Original articles
Analysis of tumor cells in the absence and presence of chemotherapeutic treatment: The case of Caputo-Fabrizio time fractional derivative

https://doi.org/10.1016/j.matcom.2021.05.007Get rights and content

Abstract

In this work, a study of the mathematical model of uncontrolled growth of tumor cells in the presence of chemotherapy is proposed. The fractional form of the model with the non-singular exponential kernel is considered. This model consists of four coupled partial differential equations (PDEs) which depict the relationship among the normal, tumor, immune cells, and the chemotherapy parameter. The purpose is to investigate the behavior of all types of cells with a change in the fractional order parameter and to show the effect of chemotherapeutic treatment on tumor cells with different levels of immune system. Before applying the proposed numerical method, an approximate expression for the fractional-order Caputo-Fabrizio (C-F) derivative of polynomial function xk is derived. A shifted Chebyshev operational matrix of fractional order derivative is deduced by using an approximation of the C-F fractional derivative of the function and some properties of the orthogonal polynomials. The system of four coupled fractional order PDEs is studied by using the operational matrix method. The dynamics of all the aforementioned cells with respect to different fractional order derivatives are derived and computed numerically for the prescribed values of parameters. These are depicted through graphs to study the diffusive nature of cells and the effect of chemotherapy on all types of cells, before and after applying the therapy. This study shows that tumor cell growth decreases with time when chemotherapy treatment is started. The concentration of tumor cells is more in the invasive fronts of the tumor site as compared to the center of the tumor. It is concluded that the growth of tumor cells is less due to chemotherapy treatment for a person with a strong immune system.

Introduction

The human body is constructed from cells. It produces cells and also destroys them. This process happens in a systematic way. But when this process of division of cells occurs in an uncontrolled way it becomes a disease called cancer. Tumor is also a kind of cancer where this uncontrolled growth occurs in solid tissues such as an organ, muscle, or bone [12], [13], [14]. Due to this property of cancer it is called a neoplastic disease. These damaged cells replace the healthy cells as time increases and it spreads with time. At the present time, many types of treatment and cure are in trend. In medical science, researchers and scientists are trying to find a permanent cure of this disease. Currently, therapies like targeted therapy, hormonal therapy, chemotherapy, radiation therapy and palliative care are available. Solid type of cancer is cured by surgery. These therapies leave some bad effects and reactions in the human body, but they can be ignored. Chemotherapy is popular nowadays. In this treatment, cancer cells are destroyed by the diffusion of medicine into affected cells. For different types of cancer different drugs or medicines are used and sometimes a combination of two or more drugs is also used for diffusion into cells. Generally, drugs are categorized into two categories. The first type is cytotoxic which prevents the division process of cells. The second type is cytostatic which kills the cancer cells [5]. The drugs which are used for treatment are alkylating and anti-metabolites. The response of tumor cells to different types of drugs is different. When the drug is injected into the part of the body where tumor is spreading, it diffuses to the tumor through the capillaries surrounded by tumor cells. The interaction process between the immune cells and tumor cells is a very complicated process. Mathematical modeling and prediction are useful tools to understand the complex biological behavior of physical problems [3], [27]. In the last few years, many mathematical models have been developed and modified on the basis of experimental data to explain the realistic behavior of such physical phenomena. One can find many mathematical models in the literature [1], [2], [7], [10]. By using fractional calculus the fractional version of such mathematical models is developed. Many researchers have considered the fractional model of tumor cells [8], [9], [19] to study the effect of chemotherapeutic treatment on cancer cells.

Our immune system plays a crucial role in defending our body from the cancerous cells and also limits the development of these cells. There are two types of immune cells: (a) CD4 helper tumor cells, which help other cells of the immune system to fight against cancerous cells [15]; (b) CD8 killer cells, which directly destroy the tumor cells. Even with these effective immune cells, our body is unable to protect us from these cancerous cells due to the following reasons:

  • 1.

    Our normal immune system is unable to identify cancerous cells from healthy cells. These cancerous cells are recognized as self and cannot be isolated as foreign cells. This phenomenon is known as tumor tolerance.

  • 2.

    The immune system is able to recognize these cancerous cells, but immune cells are not strong enough to fight and give strong response against the tumor cells.

  • 3.

    The ability to evade detection and the nature of tumor cells make the immune system unable to protect us from the tumor cells.

Thus the immune system must be boosted so that it can identify tumor cells and fight against them in an effective and efficient manner. There are many treatments available to deal with tumor cells such as chemotherapy, immunotherapy, radiotherapy and surgery. Each treatment is specifically applicable to different types of tumors based upon their locations and stages. The goal of these treatments is destruction or removal of tumor cells without damaging the healthy cells. Cytotoxic anti-neoplastic drug known as chemotherapeutic agent is used in the chemotherapy treatment. This treatment helps in destroying tumor cells and it also controls the division of tumor cells, which divide rapidly in the absence of treatment and therapy.

The present research contains the answers to the appropriate and relevant research questions on the dynamics of tumor model under different fractional order parameters, the behavior of tumor cells in the absence or presence of chemotherapy treatment, the distribution of tumor cells in spatial direction in tumor sites, and the effectiveness of immune system in destroying tumor cells.

The scientific report [19] gave the derivation of the integer order model of tumor cells with chemotherapeutic treatment and showed the effect of therapy on the behavior of tumor cells. The aim of the article is to extend the model to fractional order by replacing the integer order time derivative with the Caputo–Fabrizio time fractional derivative as 0CFDtαN(x,t)=DN2Nx2a3(1eU)Nc4NT+r2×N(1b2N), 0CFDtβT(x,t)=Dt2Tx2a2(1eU)Tc3NTc2IT+r1×T(1b1T), 0CFDtγI(x,t)=DI2Ix2a1(1eU)Ic1ITd1I+ρITμ+T+ε, 0CFDtζU(x,t)=DU2Ux2d2U+ϑ(t),0<α,β,γ,ζ1,0t1.Initially the integer order model is limited to chemotherapy treatment, where the other important medical treatments like immunotherapy and radiotherapy have not been considered. Later and also in the aforementioned fractional order model these therapies are taken into consideration.

The fractional order derivative with non-singular kernel is used extensively nowadays in various physical phenomena and mathematical modeling. The Caputo–Fabrizio derivative has an exponential kernel. This non-singular derivative is useful in describing the behavior of those physical phenomena which have some memories from previous stages. The integer order derivative fails to depict this type of behavior. The application of C–F derivative in natural convection flow through a vertical cylinder can be found in article [24]. The application of C–F derivative is found in double convection flow of viscous fluid over a moving vertical plate [25]. In integer order diffusion equation, the motion of particles follows the Brownian motion in which mean square displacement is consistent with the rule X2(t)t. But in the process of anomalous diffusion, the behavior of molecules is found to be inconsistent with nonlinear law of mean square displacement. In this case, the diffusion increases with time and obey the rule X2(t)tα. This motion of molecules that follows the non-linear mean square displacement law is known as fractional Brownian motion. While considering this non-linear rule the integer order diffusion equation is changed into fractional diffusion equation, αutα=2ux2 with α as anomalous diffusion exponent. This is connected with the Montroll-Weiss theory for continuous random walk which describes the physical meaning of time-fractional diffusion equation.

In this article, a fractional order system involving four PDEs in the Caputo-Fabrizio sense is developed. Chebyshev polynomials have been introduced into the collocation method to study the C–F fractional order non linear system of PDEs. After finding the operational matrix for fractional order differentiation, the non-linear fractional order system and initial conditions are collocated at Chebyshev nodes. By collocating a non-linear system of algebraic equations is obtained which is solved by using an iterative Newton method.

The article is organized as follows. In Section 2, the definitions and mathematical preliminaries of fractional calculus [16], [17], [18], [20] are given. In Section 3, an approximate formula for the C–F derivative of the function xk and the operational matrix of C–F differentiation are derived. To solve this mathematical model, the operational matrix method is used in Section 4. In Section 5, the integration of the coupled C–F fractional PDEs (FPDE) is given. The numerical validation and study of the proposed model for different parameter values are presented in Section 6. Section 7 is devoted to the conclusion of the overall work.

Section snippets

Basic definitions and properties

In the literature, a few fractional operators are available with different kernels and different limits of integration. The Caputo and Riemann–Liouville (R–L) definitions are widely used with power law kernel and non-singular kernel like the exponential kernel and Mittag-Leffler kernel. In this section, the definitions of fractional order derivative and integration using exponential kernel law are presented which are widely known as the Caputo–Fabrizio fractional order derivative and

Approximate expression of C–F derivative of simple polynomial function

The derivation of operational matrix of fractional integration and differentiation based upon Caputo and Riemann–Liouville can be found in [21]. In this section, the operational matrix of fractional order differentiation in C–F sense based upon shifted Chebyshev polynomial has been derived.

Theorem 1

The numerical approximation of C–F derivative of order ν of a function f(ξ)=ξσ with σν can be determined by 0CFDxνξσ=B(ν)Γ(1+σ)νν(r=0σn1(1)rξσn1rΓ(σnr)(ννν)r+1+(1)σn(ννν)σnexp(ννν

Description of model representing the chemotherapy effect on behavior of tumor cells

In this present era, mathematical modeling is an important tool for understanding and analyzing the dynamical behavior and physical properties of a physical phenomenon. In this section the model of tumor cells is discussed in the presence of chemotherapeutic effect. The fractional order model contains the four coupled PDEs defined through Eqs. (1)–(4). In this model let N, T and I be the number of normal, tumor and immune cells respectively. The model represents the diffusion and reaction

Solution of the problem

In this section, after deriving a new shifted Chebyshev C-F operational matrix, the collocation method is applied to investigate the behavior of the following model (25) and to solve this model mathematically with the help of initial conditions (24).

Assuming λ1=r2a3, λ2=r1a2, λ3=d1a1, ω1=b2r2, and ω2=b1r1, our model (1)–(4) reduce to 0CFDtαN(x,t)=DN2Nx2a3(1eU)Nc4NTω1N2+λ1N,0CFDtβT(x,t)=Dt2Tx2a2(1eU)Tc3NTc2ITω2T2+λ2T,0CFDtγI(x,t)=DI2Ix2+a1(eU)Ic1ITλ1I+ρITμ+T+ε,0CFDtζU(x,t)=

Numerical simulation and results

In this section, first the validation of our proposed method is shown by using it in a particular form of the concerned model. The accuracy of the method is shown by obtaining the absolute error between exact and numerical solutions. After the validation of the method, it is applied to solve the concerned fractional order model for different particular cases. It is shown how the tumor cells increase with time without chemotherapeutic treatment. The effects of fractional exponents on normal,

Conclusion

In this article, an approximation formula is developed for the C–F derivative of function f(x)=xk. By using this approximation, the C–F operational matrix of shifted Chebyshev polynomials is derived. The Chebyshev collocation method is used to find the numerical solutions of FPDEs. The newly derived operational matrix is successfully implemented to find the solutions of C–F time fractional system of FPDEs having four coupled equations. The validity and accuracy of the considered method can be

Acknowledgments

The authors are thankful to the revered reviewers for their valuable suggestions towards improvement of the quality of the manuscript.

S.H. Ong is partially supported by Ministry of Higher Education, Malaysia grant FRGS/1/2020/STG06/SYUC/02/1.

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