Cancer has been the leading cause of death globally; it is estimated that 13 million people will have succumbed to cancer in the next ten years (Bruno et al., 1788). The rate is so high that a lot of od developments have been done in the field of medicine to determine a better way to deal with the monster. Although there have been a lot of developments in the treatment of cancer, for example, chemotherapy, virotherapy, and surgery to remove the tumor growth. These are some of the cancer therapies that are used to combat the disease. The experimental approaches were conducted during the last decade as well as the recent years. The main idea of the study and the experiments was to come up with interventions that will see a reduction in the cases of cancer by a larger margin.
Along with these interventions, advances in engineering were also seen to come up with some tactics to understand the dynamics of cancer. One of the widely used and most basic scientific approaches to understanding the dynamics of cancer is mathematical modeling. This approach of mathematical modeling involves identifying the cell that contributes more to the propagation of cancer, the interaction between the two cells, and it also describes the dynamics under which these cells come together and help in the propagation of cancer (Chimento et al., 807). The model has been of assistance because of the role it plays in the treatment of cancer, it estimates the parameters, performs stability analysis, as well as it predicts the changes and the dynamics cancer will take. The model is seen as a breakthrough toward understanding the cancer dynamics, and the correlation between the cells and the cancer cells. This also clarifies how these cells propagate cancer.
The mathematical models used are coupled systems of describing the dynamic of every cell that interacts with each other using a governed differential equation (Greene et al., 501). The linear equations used do not admit the real solution of the problem or the treatment. Therefore, computational methods are used in solving all of them. Although the mathematical modeling gives crucial information about the immunity system in the control of the growth of tumors (Hartung et al., 6400). Numerous numbers of research should also be done to make sure the mathematical models are incorporated well with the clinical interventions that are present. Some clinical interventions that are used in treating tumor growth can as well work well with the mathematical model used to come up with a good result and prediction of how long the treatment may last by determining the rate at which the treatment takes place.
Among the many clinical interventions used in tumor treatment, the most used method is microenvironment drug therapy for the tumor, to understand the consequence of the drug on the cells of the tumor, the approach needs to be included in the model (Haris et al., 25). The effect of the drug will be calculated and the prediction of how the cell will be eradicated will be known. In this study, we developed a mathematical model that combines the essential interaction between the immune system and the cells in the tumor (Harmouth et al., 9). The goal of the study is aimed at developing an effective study of cancer treatment that will reduce tumor growth. The study will be fruitful if we attempt to come up with new contributions. This will be developing coupled models that will combine the growth of tumors, and interaction between the dendritic cells and of the T cells.
In this study research, we will look at a model used and the four important cell populations; being, dendritic cells (D(d)), killer cells (natural) (N(t)), tumor cells (T(t)), cytotoxic CD8 (L(t)). The changes in these four important cells include the interaction between the four of them and the dynamics after the treatment using cancer therapies like chemotherapy and immunotherapy drug concentration in the human bloodstream (Jarett et al., 1275). A standard approach s used in the development is used in the application of conservation of mass, with activation and diffusion. The equation for the various cells is:
+. (delD [.] = f (.) -g(.)-k[.]z (M)[.]
The main role of g and f is basically on proliferation rates. In all the mathematical models’ chemotherapy is considered the equation z(M)= 1 − e− M is used to describe chemotherapy as the renowned effective drug at some of the stages of tumor growth (Kim et al., 90). The kill factor is equated as K[·]z(M)[·]. The value of the kill factor is the ability of the treatment to stop the division as well as the growth of the tumor in all the four-cell populations. This equation includes diffusion and advection terms due to the velocity of the blood.
d[·] /dt= f(·) − g(·) − K[·]z(M)[·].
The above equation considers the temporal dynamics only, and the ordinary differential equation.
Modeling tumor cells. This model assumes it has the proliferation rate as that is modeled by the law of logic. aT(1-bT) where b and a denotes the per capita rate of growth (Yin et al., 730). The growth of tumor cells is affected by competitive interactions between three cells, tumor cell, dendritic cell, and killer cell (Kumar et al., 30). Tumor cell dynamic can be represented by an ordinary differential equation:
=aT(1 – b T) – (c1N + j D + k L )T − KTz(M)T.
The natural killer model assumes that the cells have a constant source with a recruitment term majorly used to control the interactions between cells.
g 1.N
g1 represents the maximum natural killer cell recruitment rate by the cells from the tumor, and h1 denotes the steepness of the recruitment curve. the growth of natural killer cells is affected by two main interactions, the natural killer cells, and the tumor cells, the next interaction is between the governing dynamic of the natural killer cells differentia equation is described below:
=s1+ -d1D (N − KNz(M)N – eN
They handle a significant role in the response of the immunity system. Also, they perform the task of controlling the growth of tumors (Lee et al., 20). They also act as antigens and where they update and present the antigens to T cells. The CD8 suppresses tumor growth. They stimulate the natural killer cells which may be at rest.
=s2 − f1L + d2N − d3T D − KDz(M)D − gD.
The CD8 has been known to be an important factor when it comes to killing tumor cells. Many factors impact tumor cell growth (Xu et al., 3270). A cytotoxin is a very significant constituent of the immune system, responsible for controlling and making the sure majority of the tumor cells are dead. CD8+ has also been discovered to be produced and used by the debris from the cell of the tumor. An added name that will explain the governing and suppression of CD8+ .
=f2DT − hLT − uNL2 + r1NT + -Klz(M)L-il
The incorporation of any external intervention options. TIL drug intervention is seen as the best immunotherapy approach where CD8+ is promoted through the antigen immune cells. The equation is added vL in the cytotoxin equation.
=f2DT − hLT − uNL2 + r1NT + -Klz(M)L-il+vL
This is the general model from the first to the cytotoxin model.
T=aT(1− bT)− (c1N+jD+kL) T− KTz(M)T
N= s1+ -d1D (N − KNz(M)N – En
D= s2 − f1L + d2N − d3T D − KDz(M)D − gD.
L= f2DT − hLT − uNL2 + r1NT + -Klz(M)L-il+vL
M= vM(t)-d4M
I=v1(t)-d5l
In the analysis of this work, we employed mathematical procedure of analysis to determine environments that can assist destroy the tumor cell (Wei et al., 6515). Also, the conditions that the tumor is not unstable, the tumor grows without any limit.
Equations at equilibrium point.
=0
Jacobian matrix; it isused for linearization of system.
a11 a12 a13 a14
a21 a22 a23 a24
a31 a32 a33 a34
a41 a42 a43 a44
where each symbol represents something
a11 = a − 2abT∗− c1N∗− jD∗− kL∗,
a12= − c1T∗,
a13 = − jT∗,
a14 =− kT∗,
a21 = 2g1N∗h1T∗ /(h1 + T 2) − c2N∗,
a22 = − c2T∗+ d1D∗− e,
a23 = d1N∗, a24 = 0,
a31 = d3D∗, a32 = − d2D∗,
a33 = − f1L∗+ d2N∗− d3T∗+ g ?,
a34 = − f1D∗,
a41 = f2D∗− hL∗+ r1N∗,
a42 = − uL∗2 + r1T∗,
a43 = f2T∗,
a44 =− (hT∗+ 2uN∗L∗+i)
Computational experiments
This section considers the Runge-Kutta algorithm with the estimate, values, and units used in the calculation;
In the first cells’ calculation, there is an assumption that the cytotoxin had no additional recruitment, unlike other cells. Hence removing some of the regulation and suppression (Rojas et al., 402). Next, the study considers the effect of the term in the system. the tumor cells increase at first before the immune system started the full clearance. The cell population change in dynamic as the rate of proliferation increases. Although the dendritic cells are affected, the CD8+ are affected the most by changing it dynamic as the proliferation rate doubles (Lopez et al., 2885). The influence of the source is a matter to check as well, when the foundation term of Dendritic cell is improved it also surges the CD8+ cells and the natural killer cells as well.
To study the impact of TIL, the CD8+ is boosted when the drug intervention term is used. The addition of antigens to boost the immune system by use of immunotherapy intervention is an advantage to the increase of cytotoxic cells where they generate more cells to help in fighting and eliminating the tumor (Mahlbacher et al., 50). When the non-linear term was introduced as an activation term, which is used to explain the suppression and the inactivation. This results in a drop in the cytotoxic cell when the number of tumor cells increases.
Our computational suggests a combination of two common treatment interventions that is chemotherapy and immunotherapy. This study suggests that the two-drug intervention together with the models used will give a better way to reduce tumor growth.
This section majors on the estimation of the parameters used, which also depends on the measurement of the tumor (Mistry et al., 55). The aim is to describe the tumor growth dynamics, specifically in an individual to determine and predict measures that can be taken in order to provide a better intervention. To explain this, two parameters will be used in the model, the competition rate that affects the dynamic of the tumor growth, and the parameter that is related to the proliferation of dendritic cells originating from tumor cells. The purpose is to estimate and identify values for a given data from the experiment.
The next procedure is to guess the values c1 and d3 which are completely different from the data given to create the experiment. An error expression is then set up, which is the sum of the squared difference (Nagle et al., 50). If the error used is found to be the one of the user-prescribed, the experiment stops and the values are used for the entire experiment. With the same condition with poor guesses, the algorithm is used to estimate the parameter. The algorithm is proven to give values close to the original values.
Discussion and Conclusion
A mathematical model was developed in this study, the model developed incorporated four-cell population and their dynamics (Philips et al., 900). The cells were dendritic cells, natural killer cells, cytotoxic cell and tumor cells. This model is very important in how it has incorporated the four key cells that are used in the elimination and destruction of tumor cells. The most important aspect is how to understand the dynamics of every cell and how they work together to boost the immunity of an individual. the effect of drug interventions was also discussed in the study, the two main interventions that are the use of chemotherapy, and immunotherapy. These two treatment interventions worked well with the mathematical model that was applied in this study.
Computation experiments proved the numerical solution of the two-drug intervention. The analysis showed the parameters can be used to predict tumor cell growth as well as the ways of eradicating it (Pinho et al., 370). This model is aimed at determining the dynamics of an individual cell, this will help to determine the proper intervention and predict the way forward for the individual. Future studies showed improvement in using mathematical model of tumor growth in response to treatment. This will help in treating other diseases other than cancer. Cancer has caused clinicians a lot of pressure due to the increasing case of cancer, and ways of treating the tumor have caused turmoil in the health sector at large.
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