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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.

There have been extensive studies regarding cancer as it is one of the leading causes of death [

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 [

To eradicate cancer, oncologists use anticancer drugs, which either slow down or block the cell division cycle causing cell death [

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 [

In our mathematical model, we add complexity to the PDEs representing the drug-cancer interaction (with the same assumptions) [

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 [

The mathematical model is given by

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

Before we numerically solve the model, we nondimensionalize the system to determine the key parameters. Thus, we get

We assume that cancer cells depend on the closest blood vessel, which has dimensionless radius

After a long time of treatment, the cancer cells will be saturated with the drug and the death rate becomes a constant. Since

We numerically simulate (

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

We numerically solve (

(a) Numerical simulations of (

Normalized cancer density

Normalized drug concentration

The ratio of the viable cancer mass to the initial mass

Now we vary the parameters BVF,

Temporal evolution curves of the ratio of the viable cancer mass to the initial mass calculated numerically from (

BVF = 0.005,

BVF = 0.005,

BVF = 0.005,

BVF = 0.01,

BVF = 0.01,

BVF = 0.01,

BVF = 0.05,

BVF = 0.05,

BVF = 0.05,

We have added a proliferation term to the PDE representing the interaction between cancer density and drug concentration. Then we performed numerical simulations for different values of the parameters: proliferation rate, radius of the blood vessel, diffusion length of the drug, and blood volume fraction. We found that a continuously administered drug is more effective if the tumor is highly vascularized (which means more exposure to the treatment) or if the penetration length of the drug is high. In this case, the drug overcomes proliferation and the cancer is killed in a short time. This result suggests increasing angiogenesis or perfusion. This is similar to the case where proliferation is neglected because the continuous application of the drug outweighs the effect of cancer growth.

From our result, it seems that when BVF is high and

Temporal evolution curves of the ratio of the viable cancer mass to the initial mass calculated numerically from (

Future work could also include adding physiological or biological complexity to the coupled PDEs. For example, instead of choosing the proliferation rate as a constant, it could depend on the size of the tumor [

The author declares that there are no conflicts of interest regarding the publication of this paper.