Lupine Publishers| A Model for Metastasis for Hybrid Cancer Cells

Lupine Publishers| Journal of Biomedical Engineering and Biosciences

Abstract

Hybrid cancer cells have been recently discovered. They have greater ability to form metastasis. Here a simple mathematical model is given for this phenomenon. Some comments about the possibility of their reaching brain are given.

Hybrid Tumor Cells

Recently [1,2,3,4] hybrid tumor cells have been discovered. They have the following properties:

i. They circulate more than ordinary tumor cells.

ii. They have greater ability to migrate and invade other tumors.

iii. They have greater ability to form metastasis.

The Metastasis Model

Metastasis comprises a sequence of linked steps leading to the dissemination of cancer cells from a primary tumor to other distant tissues the overwhelming majority of cancer-related deaths still result from the progressive growth of metastasis that are resistant to conventional therapies [1,2].

Motivated by this the following model is presented for the metastasis of hybrid cancer cells:

Let T1, H1 be the ordinary and hybrid tumor cells respectively of the first tumor. Let N=T1+H1. The second tumor is assumed to contain ordinary tumor cells T2. Hence the model can be represented by

dH1/dt=a1H1-N-c2H1, dT1/dt=b1T1^(2/3)-N, dT2/dt= (b2-1)T2+c2H1    (1)

where a1,b1,b2,c2 are positive constants. The metastasis term is c2H1.

The reason for the power 2/3 is that ordinary tumor cells grow due to surface cells [3,4].

The equilibrium solution for the coexistence of both tumors is:

H1eq=T1eq/(a1-c2-1)

T1eq= [b1(a1-1-c2)/(a1-c2)]^3    (2)

T2eq=c2H1/(1-b2)

It is locally asymptotically stable if:

b2<1,

[1-(2/3)(a1-c2)/(a1-c2-1)][1+c2-a1]-1>0    (3)

[1-(2/3)(a1-c2)/(a1-c2-1)]+[1+c2-a1]>0

Since hybrid cells have a greater ability to invade other cells, it is expected that they will invade brain cells. Hence brain tumors can be a good source for identifying them. Moreover trying to attract them to less important sites can be a feasible strategy to deal with them. It may be difficult to test this idea experimentally, because the hybrid state, in general, is unstable [5].

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