below: (1) normal tissue wound healing, (2) unperturbed tumor growth, and (3) tumor growth in a radiation damaged environment, a phenomena termed the Tumor Bed Effect (TBE). The TBE is based upon the observation that tumor cells implanted into such a region display a significant, dose-dependent inhibition of growth. 3. Candidate Functions. For our purposes, we redefine "autocrine response" to mean an action locally induced by a growth factor on cells identical to those secreting the factor. This definition is a slight generalization of the classical definition which defines "autocrine responses" in terms of a response induced by a cell upon itself. By a paracrine response, we mean any modification of growth dynamics that one cell type exerts upon another. Using our new definitions, both autocrine and paracrine responses are measured across a population as a whole, rather than on a per cell basis, and given these
642
SETH M I C H E L S O N AND J O H N LEITH
definitions, we can define a set of"candidates" for r and Kbased upon the levels of the TGFs in the local environment. The structure of each candidate depends entirely upon the type of growth being modeled.
3.1. Normal wound healing.
Let the pre-determined volume of a given tissue be T. Assume that the local environment could carry a greater volume, say K. In some cases, K~> T. The question becomes, how do tissues regenerate a partially destroyed segment of their own population, and how is that growth regulated and controlled? Normal tissue must regenerate and differentiate as opposed to unrestricted tumor growth wherein some cells may differentiate, but a continuously renewing stem-cell compartment persists (Ciampi et al., 1986). Thus, the logistic appearance of the regrowth curve is not due to the competition for resources, but rather to a decrease of population-wide cycling due to differentiation. Therefore, a reasonable candidate function for the Malthusian parameter, r, is one in which both r ~ 0 and dr/dt~O continuously as V ~ T. By requiring dr/dt~O we try to mimic the development of a non-dividing, differentiated subpopulation. We will discuss this aspect of cellular growth and control in more detail below, but given this simple situation, we are assuming that the average division time approaches infinity (r~0), and that its rate of deceleration is a smooth function of time (dr/dt~0). Biologically, this means that the tissue is restoring the wound, but that the restoration is gradual and its termination is not abrupt. Assume that within reasonable limits the tissue healing response is proportional to the size of the wound. Then, let r(V) = g [ ( T - V)/T]r.o m
(2)
where g is a proportionality constant (g > 0), and rnom is the nominal average per cell growth rate for stem cells. Then r(V)~>0 for all V 4 T
dr/dt = (dr/d V) (d V/dt) = { -gVr,om/T } dV/dt.
(3)
The term in braces ({ }) is less than 0 for all V, and d V/dt is greater than or equal to 0 for all t. Therefore, dr/dt
Printed in Great Britain.
0092 8240/91S3.00+ 0.00 Pergamon Press plc © 1991 Society for Mathematical Biology
A U T O C R I N E AND PARACRINE G R O W T H FACTORS IN TUMOR GROWTH: A MATHEMATICAL MODEL •
SETH
MICHELSON*
Institute for Research Data Management, Syntex Corporation, Palo Alto, CA 94303, U.S.A. •
JOHN
LEITH~
Brown University, Radiation Research Laboratories, Providence, RI 02912, U.S.A. A mathematical model of tumor growth including autocrine and paracrine control has been developed. The model starts with the logistic equation of Verhulst: dV/dt=rV(1-V/K). Autocrine controls are described as modifiers of the Malthusian growth rate (r), while paracrine controls modify the carrying capacity (K) of the system. The control mechanisms are expressed in terms of "candidate" functions, which are based upon the dynamic distribution of TGF-alpha and TGF-beta in the local tumor environment. Three paradigms of tissue growth have been modeled: normal tissue wound repair, unrestricted, unperturbed tumor growth, and tumor growth in a (radiation) damaged environment (the Tumor Bed Effect, TBE). These scenarios were used to test the dynamics of the system against known phenomena. Computer simulations are presented for each case. The model is being extended to include the description of heterogeneous tumors, within which subpopulations can express differential degrees of growth activity. Heterogeneous tumor models, with and without emergent subpopulations, and models of terminal differentiation are also discussed.
1. Introduction. There is overwhelming evidence showing that tumor growth depends not only upon the tumor cells and on their environment, but also upon the reciprocal interactions of each with the other (Keski-Oja et al., 1988; Goustin et al., 1986). Mechanisms by which a tumor can manipulate its environment have been identified previously (Ignotz and Massague, 1986; Keski-Oja et al., 1988). Endogenous compounds which promote the growth of cell populations have been identified for both normal and malignantly transformed cells. Of particular interest in this regard are the Transforming Growth Factors (TGFs), alpha and beta (Tucker et al., 1983; Shipley et al., 1984; Coffey et al., 1986, 1987). The TGFs are augmented by other growth factors, e.g. Epidermal Growth Factor (EGF), Tumor Necrosis Factor (TNF), Platelet Derived Growth Factor (PDGF), and the interleukins, to provide a complex system of regulatory controls. * T o w h o m c o r r e s p o n d e n c e s h o u l d be a d d r e s s e d . t S u p p o r t e d in p a r t b y N I H G r a n t N o . C A 5 0 3 5 0 . 639
640
SETH M I C H E L S O N A N D J O H N L E I T H
Two complementary pathways for the involvement of growth factors in the control of cell growth have been proposed (Sporn and Todaro, 1980; Sporn and Roberts, 1985; Loef et al., 1986). The classical pathway states that if a cell secreting a particular growth factor also has a surface receptor for that factor, a positive feed-forward (autocrine) control loop may be established. Therefore, any growth advantage conferred by the growth factor upon a target cell would also act upon the producer cell itself. Recently, this concept has been modified to include negative or inhibitory control mechanisms (Roberts et al., 1985). The evidence for negative autocrine response suggests that TGF-beta actually slows and stops the growth of cells of epithelial origin (King and Sartorelli, 1989; Howe et al., 1989; Sonnenschein and Soto, 1989). The paracrine pathway describes a control mechanism in which local release of growth factors affects other types of cells in the local milieu. The primary targets for the paracrine factors are the cells of the stroma, and the primary result of paracrine stimulation is an increase in the stroma's ability to support the tumor. It is through this pathway that a tumor manipulates its environment to its own advantage (Goustin et al., 1986). In this paper we modify a basic equation of logistic growth to include these adaptive strategies. We believe that these modifications are necessary for a number of reasons. Michelson et al. (1987a) showed that even with liberal measures of statistical goodness-of-fit, the simple logistic equations of Verhulst and Gompertz do not provide sufficient modeling flexibility to precisely fit t u m o r growth in vivo. Additionally, a simple logistic model ignores any dynamic evolution in the tumor population, and tries to describe adaptive tumor growth through a phenomenological function with constant parameters. To overcome these limitations we have modeled the actions exerted by the TGFs on the Malthusian growth rate (the autocrine pathway) and the competition rate (the paracrine pathway) of the Verhulst model as functions of TGF concentration and receptor activity. To test the theory, we have modeled three scenarios of tissue growth: normal wound healing, unrestricted, unperturbed tumor growth, and the Tumor Bed Effect (TBE). 2. The Basic Model. A variety of mathematical models have been used to describe logistic growth (e.g. historically, Mayneord, 1932; Delthlefsen et al., 1968; Steel and Lamerton, 1966). The simplest are model variations on a theme first presented by Gompertz and Verhulst. Additionally, various modifications of each have been developed. Each model attempts to ascribe the deceleration of population growth rate to an ultimate environmental carrying capacity. The Gompertz equation describes this deceleration phenomenologically. On the other hand, the Verhulst model, described as follows: dV/dt=rV(1-
V/K)
(1)
GROWTH FACTORS IN TUMOR GROWTH
641
includes a specific competition term to account for environmental limits. We term this equation the Basic Equation. The model depends upon three parameters: (1) r, the Malthusian growth rate of the population; (2) K, the carrying capacity of the environment (i.e. the size of the population at which the growth and death rates equilibrate due to intraspecific competition for resources, represented by the t e r m -rV2/K); and (3) Vo, the size of the population at an initial observation time. The model is intuitively more descriptive than the Gompertz equation because both r and K can be associated with explicit biological dynamics (cell cycle and competition for resource). That these mechanisms can be controlled by the autocrine and paracrine pathways make the Basic Equation a natural starting point for this research. For most tumor volume curves, the Basic Equation with constant parameters r and K does not explain the observed growth phenomena well (Michelson et al., 1987a). Adaptive strategies, such as angiogenesis, are displayed across a wide range of mammalian tumors and these mechanisms suggest that both autocrine and paracrine systems work together to alter the growth dynamics of the tumor. One can consider r to be some average measure of cell division time. As r increases, the average intermitotic period decreases. In this very simple model, if r remains positive, the average cell continues to divide, and logistic growth across the population is achieved by competition induced by limited resources. Therefore, we have modified the Basic Equation as follows: we assume that the autocrine and paracrine responses can be represented strictly by r and K in the equation. We derive a series of functions, termed "candidates", for r and K which represent autocrine and paracrine dynamics in terms of TGFalpha/EGF and TGF-beta concentrations at the tumor site. The functions depend explicitly upon the cell type and type of growth being modeled. Three typical growth situations are described below: (1) normal tissue wound healing, (2) unperturbed tumor growth, and (3) tumor growth in a radiation damaged environment, a phenomena termed the Tumor Bed Effect (TBE). The TBE is based upon the observation that tumor cells implanted into such a region display a significant, dose-dependent inhibition of growth. 3. Candidate Functions. For our purposes, we redefine "autocrine response" to mean an action locally induced by a growth factor on cells identical to those secreting the factor. This definition is a slight generalization of the classical definition which defines "autocrine responses" in terms of a response induced by a cell upon itself. By a paracrine response, we mean any modification of growth dynamics that one cell type exerts upon another. Using our new definitions, both autocrine and paracrine responses are measured across a population as a whole, rather than on a per cell basis, and given these
642
SETH M I C H E L S O N AND J O H N LEITH
definitions, we can define a set of"candidates" for r and Kbased upon the levels of the TGFs in the local environment. The structure of each candidate depends entirely upon the type of growth being modeled.
3.1. Normal wound healing.
Let the pre-determined volume of a given tissue be T. Assume that the local environment could carry a greater volume, say K. In some cases, K~> T. The question becomes, how do tissues regenerate a partially destroyed segment of their own population, and how is that growth regulated and controlled? Normal tissue must regenerate and differentiate as opposed to unrestricted tumor growth wherein some cells may differentiate, but a continuously renewing stem-cell compartment persists (Ciampi et al., 1986). Thus, the logistic appearance of the regrowth curve is not due to the competition for resources, but rather to a decrease of population-wide cycling due to differentiation. Therefore, a reasonable candidate function for the Malthusian parameter, r, is one in which both r ~ 0 and dr/dt~O continuously as V ~ T. By requiring dr/dt~O we try to mimic the development of a non-dividing, differentiated subpopulation. We will discuss this aspect of cellular growth and control in more detail below, but given this simple situation, we are assuming that the average division time approaches infinity (r~0), and that its rate of deceleration is a smooth function of time (dr/dt~0). Biologically, this means that the tissue is restoring the wound, but that the restoration is gradual and its termination is not abrupt. Assume that within reasonable limits the tissue healing response is proportional to the size of the wound. Then, let r(V) = g [ ( T - V)/T]r.o m
(2)
where g is a proportionality constant (g > 0), and rnom is the nominal average per cell growth rate for stem cells. Then r(V)~>0 for all V 4 T
dr/dt = (dr/d V) (d V/dt) = { -gVr,om/T } dV/dt.
(3)
The term in braces ({ }) is less than 0 for all V, and d V/dt is greater than or equal to 0 for all t. Therefore, dr/dt
