Received 2 May 2016; accepted 27 June 2016; published 30 June 2016
1. Introduction
1.1. Summary of the Paper
We continue the study of the cancer model from Larsen (2016) [1] . The model is
where
are birth rates and T denotes transpose. Here is chemotherapy
and is immune therapy. The parameters, , ,. We have shown previously Larsen (2016) [1] , that there are affine vector fields on, such that their time one map is T, when the eigenvalues of A have positive real part. This enables you to find a formula for the rate of change of cancer growth in. The characteristic polynomial of A is
when The discriminant of this polynomial is
The eigenvalues are
In section two we prove the Bistability Theorem for a mass action kinetic system of metastatic cancer and primary cancer C. The model also has growth factors and growth inhibitors. We show that for some values of the parameters there are exactly two positive singular points where We prove that is unstable and is stable, when one of the rate constants is small.
For we have: if the eigenvalue of A has then one can find an affine vector field, whose time one map is. Similarly, when and the eigenvalues of the cha- racteristic polynomial of A are nonzero, then one can find an affine vector field on, whose time one map is. This enables us to find a formula for the rate of change of cancer growth in This is the subject of Section 3.
The phase space of our model T is. In section four we show, that when, , , orbits of the vector field associated to T will escape phase space for both and. We obtain a formula for the first escape time. There is a similar treatment for
1.2. The Litterature
uPAR (urokinase plasminogen activator receptor) is a cell surface protein, that is associated with invasion and metastasis of cancer cells. In Liu et al. (2014) [2] a cytoplasmic protein Sprouty1 (SPRY1) an inhibitor of the (Ras-mitogen activated protein kinase) MAPK pathway is shown to interact with uPAR and cause it to be degraded. Overexpression of SPRY1 in HCT116 or A549 xenograft in athymic nude mice, led to great suppression of tumor growth. SPRY1 is an inhibitor of the MAPK pathway. Several cancer cells have a low basal expression of SPRY1, e.g. breast, prostate and liver cancer. SPRY1 promotes the lysosomal mediated degradation of uPAR. SPRY1 overexpression results in a decreased expression of uPAR protein. This paper suggests that SPRY1 regulates cell adhesion through an uPAR dependant mechanism. SPRY1 inhibits proliferation via two distinct pathways: 1) SPRY1 is an intrinsic inhibitor of the Raf/MEK/ERK pathway; 2) SPRY1 promotes degradation of uPAR, which leads to inhibition of FAK and ERK activation.
According to Luo and Fu (2014), [3] EGFR (endoplasmic growth factor receptor) tyrosine kinase inhibitors (TKIs) are very efficient against tumors with EGFR activating mutations in the EGFR intracytoplasmic tyrosin kinase domain and cell apoptosis was the result. However some patients developed resistance and this reference aimed to elucidate molecular events involved in the resistance to EGFR-TKIs. The first EGFR-TKI s to be approved by the FDA (Food and Drug Administration, USA) for treatment of NSCLC (non small cell lung cancer) were gefitinib and erlotinib. The mode of action is known. These drugs bind to the ATP binding site of EGFR preventing autophosphorylation and then blocking downstream signalling cascades of pathways RAS/ RAF/MEK/ERK and PI3K/AKT with the results, proliferation inhibition, cell cycle progression delay and cell apoptosis.
There are several important monographs relevant to the present paper, see Adam & Bellomo (1997), [4] , Geha & Notarangelo (2012), [5] , Murphy (2012), [6] , Marks (2009), [7] , Molina (2011), [8] .
2. A mass Action Kinetic Model of Metastatic Cancer
The main result of this section is Theorem 1 below that proves the bistability of the mass action kinetic system (1) to (8). Consider then the mass action kinetic system from Larsen (2016), [9] , in the species primary cancer cells, metastatic cancer cells, growth factor, growth inhibitor respectively.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
The complexes are And this defines the rate constants. With mass action kinetics the ODE s become
all We shall now find the singular points of this vector field denoted
But first we state a theorem, we shall next prove. A positive (nonnegative) singular point
of f is a singular point of f, such that Define
Theorem 1 Assume When there are exactly two positive singular points
where is unstable. Given such that
and, then there exists such that
is stable when
Consider a singular point of f and linearize
Setting the last coordinate of f equal to zero gives
when Now insert this into the first and second coordinates of f to get
(9)
and
(10)
When we get from (9)
and from (10) we get
This means that B simplifies to
Let denote the matrix you obtain by deleting row three and column three in B. Then
Also
The characteristic polynomial of is denoted
Finally
In Larsen (2016) [9] , we found two cubic polynomials such that
whenever is a nonnegative singular point of f. We shall need the following lemma.
Lemma 1 Assume Then
where
Proof. The coefficient to is according to Larsen (2016), [9]
and The coefficient to is according to Larsen (2016), [9]
Everything cancels out and leaves a zero. The coefficient to is according to Larsen (2016), [9]
Square and multiply to get
Everything cancels out except
The coefficient to is according to Larsen (2016), [9]
Multiply
Everything cancels out except
Finally the constant term is
The lemma follows.
Theorem 2 Assume When there are exactly two positive singular points of f
where
Proof. We have
where
and
due to symmetry of When P and have two positive roots
in P and
in, see (15) and (16) below. We are going to verify that
(11)
are singular points of f and that
(12)
are not singular points of f. Here
and
Also
(13)
(14)
We have
(15)
and logically equivalent
(16)
where To see (15) compute
So
and from this the formula follows. And (16) is a similar computation.
We shall insert (15), (16) in the first coordinate of f, multiplied with
Now abbreviate and find
Multiply with to get
But this amounts to
and this vanishes due to the formula for roots of quadratic polynomials. That the second coordinate vanishes is logically equivalent. So (11) are singular points of f.
We shall now argue, that
is not a singular point of f. To this end define
Insert the formulas (15), (16) for in the first coordinate of f multiplied with to get
Multiply with to find
(17)
(18)
But (17) is zero by the above and (18) is nonzero. So is not a singular point. That is not a singular of f is logically equivalent. The theorem follows.
In the remainder of the proof of Theorem 1, we assume, that
We shall now verify that is unstable. We shall show that
But we have
Simply insert (15) and (16) in the numerator
Now we use that
so
is equivalent to
The right hand side here is negative and the left hand side is positive. Thus has a positive eigenvalue. So is unstable.
We shall now show that is stable, when is small. We shall use the Routh Hurwitz criterion. So we start by showing, that But similarly to the above
But this amounts to
which is equivalent to
and this again is equivalent to
and from this it follows that We have the following formula for
And a formula for
Define
so that
Now introduce these two formulas in the formulas for
Notice that for small Also
is negative for small The Routh Hurwitz criterion says in our framework, that
is equivalent to stability of But is equivalent to
because our assumptions imply So is equivalent to
This equation holds for small. So is stable for small. This follows by writing
where and h is smooth. This is the standard trick from singularity theory. Then
And from this it follows that is stable for small. To be precise, given such that and, then there exists such that is stable when Theorem 2 follows.
Consider the mass action kinetic system in the species cancer cells, growth factor, growth inhibitor and a protein, respectively.
(19)
(20)
(21)
(22)
(23)
(24)
(25)
The complexes are And this defines the rate constants. With mass action kinetics the ODE s become
see Horn and Jackson (1972), [10] . Notice that (24), (25) are the Brusselator, which is known to have oscillating solutions for some values of the parameters, see Sarmah et al. (2015), [11] . Subtracting on both sides of (25) gives the reaction Let With these parameter values and initial conditions the system oscillates, see Figure 1.
3. Eigenvalues with Negative Real Part
In this section in the discrete model T of the introduction. The purpose of this section is to find a formula for the rate of change of cancer growth
Figure 1. The oscillating mass action kinetic system. I have plotted P versus C.
on the hyperplane Here is an integral curve of the vector field Y, defined below. There are four cases to consider. First assume, that Let We shall assume that Define
and compute, when
If has negative real part we might be able to find an affine vector field whose time one map is. Notice that
By Larsen (2016), [1] ,
Then
Define the vector field
(26)
and let
where The flow of X is
(27)
(28)
where Also
If
then
Assume that Then we can let
But this means that
because we have
So we get
i.e. Consider first the immune therapy model
So assuming
We want to have
and
such that
Here denotes the time one map of X and Define
Then
Thus
Now
Define
Let denote the first row in U. Compute letting
where is an integral curve of Y through And, because this is equal to
Now suppose and distinct and define
Then
when because the columns of D are eigenvectors of A corresponding to eigenvalues respectively. Compute, when the inverse
Then
Define the vector field
(29)
X has flow
(30)
and the time one map is
and we want this to be
Then define the vector field
This vector field has time one map
Then arguing as before
and
We can now find
Next consider the chemo therapy model
and initially, that Define the vector field X by (26). It has flow (27), (28). Define the vector field
We want this vector field to have time one map
Then we find
Now compute arguing as above
Finally we can find
and this becomes
Now consider the chemo therapy model, when and distinct. Define the vector field X by (29). It has flow (30). Here
The second coordinate here should be equal to
while the third coordinate should be equal to
in order that the time one map of is. Now we can find
and this is simplified to
Remark 1 When then that is So
by the above you can find an affine vector field whose time one map is. Similarly when
then and So by the above, you have a formula for on
4. Escaping Phase Space
In this section The phase space of our model T of the introduction is. When integral curves of B from theorem 1 in Larsen (2016), [1] , starting in will always escape phase space for both and Here
and where
U as in section 3. This vector field, B, has time one map T, see Larsen (2016), [1] , or argue as in Section 3.
The purpose of this section is to prove, that there exists a first escape time, i.e. the existence of a smallest, such that
When we prove, that either
or there exists a smallest such that
Proposition 3 Suppose Given then there exists such that
Proof. We have the following formula for the flow of B
Here
and
Define
Since we can define by
It follows that we have the following formula
Since the proposition follows.
Remark 2 By the proof we have
implies Here. Let denote the smallest positive solution to
When we have the following proposition using the definitions
These formulas are explained in the proof of Proposition 4.
Let where
D as in section 3. B has time one map T, see Larsen (2016), [1] , or argue as in section three.
Proposition 4 Suppose Let be given. (i) If then there exists a unique such that
If then
for all.
(ii) If then there exists a unique such that
If then
for all.
Proof. First of all the flow of F is
We have the following formula
where is the first row of D. From this equation, (i) follows. For (ii) write
From this formula, (ii) follows.
Remark 3 In case (i) of the proposition, if we have
implies
In case (ii) of the proposition, if we have
implies
We shall now derive a formula for the first escape time To start with, assume that Notice that
and
where
i.e.
Compute
where
If let If define by
Then we have the following formulas
(31)
(32)
Assume that Then there exists such that
for If there exists such that
we claim that there are atmost finitely many such solutions and hence that there exists a smallest such that
Assume for contradiction, that there are infinitely many solutions to
By (31) there are exactly solutions to
Since there are infinitely many solutions to there exist
in such that
By the mean value theorem, there exists such that
Hence
A contradiction and there are only finitely many solutions to If there exists a such that let denote the smallest such number, and otherwise let
If then
Since then Define by
(33)
so
By denote the smallest positive solution to Suppose and if let otherwise write (33). If
let otherwise let
so that
By denote the smallest positive. Here
Suppose If let otherwise write (33). Then there exists such that By denote the smallest positive solution to arguing as above.
If for all let otherwise denote by the smallest positive solution to Now define the first escape time by
We shall now find the first escape time when Then we have
and
where
i.e.
Assume in the notation of Proposition 4, that and let
If let Now compute
and
There are atmost two solutions to If there exists such that let denote the smallest such solution, otherwise let If there exists such that let denote the smallest such solution, otherwise let Now define the first escape time, when
5. Summary and Discussion
In this paper we proved that the model of primary and metastatic cancer in Section 2 is bistable, in the sense, that there are exactly two positive singular points. One of them is unstable, and when one of the rate constants is small the other is stable. Then we found formulas for the rate of change of cancer growth for the model T of the introduction, when for the eigenvalues are nonzero and for when In section four we proved that there is a first escape time for the flow of the affine vector field associated to T when A similar result when was also treated.
It would be interesting to figure out what happens if the polynomials of section 2 are cubic polynomials and not quadratic as in Theorem 1.
How do cancer cells coordinate glycolysis and biosynthesis. They do that with the aid of an enzyme called Phosphoglycerate Mutase 1. In the reference [12] , the authors suggest a dynamical system for their findings in a figure at the end of the paper. In the reference [13] , A. K. Laird showed that solid tumors do not grow exponentially, but rather like a Gompertz function. The publications of the author are concerned with semi Riemannian dynamical systems, e.g. Lorentzian Geodesic Flows, see [14] and electrical network theory of countable graphs, see [15] , [16] .