Cancer Dynamics: Integrating Immune System and the Chemotherapy

Transcription

1 Applied Mathematical Sciences, Vol. 12, 2018, no. 32, HIKARI Ltd, Cancer Dynamics: Integrating Immune System and the Chemotherapy Oscar A. Manrique Arias 1, Valentina Zuluaga Zuluaga 2, Carlos A. Abello M. 3, Juan C. Jamboos T. 4, Dalia M. Muñoz P. 5 and Anibal Muñoz L. 6 2 Grupo de estudio y desarrollo de software educativo GEDES) 1,3,4,5,6 Grupo de Modelación Matemática en Epidemiología GMME) Facultad de Educación, Maestría en Biomatemáticas Facultad de Ciencias Básicas y Tecnologías Universidad del Quindío, Quindío, Colombia Copyright c 2018 Oscar A. Manrique Arias et al. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract A mathematical model based on nonlinear ordinary differential equations is presented, interpreting the cancer dynamics in a human host. For this, in the mathematical model the tumor and the immunological cells, are considered. In addition, the chemotherapy treatment in the process, is considered. The local stability analysis of the equilibrium points and simulations of the system, are performed. Keywords: Mathematical model, immunological cells, chemotherapy treatment, cancer dynamics 1 Introduction Cancer is a disease that can occur anywhere in the body, which starts when the cells grow uncontrollably exceeding the size of the normal cells, which hinders proper functioning of the body. The normal cells divide in an orderly manner, they die when are damaged and replaced by new cells. On the other hand, the cancerous cells grow out of control causing the normal cells to move, what causes problems in the affected area. These cancerous cells form masses that

2 1688 Oscar A. Manrique Arias et al. are called tumors, but not all of them are malignant [8]. Since cancer can occur anywhere in the body, there are different types of cancer as: Colon or rectum cancer, neck uterine, endometrium, stomach, esophagus, bone, liver, larynx and hypopharynx, leukemia, lymphoma, multiple myeloma, eye, oropharynx and oral cavity, ovary, penis, skin, prostate, lung, pancreas, kidney, breast, testicle, thyroid, vagina, bladder, biliary vesicle and vulva [7]. These are caused by several factors: Genetics, lifestyles such as smoking, poor nutrition and lack of exercise, infections and aspects related to the environment such as radiation and exposure to chemical substances. There are several treatments for cancer such as: Surgery, chemotherapy and radiotherapy, to mention the most common. Surgery is used in order to remove or to extirpate the tumor or a part of it. This type of treatment is not feasible in all cases, one of these is leukemia blood cancer), which is due perform another type of treatment. On the contrary, chemotherapy is the use of medicines that help decrease the growth and proliferation of cancer cells or in the best case kill them. This kind of treatment is useful for cancer that spreads and radiotherapy use ionizing radiation, generally as part of cancer treatment to control o kill malignant cells. It works in a similar way to chemotherapy reduces or kills the cancerous cells) [8]. For this reason different authors have focused their attention on the study of this disease. Among them are: The National Institute of Cancerology Health Ministry and Social Protection), with his work entitled Incidence, mortality and prevalence of Cancer in Colombia , which presents the modeling and validation results for estimating incident cases, mortality and prevalence of cancer, where they obtained the estimated 41,366 cases prevalent at one year in Colombia; 18,458 in men and 22,908 in women. Among men, the most prevalent locations were prostate, stomach, colon-rectum and anus. In women, the more locations prevalent were breast, uterus neck and thyroid [4]. Marion Piñeros and Raúl Hernando Murillo, with his study: Incidence of cancer in Colombia: Importance of the information sources in the collection of estimative figures, in which they estimated the cancer incidence in Colombia and compared the figures obtained by varying the information sources for the adjustment by sub-registration of mortality and the basic information for the model, where they obtained that the adjustment of mortality data and the cancer registers that are used in the estimation process influenced significantly and highlight the importance of considering regional differences [5]. Jair Zapata Peña and Alba Cristina Ortiz, with their research: Use of mathematical models for the description of the growth of cancerous tumors, in which they shown several models that use ordinary differential equations, partial differ-

3 Cancer dynamics: integrating immune system and the chemotherapy 1689 ential equations, discrete-stochastic models, statistical and numeric analytical models to describe the growth of cancerous tumors, where are shown graphic results of simulations for cancer and dead cell populations [6]. Some antecedents about cancer modeling are: Collantes L., et al. with their research: Study and development of mathematical models of cellular resistance to chemotherapy, makes a model based on differential equations describing the transfer behavior of resistance to chemotherapy in mixtures of two tumor cells populations: resistant and sensitive [12]. Romero M., et al. present the study of a mathematical model for the inhibition of angiogenesis in secondary tumors. The authors analyze a partial differential equation model already proposed, where the inhibition depends on a substance that secretes the primary tumor [13]. Villalobos M., et al. in their research: Computational model of tumor growth for multicellular spheroids of the MCF7 cell line, development a general model for the kinetics of tumor spheroids, by means of simulations, from which concluded that it is possible to perform a successful simulation with reduced parameters [14]. Rivas M., et al. with their work: Multi-phase model for tumor growth in the neoplastic avascular phase: A quorum sensing hypothesis in the tumor autoinducers, verify that the tumor occasionally reaches an equilibrium state prior to the induction of vasculogenesis [15]. The writing is divided into three sections: The first of them is an introduction about cancer, its treatment and an antecedent on the modeling of the disease. The second section consist of the description of the mathematical model and a reformulation of it. Finally, in the third section correspond to the mathematical model simulation and the conclusions. 2 Mathematical Model It is considered a model based on nonlinear differential equations, which interprets the cancer dynamics in a human host and its reaction in the immune system with chemotherapy. The proposed model considers the following state variables: T t) : Average number of tumor cells, It) : Average number of cells of the immune system and Qt) : Level of chemotherapy in the immune system, at a time t, respectively.

4 1690 Oscar A. Manrique Arias et al. The parameters of the model are: r : Growth rate of the tumor cells; ρ : Immunological response rate due to the presence of the tumor; γ : Antigenicity rate substance that causes the immunological system produces antibodies against itself); k : Load capacity of the tumor cells; α i, where 1 i 2, i Z: Elimination rate of the tumor cells by the immunological system interaction; ɛ : Mortality rate of the immune cells; a, b: Velocity reaction of each population cell to the drug used in the chemotherapy; µ : Mortality rate of tumor cells due to chemotherapy; s : Increase of external immunological cells to the system; λ : Decay rate of the chemotherapy and δ : Elimination rate of cells from immune system by the chemotherapy. The dynamic system that describes the dynamics is: dt 1 dt = rt T ) α 1 IT k di dt = s ɛi + ρit γ + T α 2IT µt Q a + Q δi Q b + Q d Q dt = q λ Q 3) where, r, a, b, k, s > 0, 0 < ρ, γ, α 1, α 2, ɛ, µ, λ, δ < 0 and with initial conditions T 0) = T 0, I0) = I 0, Q0) = Q 0 not negative. The flow chart is shown in the following figure: 1) 2) rt 1 T ) k T ρit γ + T I s µt Q α 1 IT + a + Q δi Q ɛi + α 2 IT + b + Q q Q λ Q Figure 1: Flow diagram of the dynamics. The region of epidemiological significance for system 1-3 is given by: Ω = { T, I, Q) R 3 + : T 0, I > 0, Q > 0 }

5 Cancer dynamics: integrating immune system and the chemotherapy 1691 In the study Um modelo matemático de câncer com quimioterapia e imunoterapia by Martin, N.A. et al. 2015), propose an ordinary differential equation model in order to analyze the immunology system response on tumor cells versus different treatments chemotherapy and immunotherapy), where by means of simulations they obtained that according to the treatment used to counteract the cancerous tumor, control of the tumor can be achieved, prolonging the treatment. For this research, the ordinary differential equation 3) is used, which interprets treatment by chemotherapy applied to the system [2]. 2.1 Dynamic system reduction The system described in 1), 2) and 3) can be reduced because it is a system uncoupled one way. Equation 3) is a linear differential equation and its solution is given by: Qt) = q λ + e λt Q 0 q ) λ If t then Qt) q, that is, Q = q presents a dynamic asymptomatic λ λ stable to q. Considering Qt) = Q, where Q = q the system 1)-3) presents λ λ an asymptotic dynamics to: dt 1 dt = rt T ) k di dt α 1 IT µt Q a + Q = s ɛi + ρit γ + T α 2IT δiq b + Q 4) 5) The region of epidemiological significance for the system 4)-5) is given by: Ω = { T, I) R 2 + : T 0, I > 0 } 3 Mathematical Model Analysis Starting with the calculation of the infection and prevalence-free equilibrium points, which are determined by equating to zero the equations 4) and 5), obtaining the following algebraic system: rt 1 T ) α 1 IT µt Q k a + Q = 0 6) s ɛi + ρit γ + T α 2IT δiq b + Q = 0 7) To solve this system, we factorize 6), having:

6 1692 Oscar A. Manrique Arias et al. T = 0 y I = 1 r r α 1 k T µq ) a + Q Having T = 0 is substituted in 7), then we obtain: E 0 = 0, ) sb + Q) ɛb + Q) + δq The point E 0 is called the infection-free point since in this case does not find tumor cells. That is, T = 0 therefore, there is a presence of cancerous cells. Now with I = 1 α 1 r r T ) µq k a+q and substituting in 7), is obtained: where, c 1 = α 2r α 1 k c 2 = α 2r α 1 + α 2µQ α 1 a + Q) + ɛr α 1 k c 1 T 3 + c 2 T 2 + c 3 T + c 4 = 0 8) c 3 = s ɛr + ɛµq α 1 α 1 a + Q) ρr α 1 k + α 2γr α 1 k + δqr α 1 kb + Q) + ɛr α 1 k + δqγr α 1 kb + Q) c 4 = ρr ρµq α 1 α 1 a + Q) α 2ρr + α 2ρµQ α 1 α 1 a + Q) δqr α 1 b + Q) + δµq 2 α 1 a + Q)b + Q) + ρ ɛr + ɛµq α 1 α 1 a + Q δqρr α 1 b + Q) + δρµq 2 α 1 a + Q)b + Q) To show that equation 8) with the parameter values presented in table 1 has real roots, we applied the formula of the cubic equation of Cardano [10]. First the equation 8) is rewritten: T 3 + a 2 T 2 + a 1 T + a 0 = 0 where: a 0 = c 4 c 1, a 1 = c 3 c 1, a 2 = c 2 c 1. Cardano says that for a cubic equation as shown previously, the roots are all real if it satisfies the following inequality: Z 3 + R 2 < 0 with Z = 1 3 a a2 2 y R = 1 6 a 1a 2 3a 0 ) 1 27 a3 2. For this case Z = ) 20 and R = ) 30, for which Z 3 + R 2 = ) 55 < 0, what allows conclude the equation 8)presents

7 Cancer dynamics: integrating immune system and the chemotherapy 1693 all its real roots. In addition to analyzing the sign of the roots, is used of Descartes rule, which establishes that the possible number of positive roots of a polynomial is equal to the number of changes of signs in the term coefficients or less than the changes of signs by a multiple of 2 [11]. In the case of the polynomial T 3 +a 2 T 2 +a 1 T +a 0 = 0 there are three changes of signs, what determines that the possible number of positive roots of the polynomial are 3 or 1. Finally point E 1 is given by: E 1 T, I ) where, T is a positive real root of the equation 8) and I = 1 α 1 r r T ) µq k a+q. In the linearization process of the system 4) - 5) the Jacobian matrix was calculated for a generic point E of the system. r 2 T r JE) = k α 1I µq a + Q α 1 T γρi ρ + T ) α 2I ɛ + ρt 2 γ + T α 2T From the Jacobian matrix evaluated at point E 0, we have: δq b + Q JE 0 ) = ρ γ r α 1sb + Q) ɛb + Q) + δq µq a + Q ) ) sb + Q) sb + Q) α 2 ɛb + Q) + δq ɛb + Q) + δq 0 ɛ δq b + Q The matrix JE 0 ) is a lower triangular matrix, due to this the eigenvalues of JE 0 ) are: λ 1 = r α 1sb + Q) ɛb + Q) + δq µq a + Q y λ 2 = ɛ + δq ) b + Q We have so that λ 2 < 0, now if we consider λ 1 < 0, we obtain rɛb + ɛq + δq)a + Q) α 1 sb + sq)a + Q) + µqɛb + ɛq + δq) < 1 Therefore, it is concluded: E 0 is local and asymptotically stable if and only if rɛb+ɛq+δq)a+q) α 1 sb+sq)a+q)+µqɛb+ɛq+δq) < 1 In the analysis of the equilibrium point E 1 the parameter values shown in table 1 are substituted with q = 10, s = 10 8, ρ = 0.235, obtaining:

8 1694 Oscar A. Manrique Arias et al. E 1 = 53, ) Now the Jacobian matrix evaluated at point E 1 is: JE 1 ) = ) The eigenvalues of JE 1 ) are : λ 1 = and λ 2 = 0.11 of which concludes that the equilibrium point is local and asymptotically stable for the parameter values given. 4 Simulations and conclusions The simulations of the system 4) - 5) are carried out in the MATLAB software using the routine ode45. In these simulations the following conditions: T 0) = 10 3, I0) = 10 7 and the values of the parameters show in table 1, are considered. Parameter Value Reference r 10 2 [1] ρ [2] γ 10 2 [9] k 10 7 [1] α [3] α [3] ɛ [9] a [1] b [1] µ 1.3 [2] s [2] λ 4.16 [1] q Variable - δ 10 1 [2] Table 1: Parameter values of the model. Figure 2 illustrates the dynamics of the model influenced by chemotherapy, in which similar behavior to figure 3, is observed, but the action of the immune system and the treatment by chemotherapy is much more efficient, due to a more intense treatment, which allow tumor cells stabilize them at zero.

9 Cancer dynamics: integrating immune system and the chemotherapy x Células Tumorales Células Inmunologícas Tiempo Tiempo Figure 2: Dynamics of tumor and the immunological system cells, with q = 120, s = 10 8, ρ = 0.235, it stabilizes in E 0 = 0, rɛb+ɛq+δq)a+q) ) and α < 1. 1sb+sQ)a+Q)+µQɛb+ɛQ+δQ) Figure 3 illustrates the dynamics of the model influenced by chemotherapy, observing over time the tumor cells are affected by the chemotherapy treatment and the efficient action of the immunological system, considerably stabilizing the tumor cells without eradicating them) x Células Tumorales Células Inmunologícas Tiempo Tiempo Figure 3: Dynamics of the tumor cells and the cells of the immunological system, with q = 10, s = 10 8, ρ = 0.235, it stabilizes at E 1 = 53, ) and rɛb+ɛq+δq)a+q) α > 1. 1sb+sQ)a+Q)+µQɛb+ɛQ+δQ) When treatment as chemotherapy is considered, it presents a positive effect in relation to the reduction of tumor cells, since it performs adequately work to reduce or eliminate these cancerous cells. In the same way when the immunological response rate due to the presence of tumor is ρ = an efficient response of the immune system, is observed. References [1] D. S. Rodrigues, Modelagem Matemática em Câncer: Dinâmica Angiogênica e Quimioterapia Anti-Neoplásica, Diss., Instituto de Biociências, Unesp, [2] N. A. Martin, G. Cruz-Pacheco, P. F. Mancera, Um modelo matemático de câncer com quimioterapia e imunoterapia, Proceeding Series of the

10 1696 Oscar A. Manrique Arias et al. Brazilian Society of Computational and Applied Mathematics, 2015). [3] V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor, A. S. Perelson, Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis, Bulletin of Mathematical Biology, ), no. 2, [4] Constanza Pardo Ramos, Ricardo Cendales Duarte, Incidencia, mortalidad y prevalencia de Cáncer en Colombia. Ministerio de salud y Protección social, Instituto Nacional de Cancerología, 2015). Recuperado de: incidencia1.pdf [5] Marion Piñeros, Raúl Hernando Murillo, Incidencia de cáncer en Colombia: Importancia de las fuentes de información en la obtención de cifras estimativas, Revista Colombiana de Cancerología, ), no. 1, [6] J. Z. Peña, A. C. Ortiz, Uso de modelos matemáticos para la descripción del crecimiento de tumores cancerosos, Nova, ), [7] American Cancer Society, Aspectos básicos sobre el cáncer Qué es el cáncer?, 2017). Recuperado de: cancer/aspectos-basicos-sobre-el-cancer/que-es-el-cancer. html [8] Instituto Nacional del Cáncer, Naturaleza del Cáncer Qué es el Cáncer?, 2015). Recuperado de: naturaleza/que-es [9] L. G. De Pillis, A. Radunskaya, A mathematical tumor model with immune resistance and drug therapy: an optimal control approach, Journal of Theoretical Medicine, ), no. 2, [10] E. W. Weisstein, Cubic formula, Omega, ), 87. [11] Varsity tutors, Regla de los signos de Descartes, 2017). Recuperado de spanish/topics/descartes-rule-of-signs [12] Lucía Olvera Collantes, Estudio y Desarrollo de Modelos Matemáticos de Resistencia Celular a la Quimioterapia, Diss., Universidad de Cádiz, Puerto Real, 2016.

11 Cancer dynamics: integrating immune system and the chemotherapy 1697 [13] Mónica Romero López, Estudio de un modelo matemático para la inhibión de angiogénesis en tumores secundarios, 2012). Recuperado de: Tesis-MonicaRomero-VersionFinal.pdf [14] Samuel Ruiz Arrebola, Modelo computacional de crecimiento tumoral para esferoides multicelulares de la línea celular MCF7 Tesis de maestría), 2013). Recuperado: Ruiz_Arrebola.pdf [15] Manolo Rivas Cañas, Modelo Multi-Fase Para el Crecimiento Tumoral en la Fase Avascular Neooplástica: Una Hipóteis Quorum Sensing en Los Auto-Inductores Tumorales, Diss., Universitat Politecnica de Catalunya, Received: September 5, 2018; Published: December 28, 2018