Bulletin of Mathematical Biology (2006) 68: 1495–1526 DOI 10.1007/s11538-005-9042-z
ORIGINAL ARTICLE
A Mathematical Model for the Regulation of Tumor Dormancy Based on Enzyme Kinetics Khalid Boushabaa,∗ , Howard A. Levinea , Marit Nilsen-Hamiltonb a b
Department of Mathematics, Iowa State University, Ames, Iowa 50011, USA Department of Biochemistry, Biophysics and Molecular Biology, Iowa State University, Ames, Iowa 50011, USA
Received: 13 December 2004 / Accepted: 29 April 2005 / Published online: 28 July 2006 C Society for Mathematical Biology 2006
Abstract In this paper we present a two-compartment model for tumor dormancy based on an idea of Zetter [1998, Ann. Rev. Med. 49, 407–422] to wit: The vascularization of a secondary (daughter) tumor can be suppressed by an inhibitor originating from a larger primary (mother) tumor. We apply this idea at the avascular level to develop a model for the remote suppression of secondary avascular tumors via the secretion of primary avascular tumor inhibitors. The model gives good agreement with the observations of [De Giorgi et al., 2003, Derm. Surgery 29, 664–667]. These authors reported on the emergence of a polypoid melanoma at a site remote from a primary polypoid melanoma after excision of the latter. The authors observed no recurrence of the melanoma at the primary site, but did observe secondary tumors at secondary sites 5–7 cm from the primary site within a period of 1 month after the excision of the primary site. We attempt to provide a reasonable biochemical/cell biological model for this phenomenon. We show that when the tumors are sufficiently remote, the primary tumor will not influence the secondary tumor while, if they are too close together, the primary tumor can effectively prevent the growth of the secondary tumor, even after it is removed. It should be possible to use the model as the basis for a testable hypothesis. Keywords Melanoma · Plasmin · Tumor dormancy · Growth factors and inhibitors · TGFβ · Half lives · Compartment model 1. Introduction In many cases, the surgical removal of a malignant tumor from a host is insufficient to ensure that the cancer will not reoccur in the host. Indeed, in Leaf (2004), the author reports statistics that show to when the probability of metastatic tumors is small, the survival rate in cancers such as prostate, breast, colorectal and ∗ Corresponding
author. E-mail address:
[email protected] (K. Boushaba).
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lung is relatively large (80% or more in the first three cases, 40% in the case of lung cancer), whereas in patients with distant metastases, the survival rate is fairly small (0–35% in the first three cases and less than 5% in the case of lung cancer). See also Sporn (1996) as quoted in Glotzman et al. (2004) where it is remarked that 90% of all cancer deaths can be associated with metastasis formation. Thus, in the belief that a good mathematical understanding of distance-controlled regulation of the growth of secondary, presumably metastatic, tumors will help the clinician to better quantify this phenomenon, we present a model for this regulation based upon our current understanding of the modern theory of chemotaxis and the biochemical events surrounding the process by which tumors may grow. By this statement, we do not mean to imply that this is the first such attempt to model metastasis. Indeed, we refer to Yorke et al. (1993) where a Gompertzian growth model was used to model metastasis in prostate cancer. The primary motivation for our model comes from an article of Zetter (1998) who suggested that the vascularization of secondary metastatic tumors may be inhibited from development beyond the avascular stage, by growth inhibitors, secreted by a primary tumor. The idea being that secreted growth factors from the primary tumor have a rather short half life and thus cannot diffuse very far without being degraded or binding to the ECM and becoming deactivated whereas secreted inhibitors have a much longer half life and thus are able to diffuse over longer distances. For example, inhibitors such as TGFβ in the inactive form have a long half life and are able to diffuse relatively long distances through the ECM as well travel through the existing vasculature to suppress remote tumors. Experimental evidence for this is described in Yoon et al. (1999), Kirach et al. (2000), Guba et al. (2001), Jung et al. (2002), and De Giorgi et al. (2003). Latent TGFβ, as its name suggests, is the inactive form of an inhibitor of cell proliferation. The active form of this inhibitor, TGFβ, is created from latent TGFβ by proteolysis that is catalyzed by a protease such as plasmin. Plasmin cleaves a peptide bond in latent TGFβ to release TGFβ, which inhibits the proliferation of certain cell types. As is true for many important biological activities, plasmin activity is also regulated by a transition from an inactive (plasminogen) to an active (plasmin) state. This transition is catalyzed by the protease plasminogen activator, which cleaves a peptide bond in plasminogen to release active plasmin. The production of plasminogen activator by cells is in turn regulated by growth factors such as FGF2 as a consequence of a specific interaction with their cell surface receptor. FGF2 binds to and activates its receptor, thus initiating a signal transduction cascade inside the cell that results in the synthesis of plasminogen activator at a higher rate than prior to activation by the growth factor. Thus, activation of its receptor by FGF both stimulates cell proliferation and initiates a cascade of events that result in the production of active TGFβ, an inhibitor of cell proliferation. 1.1. Outline
r Section 2 is concerned with the relevant biochemical kinetics for the model. r In Section 3, the modeling of tumor cell proliferation and apoptosis is discussed.
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Table 1 Notation. Species
Notation
Concentration
Receptor Fibroblast growth factor, FGF Urokinase plasminogen activator, uPA Tissue growth factor beta TGFβ Latent TGFβ , Plasminogen Plasmin
R G C Ia Ii Pg Pm
[R] [G] [C] [Ia ] [Ii ], [Pg ] [Pm]
r In Section 4, the partial differential equations are replaced by a twocompartment model of 12 ordinary differential equations.
r Section 5 is devoted to a discussion of the equilibrium states. r Section 6 includes a discussion of the numerical simulations and the conclusions one can draw from them.
r In Section 7, some experimental evidence from the literature is given that could be explained by the model.
r In Section 8, a summary and an interpretation of our results is given. 2. Biochemistry The overall idea is as follows (see Table 1 for notation): Suppose Ra is a receptor on a tumor cell capable of being activated by a growth factor. Let G be a molecule of growth factor, for example F GF2, expressed by the tumor cell. Then, the signal transduction pathway by which G induces the tumor cell to express a plasminogen degrading enzyme, C can be modeled in terms of enzyme kinetics thusly: k1
G + Ra {Ra G}, k−1
k2
{Ra G} → C + Ra .
(1)
While the actual biochemical pathway is long and involved, this simplification suffices for our purpose.1 The example we have in mind here is uP A, urokinase plasminogen activator, for the enzyme and F GF2 for the growth factor. In its turn, the enzyme uP Awill degrade the matrix plasminogen Pg by hydrolysis of a certain peptide bond: k3
C + Pg {C Pg }, k−3
k4
{C Pg } → C + Pg + Pm
(2) k2
is entirely reasonable that {Ra V} → mC + Ra where m ≥ 1 reflects the turnover rate of growth factor to protease during the cell cycle. For simplicity as well as for the lack of any experimental evidence to the contrary, we take m = 1 here. 1 It
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where now Pm is a second enzyme, here plasmin. Here Pg is a symbol which represents the product of proteolysis. The tumor cells also express TGFβ , the latent form of TGFβ, a stable protein. We shall denote it by the shorter notation Ii . The plasmin degrades the latent form of TGFβ, to produce the active form, TGFβ, which will be denoted by Ia , via the mechanism: K5
Ii + Pm {Ii Pm}, k−5
k6
{Ii Pm} → Ia + L + Pm.
(3)
Here L denotes LAP, the latency-associated peptide. Experimental evidence for this is found in Sato and Rifkin (1989), Lyons et al. (1990), Lyons et al. (1998), Sato et al. (1990), and Tsuzuki et al. (2000). TGFβ functions to block receptor signals, inhibiting the production of uP A via the equilibrium: νe
Ra + Ia {Ri },
(4)
where Ri represents the receptor in the inactive or inhibited state (Ra : Ia ) and where νe is the equilibrium constant. These four chemical equations constitute a positive–negative feedback loop. As more enzyme is produced from the matrix plasminogen in response to growth factor, more plasmin is created. This in turn activates more TGFβ from the latent form TGFβ . This in turn inactivates the cell receptor signals. As more cell receptor signals are blocked, there are fewer available to catalyze the first reaction and hence the production of enzyme falls. This in turn slows the degradation of the plasminogen which in turn causes a drop in the production of plasmin via the second reaction. This in turn results in a drop in the production of active TGFβ as the plasmin concentration falls. The equilibrium in the fourth equation is then driven to the left and more receptors are returned from the inactive to the activated state. The biochemical pathway is summarized in Fig. 1. Remark 1. The model can be extended to include extracellular matrix (ECM) proteolysis. For example, plasmin is a rather nonspecific enzyme that degrades matrix collagen and other ECM proteins (denoted here by F) via k7
Pm + F {Pm F}, k−7
k8
{Pm F} → F + Pm.
(5)
Where F denotes the products of this degradation. Under certain circumstances, enzymatic breakdown of fibronectin degradation by growth factor-induced proteases leads to products that act as inhibitors of growth factors (GF) or their receptors (GFR). One such angiogenic inhibitor is endostatin. This also leads to a
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Overview of the biochemical pathway. Cell based protein Extracellular protein Competing species
+
C
Pg Pm
+
C
+ G
+
Ii
Ra + Ia + Pm
Ri
Fig. 1 Summary of biochemical pathway in the text, chemical Eqs. (1)–(4). Notice that growth factor (G) in effect competes with TGFβ (I a ) for the active receptor Ra . This is a typical example of a negative feedback loop.
negative feedback loop. However, in order to keep things as simple as possible, we have not included fibronectin proteolysis here. In accordance with the usual chemical conventions, we will let [A](t) denote the local concentration of species A in micromoles per liter (micromolarity) at time t. We may write [R](t) = [Ra ](t) + [{Ri }](t) + [{Ra G}](t), [{Ri }](t) = νe [Ra ](t)[Ia ](t), [Nj ](t) = κ[Rj ](t), where j = i, a.
(6)
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Here [N] denotes the local concentration of tumor cells. It is assumed that the total number of available receptors per unit volume is proportional to the number of receptors capable of initiating protease expression in response to growth factor, κ being the proportionality constant. This constant has been estimated in Levine et al. (2001b) to be of order unity. However, before writing down the laws of mass action for (1)–(4), one must take into account some experimental observations: 1. Tumor cells are capable of expressing growth factors. 2. The rate of tumor cell mitosis is dependent on the local concentration of growth factor. 3. Tumor cells express latent TGFβ. 4. Plasminogen is in such excess and is so stable that we may take its concentration to be constant. 5. The half life of TGFβ is shorter than that of the latent form so that its diffusion in the ECM may be neglected. 6. F GF2 and latent TGFβ diffuse through the surrounding extracellular matrix (ECM) with different diffusivities. 7. F GF2 has a shorter half life than stabilized inhibitors such as latent TGFβ. 8. Tumor cell movement is very slow except when it is by convection (as it is in the case of metastasis). In principle, we must account for these observations in the full set of kinetic equations arising from the chemical equations (1)–(4). Doing this, we have ∂[G] = k−1 [{Ra G}] − k1 [G][Ra ] + σg [Ra ] − µg [G] + Dg [G], ∂t ∂[{Ra G}] = −(k−1 + k2 )[{Ra G}] + k1 [G][Ra ], ∂t ∂[C] = k2 [{Ra G}] − µc [C] + Dc [C], ∂t ∂[{Pg C}] = −(k−3 + k4 )[{Pg C}] + k3 [Pg ][C], ∂t ∂[Pm] ∂t ∂[{Ii Pm}] ∂t ∂[Ia ] ∂t ∂[Ii ] ∂t
= k4 [{C Pg }] − µp [Pm] + Dp [Pm]), = −(k−5 + k6 )[{Ii Pm}] + k5 [Pm][Ii ], = k6 [{Ii Pm}] − µa [Ia ] + Da [Ia ], = k−5 [{Ii Pm}] − k5 [Ii ][Pm] + σi [Ra ] − µi [Ii ] + Di [Ii ],
(7)
where denotes the n-dimensional diffusion operator (n = 1, 2, or 3). Here and throughout the remaining paper we reserve the constants σz, Dz and µz for cell
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expression of protein, diffusion coefficient of protein and decay rate of protein Z, respectively. The constants σz are proportional to the number of mRNA molecules per cell that lead to the translation of Z. As the rate of production of active receptors is tied to the movement and proliferation of tumor cells, we postpone the discussion of the rate equation for [Ra ] to the next section. If we assume that the concentrations of the intermediate species {Ra G}, {Pg C}, {Ii Pm} are nearly constant, then we can combine the resulting expressions obtained by setting the right-hand sides of the second, fourth and sixth equations in (7) to zero, with Eq. (6) to obtain [Ra ] =
[R] . 1 1 + νe [Ia ] + [G]/Km
(8)
In order to simplify the notation a bit as well as to emphasize the space and time distribution of the proteins, we write [G] = g(x, t),
[N] = η(x, t),
[C] = c(x, t),
[Ia ] = ιa (x, t),
[Pm] = pm(x, t),
[Ii ] = ιi (x, t).
(9)
Then [Ra ] =
[R] . 1) (1 + νe ιa + g/Km
i i = (k2i + k−(2i−1) )/k2i−1 and λi = k2i /Km . Since [Pg ] is We employ the notation Km p in excess, we may write λ2 = λ2 [Pg ]. There result the following partial differential equations of transport for G, C, Pm, Ia , Ii :
∂t g = Dg g − µg g +
σg − λ1 g η , 1 η 1 + νe ιa + g/Km 0
∂t c = Dc c − µc c +
η λ1 g , 1 η 1 + νe ιa + g/Km 0 p
∂t pm = Dp pm − µp pm + λ2 c, ∂t ιa = Da ιa − µa ιa + λ3 ιi pm, ∂t ιi = Di ιi − µi ιi − λ3 ιi pm +
σi η 1 1 + νe ιa + g/Km η0
(10)
where we have renormalized the constants so that λ1 = k2 η0 κ where η0 is the carrying capacity of the tumor cells, while σzκη0 is replaced by σz for z as one of the indices g, i. Notice that in this mechanism, the first Michaelis-Menten constant is increased by the presence of the inhibitor. Such inhibitory mechanisms are termed “competitive” (Voet and Voet, 1995, p. 360).
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3. Tumor cell movement A description of cell movement is a bit more complicated. However, an easy derivation can be given on the basis of the continuity equation and chemotactic considerations. First, it is reasonable to assume that tumor cell mitosis depends on the concentration of growth factor and that tumor cell apoptosis is linear in cell density. These observations lead us to modify the continuity equation for cell density to write ∂η = −∇ · Jη + φ(g)η(1 − η/η0 ) − µη η ∂t
(11)
where Jη is the flux of tumor cells. The coefficient of the logistic term φ(g) is a measure of how growth factor influences mitosis. Typically, it has the form φ(g) = λg/(K + g) where λ, K are empirical constants and λ > µn u. The idea is that sufficient growth factor is needed for the birth rate to exceed the death rate, but the effect of growth factor upon the former at saturation is limited to a maximum value of λ. If the biology dictates that excess growth factor has a detrimental effect upon cell proliferation, then it makes better sense to use a φ of the form φ(g) = gλ exp (−g/g0 )/(K + g). The flux of cells has the form Jη = −Dη {∇η − [ψ1 (g)∇g]η}. The first gradient, ∇η, is the term that is responsible for unbiased random cell movement. The quantity in square brackets reflects the chemotactic (ψ1 > 0) influence of growth factor on cell movement. By rewriting this in the form ψ1 (g) = d ln[τ1 (g)]/dg where τ1 is easily found by quadrature, we can rewrite the flux vector in the form η . Jη = −Dη ∇ η ln τ1 (g) In this formulation, one sees that η should follow τ1 (g).2 2 More
accurately, the flux of cells depends not only upon the local gradient of growth factor, but also upon the local variation of fibronectin or other ECM collagen density vis: Jη = −Dη {∇η − [ψ1 (g)∇g + ψ2 ( f )∇ f ]η} where f is the local ECM density through which the cells must move mechanically. Such mechanical dependence of cell movement is often termed “haptotaxis.” However, from a mathematical point of view, chemotaxis and haptotaxis are really the same phenomena. That is, they represent gradient-influenced movement. In both cases, it is known that cells will move up a growth factor gradient for small concentrations of growth factor and then either move away or not respond at all to large concentrations of growth factor. Likewise, cells will follow increasing concentrations of fibronectin or matrix collagen (because there is an improved cell-matrix contact as the level of fibronectin or matrix collagen rises). However, as the fibronectin or matrix collagen density increases, the cells lose the ability to move presumably because the fibronectin or matrix collagen density has become so large that the cells cannot penetrate it or because excess contact prohibits movement.
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Combining these, we obtain the equation for tumor cell movement: η ∂η = ∇ · Dη ∇ η ln + φ(g)η(1 − η/η0 ) − µη η. ∂t τ1 (g)
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(12)
One can give a probabilistic derivation of this form of the equation based on the ideas of Davis (1990) as was done in Othmer and Stevens (1997) in one space dimension where it was proposed as a model of D. discoideum movement. This form has also been used to model the movement of endothelial cells in angiogenesis (Levine et al., 2001a,b, 2002a,b).3 In general, we will have Da > Di Dη while µg , µa µi . The latter condition is the mathematical expression that growth factor and TGFβ have much shorter half lives than latent TGFβ.4 4. Compartment model To Eqs. (10) and (12), must be appended five initial conditions and boundary conditions.5 The boundary conditions will depend upon the geometry and space dimension of the underlying problem. For example, in the case that a small avascular daughter tumor is located several microns from the mother tumor, one might imagine both to be embedded in small spheres in three dimensions and that the flux of the biochemicals in and out of the sphere is only diffusion driven. In order to simulate such a situation without going through excessive computations we turn to a compartment model for (10), (12). We envisage two tumors, one with an initial cell mass given by ηp (0) = η0 Np (0) and one with an initial cell mass η(0) = η0 Ns (0) where Np (0), Ns (0) denote the volumes of these two tumors. We will assume that 1 ≥ Np (0) > Ns (0) > 0 initially so that the subscripts refer to the primary and secondary tumors respectively. We neglect cell movement, as we assume each tumor stays in its own compartment. However, we imagine the two tumors to be separated by a distance L and that they communicate only through diffusion of the bio-chemicals. we let gp , cp , pm, p , ιa, p , ιi, p , Np denote the concentrations of the biochemical and gs , cs , pm,s , ιa,s , ιi,s , Ns cellular species at the primary tumor and denote the corresponding quantities at the secondary tumor. Then in the primary tumor compartment, we have the evolution: ∂t gp = (Dg /L2 )(gs − gp ) − µg gp +
Np σg − λ1 gp , 1 η 1 + νe ιa, p + gp /Km 0
way to view τ (g) is as a correction factor for cell movement in the “cellular free energy” in much the same way that activity coefficients replace concentrations in classical chemical thermodynamics. Then Jη may be viewed as the “flux of cellular free energy.” (The authors thank James Keener for this observation. See Wall (1958) or any good text on chemical thermodynamics for details.) 4t 1/2 ≡ ln 2/µ. 5 Boundary conditions are not needed for those variables for which diffusion may be neglected. For example, if the half life of a protein is very small, it will be degraded or transformed into another protein long before it has a chance to move via diffusion. In such cases, we may neglect transport via diffusion. 3 Another
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∂t cp = (Dc /L2 )(cs − cp ) − µc cp +
Np λ1 gp , 1 1 + νe ιa, p + gp /Km η0 p
∂t pm, p = (Dp /L2 )( pm,s − pm, p ) − µp pm, p + λ2 cp ,
(13)
∂t ιa, p = (Da /L2 )(ιa,s − ιa, p ) − µa ιa, p + λ3 ιi, p pm, p , ∂t ιi, p = (Di /L2 )(ιi,s − ιi, p ) − µi ιi, p − λ3 ιi, p pm, p + ∂t Np =
Np σi 1 η 1 + νe ιa, p + gp /Km 0
λgp Np (1 − Np /η0 ) − µη Np K + gp
while in the secondary compartment: ∂t gs = (Dg /L2 )(gp − gs ) − µg gs +
σg − λ1 gs Ns , 1 1 + νe ιa, p + gs /Km η0
∂t cs = (Dc /L2 )(cp − cs ) − µc cs +
Ns λ1 gs , 1 η 1 + νe ιa, p + gs /Km 0 p
∂t pm,s = (Dp /L2 )( pm, p − pm,s ) − µp pm,s + λ2 cs ,
(14)
∂t ιa,s = (Da /L2 )(ιa, p − ιa,s ) − µa ιa,s + λ3 ιi,s pm,s , ∂t ιi,s = (Di /L2 )(ιi, p − ιi,s ) − µi ιi,s − λ3 ιi,s pm,s + ∂t Ns =
Ns σi , 1 1 + νe ιa, p + gs /Km η0
λgs Ns (1 − Ns /η0 ) − µη Ns . K + gs
Thus, our system of 6 partial differential equations has been “reduced” to a system of 12 ordinary differential equations.6 Notice that the separation distance L is now incorporated into the diffusion coefficients. In order to mimic the removal of the primary tumor by the surgeon’s knife at some time T > 0, we replace Np by H(T − t)Np for t ≥T where H(x) is the Heaviside function wherever Np appears in (13). argument for replacing the term Dg f by (Dg /L2 )(gs − gp ) in the primary compartment in the first equation for growth factor and by (Dg /L2 )(gp − gs ) in the secondary compartment (with similar replacements in the inhibitor equations) is as follows: Suppose the primary compartment is at x = 0 and the secondary compartment is at x = L. Then the flux of growth factor out of the primary compartment to the secondary compartment is Jp = −Dg (gp − gs )/L, while the flux of growth factor from the secondary compartment to the primary compartment is Js = −Dg (gs − gp )/L. The continuity equation at x = 0 then states that ∂t gp = −Jp /L = D f (gs − gp )/L2 while at x = L, ∂t gs = −Js /L = Dg (gp − gs )/L2 . This is reasonable, as if gp > gs we would expect growth factor to diffuse from the primary tumor to the secondary tumor i. e., ∂t gp < 0 and ∂t gs > 0.
6 The
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5. Equilibrium states for (13) and (14) Observe that the right-hand side of (13) can be obtained from that of (14) by interchanging the roles of the subscripts p, s. In order to understand how the primary tumor affects the secondary tumor, we need to solve the algebraic system in the first compartment: 0 = (Dg /L2 )(gs − gp ) − µg gp +
Np σg − λ1 gp , 1 η 1 + νe ιa, p + gp /Km 0
0 = (Dc /L2 )(cs − cp ) − µc cp +
Np λ1 gp , 1 η 1 + νe ιa, p + gp /Km 0 p
0 = (Dp /L2 )( pm,s − pm, p ) − µp pm, p + λ2 cp ,
(15)
0 = (Da /L2 )(ιa,s − ιa, p ) − µa ιa, p + λ3 ιi, p pm, p , 0 = (Di /L2 )(ιi,s − ιi, p ) − µi ιi, p − λ3 ιi, p pm, p + 0=
Np σi , 1 1 + νe ιa, p + gp /Km η0
λgp Np (1 − Np /η0 ) − µη Np . K + gp
There is also the dual system in which we interchange the roles of s and p which tells us how the secondary tumor influences the primary tumor: 0 = (Dg /L2 )(gp − gs ) − µg gs +
σg − λ1 gs Ns , 1 η 1 + νe ιa,s + gs /Km 0
0 = (Dc /L2 )(cp − cs ) − µc cs +
Np λ1 gs , 1 η 1 + νe ιa,s + gs /Km 0 p
0 = (Dp /L2 )( pm, p − pm,s ) − µp pm,s + λ2 cs ,
(16)
0 = (Da /L2 )(ιa, p − ιa,s ) − µa ιa,s + λ3 ιi,s pm,s , 0 = (Di /L2 )(ιi, p − ιi,s ) − µi ιi,s − λ3 ιi,s pm,s + 0=
Ns σi , 1 η 1 + νe ιa,s + gs /Km 0
λgs Ns (1 − Ns /η0 ) − µη Ns . K + gs
Suppose that Np > Ns > 0. Then we can write, from the last of the equations in (15) and (16) gp = η(Np ) =
Kµη , λ(1 − Np /η0 ) − µη
gs = η(Ns ) =
Kµη . λ(1 − Ns /η0 ) − µη
(17)
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From the first equation in each of (15) and (16) we obtain a simple linear system in gp , gs which we can solve to write −1 Dg Dg µg 2 + µ + µ g R(Np , ιa, p ) g L2 L2 Dg R(Ns , ιa,s ) , + L2
gp = η(Np ) =
= G(Np , Ns , ιa, p , ιa,s ),
(18)
gs = η(Ns ) = G(Ns , Np , ιa,s , ιa, p ), where R(x, y) =
σg − λ1 η(x) x . 1 η 1 + νe y + η(x)/Km 0
(19)
Thus, the two equations η(Np ) = G(Np , Ns , ιa, p , ιa,s ) and η(Ns ) = G(Ns , Np , ιa,s , ιa, p ) constitute a system through which the tumor sizes at both sites are controlled by the local concentration of TGFβ at both sites or conversely. From the second equation in each of (15) and (16) we can write −1 Dc µc Dc Dc 2 cp = + µ + µ , ι ) + , ι ) , S(N S(N c p a, p s a,s c L2 L2 L2 = C(Np , Ns , ιa, p , ιa,s ), cs = C(Ns , Np , ιa,s , ιa, p ),
(20)
where S(x, y) =
x λ1 η(x) . 1 η 1 + νe y + η(x)/Km 0
(21)
From the third equation in each of (15) and (16) we can write p
pm, p = λ2
Dp µp + µ2p L2
−1
Dp + µ p C(Np , Ns , ιa, p , ιa,s ) L2
Dp + C(N , N , ι , ι ) s p a, p a,s , L2 = Pm(Np , Ns , ιa, p , ιa,s ), pm,s = Pm(Ns , Np , ιa,s , ιa, p ).
(22)
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Finally, by adding the fourth and fifth equation in each of (15) and (16) again obtain a linear system for which we can write ιi, p =
Di µi + µi2 L2
−1
Di Di + µ , ι , ι ) + , ι , ι ) J (N J (N i p a, p a,s s a,s a, p , L2 L2
= I(Np , Ns , ιa, p , ιa,s ), ιi,s = I(Ns , Np , ιa,s , ιa, p )
(23)
where J (x, y, z) = (Da /L2 )z − ((Da /L2 ) + µa )y +
x σi . 1 1 + νe y + η(x)/Km η0
(24)
Thus, all the stationary variables may be expressed in terms of either (Np , Ns ) or (ιa, p , ιa,s ). Remark 2. In order that ιa, p , ιa,s are both biologically meaningful, i.e. nonnegative, it is necessary and sufficient that both Di Di + µ , ι , ι ) + J (N J (Ns , ιa,s , ιa, p ) ≥ 0, i p a, p a,s L2 L2 Di Di , ι , ι ) + + µ (25) J (N J (Ns , ιa,s , ιa, p ) ≥ 0. p a, p a,s i L2 L2 These two conditions may or may not impose further conditions on Ns , Np . To see this, suppose that σi , the rate of tumor cell expression of Ii , or latent TGFβ in our example, is very small so that for all intents and purposes J (x, y, z) ≈ (Da /L2 )z − ((Da /L2 ) + µa )y. The two inequalities then reduce to the requirements that
Di µa Di µa Da µi Da µi ιa,s − ιa, p ≥ 0, − + µa µi + L2 L2 L2 L2
Di µa Da µi Di µa Da µi − − + µ µ + ι ιa,s ≥ 0. a, p a i L2 L2 L2 L2
Neither of these can hold if µi /µa < Di /Da . For our case, the diffusion coefficient of latent TGFβ is larger than for TGFβ, which is a statement that follows from the Stokes–Einstein equation as the molecular weight of the former is larger than that of the latter. On the other hand, the half life of TGFβ is much smaller than that of the latent form. Therefore, µi /µa < 1 < Di /Da and the constraint conditions (25) are not automatically fulfilled.
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Clearly, J (Np , ιa, p , ιa,s ) ≥ 0 will hold if the pair (Np , ιa, p ) satisfies − ((Da /L2 ) + µa )ιa, p +
Np σi ≥0 1 η 1 + νe ιa, p + η(Np )/Km 0
(26)
with J (Ns , ιa,s , ιa, p ) ≥ 0 holding if the pair (Ns , ιa,s ) satisfies the analogous inequality. When both of these hold, both of (25) will hold and when both fail, so do both of (25). However, one may hold and the other may fail with both of (25) holding. Remark 3. variables:
We can glean a bit more from (26) as follows. Define the following
α = Da /L2 + µa ,
ξp = 1 + η(Np )η0 ,
βp =
σi Np νe . η0
Then (26) will hold if and only if 0 ≤ ιa, p ≤
2βp /(αξp ) 4βp / αξp2 + 1 + 1
(27)
holds. Inequality (27) contains some further information. It tells us that the concentration of the active inhibitor at the primary tumor site is controlled by both diffusion and decay. We can see this more clearly as follows. Suppose the two tumors are very far apart. Then one sees that when µa is very large:
0 ≤ ιa, p ≤
βp . µa
On the other hand, if the two tumors are close to each other (which would be the case if L is small), then 0 ≤ ιa, p ≤
βp L2 . ξp Da
6. Simulations By introducing the change of variables with dimensionless time and length scales in Table 2, we may renormalize the system (10)–(12) and consequently the twocompartment model as follows. Then the system of differential equations may be written in the form: ∂τ G = DGG − µG G +
σG − 1 G N, 1 + Ia + G
Bulletin of Mathematical Biology (2006) 68: 1495–1526
1509
Table 2 Dimensionless variables. Variable
Dimensionless variable
x t g c pm ιa ιi η 1 λ1 = k2 /Km
ξ = x/L τ = t/T 1 G = g/Km 1 C = c/Km Pm = pm/[Pg ] Ia = νe ιa Ii = νe ιi N = η/η0 1 = Tλ1
2 λ2 = k4 /Km
1 )/[P ] = Tλ K 1 2 = (Tλ2 Km g 2 m
3 λ3 = k6 /Km
3 = Tλ3 [Pg ]
σg σi Dg Dc Dp Da Di Dη µg µc µp µa µi µη λ K
1 σG = Tσg /Km σ I = (Tνe σi ) DG = T Dg /L2 DC = T Dc /L2 D P = T Dp /L2 D A = T Da /L2 D I = T Di /L2 Dη = T Dη /L2 µG = Tµg µC = Tµc µ P = Tµp µ A = Tµa µ I = Tµi µN = Tµη 1 = Tλ/Km 1 K = K/Km
∂τ C = Dc C − µC C +
p
1 G N, 1 + Ia + G
∂τ Pm = D P Pm − µ P Pm + 2 C,
(28)
∂τ Ia = D AIa − µ AIa + 3 Ii Pm, σI N, 1 + Ia + G N ∂N G = ∇· Dη ∇ N ln + N (1 − N ) − µN N . ∂τ τ1 (G)τ2 (Ii ) K+G ∂τ Ii = Di Ii − µ I Ii − 3 Ii Pm +
We have included a number of figures to illustrate the sensitivity of the outcomes to the various parameters in the model. In Figs. 2–6, the vertical axes are in dimensionless units. In the remaining figures, the vertical axis units are time in days. The first set of figures are to be understood in the following context: Two tumors are implanted, one in a compartment at x = 0 and a second in a compartment at x = L, the initial mass of the former being larger than that of the latter. We
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Primary and secondary tumor properties for L=7, The secondary tumor decays without surgery. (Time in days on horizontal axes.) -4 x 10 10
0.03
secondary site time course FGF2 (g)
primary site time course
0.02 5 0.01
uPA(c) 0
0 0
200
400
600
800
0
200
400
600
800
-3
20
x 10
8 6
15 plasmin (p) 10
4
5
2
TGFβ (Ia)
0
0 0
200
400
600
0
800
200
400
600
800
600
800
1
10000
cell density (N)
LTGFβ (Ii)
0.5
5000
0
0 0
Fig. 2
200
400
600
800
0
200
400
Spontaneous decay of secondary tumor in the presence of the primary tumor.
therefore refer to the larger as the primary tumor and the smaller as the secondary tumor. For each set of parameters given in Table 2, and for each pair of primary and secondary tumor masses, there are two mutually exclusive possibilities; the primary tumor inhibitor output will cause the secondary tumor to shrink, or the primary tumor inhibitor output will be insufficient to prevent the growth of the secondary tumor to some larger size. We might expect therefore, the existence of a threshold distance L∞ such that if we wait a very long time and if L < L∞ , the secondary tumor dies out while if L > L∞ , the secondary tumor will grow to a larger mass. In the former case, we will be left with a single solid tumor, while in the latter case we will have two tumors of equal mass. Our goal is to interrupt this process at some fixed time by removing the primary tumor at some finite time τ . Again, one of the two situations described earlier will again occur for each such τ , i.e., the secondary tumor will die out or the secondary tumor will grow to some finite size. We let Lτ denote the value of this threshold distance if the primary tumor is removed at time τ (see Fig. 7 for the two cases τ = 10 days and τ = ∞).
Bulletin of Mathematical Biology (2006) 68: 1495–1526
1511
Primary and secondary tumor properties for L=7, for early times (t60) after surgery at t=10 days. (Time in days on horizontal axes.) -3
x 10
secondary site time course primary site time course
4
0.04 FGF2 (g)
0.02
2 uPA(c) 0
0 200
400
600
800
200
400
600
800
2.5 0.06
2 1.5
0.04 plasmin (p)
TGFβ (Ia)
1
0.02
0.5
0
0 200
400
600
800
5000
200
400
600
800
1
4000
cell density (N)
3000
LTGFβ (Ii)
0.5
2000
secondary tumor recovery
000 0
0 200
400
600
800
200
400
600
800
Fig. 4 Primary and secondary time courses after primary tumor is removed at 10 days. Secondary tumor recovers some 300 days after removal of the primary tumor.
of the former primary tumor. Likewise, the secondary environment will experience an increase in the concentration of growth factor and inhibitor, while the collagen matrix will degrade. However, as we see in Fig. 5, if the secondary tumor is very close to the primary tumor (L = 1.0 cm), then even removal of the primary tumor will not induce the growth of the secondary tumor. On the other hand, as we see in Fig. 6, if the secondary tumor is very remote from the primary tumor, L = 50 cm, the growth of the secondary tumor is not inhibited by that of the primary tumor. This is illustrated in Fig. 7. If L < 5.5, the secondary tumor will die out regardless of the time of surgical removal of the primary tumor, although the time to die out to less than 1% of its original size increases as L increases to L∞ from below. When L is in the window (5.5 < L < 8.5), the secondary tumor will die away if we do not remove the primary tumor, while it will grow to the size of the former primary tumor if we do remove it. If L > L∞ , then the secondary tumor will grow regardless of whether or not we remove the primary tumor, the time of growth being smaller if we remove the primary tumor than if we do not. These times approach each other as L → ∞. Put another way, we see from Fig. 7 that surgery always favors secondary tumor growth when 5.5 < L < 8.5, while it cannot prevent secondary tumor
Bulletin of Mathematical Biology (2006) 68: 1495–1526
1513
Primary and secondary tumor properties for L=1 with primary tumor removed at t=10 days, secondary tumor decays. (The primary tumor density is off scale in the cell density plot. Time in days on horizontal axes. ) -4 x 10 20 0.03 secondary site time course
15
0.02 FGF2 (g)
primary site time course
10 uPA(c)
0.01
5
0
0 0
20
40
60
0
20
40
60
6 0.2 4
plasmin (p)
TGFβ (Ia)
0.1
2
0
0 0
20
40
60
4000
0
20
40
60
0.03 cell density (N)
3000
0.02
LTGFβ (Ii)
2000
0.01
1000
0
0 0
20
40
60
0
50
100
150
200
Fig. 5 Primary tumor and secondary tumors are in close proximity (1.0 cm). Removal of the primary tumor does not permit growth of the secondary tumor.
growth (although it can delay it) if the primary tumor is sufficiently remote from it (L > 8.5) and can favor secondary tumor growth if the primary and secondary tumors are too close to one another. When the tumors are close, the removal of the primary tumor lengthens the extinction time of the secondary tumor. In Fig. 9, we fixed all the parameters except L and computed how the efficiency depends upon the rate of cellular expression of inhibitor (σι ). Among the various trial values for (σi ) in the figure, we used the value of σi from the table in panel 9(c). Because the number of mRNAs per cell is not known precisely (it could be as few as one per cell), we have included this figure to test how sensitive the results are to the value of σi . We see that for low levels of σi , the secondary tumor can grow quite far from the tumor regardless of whether or not the primary tumor is removed 9(a). However, at high levels, the secondary tumor cannot survive whether or not the primary tumor is removed even if the secondary site is as much as 20 cm from the primary cite. In Fig. 10, we fixed the intermediate tumor distances at 3, 7, and 20 and computed how the window depends upon σg . Clearly, the closer the tumors are, the larger is the range of values of σg for which surgery plays some role in the
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Bulletin of Mathematical Biology (2006) 68: 1495–1526 Primary and secondary tumor properties for L=50, no surgery, secondary tumor grows. (Time in days on horizontal axes.) -3
x 10 3
0.04
2
FGF2 (g)
0.02
uPA(c)
1 0
0
0
200
400
600
800
0
200
6
0.04
2 0
600
800
secondary site time course primary site time course
4
plasmin (p)
0.02
400
TGFβ (Ia)
0 0
200
400
600
800
0
200
400
600
800
1 10000 5000
0.5
LTGFβ (Ii)
cell density (N)
0 0
200
400
600
800
0
0
200
400
600
800
Fig. 6 Primary and secondary tumor are relatively remote from one another. The secondary tumor grows independently of the primary tumor.
suppression of secondary tumor growth. (When L = 3 the σ interval is roughly [0.31, 0.39] but when L = 20 this shrinks to [0.4744, 0.4772].) In Fig. 11, we see how the efficiency varies for several values of λ3 , the TGFβ production rate. Here we see that for high TGFβ production rates, the secondary tumor will die out at distances fairly remote from the primary tumor independently of the removal of the primary. On the other hand, when this rate is low, the secondary tumor will grow relatively close to the primary tumor whether or not the latter is removed surgically. In Fig. 12, we vary λ3 for intermediate tumor distances 3, 7 and 20. Clearly, the closer the tumors are, the larger is the range of values of λ3 for which surgery plays some role in the suppression of secondary tumor growth. (When L = 3 the λ3 interval is roughly [0.44, 0.63] but when L = 20 this shrinks to [0.793, 0.797]). In Fig. 13, we vary L for variable K. For small values of K, the secondary and the primary tumor will always grow. For larger values of K, the secondary will always be suppressed even at distances quite remote from the primary whether or not the primary is removed. This is to be expected because the larger K is, the more negative is the quantity λ/K − µη. Notice that for the largest K used here (in Fig. 13(e)), λ/K − µη = 6.25(10−3 )/1.7(10−3 ) − 1.010−3 > 0 so that cells do retain their ability to proliferate over quite a wide range of K.
Bulletin of Mathematical Biology (2006) 68: 1495–1526
1515
Comparison of extinction and growth times for the secondary tumor with and without removal of the primary tumor. 800
700 vertical asymptote at L∞=11.7
Time in days
600
vertical asymptote at L 10 =7.7
500
400
300
growth time without surgery extinction time without surgery growth time, after surgery extinction time, after surgery
200
100
2.8
5.6 8.4 11.2 14 16.8 Distance between primary and secondary tumors, L, in cm.
19.6
Fig. 7 The effect of surgery on growth and extinction. There is a window of positions from the primary tumor where secondary tumors cannot grow if the primary tumor is present and will grow if the primary tumor is removed.
In Fig. 14 we fixed the intermediate tumor distances at 3, 7 and 20. Again, the closer the tumors are, the larger is the range of values of K for which surgery plays some role in the suppression of secondary tumor growth. The numerical values given in Table 3 were found as follows: 1. The diffusion constants Dp , Dc , Di , Da were computed from the Stokes– Einstein relationship and a knowledge of the molecular weights of F GF2, plasmin, uPA, TGFβ and TGFβ using the value of Dg found from (Filion and Popel, 2004, p. 652) at 37◦ C. (The diffusion coefficient is roughly proportional −1/3 to Mw for large proteins where Mw is the molecular weight of the protein in question. However, one has to be careful about the use of this assumption (see He and Niemeyer, 2003). There the authors remark that the shape of the molecule can have a very significant effect on the magnitude of the diffusion coefficient.) 2. The rate constants k1 , k−1 were taken from (Filion and Popel, 2004, p. 650) (the authors took these from Nugent and Edelman, 1992.) 3. The catalytic constant k2 was estimated as follows: The number of amino acids (432) in uP A is divided by 20 as it is known that the overall cellular transcription–translation rate for proteins is roughly 10–20 amino acids per
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Fig. 8 The dependence of the window described in the previous figure caption on the surgery removal time.
second (Anderson et al., 1983; Pavlov et al., 1996). This gives an assembly time per mRNA molecule per cell of a single uPA molecule of 21.6 s or 0.012 h. That is, 1.666 protein molecules are assembled from a single mRNA molecule in an hour. From Castello et al. (2002), there are around 45 mRNA molecules of uPA per cell in breast cancer cells (as apposed to 30 per normal breast cell). Thus, there are 45 × 1.6666 = 75.0 protein molecules produced every hour per malignant cell. There are roughly 1012 –1013 cells per liter. From this, there are between 75.0 × 106+12 /(6 × 1023 ) = 1.25 × 10−4 and 1.25 × 10−3 µM of uPA being produced per hour per liter of cells. Suppose the enzyme equation is at steady 1 state, with no inhibitor present and the growth factor concentration is at Km /2. −1 −4 With [C] = 6.25 × 10 µM and µc = 8.38 h (the half life for uPA is 5 min), we see that µc [C] = k2 /3 = k2 δη0 /3 = 5.30 × 10−3 h−1 or k2 = 1.59 × 10−2 h−1 .7 4. The constant λ is roughly 1/32 of an hour, the maximum growth rate for malignant cells being roughly 32 h, although it can vary quite a bit depending on the particular tumor cells. The estimate for η0 is based on a cell volume of 102 –103 µm3 . The apoptosis rate µη is taken from Levine et al. (2001a). 7 The units are correct here because we have written the cell concentration as [R] = η × δ where T 0 δ is the number of cell receptors and η0 is the carrying capacity. Using 105 for the former and 5 × 1012 for the latter we see that the product is roughly 5 × 1017 receptors per liter or 1 µM.
Bulletin of Mathematical Biology (2006) 68: 1495–1526
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Variation of efficiency with inter tumor distance and various σ
I
(Inter tumor distance on horizontal axis, time on vertical axis.) 300 250 200
800
800
600
600 (b) σ =0.2250 I
(a) σ =0.0225 I
400
400
200
200
150 100
4
8
12
1
2
3
4
(c) σ =0.45 I
5
800
5
500
600
growth time without surgery
10
15
(e) σI=0.5625
400
extinction time without surgery growth time, after surgery
400
300
extinction time, after surgery
(d) σ =0.47 I
200
200 5
10
15
20
100
5
10
15
20
Fig. 9 Sensitivity study for the rate of tumor expression of latent TGFβ. The rate is fixed in each panel.
5. From the literature, Lijnen et al. (1996), and Ploplis et al. (1995), we have found estimates of 84–130 µg/l of plasminogen in plasma. A good estimate for the molecular weight of plasma is around 88 µg/µmole. We therefore used a value for the concentration of plasminogen in plasma of 1.0 µM. 6. The half life for TGFβ may found in Wakefield et al. (1990), Beck et al. (1993), and Zioncheck et al. (1994), while those for TGFβ are from Wakefield et al. (1990). Half lives estimates for FGF2 in plasma were found to be in the range of 1.5–3 min (Whalen et al., 1989; Edelman et al., 1993). These should be close to the values in tissue. We took the uP A half life from the data in Edelman et al. (1993), Whalen et al. (1989), and Lazarous et al. (1997). Half life estimates for plasmin may be found in Chandler et al. (2002) and Takeda and Nakabayashi (1974). The value of σg cannot be found directly. We estimated the value of σg as follows: Using the data in Mansbridge et al. (1999), our best estimate is that there are roughly 2–5 FGF mRNA molecules per cell. Using a translation rate of 20 amino acids per second and approximately 160 amino acids per molecule of FGF, we find a production rate of 2(1/8)3600 = 900 FGF molecules per cell per hour. Using an estimated cell volume of 100 µm3 , we see that the micromolarity rate is [900 h−1 /(6 × 1023 )(M)−1 ] × 106 µM/[102 (µm)3 × 10−12 cm3 × (µm)−3 × 10−3 l × cm−3 ] = 0.015 µM/h. As we do not know the value of σi we must content ourselves with the sensitivity analysis illustrated in Fig. 5. This presupposes an mRNA/cell range of around three to eight molecules per cell for TGFβ.
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Bulletin of Mathematical Biology (2006) 68: 1495–1526 Variation of efficiency with σg for various inter tumors distances. (Time on the vertical axes). growth time, after surgery extinction time,after surgery extinction time without surgery growth time without surgery
3000 L=4.2 2000 1000 0 0.03
3000
0.04
0.05
0.06
0.07
0.08
L=28.2
2000 1000 0
0.032
0.034
0.036
0.038
0.04
0.042
0.044
4000 L=71.1 3000 2000 1000 0.036
.0365
0.037
0.0375
0.038
σg
Fig. 10 Sensitivity study for the rate of tumor expression of latent TGFβ. The intertumor distance is fixed in each panel.
7. We estimated the equilibrium constant, νe , for the TGFβ inhibition of FGF receptor signaling was estimated using the results of De Crescenzo et al. (2003) and De Crescenzo et al. (2005). There it is shown that when TGFβ is fully bound to its own receptors, one can expect an equilibrium constant in the range 5 × (10−4 µM ≤ (1/νe ) ≤ 5 × 10−3 µM. 2 8. From Ellis et al. (1991) we obtained k4 , Km (kcat , Km) for the enzyme reaction Pg + uP A [Pg uP A] → Pm + uP A in several cases. Two that interest us here are when the uPA is bound to the human monocyte cell line U937 receptors and when the uPA is in solution. In the first instance, the authors report (kcat , Km) ≈ (0.12/s, 0.67 µM) with experimental errors of around 50%. In solution, they give (kcat , Km) ≈ (0.73/s, 25 µM). We used the latter values (converted to reciprocal hours) for the tabular entries. We used the Matlab solver ODE15s to solve our system of ordinary differential equations. We choose this solver because the system of differential equations are stiff. Stiff problems often occur when conditions are initially changing rapidly, as is the case in many kinetic reactions involving enzyme kinetics (see Murray, 1989). The cause of stiffness in our problem can be related to the fact that the individual equations in the system have widely varying time scales.
Bulletin of Mathematical Biology (2006) 68: 1495–1526
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Variation of efficiency with inter tumor distance and various λ3 values (Inter tumor distance on horizontal axis, time on vertical axis.)
800
800
240 (a) λ =0.0149
(c) λ3=0.745
(b) λ3=0.3725
3
220
600
600
400
400
200
200
200
180
160
1
2
3
4
5
1
2
3
4
5
5
10
15
800 (d) λ3=0.79
400
600
(e) λ3=1
growth time without surgery extinction time without surgery growth time, after surgery
400
300
extinction time, after surgery
200
200 10
20
30
5
10
15
20
Fig. 11 Sensitivity study for the rate of plasmin release of active TGFβ from the latent form. The rate is fixed in each panel.
For such problems, the time steps must be small enough to accommodate the equation with the most rapid growth of the right-hand side (in absolute value) so that the approximate numerical solution does not drift too far from the exact solution. But once the more rapid transients have died out, larger time steps are more efficient for following the remaining, slower reactions to their steady states. However, when to make the switch from smaller to larger time steps may depend upon several parameters of the system. The sensitivity analysis the curves in Figs. 10–14 appear to be somewhat oscillatory about some mean curve; For example, in Fig. 5, the calculation of each point of the curve requires an integration of the entire system for a fixed value of L. Since the system is stiff, the increase of step size for each fixed value of L from a small value required to deal with the rapidly growing right-hand side(s) of some of the equations to a larger step size as we near the equilibrium, depends upon L. In turn, this leads to the numerical artifact.
7. Comparison with experiment In De Giorgi et al. (2003), the authors report on the removal of a “polypoid lesion at the cutaneous level, measuring about 15 mm and with a diameter of 10 mm in the upper part . . . .” The authors further report “Histopathologically, a diffuse proliferation of epitheliod, pigmented atypical melanocytes was observed beneath the dermis” (see Fig. 15). Thirty days after removal of the lesion, the authors report
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Bulletin of Mathematical Biology (2006) 68: 1495–1526 Variation of effiency with λ3 for various inter tumor distances. (Time in days on the vertical axes).
600 L=4.2 400 200 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
700 600
growth time without surgery extinction time without surgery growth time, after surgery extinction time, after surgery
L=9.9
500 400 300 0.6
0.65
0.7
0.75
0.8
0.85
0.79
0.8
0.81
0.9
700 600 L=28.2 500 400 0.76
0.77
0.78 λ
3
Fig. 12 Sensitivity study for the rate of plasmin release of active TGFβ from the latent form. The inter tumor distance is fixed in each panel.
“After approximately one month, the appearance within the radius of approximately 5 to 7 cm from the operation scar, a typical satellitosis with numerous rounded nodular lesions with gray-bluish pigmentation and of a hard consistency was observed . . . ” (see Fig. 16).
8. Discussion Our model supports the notion if a small secondary (avascular) tumor is sufficiently remote from a larger primary avascular tumor, the growth of the latter will be essentially independent from the growth of the former. On the other hand, if a small secondary tumor is initially imbedded relatively close to a large primary, the inhibitor from the latter should prevent the growth of the secondary tumor. Our simulations suggest an experimental protocol that could be used to verify, in a controlled fashion, the model predictions. The astute reader will ask “Why do they suppose the primary tumor is making a growth inhibitor and why does not it affect its own growth.” The tumor released inhibitor does indeed control the size of the primary tumor, but it does not eliminate the primary tumor. In our model, the effect of self regulation of tumor derived inhibitor can be seen in the panel labeled “cell density” in Fig. 2. The solid line shows
Bulletin of Mathematical Biology (2006) 68: 1495–1526
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Variation of efficiency with inter tumor distance and various values of K. 800
800
52
(b) K=0.016
(a) K=0.084 48
600
600
400
400
200
200
44 40 36 2
4
6
8
0.5
1
1.5
(c) K=0.0165
0.5
2
1
1.5
2
800 400
growth time without surgery
(e) K=0.017
extinction time, after surgery
600
extinction without surgery
300
extinction without surgery
400 200 (d) K=0.0168 200
100 5
10
15
5
10
15
20
Fig. 13 Sensitivity study for the “Michaelis-Menten constant” K for the mitosis rate for tumor cells. The rate is fixed in each panel. Variation of effiency with K for various inter tumors distances (time on the vertical axes). 650 L=4.2 500 350
1.67
1.672
1.674
1.676
1.678
-2 1.682 x10
1.68
800 650
L = 9.9
500 x10-2
350 1.678
1.6785
1.679
1.6795
1.68
1.6805
1.681
1.6815
1.682
growth time without surgery extinction time withiout surgery extinction time, after surgery growth time, after surgery
650 L=28.2 400
250 1.6804
1.6805
1.6805
1.6805
1.6806
1.6807
1.6807
1.6807
1.6808
1.6809
1.6809
x10-2
K
Fig. 14 Sensitivity study for the the “Michaelis-Menten constant” K for the mitosis rate for tumor cells. The inter tumor distance is fixed in each panel.
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Table 3 Numerical values used in simulations. Equation
Constants
Values
Time scale (4) (10)
T νe Dg Dp Dc Di Da µg µp µc µi µa σg σi [Pg ] k1
1.0 h 1.1 × 103 (µM)−1 7.92 × 10−3 cm2 h−1 6.48 × 10−3 cm2 h−1 7.73 × 10−3 cm2 h−1 6.32 × 10−3 cm2 h−1 9.43 × 10−3 cm2 h−1 14.0–28.0 h−1 0.154–0.0495 h−1 8.38 h−1 0.385 h−1 6.93 h−1 1.50 × 10−2 µM h−1 0.45 µM h−1 1.01 µM 2.5 × 102 (µM h)−1 –1.5 × 104 (µM h)−1 0.048 min−1 = 2.88 h−1 1.59 × 10−2 h−1 (k−1 + k2 )/k1 1.5 × 104 (µM)−1 h−1 2.68 × 103 h−1 25.0 µM 0.745 (h µM)−1 1012 /l= 109 /(cm)3 6.25 × 10−3 h−1 1.68 × 10−2 µM 1.0 × 10−3 h−1
(12)
k−1 k2 1 Km λ1 k4 2 Km 3 λ3 = k6 /Km η0 λ K µη
Notes and comments
(See notes in the text) (See notes in the text) (See notes in the text) (See notes in the text) (See notes in the text) (See notes in the text) (See notes in the text) (See notes in the text) (See notes in the text) (See notes in the text) (See notes in the text) (See notes below) (Trial value, no data found) (See notes below) (See notes in the text) (See notes in the text) (See notes in the text) (See notes in the text) (See notes in the text) (See notes in the text) (See notes in the text) (Trial value, no data found) (See notes in the text) (See notes in the text) (Trial value, no data found) (See notes in the text)
the drop in primary tumor size to steady state before surgery. It has been demonstrated that tumors produce a variety of growth factors and growth inhibitors. Often the growth inhibitors are, as presented herein, released as the matrix in a latent form (L-TGFβ) and are activated by enzymes such as plasmin. On the other hand, although we have not considered the case here, tumor released proteases can release other inhibitors of growth such as endostatin, a byproduct of collagen degradation. The model we propose here is only one possible explanation for the growth of secondary tumors after the surgical removal of a primary tumor. One of the referees suggested that “perhaps the growth of secondaries was promoted by a large number of factors produced during wound healing following surgical resection.” This is also possible. However, most of the growth factor induced by the surgeon’s knife would be at or very near the wound site. This wound induced growth factor should induce secondary tumor growth very near the wound site and not remote from it as is observed. Moreover, in order for there to appear secondary tumors at a more remote distance from the primary tumor site, there must have been some
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Fig. 15 Primary tumor before removal. No secondary tumors are observed nearby. The base has a diameter of 1.0 cm, while the overall diameter is 1.5 cm. A 1.5 cm margin of healthy tissue was left at the base after excision (from De Giorgi et al., 2003, by permission).
Fig. 16 Thirty days (720 h) after primary tumor excision. Secondary tumors form some 5–7 cm from the excision site (from De Giorgi et al., 2003, by permission).
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secondary tumor cells at that remote site to begin with unless some mutational events occurred as a consequence of the resection. Finally, although the motivation for our model was suggested by Zetter’s hypothesis that the vascularization of metastatic tumors is suppressed by primary tumor expressed inhibitor, we decided to keep the model simple by considering the earlier stage in which small avascular tumors are presumed to be regulated in size by primary tumor. Thus, inhibitors of vascularization are not considered in this model. References Anderson, S.G., Buckingham, R.H., Kurland, C.G., 1983. Does codon composition influence ribosome function? EMBO 3, 91–94. Beck, L.S., DeGuzman, W.P.L., Lee, Y.X., Siegel, M.W., Amento, E.P., 1993. One systemic administration of transforming growth factor-beta 1 reverses age- or glucocorticoid-impaired wound healing. J. Clin. Invest. 92, 2841–2849. Castello, R., Estelles, A., Vazquez, C., Falco, C., Espana, F., Almenar, S.M., 2002. Quantitative real-time reverse transcription-pcr assay for urokinase plasminogen activator, plasminogen activator inhibitor type 1, and tissue metalloproteinase inhibitor type 1 gene expressions in primary breast cancer. Clin. Chem. 48, 1288–1295. Chandler, W.L., Alessi, M.C., Aillaud, M.F., Vague, P., Juhan-Vague, I., 2002. Formation, inhibition and clearance of plasmin in vivo. Haemostasis 30, 204–218. Davis, B., 1990. Reinforced random walks. Probal. Theory Relat. Fields 84, 203–229. De Crescenzo, G., Grothe, S., Zwangstra, J., Tsang, M., O’Connor-McCourt, M.D., Real time monitoring of the interactions of transforming growth factor-β (tgf-βisoforms with latency associated protein and the ectodomains of the (tgf-β). J. Biol. Chem. 276, 29632–29643. De Crescenzo, G., Pham, P.L., Durocher, Y., O’Connor-McCourt, M.D., 2003. Transforming growth factor-beta(tgf-β binding to the extracellular domain of the type ii (tgf-β receptor: Receptor capture on a biosensor surface using a new coiled-coil capture system demonstrates that avidity contributes significantly to high affinity binding. J. Mol. Biol. 328, 1173– 1183. De Giorgi, V., Massai, D., Gerlini, G., Mannone, F., Quercioli, E., Carli, P., 2003. Immediate local and regional recurrence after the excision of a polypoid melanoma: Tumor dormancy or tumor activation? Dermatol. Surg. 29, 664–667. Edelman, E.R., Nugent, M.A., Karnovsky, M.J., 1993. Perivascular and intravenous administration of basic fibroblast growth factor: Vascular and solid organ deposition. Proc. Natl. Acad. Sci. 90, 1513–1517. Edelman, E.R., Nugent, M.A., Karnovsky, M.U., 1993. Perivascular and intravenous administration of basic fibroblast gowth factor: Vascular and solid organ deposition. Proc. Natl. Acad. Sci. 90, 1513–1517. Ellis, V., Behrendt, N., Dano, K., 1991. Plasminogen activation by receptor-bound urokinase. a kinetic study with both cell-associated and isolated receptor. J. Biol. Chem. 266, 12752– 12758. Filion, R.J., Popel, A.S., 2004. A reaction-diffusion model of basic fibroblast growth factor integrations with cell surface receptors. Ann. Biochem. Eng. 32, 645–663. Glotzman, J., Mikula, M., Andreas, E., Schulte-Hermann, R., Foisner, R., Beug, H., Mikulits, W., 2004. Molecular aspects of epithelial cell plasticity;implications for local tumor invasion and metastasis. Mutat. Res. 566, 9–20. Guba, M., Cernaianu, G., Koehl, G., Geissier, E.K., Jauch, K., Anthuber, M., Falk, W., Steinbauer, M., 2001. A primary tumor promotes dormancy of solitary tumor cells before inhibiting angiogenesis. Cancer Res. 61, 5375–5379. He, L., Niemeyer, B., 2003. A novel correlation for protein diffusion coefficients based on molecular weight and radius of gyration. Biotechnol. Prog. 19, 544–548. Jung, S.P., Siegrist, B., Hornick, C.A., Wang, Y.-Z., Wade, M., Anthony, C.T., Woltering, E.A., 2002. Effect of humen recombinant endostatin protein on human angiogenesis. Angiogenesis 5, 111–118.
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