The Bistability Theorem in a Model of Metastatic Cancer

The main theorem of the present paper is the bistability theorem for a four dimensional cancer model, in the variables M C C GF GI , , , representing primary cancer C, metastatic cancer M C , growth factor GF and growth inhibitor GI, respectively. It says that for some values of the parameters this system is bistable, in the sense that there are exactly two positive singular points of this vector field. And one is stable and the other unstable. We also find an expression for ( ) C t d 0 d for the discrete model T of the introduction, with variables ( ) C GF GI , , , where C is cancer, GF GI , are growth factors and growth inhibitors respectively. We find an affine vector field Y whose time one map is T2 and then compute ( ) C t d 0 d , where ( ) ( ) ( ) ( ) C t GF t GI t , , is an integral curve of Y through ( ) GF GI  0 0 0, , ∈ . We also find a formula for the first escape time for the vector field associated to T, see section four.


. Summary of the Paper
We continue the study of the cancer model from Larsen (2016) [1].The model is ∈  .We have shown previously Larsen (2016) [1], that there are affine vector fields on 3   , 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 0 C = .The characteristic polynomial of A is In section two we prove the Bistability Theorem for a mass action kinetic system of metastatic cancer M C and primary cancer C. The model also has GF growth factors and GI growth inhibitors.We show that for some values of the parameters there are exactly two positive singular points

< <
We prove that * c + is unstable and * c − is stable, when one of the rate constants is small.For 0 ∇ < we have: if the eigenvalue a ib + of A has 2   2   a b > then one can find an affine vector field, whose time one map is 2  T .Similarly, when 0, 0 αδ βσ ∇ > + < and the eigenvalues , λ λ − + of the characteristic polynomial of A are nonzero, then one can find an affine vector field on 3   , whose time one map is 2 T .This enables us to find a formula for the rate of change of cancer growth in 0. C = This is the subject of Section 3.
The phase space of our model T is 3 +  .In section four we show, that when 0 > orbits of the vector field associated to T will escape phase space for both 0 t > and 0 t < .We obtain a formula for the first escape time.There is a similar treatment for 0. ∇ > 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.

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 , , , M C C GF GI primary cancer cells, metastatic cancer cells, growth factor, growth inhibitor respectively.

GF C
→ The complexes are 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 ( ) , , , , where * * * * , .
( ) Setting the last coordinate of f equal to zero gives This means that B simplifies to ( ) Let B  denote the matrix you obtain by deleting row three and column three in B. Then In Larsen (2016) [9], we found two cubic polynomials , M P P such that ( ) ( ) where ( ) Larsen (2016), [9]    , , .

M
C is according to Larsen (2016), [9]   ( ) ( ) ( )( ) Everything cancels out and leaves a zero.The coefficient to 2 M C is according to Larsen (2016), [9]   ( ) ( ) ( Everything cancels out except and ( 16) below.We are going to verify that ( ) ( ) are singular points of f and that are not singular points of f.Here We have ( ) and logically equivalent ( ) where ( )  2 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 x k = ± ∇ and find ( ) ( ) Multiply with ( ) 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 Multiply with ( ) But ( 17) is zero by the above and (18) is nonzero.So ( ) 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 ( ) 15) and ( 16) in the numerator The right hand side here is negative and the left hand side is positive.Thus B  has a positive eigenvalue.So ( ) is unstable.
We shall now show that ( ) is stable, when 59 k is small.We shall use the Routh Hurwitz criterion.So we start by showing, that det 0.

B < 
But similarly to the above ( ) And a formula for σ Now introduce these two formulas in the formulas for , σ τ The Routh Hurwitz criterion says in our framework, that 0, 0, 0, This equation holds for small 59 k .So ( ) is stable for small 59 k .This follows by writing ( ) and h is smooth.This is the standard trick from singularity theory.Then And from this it follows that ( ) is stable for small 59 k .To be precise, given ( ) Consider the mass action kinetic system in the species , , , C GF GI P cancer cells, growth factor, growth inhibitor and a protein, respectively.

GF C
The complexes are

Eigenvalues with Negative Real Part
In this section 0, 0 If λ + has negative real part we might be able to find an affine vector field whose time one map is 2 T .Notice that ( ) T y A y Ac c = + + By Larsen (2016), [1], Define the vector field ( ) where But this means that sin tan 1 We want to have ( ) Here 1

X
Φ denotes the time one map of X and 3 .
C GF GI t is an integral curve of Y through ( ) 3 0 0 0, , .
GF GI ∈  And, because ( ) ) ) because the columns of D are eigenvectors of A corresponding to eigenvalues 1, , Define the vector field ( ) 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 initially, that Then we find ( Now compute arguing as above Finally we can find ) ) ) ) ) The second coordinate here should be equal to

Escaping Phase Space
In this section 0, 0.
The phase space of our model T of the introduction is > integral curves of B from theorem 1 in Larsen (2016), [1], starting in 3 +  will always escape phase space for both 0 t > and 0. t < Here 1 1 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 0 t > , i.e. the existence of a smallest 0 t > , such that ( ) ( ) we prove, that either ( ) ( ) , , e 1 cos sin sin cos ( ) It follows that we have the following formula ( ) ( ) the proposition follows.
Remark 2 By the proof we have D as in section 3. B has time one map T, see Larsen (2016), [1], or argue as in section three.Proposition 4 Suppose det 0, 0, , 0, 0.
, exp ln , , , exp ln ( ) In case (ii) of the proposition, if 0 z < we have ln ln ln We shall now derive a formula for the first escape time .FET + ∈  To start with, assume that 3 , c , cos , sin y z y z

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 0 ∇ > the eigenvalues , λ λ + − are nonzero and for 0 ∇ < when 2 2 0. a b − > In section four we proved that there is a first escape time for the flow of the affine vector field associated to T when 0.

∇ <
A similar result when 0 ∇ > was also treated.
this vector field.And one is stable and the other unstable.We also find an expression for model T of the introduction, with variables () and growth inhibitors respectively.We find an affine vector field Y whose time one map is T 2 and then compute Insert the formulas(15),(16) for * * , M C C − + in the first coordinate of f multiplied with 54 * 56 * 59

Figure 1 .
Figure 1.The oscillating mass action kinetic system.I have plotted P versus C.
Define the vector field X by (26).It has flow (27), (28).Define the vector field 1 Y DU X U − =   We want this vector field to have time one map the vector field X by (29).It has flow (30).Here

order that the time one map of 1 DXD − is 2 T
. Now we can find Soby the above you can find an affine vector field whose time one map is2  T .Similarly when 9 So by the above, you have a formula for We have the following formula for the flow of B ( ) Here p ∈  .Let 1 s denote the smallest positive solution to

F
otherwise denote by I s the smallest positive solution to ( ) 0.
We shall now find the first escape time when det 0 denote the smallest such solution, otherwise let .I s = +∞ Now define the first escape time, GI is a nonnegative singular point of f.We shall need the following lemma.