Computational and Mathematical Methods in Medicine

Volume 2017 (2017), Article ID 3676295, 7 pages

https://doi.org/10.1155/2017/3676295

## Mechanistic Model for Cancer Growth and Response to Chemotherapy

Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Correspondence should be addressed to Eman Simbawa; as.ude.uak@awabmise

Received 9 May 2017; Accepted 19 July 2017; Published 27 August 2017

Academic Editor: Jan Rychtar

Copyright © 2017 Eman Simbawa. This is an open access article 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

Cancer treatment has developed over the years; however not all patients respond to this treatment, and therefore further research is needed. In this paper, we employ mathematical modeling to understand the behavior of cancer and its interaction with therapy. We study a drug delivery and drug-cell interaction model along with cell proliferation. Due to the fact that cancer cells grow when there are enough nutrients and oxygen, proliferation can be a barrier against a response to therapy. To understand the effect of this factor, we perform numerical simulations of the model for different values of the parameters with a continuous delivery of the drug. The numerical results showed that cancer dies after short apoptotic cycles if the cancer is highly vascularized or if the penetration of the drug is high. This suggests promoting angiogenesis or perfusion of the drug. This result is similar to the situation where proliferation is not considered since the constant release of drug overcomes the growth of the cells and thus the effect of proliferation can be neglected.

#### 1. Introduction

There have been extensive studies regarding cancer as it is one of the leading causes of death [1]. The main goal of these studies is to find the most effective therapy with minimal patient suffering. One aspect of research includes mathematical modeling which offers a platform to study cancer without losing patients’ lives [2–6]. It provides an insightful tool to explore and predict the growth of cancer as well as the response to therapy by using biological and physical properties. These models are then validated using in vivo and in vitro experiments as well as patients’ data. The results help oncologists customize therapy for each patient by understanding the physical and biological barriers that make some cancer patients not respond to therapy.

In light of cell population, one could use ordinary differential equations (ODEs) to describe the evolution of the total number of cancer cells with and without chemotherapy [7]; however, since cancer may invade the surrounding tissue and spread, one could subsequently incorporate spatial effects by studying partial differential equations (PDEs) [5, 8]. Cancer cells grow exponentially in early stages due to sufficient supply of oxygen and nutrients [9–11]. Then growth decreases until the population size reaches its carrying capacity after nutrient supply is no longer enough, which is represented by the logistic [4, 12] and Gompertz models [13, 14]. These ODEs can be used to describe the interaction between cancer growth and therapy by adding an anticancer treatment term. With constant drug concentration, the exponential model predicts that cancer will continuously grow. However, the logistic and Gompertz models show that therapy will hold the cancer to some maximum size depending on the values of the parameters [7].

To eradicate cancer, oncologists use anticancer drugs, which either slow down or block the cell division cycle causing cell death [15]. These drugs are considered toxic because they attack rapidly growing cells including skin [16], gut [17], and bone marrow [18]. One anticancer treatment protocol includes a series of scheduled doses (conventional bolus treatment) administered intravenously into the blood stream [19]. Another protocol releases a drug at a constant rate through, for example, nanoparticles [20]. Mathematical modeling suggests that the effect of this constant delivery depends on the initial size of the cell population when the drug is first given [10]. Moreover, a continuous infusion is more effective than bolus applications because of the higher uptake rate [21] and because cancer cells proliferate between doses [22]. This kind of drug delivery exposes the healthy tissue to an extensive amount of toxicity without allowing them to regrow. This can be avoided by developing drugs targeting only cancer cells. Choosing the therapeutic strategy depends on the type of cancer. If the cancer has drug-resisting cells, then mathematical modeling indicates that a bolus dose is more effective as the cancer responds to it faster than a drug given continuously. The two regimes yield the same result for cancer with drug susceptible cells [8].

Most of the mathematical models describe the evolution of cancer as a spatially uniform mass, which grows at a fixed rate. In this paper, we consider the spatial influences on the dynamics between cancer and chemotherapy with constant drug delivery. Specifically, we develop the coupled PDE for drug-cell interaction and drug diffusion and perfusion [23] by considering an extra biological effect, which is cell proliferation. These equations represent a more realistic situation since highly vascularized cancers can proliferate between doses. Model predictions are given through numerical simulations for different values of the key biological parameters (proliferation rate, radius of the blood vessel, diffusion length of the drug, and blood volume fraction) along with the ratio of the viable cancer mass to its initial mass after giving the drug. These simulations represent cancer response for a continuous drug delivery but are not limited to this kind of drug method. Our results provide the opportunity to understand the interaction between cancer and chemotherapy. They can be used as a basis to model more complicated situations or as an alternative therapeutic strategy such as immunotherapy.

#### 2. Methods

##### 2.1. Mathematical Model

In our mathematical model, we add complexity to the PDEs representing the drug-cancer interaction (with the same assumptions) [23] by adding a proliferation term. We assume that the cancer is vascularized with enough nutrients and oxygen creating an ideal environment for cancer to grow at a rate proportional to its density (with and without treatment).

The first equation in the coupled PDEs represents diffusion of the drug into the cancer after it is delivered through the blood vessel and the binding rate to cancer cells. The second equation represents the death rate caused by the drug and the growth rate of cancer cells. The death rate is proportional to the history of drug uptake by cancer cells. After the cells uptake the drug, it will typically damage the DNA. Thus the increasing uptake over time causes more damage across the cell population and an increase in cell death [21, 23]. This represents the only death mechanism caused by the drug. We assume that the growth rate is a constant, although it may depend on the type of cancer or its density; therefore the tumor grows exponentially without treatment at a constant rate. We will study the model for a cylindrically symmetric domain with an infinite radius, where the cancer is initially homogenous and the drug has a constant concentration at the blood vessel with no flux at infinity.

The mathematical model is given bywhere is the drug concentration, is the density of cancer cells, is the drug diffusivity, is the cellular uptake rate of drug per-volume, is the death rate of tumor cells per unit cumulative drug concentration, and is the growth rate of cancer cells. Since the diffusion rate of the drug is faster than the cell cycle, then the time derivative in (1) is replaced by zero (because it does not depend on time). Therefore, we need to find the quasi-steady state solution of (1) given by . Thus we need boundary conditions for (1) and an initial condition for (2).

We assume that the domain surrounding the blood vessel is cylindrically symmetric. This means that the system depends on two parameters: time and radial distance . At the blood vessel, there is a constant rate of drug release , for example, through nanocarriers. If there is no flux (the tumor is infinitely sized). Accordingly, we have the following initial and boundary conditions: where is the radius of the blood vessel and is initially homogenous.

##### 2.2. Nondimensionalizing

Before we numerically solve the model, we nondimensionalize the system to determine the key parameters. Thus, we getwhere the dimensionless variables are , , , , , , and . is the time of the apoptotic cycles caused by the drug [21] and is the diffusion length of the drug.

We assume that cancer cells depend on the closest blood vessel, which has dimensionless radius . Therefore, we estimate the dimensionless radius of the cylindrical region supported by the blood vessel by [6, 23]. BVF is the blood volume fraction (that is the ratio of the volume of blood to the volume of the tumor), which is less than 1. A higher value of BVF represents a highly vascularized tumor; this means that there are more blood vessels and therefore more treatment will be delivered to the tumor. Therefore, (8) can be rewritten as Note that we will drop the dash for simplicity.

##### 2.3. Long-Term Response

After a long time of treatment, the cancer cells will be saturated with the drug and the death rate becomes a constant. Since is a finite continuous series of treatments, then by taking , the time integration of drug uptake is ; and hence from (5) we get . This means that the tumor will grow or decay exponentially depending on the values of and . If , then cancer will continuously proliferate. On the other hand, if , then cell death overcomes cell growth. Otherwise, if , we have a quiescence state since cancer progression is balanced with cancer death.

##### 2.4. Numerical Solution

We numerically simulate (4) and (5) with the initial and boundary conditions given by (6), (7), and (9). After nondimensionalizing, the only parameters in theses equations are , , and BVF which are biological parameters. Small values of represent large diffusion of the drug if we fix ; and large values of BVF represent tumor with high vascularization. First, we discretize and in space and then we solve (5) at each time step using fourth-order Runge-Kutta Method [24], where is given. The latter is calculated by solving (4) (using finite difference method [25]), where is known from the previous time step.

##### 2.5. Calculating the Ratio of the Viable Cancer Mass to the Initial Mass

First, we integrate the density of the viable cancer cells at each time step over the cylindrically symmetric domain around the blood vessel (after drug diffusion). This is done during the numerical simulation (explained in the previous section). Then, we calculate the ratio of the viable cancer mass to the initial mass as follows:The initial mass is equal to the initial volume of the tumor, since at , which is given by .

#### 3. Results

We numerically solve (4)–(7) and (9), for BVF = 0.01, (same as in [23] to compare the results), and for 10 apoptotic cycles (caused by the drug). Note that with we get the same model as in [23]. The solution in Figure 1(a) shows that, at the beginning of the simulation, cancer cells near the blood vessel wall die (due to drug penetration) and further away cells proliferate. Then, we get a similar result as in the case for , where the killing of cancer cells increases causing also an increase in the drug concentration (Figure 1(b)) killing more cells. In Figure 1(c) (for ), the ratio of the viable cancer mass to the initial mass increases at the beginning of the treatment due to proliferation; then after a short time, the drug overcomes proliferation and all cancer cells die after 6 apoptotic cycles, which is similar to the number of cycles needed for .