Mathematical model for pulsed chemotherapy

A pulsed chemotherapeutic treatment model is investigated in this work. We prove the existence of nontrivial periodic solutions by the mean of Lyapunov-Schmidt bifurcation method of a cancer model. In this model we consider the case of application of two drugs, the first one P with continuous effect, it appears in the differential equations, and the second one T with instantaneous effects expressed by impulse equations. The existence of bifurcated nontrivial periodic solutions are discussed with respect to the competition parameter values.


Introduction
In this work a mathematical model for cancer chemotherapy is studied by considering interactions between tumor and normal cells.This model consists of three nonlinear ordinary differential equations describing the dynamic of the cancer under the continuous effect of a drug P, and three discrete equations describing the instantaneous effects of a drug T on the different types of cancerous cells, it is called pulsed-therapy.When the drug P is used, a fraction of normal and sensitive cells are killed, the effect of P is expressed by the rates p 1 and p 2 in the differential equations describing the dynamics of normal and sensitive cells.The drug T has an instantaneous effect described by impulse effects on all kinds of cells.The model studied in this work is a generalization of those considered in [1,[5][6][7] and [10], where only one drug with impulse effects is considered.We use similar approach to that used in [7] to find conditions of stability of trivial solution and bifurcation of nontrivial periodic solutions corresponding to eradication of the tumor and its persistence, respectively.Some recent works have considered models similar to our model, see for instance [3,5,8] and [11].
Our work is organized as follow, in the next two sections we define variables and parameters of the model and the approach of fixed point.In section four we study the stability of trivial solution and the bifurcation of periodic nontrivial solutions.In the last section we give some concluding remarks.a e-mail: lakahmed2000@yahoo.frb e-mail: mhelal_abbes@yahoo.frc e-mail: lakmeche@yahoo.frITM Web of Conferences

The model
The model studied in this work is the following where t i+1 − t i = > 0, ∀i ∈ N, and for j = 1, 3, T j , R are positive constants.
The variables and parameters are : period between two successive T drug treatment, x j : normal (resp.sensitive tumor and resistant tumor) cell biomass for j = 1 (resp.2,3), r j : growth rate of the normal (resp.sensitive tumor and resistant tumor) cells for j = 1 (resp.2,3), k j : carrying capacity of the normal (resp.sensitive tumor and resistant tumor) cells for j = 1 (resp.2,3), j : competitive parameter of the normal (resp.sensitive tumor and resistant tumor) cells for j = 1 (resp.2,3), T j : survival fraction of the normal (resp.sensitive tumor and resistant tumor) cells for j = 1 (resp.2,3), their values are completely determined by the quantity of injected T drug dose, R: fraction of cells mutating due to the dose of the T drug which is less than T 2 , m: acquired resistance parameter usually it is very small (see [9]), p 1 (resp.p 2 ): death rate fraction of the normal (resp.sensitive tumor) cells, due to the drug P which is less than r 1 (resp.r 2 − m).

The fixed point approach
We study bifurcation of nontrivial periodic solution, which occurs at e 0 = 0 = i 0 or e 0 = 0 = i 0 .To this purpose, we shall employ a fixed point argument.
We reduce the problem of finding a periodic solution of ( 1)-( 6) to a fixed point problem.Then = (., X 0 ) is a -periodic solution of ( 1)-( 6) if and only if its initial condition X 0 is a fixed point for = (., X 0 ).A fixed point X 0 of ( , .) is the initial state of (1)-( 6) which gives a -periodic solution verifying (0) = X 0 .Consequently, to establish the existence of nontrivial periodic solutions of ( 1)-( 6), we need to prove the existence of nontrivial fixed points of by the mean of bifurcation analysis.

Stability of the trivial fixed point
We have the following results.

Theorem 3.2:
The trivial periodic solution of ( 1)-( 6) is exponentially stable if and only if , and The solution is exponentially stable if and only if the spectral radius of D X ( 0 , 0 ) is less than one, that is and In view of the fact that 2 K 1 < 1 and 3 K 1 < 1 (see [10]), we have and Then is stable as an equilibrium for the full system (1)-( 6) if and only if

Critical cases
In this section, we are interested in the bifurcation of nontrivial periodic solutions from = (x S , 0, 0).To find a nontrivial periodic solution of period with initial condition X, we need to solve the fixed point problem Let ¯ and X such that = 0 + ¯ and X = X 0 + X.The Eq. ( 16) is equivalent to where Since is a trivial 0 -periodic solution of ( 1)-( 6), then it is associated to the trivial fixed point 0 of ( 0 , .). where and A necessary condition for the bifurcation of nontrivial zeros of the function M is that the determinant of D X M(0, (0, 0, 0)) is equal to zero, i.e. a 0 .e0 .i0 = 0. From stability of x s we have (10), i.e. a 0 = 0, it follows that e 0 .i0 = 0.
We have e 0 = 0 for and It remains to find sufficient conditions to obtain bifurcation.We have the following results.
We have the following results.

Concluding remarks
In this work, we have studied a nonlinear mathematical model describing evolution of cell population constituted by three kinds of cells (normal cells, sensitive tumor cells and resistant tumor cells) under chemotherapeutic treatment by two drugs, the first one P with continuous effect and the second one T with instantaneous effects.This model is a generalization of those studied by [1], [5][6][7] and [10].
We have found sufficient conditions for exponential stability of trivial periodic solutions corresponding to eradication of the tumor, after that we have studied conditions of bifurcation of nontrivial periodic solutions which corresponds to the onset of the tumor.Bifurcation of nontrivial periodic solutions are studied and sufficient condition for bifurcation are found.