A MATHEMATICAL MODEL FOR THE REGULATION OF TUMOR DORMANCY
BASED ON ENZYME KINETICS
KHALID BOUSHABA & HOWARD A. LEVINE & MARIT NILSEN-HAMILTON
Abstract. In this paper we present a two compartment model for tumor dormancy based on an idea of
Zetter [45] to wit: The vascularization of a secondary (daughter) tumor can be suppressed by 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 [8]. 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 five to seven centimeters from the primary site within a period of one
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.
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 [19], the author reports statistics that show that when the
probability of metastatic tumors is small, the survival rate in cancers such as prostate, breast, colorectal
and 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 [35] as quoted in [13] 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 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 [44] 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 [45] 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 T GF β in the incative form have a long half life and are able to diffuse
relatively long distances through the ECM as well as to travel through the existing vasculature to suppress
remote tumors. Experimental evidence for this is described in [8, 14, 17, 43, 16].
Date: April 15, 2005.
1
2
KHALID BOUSHABA & HOWARD A. LEVINE & MARIT NILSEN-HAMILTON
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.
Outline
• Section 2 is concerned with the relevant biochemical kinetics for the model.
• In Section 3 the modeling of tumor cell proliferation and apoptosis is discussed.
• In Section 4 the partial differential equations are replaced by a two compartment model of twelve
ordinary differential equations.
• Section 5 is devoted to a discussion of the equilibrium states.
• Section 6 includes a discussion of the numerical simulations and the conclusions one can draw from
them.
• In Section 8 some experimental evidence from the literature is given that could be explained by the
model.
• Section 7 contains a discussion of some extensions of the model to other types of geometries and
systems.
2. Biochemistry
The overall idea is the following: 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 GF 2, 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
(2.1)
k
{Ra G} →2 C + Ra .
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 GF 2 for the
growth factor. In its turn, the enzyme uP A will degrade the matrix plasminogen Pg by hydrolysis of certain
peptide bonds:
k3
C + Pg {CPg },
(2.2)
k−3
k
{CPg } →4 C + Pg0 + Pm
where now Pm is a second enzyme, here plasmin. Here Pg0 is a symbol which represents most of the products
of proteolysis. The tumor cells also express T GF β` , the latent form of T GF β, a stable protein. We shall
k2
1It is entirely reasonable that {R V } →
mC + Ra where m ≥ 1 reflects the turnover rate of growth factor to protease during
a
the cell cycle. For simplicity as well as for the lack of any experimental evidence to the contrary, we take m = 1 here.
A MODEL FOR THE REGULATION OF TUMOR DORMANCY
Table 1.
3
Notation
species
notation
concentration
receptor
fibroblast growth factor, FGF
urokinase plasminogen activator, uPA
tissue growth factor beta T GF β
latent T GF β` ,
plasminogen
plasmin
R
G
C
Ia
Ii
Pg
Pm
[R]
[G]
[C]
[Ia ]
[Ii ],
[Pg ]
[Pm ]
denote it by the shorter notation Ii . The plasmin degrades the latent form of T GF β, to produce the active
form, T GF β, which will be denoted by Ia , via the mechanism:
k5
Ii + Pm {Ii Pm },
(2.3)
k−5
k
{Ii Pm } →6 Ia + L + Pm .
Here L denotes LAP , the latency associated peptide. Experimental evidence for this is found in [25, 26, 33,
34, 37]. T GF β functions to block receptor signals, inhibiting the production of uP A via the equilibrium:
(2.4)
νe
Ra + Ia {Ri },
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 T GF β from the latent form T GF β` . This in turn inactivates cell receptors. As more cell receptors 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 T GF β 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 Figure 1.
Remark 1. The model can be extended to include extracellular matrix (ECM) proteolysis. For example,
plasmin is a rather non specific enzyme which degrades matrix collagen and other ECM proteins (denoted
here by F ) via
k7
Pm + F {Pm F },
(2.5)
k−7
k
{Pm F } →8 F 0 + Pm .
where F 0 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 negative feedback loop. However, in order to keep things as simple as possible, we have not included
fibronectin proteolysis here.
4
KHALID BOUSHABA & HOWARD A. LEVINE & MARIT NILSEN-HAMILTON
Overview of the biochemical pathway.
Cell based protein
Extracellular protein
Competing species
+
C
Pg
Pm
+
C
+
G
+
Ii
Ra
+
Ia
+
Pm
Ri
Figure 1. This figure summarizes the biochemical pathway in the text, chemical equations
(2.1), (2.2), (2.3) and (2.4). Notice that growth factor (G) competes with TGFβ (Ia ) for
the active receptor Ra . This is a typical example of a negative feedback loop.
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),
(2.6)
[{Ri }](t) = [Ra ](t)[I](t)/νe ,
[Nj ](t) = κ[Rj ](t), where j = i, a.
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 [21]
to be of order unity.
However, before writing down the laws of mass action for (2.1)-(2.4), 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 T GF β.
4. Plasminogen is in such excess and is so stable that we may take its concentration to be constant.
5. The half life of T GF β is shorter than that of the latent form so that its diffusion in the ECM may
be neglected.
A MODEL FOR THE REGULATION OF TUMOR DORMANCY
5
6. F GF 2 and latent T GF β diffuse through the surrounding extracellular matrix (ECM) with different
diffusivities.
7. F GF 2 has a shorter half life than stabilized inhibitors such as latent T GF β.
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 (2.1)-(2.4). Doing this, we have
(2.7)
∂[G]
∂t
∂[{Ra G}]
∂t
∂[C]
∂t
∂[{Pg C}]
∂t
∂[Pm ]
∂t
∂[{Ii Pm }]
∂t
∂[Ia ]
∂t
∂[Ii ]
∂t
= k−1 [{Ra G}] − k1 [G][Ra ] + σg [Ra ] − µg [G] + Dg ∆[G],
= −(k−1 + k2 )[{Ra G}] + k1 [G][Ra ],
= k2 [{Ra G}] − µc [C] + Dc ∆[C],
= −(k−3 + k4 )[{Pg C}] + k3 [Pg ][C],
= k4 [{CPg }] − 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 ],
where ∆ denotes the n-dimensional diffusion operator (n = 1, 2, or 3). Here and throughout the remainder
we reserve the constants σz , Dz and µz for cell expression of protein, diffusion coefficient of protein and
decay rate of protein Z respectively. The constants σz are proportional to the number of mRN A 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 (2.7) to zero, with the equations (2.6) to obtain
(2.8)
[Ra ] =
[R]
.
1
1 + νe [Ia ] + [G]/Km
In order to simplify the notation a bit as well as to emphasize the space and time distribution of the proteins,
we write
(2.9)
[G] = g(x, t),
[N ] = η(x, t)
[C] = c(x, t),
[Ia ] = ιa (x, t)
[Pm ] = pm (x, t),
[Ii ] = ιi (x, t).
. Then
[Ra ] =
[R]
.
1 )
(1 + νe ιa + g/Km
6
KHALID BOUSHABA & HOWARD A. LEVINE & MARIT NILSEN-HAMILTON
i
i
We employ the notation Km
= (k2i + k−(2i−1) )/k2i−1 and λi = k2i /Km
. Since [Pg ] is in excess, we may
p
write λ2 = λ2 [Pg ]. There results following partial differential equations of transport for G, C, Pm , Ia , Ii :
σg − λ1 g
η
,
1 η
1 + νe ιa + g/Km
0
λ1 g
η
∂t c = Dc ∆c − µc c +
,
1 η
1 + νe ιa + g/Km
0
∂t pm = Dp ∆pm − µp pm + λp2 c,
∂t g = Dg ∆g − µg g +
(2.10)
∂t ιa = Da ∆ιa − µa ιa + λ3 ιi pm ,
η
σi
1
1 + νe ιa + g/Km η0
where we have renormalized the constants so that λ1 = k2 η0 κ where η0 is the carrying capacity of the tumor
cells while σz κη0 is replaced by σz for z as one of the indices g, i.
Notice that in this mechanism, the first Michealis constant is increased by the presence of the inhibitor.
Such inhibitory mechanisms are termed ”competitive” [38], p.360.
∂t ιi = Di ∆ιi − µi ιi − λ3 ιi pm +
3. Tumor cell movement
A description of cell movement is a bit more complicated. However, an easy derivation can be given based
on 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
(3.1)
= −∇ · Jη + φ(g)η(1 − η/η0 ) − µη η
∂t
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 reflect 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
2More 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
A MODEL FOR THE REGULATION OF TUMOR DORMANCY
7
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)
(3.2)
One can give a probabilistic derivation of this form of the equation based on the ideas of [5] as was done in
[30] 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 [21, 22, 20, 23].3
In general, we will have Da > Di >> Dη while µg , µa >> µi . This latter condition is the mathematical
expression that growth factor and T GF β have much shorter half lives than latent T GF β.4.
4. Compartment Model
To the equations (2.10), (3.2), must be appended five initial conditions and boundary conditions5 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 bio-chemicals 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 (2.10),(3.2). We envisage two tumors, one with an initial cell mass given by ηp (0) = η0 Np (0)
and one with an initial cell mass ηs (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, since 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:
σg − λ1 gp
Np
∂t gp = (Dg /L2 )(gs − gp ) − µg gp +
,
1
1 + νe ιa,p + gp /Km η0
Np
λ1 gp
,
∂t cp = (Dc /L2 )(cs − cp ) − µc cp +
1
1 + νe ιa,p + gp /Km η0
(4.1)
∂t pm,p = (Dp /L2 )(pm,s − pm,p ) − µp pm,p + λp2 cp ,
∂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 =
σi
Np
1
1 + νe ιa,p + gp /Km η0
λgp
Np (1 − Np /η0 ) − µη Np .
K + gp
there is 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.
3Another 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 [40] 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.
8
KHALID BOUSHABA & HOWARD A. LEVINE & MARIT NILSEN-HAMILTON
while in the secondary compartment:
σg − λ1 gs
Ns
,
1 η
1 + νe ιa,p + gs /Km
0
Ns
λ1 gs
,
∂t cs = (Dc /L2 )(cp − cs ) − µc cs +
1 η
1 + νe ιa,p + gs /Km
0
∂t gs = (Dg /L2 )(gp − gs ) − µg gs +
(4.2)
∂t pm,s = (Dp /L2 )(pm,p − pm,s ) − µp pm,s + λp2 cs ,
∂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 η
1 + νe ιa,p + gs /Km
0
λgs
Ns (1 − Ns /η0 ) − µη Ns .
K + gs
Thus our system of six partial differential equations has been ”reduced” to a system of twelve 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 (4.1).
5. Equilibrium states for (4.1), (4.2).
Observe that the right hand side of (4.1) can be obtained from that of (4.2) 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:
σg − λ1 gp
Np
,
1 η
1 + νe ιa,p + gp /Km
0
λ1 gp
Np
0 = (Dc /L2 )(cs − cp ) − µc cp +
,
1 η
1 + νe ιa,p + gp /Km
0
0 = (Dg /L2 )(gs − gp ) − µg gp +
(5.1)
0 = (Dp /L2 )(pm,s − pm,p ) − µp pm,p + λp2 cp ,
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=
σi
Np
,
1 η
1 + νe ιa,p + gp /Km
0
λgp
Np (1 − Np /η0 ) − µη Np .
K + gp
6The argument for replacing the term D ∆f by (D /L2 )(g − g ) in the primary compartment in the first equation for
g
g
s
p
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 = Df (gs − gp )/L2 while at x = L, ∂t gs = −Js /L = Dg (gp − gs )/L2 . This is reasonable since 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.
A MODEL FOR THE REGULATION OF TUMOR DORMANCY
9
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:
σg − λ1 gs
Ns
,
1 η
1 + νe ιa,s + gs /Km
0
Np
λ1 gs
0 = (Dc /L2 )(cp − cs ) − µc cs +
,
1 η
1 + νe ιa,s + gs /Km
0
0 = (Dg /L2 )(gp − gs ) − µg gs +
(5.2)
0 = (Dp /L2 )(pm,p − pm,s ) − µp pm,s + λp2 cs ,
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 (5.1) and (5.2)
Kµη
,
λ(1 − Np /η0 ) − µη
Kµη
gs = η(Ns ) =
.
λ(1 − Ns /η0 ) − µη
gp = η(Np ) =
(5.3)
From the first equation in each of (5.1) and (5.2) we obtain a simple linear system in gp , gs which we can
solve to write
−1 Dg µg
Dg
Dg
2
gp = η(Np ) =
+
µ
+
µ
R(N
,
ι
)
,
R(N
,
ι
)
+
g
s
a,s
p
a,p
g
L2
L2
L2
(5.4)
= G(Np , Ns , ιa,p , ιa,s ),
gs = η(Ns ) = G(Ns , Np , ιa,s , ιa,p ),
where
R(x, y) =
(5.5)
σg − λ1 η(x)
x
.
1
1 + νe y + η(x)/Km η0
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 T GF β at both sites
or conversely.
From the second equation in each of (5.1) and (5.2) we can write
−1 Dc µc
Dc
Dc
2
cp =
+ µc
+ µc S(Np , ιa,p ) +
S(Ns , ιa,s ) ,
L2
L2
L2
= C(Np , Ns , ιa,p , ιa,s ),
(5.6)
cs = C(Ns , Np , ιa,s , ιa,p ),
where
(5.7)
S(x, y) =
λ1 η(x)
x
.
1 η
1 + νe y + η(x)/Km
0
10
KHALID BOUSHABA & HOWARD A. LEVINE & MARIT NILSEN-HAMILTON
From the third equation in each of (5.1) and (5.2) we can write
−1 Dp µp
Dp
Dp
2
pm,p = λp2
+
µ
+
µ
C(N
,
N
,
ι
,
ι
)
+
C(N
,
N
,
ι
,
ι
)
p
p
s a,p a,s
s
p a,p a,s ,
p
L2
L2
L2
(5.8)
= Pm (Np , Ns , ιa,p , ιa,s ),
pm,s = Pm (Ns , Np , ιa,s , ιa,p ).
Finally, by adding the fourth and fifth equation in each of (5.1) and (5.2) again obtain a linear system for
which we can write
−1 Di
Di
Di µi
2
+ µi
+ µi J(Np , ιa,p , ιa,s ) +
J(Ns , ιa,s , ιa,p ) ,
ιi,p =
L2
L2
L2
(5.9)
= I(Np , Ns , ιa,p , ιa,s ),
ιi,s = I(Ns , Np , ιa,s , ιa,p )
where
(5.10)
J(x, y, z) = (Da /L2 )z − ((Da /L2 ) + µa )y +
σi
x
.
1 η
1 + νe y + η(x)/Km
0
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
J(Ns , ιa,s , ιa,p ) ≥ 0
J(N
,
ι
,
ι
)
+
i
p a,p a,s
L2
L2
(5.11)
Di
Di
+
µ
J(N
,
ι
,
ι
)
+
i J(Ns , ιa,s , ιa,p ) ≥ 0.
p a,p a,s
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 T GF β 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
Da µi
Di µa
Di µa
−
ιa,s −
+ µa µi +
2
2
L
L
L2
Da µi
Di µa
Di µa
−
ιa,p −
+ µa µi +
2
2
L
L
L2
Da µi
ιa,p ≥ 0,
L2
Da µi
ιa,s ≥ 0.
L2
Neither of these can hold if µi /µa < Di /Da . For our case, the diffusion coefficient of latent T GF β is larger
than for T GF β, a statement which follows from the Stokes-Einstein equation since the molecular weight
of the former is larger than that of the latter. On the other hand the half life of T GF β is much smaller
than that of the latent form. Therefore µi /µa < 1 < Di /Da and the constraint conditions (5.11) are not
automatically fulfilled.
Clearly J(Np , ιa,p , ιa,s ) ≥ 0 will hold if the pair (Np , ιa,p ) satisfies
(5.12)
−((Da /L2 ) + µa )ιa,p +
σi
Np
≥0
1 η
1 + νe ιa,p + η(Np )/Km
0
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 (5.11) will hold and when both fail, so do both of (5.11). However one may hold and the other
fail with both of (5.11) holding.)
A MODEL FOR THE REGULATION OF TUMOR DORMANCY
11
Remark 3. We can glean a bit more from (5.12) as follows. Define the following variables:
α = Da /L2 + µa ,
ξp = 1 + η(Np )η0 ,
σi Np νe
βp =
.
η0
Then (5.12) will hold if and only if
(5.13)
2βp /(αξp )
0 ≤ ιa,p ≤ q
4βp /(αξp2 ) + 1 + 1
holds.
Inequality (5.13) 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:
s
βp
0 ≤ ιa,p ≤
.
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
re-normalize the system (2.10)-(3.2) and consequently the two compartment model as follows:
Then the system of differential equations may be written in the form:
σG − Λ1 G
∂τ G = DG ∆G − µG G +
N,
1 + Ia + G
Λ1 G
∂τ C = Dc ∆C − µC C +
N,
1 + Ia + G
∂τ Pm = DP ∆Pm − µP Pm + Λ2 C,
(6.14)
∂τ Ia = DA ∆Ia − µA Ia + Λ3 Ii Pm ,
σI
∂τ Ii = Di ∆Ii − µI Ii − Λ3 Ii Pm +
N,
1 + Ia + G
∂N
N
ΛG
= ∇· Dη ∇ N ln
+
N (1 − N ) − µN N .
∂τ
τ1 (G)τ2 (Ii )
K+G
We have included a number of figures to illustrate the sensitivity of the outcomes to the various parameters
in the model. In Figures Ia-IIb, the vertical axes are in dimensionless units. In the remaining figures, the
vertical axis units are time in days.
The first set of figures (Figures Ia-IIb) 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 therefor 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
12
KHALID BOUSHABA & HOWARD A. LEVINE & MARIT NILSEN-HAMILTON
Table 2. Dimensionless variables
Variable
x
t
g
c
pm
ιa
ιi
η
1
λ1 = k2 /Km
2
λ2 = k4 /Km
3
λ3 = k6 /Km
σg
σi
Dg
Dc
Dp
Da
Di
Dη
µg
µc
µp
µa
µi
µη
λ
K
Dimensionless variable
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
Λ1 = T λ 1
p 1
1
Λ2 = (T λ2 Km )/[Pg ] = T λ2 Km
Λ3 = T λ3 [Pg ]
1
σG = T σg /Km
σI = (T νe σi )
DG = T Dg /L2
DC = T Dc /L2
DP = T Dp /L2
DA = T Da /L2
DI = T Di /L2
Dη = T Dη /L2
µG = T µg
µC = T µc
µP = T µp
µA = T µa
µI = T µi
µN = T µη
1
Λ = T λ/Km
1
K = K/Km
secondary tumor to some larger size. We might expect therefore, the existence of a threshold distance
L∞ such that it 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 above 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 Figure IIIa for the two cases τ = 10
(days) and τ = ∞.)
Based upon the idea that the primary tumor is the source of sufficient inhibitor to suppress the secondary
tumor, it is reasonable to expect that Lτ1 < Lτ2 < L∞ , if τ1 < τ2 < ∞ i. e., the primary tumor will
control more remote secondary tumors the longer it remains in the patient. However, L∞ is the greatest
distance beyond which the primary tumor cannot control the secondary tumor. Also, on logical grounds,
limτ →+∞ Lτ = L∞ . In Figure IIIb, we have plotted φ(τ ) = Lτ as a function of τ .
A MODEL FOR THE REGULATION OF TUMOR DORMANCY
13
This leads us define the half saturation time for Lτ to be that time τm for which Lτm = 0.5L∞ . We say
that one set of parameters in the table gives rise to more efficient tumor suppression than that of a second
set of parameters if its relaxation time is smaller (assuming we begin with the same primary and secondary
tumor masses).
In Figure Ia below, in the presence of the primary tumor (solid curves) we see that the secondary tumor
(dashed curves) will shrink in size and the concentrations of its growth and inhibitor production will decline.
(The secondary tumor is L = 7cm away from the primary tumor.) In Figures Ib(i-ii), the primary tumor
was removed after 10 days (solid curves). We see (dashed curves) that the secondary tumor, again with
the same value of L, after a delay of roughly 400 days, now grows to the size 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 Figure IIa, if the secondary tumor is very close to the primary tumor (L = 1.0cm),
then, even the removal of the primary tumor will not induce the growth of the secondary tumor. On the other
hand, as we see in Figure IIb, if the secondary tumor is very remote from the primary tumor, L = 50cm,
the growth of the secondary tumor is uninhibited by that of the primary tumor.
This is illustrated in Figure IIIa. 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 it 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 tumor than if we do not. These times approach
each other as L → ∞.
Put another way, we see from Figure IIIa that surgery always favors secondary tumor growth when
5.5 < L < 8.5 while it cannot prevent 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 Figures IVa-e, we fixed all of 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 Figure IVc. Because the number of mRNA’s per cell is not known precisely
(it could be as few as 5 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 (Fig VIa). 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 Figures Va-c, 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 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 Figures VIa-e, we see how the efficiency varies for several values of λ3 , the T GF β production rate.
Here we see that for high T GF β production rates, The secondary tumor will die out at distances fairly
remote from the primary independently of 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 Figures VIIa-c, 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 [.793, .797].
14
KHALID BOUSHABA & HOWARD A. LEVINE & MARIT NILSEN-HAMILTON
Table 3.
Equation
time scale
(2.4)
(2.10)
(3.2)
constants
T
νe
Dg
Dp
Dc
Di
Da
µg
µp
µc
µi
µa
σg
σi
[Pg ]
k1
k−1
k2
1
Km
λ1
k4
2
Km
3
λ3 = k6 /Km
η0
λ
K
µη
Numerical values used in simulations
values
1.0 hour
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 − 28.0h−1
0.154 − 0.0495h−1
8.38h−1
0.385h−1
6.93h−1
1.50(10−2 )µMh−1
0.45µMh−1
1.01µM
2.5(102 )(µM h)−1 -1.5(104 )(µM h)−1
0.048 min−1 =2.88h−1
1.59(10−2 )h−1
(k−1 + k2 )/k1
1.5(104 )(µM )−1 h−1
2.68(103 )h−1
25.0µM
0.745(hµM)−1
1012 /l= 109 /(cm)3
6.25(10−3 )h−1
1.68(10−2 )µM
1.0(10)−3 h−1
Notes and Comments
(See notes in the text.)
”
”
”
”
”
”
”
”
”
”
(See notes below)
(Trial value, no data found.)
(See notes below)
(See notes in the text.)
”
”
”
”
”
”
(Trial value, no data found.)
(See notes in the text.)
”
(Trial value, no data found.)
(See notes in the text.)
In Figures VIIIa-e, 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 Figure VIIIe),
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.
In Figures IXa-c, 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 GF 2, plasmin, uPA, TGFβ` and TGFβ using the value of
−1/3
Dg found from [12], page 652 at 37o C. (The diffusion coefficient is roughly proportional to Mw
for large proteins where Mw is the molecular weight of the protein in question. However, one has to
A MODEL FOR THE REGULATION OF TUMOR DORMANCY
2.
3.
4.
5.
6.
7.
8.
15
be careful about the use of this assumption. See [15]. There the authors remark that the shape of
the molecule can have a very significant effect on the magnitude of the diffusion coefficient.)
The rate constants k1 , k−1 were taken from [12], page 650. (The authors took these from [29].)
The catalytic constant k2 was estimated as follows: The number of amino acids (432) in uP A is
divided by 20 since it is known that the overall cellular transcription-translation rate for proteins is
roughly ten-twenty amino acids per second[1, 31]. This gives an assembly time per mRNA molecule
per cell of a single uPA molecule of 21.6 seconds or 0.012 hrs. That is, 1.666 protein molecules
are assembled from a single mRNA molecule in an hour. From [3], there are around 45 mRNA
molecules 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 ) micro
moles and 1.25(10−3 ) micro moles of uPA being produced per hour per liter of cells. Suppose the
enzyme equation is at steady state, with no inhibitor present and the growth factor concentration
1
/2. With [C] = 6.25(10)−4 µM and µc = 8.38h−1 (the half life for uPA is 5 minutes, see
is at Km
below), 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
The constant λ is roughly 1/32 of an hour, the turnover rate for malignant cells being roughly 32
hours 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 cubic microns. The apoptosis rate µη is taken from [20].
From the literature, [24, 32], we have found estimates of 84-130 micrograms per liter of plasminogen
in plasma. A good estimate for the molecular weight of plasma is around 88 micrograms per micro
mole. We therefore used a value for the concentration of plasminogen in plasma of 1.0µM.
The half life for T GF β may found in [2, 39, 46] while those for T GF β` are from [39]. Half lives
estimates for FGF2 in plasma were found to be in the range of 1.5 to 3 minutes [10, 41]. These
should be close to the values in tissue. We took the uP A half life from the data in [9, 18, 42]. Half
life estimates for plasmin may be found in [4, 36].
The value of σg cannot be found directly. We estimated the value of σg as follows: Using the
data in [27], 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 cubic microns, we see that the micromolarity rate is [900h−1 /(6 ×
1023 )(M)−1 ] × 106 µM/[102 (µm)3 × 10−12 cm3 × (µm)−3 × 10−3 l × cm−3 ] = 0.015µM/h.
Since we do not know the value of σi we must content ourselves with the sensitivity analysis
illustrated in Figure V. This presupposes an mRNA/cell range of around 3 − 8 molecules per cell for
TGFβ.
We estimated the equilibrium constant, νe , for the TGFβ inhibition of FGF receptor signaling was
estimated using the results of [7, 6]. There it is shown that when T GF β is fully bound to its own
receptors, one can expect an equilibrium constant in the range 5(10−4 )µ M ≤ (1/νe ) ≤ 5(10−3 )µM.
2
From [11] 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 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/sec, 0.67µM ) with experimental errors of around 50%. In solution, they give
(kcat , Km ) ≈ (0.73/sec, 25µM ). We used the latter values (converted to reciprocal hours) for the
tabular entries.
7The units are correct here because we have written the cell concentration as [R] = η × δ where δ is the number of cell
0
T
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 .
16
KHALID BOUSHABA & HOWARD A. LEVINE & MARIT NILSEN-HAMILTON
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 many kinetic reactions involving enzyme kinetics. (See [28].)
The cause of stiffness in our problem can be tied to the fact that the individual equations in the system
have widely varying time scales.
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 Figures V-IX appear to be somewhat oscillatory about some mean
curve; For example, in Figure V, the calculation of each point of the curve requires an integrations 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. Discussion and future work
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 doesn’t 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 Figure 1a. The solid line shows 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 naturally occurring tissue proteins
such as plasminogen. On the other hand, although we have not considered the case here, tumor released
proteases can induce 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 secondary tumor cells
at that remote site to begin with unless some mutational events occurred in 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.
8. Comparison with experiment
In [8] the authors report on the removal of a ”polypoid lesion at the cutaneous level, measuring about
15mm and with a diameter of 10mm in the upper part....” The authors further report ”Histopathologically,
a diffuse proliferation of epitheliod, pigmented atypical melanocytes was observed beneath the dermis.” See
A MODEL FOR THE REGULATION OF TUMOR DORMANCY
17
Figure X (from [8] by permission.) Thirty days after removal of the lesion, the authors report: ”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 Figure XI (from [8] by permission.) below.
18
KHALID BOUSHABA & HOWARD A. LEVINE & MARIT NILSEN-HAMILTON
Figure Ia. Primary and secondary tumor properties for L=7, The secondary
tumor decays without surgery. (Time in days on horizontal axes.)
-4
10
0.03
x 10
secondary site time course
primary site time course
FGF2 (g)
0.02
5
0.01
uPA(c)
0
0
0
200
400
600
800
-3
20
x 10
0
200
400
600
800
8
6
15
plasmin (p)
TGFβ (Ia)
4
10
2
5
0
0
0
200
400
600
0
800
200
400
600
800
600
800
1
10000
cell density (N)
LTGFβ (Ii)
5000
0.5
0
0
200
400
600
800
0
0
200
400
A MODEL FOR THE REGULATION OF TUMOR DORMANCY
19
Figure Ib(i). Primary and secondary tumor properties for L=7, for early times (t<60 days)
after surgery at t=10 days. (Time in days on horizontal axes.)
-4
10
0.03
x 10
FGF2 (g)
secondary site time course
primary site time course
0.02
5
uPA(c)
0.01
0
0
0
20
40
0
60
0.2
20
40
60
8
6
0.15
plasmin (p)
0.1
4
0.05
2
TGFβ (Ia)
0
0
0
20
40
60
0
8000
20
40
60
40
60
1
6000
cell density (N)
4000
0.5
LTGFβ (Ii)
2000
0
0
0
20
40
60
0
20
Figure Ib(ii). Primary and secondary tumor properties for L=7, for late times
(t>60) 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
0.06
200
400
600
800
2
TGFβ (Ia)
0.04
plasmin (p)
1
0.02
0
0
200
400
600
800
200
400
600
800
1
4000
cell density (N)
LTGFβ (Ii)
0.5
2000
0
secondary tumor
recovery
0
200
400
600
800
200
400
600
800
20
KHALID BOUSHABA & HOWARD A. LEVINE & MARIT NILSEN-HAMILTON
Figure IIa. 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
-4
on horizontal axes. )
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)
0.1
TGFβ (Ia)
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
Figure IIb. 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
800
primary site time course
2
0
600
secondary 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
A MODEL FOR THE REGULATION OF TUMOR DORMANCY
21
Figure III, Comparison of extinction and growth times for the
secondary tumor with and without removal of the primary tumor.
800
700
vertical asymptote at L∞=8.5
vertical asymptote at L10=5.5
Time in days
600
500
400
300
growth time without surgery
extinction time without surgery
growth time, after surgery
extinction time, after surgery
200
100
2
4
6
8
10
12
Distance between primary and secondary tumors, L, in cm.
14
Figure IIIb. Plot of φ(τ)=Lτ/L∞, the ratio of transition lengths from extinction
-1
to growth for the secondary tumor. (Half saturation time, τm=φ (1/2).)
1
τ
φ(τ)=L /L
∞
0.8
0.6
Half saturation time coordinates
≈(3.5 days, 0.5)
0.4
curve given by φ(τ)=0.8τ/(τ+7.5)+0.22
data points computed from system of odes.
0.2
0
0
20
40
60
Removal time , τ, for the primary tumor (in days).
80
200
22
KHALID BOUSHABA & HOWARD A. LEVINE & MARIT NILSEN-HAMILTON
Figure IV, Variation of efficiency with inter tumor distance and various σ
I
(Inter tumor distance on horizontal axis, time on vertical axis.)
300
800
800
IV-c
σI=0.45
250
600
600
IV-b
σ =0.2250
200
I
IV-a
σI=0.0225
150
100
800
4
8
400
400
200
200
12
1
2
3
4
5
5
10
500
IV-d
σI=0.47
600
15
IV-e
σI=0.5625
400
growth time without surgery
extinction time without surgery
300
growth time, after surgery
400
extinction time, after surgery
200
200
5
10
15
100
20
5
10
15
Figure V. Variation of effiency with σI for various inter tumors distances. (Time on the vertical axes).
700
L=3
growth time without surgery
extinction time, after surgery
growth time without surgery
extinction without surgery
600
500
L=20L=3
400
300
200
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
600
L=7
500
400
300
0.4
0.41
0.42
0.43
0.44
0.45
0.46
0.47
0.48
0.49
700
600
L=20
500
400
0.472
0.473
0.474
0.475
σI
0.476
0.477
0.478
0.479
0.5
20
A MODEL FOR THE REGULATION OF TUMOR DORMANCY
23
Figure VI, Variation of efficiency with inter tumor distance and various λ values
3
(Inter tumor distance on horizontal axis, time on vertical axis.)
800
800
240
VI-c
λ =0.745
3
220
600
600
VI-b
λ =0.3725
3
200
VI-a
λ =0.0149
400
400
200
200
3
180
160
1
2
3
4
5
1
2
3
4
5
5
10
15
800
600
VI-d
λ3=0.79
400
growth time without surgery
extinction time without surgery
300
growth time, after surgery
400
extinction time, after surgery
VI-e
λ3=1
200
200
10
20
5
30
10
Figure VII. Variation of effiency with λ3 for various inter tumor distances. (Time in days on the vertical axes).
600
L=3
400
200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
700
growth time without surgery
extinction time without surgery
growth time, after surgery
extinction time, after surgery
600
L=7
500
400
300
0.6
0.65
0.7
0.75
0.8
0.85
0.79
0.8
0.81
700
600
L=20
500
400
0.76
0.77
0.78
λ
3
0.9
15
20
24
KHALID BOUSHABA & HOWARD A. LEVINE & MARIT NILSEN-HAMILTON
Figure VIII, Variation of efficiency with inter tumor distance and various values of K.
800
800
52
VIII-c
K=0.0165
VIII-b
K=0.016
48
VIII-a
K=0.084
600
600
400
400
200
200
44
40
36
2
4
6
8
0.5
1
1.5
0.5
2
1
1.5
2
800
VIII-d
K=0.0168
400
VIIII-e
K=0.017
growth time without surgery
600
extinction time, after surgery
300
growth time without surgery
extinction without surgery
400
200
200
100
5
10
15
5
10
15
Figure IX: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
K
1.6807
1.6807
1.6808
1.6809
1.6809
x10-2
20
A MODEL FOR THE REGULATION OF TUMOR DORMANCY
Figure X. (From [8], by permission.)
Primary tumor before removal. No
secondary tumors are observed nearby.
The base has a diameter of 1.0 cm while
over all diameter is 1.5 cm. A 1.5 cm
margin of healthy tissue was left at the
base after excision.
Figure XI. (From [8], by permission.) The authors
remark that thirty days (720 hours) after removal
of the primary tumor, secondary tumors formed
some 5-7 cm. from the operation scar.
25
26
KHALID BOUSHABA & HOWARD A. LEVINE & MARIT NILSEN-HAMILTON
References
[1] S. G. Anderson, R. H. Buckingham, and C. G. Kurland, Does codon composition influence ribosome function?, EMBO,
3 (1983), pp. 91–94.
[2] L. S. Beck, W. P. L. DeGuzman, Y. X. Lee, M. W. Siegel, and E. P. Amento, One systemic administration of
transforming growth factor-beta 1 reverses age- or glucocorticoid-impaired wound healing, J. Clin. Invest., 92 (1993),
pp. 2841–2849.
[3] R. Castello, A. Estelles, C. Vazquez, C. Falco, F. Espana, S. M. Almenar, F. C., and A. J., 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 (2002), pp. 1288–1295.
[4] W. L. Chandler, M. C. Alessi, M. F. Aillaud, P. Vague, and I. Juhan-Vague, Formation, inhibition and clearance
of plasmin in vivo, Haemostasis, 30 (2002), pp. 204–218.
[5] B. Davis, Reinforced random walks, Probal. Theory Related Fields, (1990), pp. 203–229.
[6] G. De Crescenzo, S. Grothe, J. Zwangstra, M. Tsang, and M. D. O’Connor-McCourt, Real time monitoring of
the interactions of transforming growth factor-β (tgf-βisoforms with latency associated protein and the ectodomains of the
(tgf-β.
[7] G. De Crescenzo, P. L. Pham, Y. Durocher, and M. D. O’Connor-McCourt, 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
(2003), pp. 1173–1183.
[8] V. De Giorgi, D. Massai, G. Gerlini, F. Mannone, E. Quercioli, and P. Carli, Immediate local and regional recurrence
after the excision of a polypoid melanoma: tumor dormancy or tumor activation?, Derm. Surgery, 29 (2003), pp. 664–667.
[9] E. R. Edelman, N. M. A., and M. J. Karnovsky, Perivascular and intravenous administration of basic fibroblast growth
factor: vascular and solid organ deposition, Proc. Natl. Acad. Sci., 90 (1993), pp. 1513–7.
[10] E. R. Edelman, M. A. Nugent, and M. U. Karnovsky, Perivascular and intravenous administration of basic fibroblast
gowth factor: vascular and solid organ deposition, Proc. Natl. Acad. Sci,, 90 (1993), pp. 1513–17.
[11] V. Ellis, N. Behrendt, and K. Dano, Plasminogen activation by receptor-bound urokinase. a kinetic study with both
cell-associated and isolated receptor, J. Biol. Chem., 266 (1991), pp. 12752–8.
[12] R. J. Filion and A. S. Popel, A reaction-diffusion model of basic fibroblast growth factor integrations with cell surface
receptors, Ann. Biochem. Eng., 32 (2004), pp. 645–663.
[13] J. Glotzman, M. Mikula, E. Andreas, R. Schulte-Hermann, R. Foisner, H. Beug, and W. Mikulits, Molecular
aspects of epithelial cell plasticity;implications for local tumor invasion and metastasis, Mutation Res., 566 (2004), pp. 9–
20.
[14] M. Guba, G. Cernaianu, G. Koehl, E. K. Geissier, K. Jauch, M. Anthuber, W. Falk, and M. Steinbauer, A primary
tumor promotes dormancy of solitary tumor cells before inhibiting angiogenesis, Cancer Res., 61 (2001), pp. 5375–5379.
[15] L. He and B. Niemeyer, A novel correlation for protein diffusion coefficients based on molecular weight and radius of
gyration, Biotechnol. Prog., 19 (2003), pp. 544–548.
[16] S. P. Jung, B. Siegrist, C. A. Hornick, Y.-Z. Wang, M. Wade, C. T. Anthony, and E. A. Woltering, Effect of
humen recombinant endostatinr protein on human angiogenesis, Angiogenesis, 5 (2002), pp. 111–118.
[17] M. Kirach, G. Schakert, and P. M. Black, Angiogenesis, metastasis and endogenous inhibition, J. Neurooncol., 50
(2000), pp. 173–80.
[18] D. F. Lazarous, M. Shou, J. A. Stiber, D. M. Dadhania, V. Thirumurti, E. Hodge, and E. F. Unger, Pharmacodynamics of basic fibroblast growth factor: route of administration determines myocardial and systemic distribution.,
Cardiovasc Res, 36 (1997), pp. 78–85.
[19] C. Leaf, Why we’re losing the war on cancer (and how to win it), Fortune, 149 (2004), pp. 76–97.
[20] H. A. Levine, S. Pamuk, B. D. Sleeman, and M. Nilsen-Hamilton, Mathematical modeling of capillary formation and
development in tumor angiogenesis: Penetration into the stroma, Bull. Math. Biol., (2001), pp. pp.801–863.
[21] H. A. Levine, B. D. Sleeman, and M. Nilsen-Hamilton, Mathematical modeling of the onset of capillary formation
initiating angiogenesis, J. Math. Biol., (2001), pp. 195–238.
[22]
, A mathematical model for the roles of pericytes and macrophages in the initiation of angiogenesis: I. the role of
protease inhibitors in preventing angiogenesis, Mathematical Biosciences, (2002), pp. 77–115.
[23] H. A. Levine, A. L. Tucker, and N.-H. M., A mathematical model for the role of cell signaling and transduction in the
initiation of angiogenesis, Growth Factors, (2002), pp. pp. 155–176.
[24] H. R. Lijnen, P. Carmeliet, A. Bouche, L. Moons, V. A. Ploplis, E. Plow, and D. Collen, Restoration of thrombolytic potential in plasminogen-deficient mice by bolus administration of plasminogen, Blood, 88 (1996), pp. 870–876.
[25] R. M. Lyons, L. E. Gentry, A. F. Purchio, and H. L. Mosesl, Mechanism of activation of latent recombinant transforming growth factor β1 by plasmin, J. Cell Biol., 110 (1990), pp. 1361–1367.
A MODEL FOR THE REGULATION OF TUMOR DORMANCY
27
[26] R. M. Lyons, J. Keski-Oja, and H. L. Mosesl, Proteyolytic activation of latent transforming growth factor-β from
fibroblast conditioned medium, J. Cell Biol., 106 (1988), pp. 1659–1665.
[27] J. N. Mansbridge, K. Liu, R. E. Pinney, R. Patch, A. Ratcliffe, and G. K. Naugnton, growth factors secreted by
fibroblasts: role in healing diabetic foot ulcers, diabetes, Obesity and Metabolism, 1 (1999), pp. 265–279.
[28] J. Murray, Mathematical Biology, Biomathematics Texts, Springer-Verlag, 1989.
[29] M. A. Nugent and E. R. Edelman, Kinetics of basic fibroblast growth factor binding to its receptor and heparan sulfate
proteoglycan: A mechanism for cooperativity, Biochemistry, 31 (1992), pp. 8876–8883.
[30] H. G. Othmer and A. Stevens, Aggregation, blow up and collapse: The abc’s of taxis and reinforced random walks,
SIAM J. Appl. Math., (1997), pp. 1044–1081.
[31] M. Y. pavlov and M. Ehernberg, Rate of translation of natural mrnas in an optimized in vitro system, Arch. Biochem.
Biophys., 328 (1996), pp. 9–16.
[32] V. A. Ploplis, P. Carmeliet, S. Vazirzadeh, I. Van Vlaenderen, L. Moons, E. F. Plow, and D. Collen, Effects of
disruption of the plasminogen gene on thrombosis, growth, and health in mice, Circulation, 92 (1995), pp. 2585–2593.
[33] Y. Sato and D. B. Rifkin, Inhibition of endothelial cell movement by perycytes and smooth muscle cells: activation of
latent tgf-beta1 like molecule by plasma during co-culture, J. Cell Biol, 109 (1989), pp. 309–315.
[34] Y. Sato, R. Tsuboi, R. Lyons, H. Moses, and D. B. Rifkin, Characterization of the activation of latent tgf-β by
co-cultures of endothelial cells and pericytes of smooth muscle cells: a self-regulating system, J. Cell Biol, 111 (1990),
pp. 757–764.
[35] M. B. Sporn, The war on cancer, Lancet, 347 (1996), pp. 1377–1381.
[36] Y. Takeda and M. Nakabayashi, Physicochemical and biological properties of human and canine plasmins, J. Clin.
Invest., 53 (1974), pp. 154–62.
[37] Y. Tsuzuki, D. Fukumura, B. Oosthuyse, C. Koike, P. Carmeliet, and R. K. Jain, Vascular endothelial growth factor
(vegf ) modulation by targeting hypoxia-inducible factor-1alpha→ hypoxia response element→ vegf cascade differentially
regulates vascular response and growth rate in tumors, Cancer Res., (2000), pp. 6248–52.
[38] D. Voet and j. Voet, Biochemistry, Second Edition, John Wiley and Sons, 1995.
[39] L. M. Wakefield, T. S. Winokur, R. S. Hollands, K. Christopherson, A. D. Levinson, and M. B. Sporn, ecombinant
latent transforming growth factor beta 1 has a longer plasma half-life in rats than active transforming growth factor beta
1, and a different tissue distribution, J Clin Invest, 86 (1990), pp. 1976–7684.
[40] F. T. Wall, Chemical Thermodynamics, Freeman, 1958.
[41] G. F. Whalen, Y. Shing, and J. Folkman, The fate of intravenously administered bfgf and the effect of heparin, Growth
Factors, 1 (1989), pp. 157–64.
[42]
, he fate of intravenously administered bfgf and the effect of heparin, Growth Factors, 1 (1989), pp. 157–64.
[43] S. S. Yoon, H. Eto, C. Lin, H. Nakamura, T. M. Pawlik, S. U. Song, and K. K. Tanabe, Mouse endostatin inhibits
the formation of lung and liver metastases, Cancer Res., 99 (1999), pp. 6251–6236.
[44] E. D. Yorke, L. Fuks, L. Norton, W. Whitemore, and C. C. Ling, Modeling the development of metastases from
primary and locally recurrent tumors: comparison with a clinical database for prostatic cancer, Cancer Res., 53 (1993),
pp. 2987–2993.
[45] B. R. Zetter, Angiogenesis and tumor metastasis, review, Ann. Rev. Me, 49 (1998), pp. 407–22.
[46] S. A. Zioncheck, T. F.and Chen, L. Richardson, M. Mora-Worms, C. Lucas, D. Lewis, J. D. Green, and J. Mordenti, pharmacokinetics and tissue distribution of recombinant human transforming growth factor beta 1 after topical and
intravenous administration in male rats, Pharm. Res., 11 (1994), pp. 213–220.
28
KHALID BOUSHABA & HOWARD A. LEVINE & MARIT NILSEN-HAMILTON
Author addresses
Kahlid Boushaba
Department of Mathematics
Iowa State University
Ames, Iowa, 50011
United States of America
boushaba@iastate.edu
and
Howard A. Levine
Department of Mathematics
Iowa State University
Ames, Iowa, 50011
United States of America
halevine@iastate.edu
and
Marit Nilsen-Hamilton
Department of Biochemistry, Biophysics and Molecular Biology
Iowa State University
Ames, Iowa, 50011
United States of America
marit@iastate.edu