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Over the last few decades, there have been significant developments in theoretical, experimental, and clinical approaches to understand the dynamics of cancer cells and their interactions with the immune system. These have led to the development of important methods for cancer therapy including virotherapy, immunotherapy, chemotherapy, targeted drug therapy, and many others. Along with this, there have also been some developments on analytical and computational models to help provide insights into clinical observations. This work develops a new mathematical model that combines important interactions between tumor cells and cells in the immune systems including natural killer cells, dendritic cells, and cytotoxic CD8^{+} T cells combined with drug delivery to these cell sites. These interactions are described via a system of ordinary differential equations that are solved numerically. A stability analysis of this model is also performed to determine conditions for tumor-free equilibrium to be stable. We also study the influence of proliferation rates and drug interventions in the dynamics of all the cells involved. Another contribution is the development of a novel parameter estimation methodology to determine optimal parameters in the model that can reproduce a given dataset. Our results seem to suggest that the model employed is a robust candidate for studying the dynamics of tumor cells and it helps to provide the dynamic interactions between the tumor cells, immune system, and drug-response systems.

Cancer is one of the leading causes of death in the world today. By 2030, over 13 million are estimated to harbor some form of the disease. While there have been many developments in cancer therapies including surgery, chemotherapy, immunotherapy, and radiotherapy, there is still a lot that is unknown about the dynamics of how cancer cells are created, propagated, and destroyed.

Over the past few decades, there have been several experimental approaches and interventions developed that have helped us to understand the dynamics of tumor growth and its interactions with the immune system [

In the last two decades, there have also been several experimental advances in developing interventional therapies for cancer such as immunotherapy, virotherapy, targeted drug therapies, and chemotherapy. Along with these experimental developments, there have been some advances in scientific and engineering solutions to capture the dynamics of cancer. One of the promising approaches includes mathematical modeling [

These mathematical models are often coupled system of governing differential equations that describe the dynamics of each of the interacting component cells. Specifically, the interactions between tumor growth and the immune system are often described using a system of coupled differential equations with prescribed initial conditions. These equations include nonlinear interactions and do not often admit an exact solution and therefore require computational methods to solve them. While these mathematical models have provided useful information regarding the importance of the immune system in controlling tumor growth, there is still a great need to continue to enhance existing models to incorporate new clinical developments and biological discoveries. For example, there have been studies suggesting the effectiveness of chemotherapy with immunotherapy and vaccine therapies [

Of the many clinical approaches that are tested for cancer therapy, one of the popular approaches includes drug therapy to the tumor microenvironment. To understand the impact of the drugs delivered to the tumor cell site, it is important to include the effect of these drugs into the models as well. Towards this end, we develop a mathematical model that will combine essential interactions between growing tumor cells and cells of the innate and specific immune system coupled with models for drug delivery to these cell sites. Our goal is to use these models developed to study the effectiveness of anticancer drugs to reduce tumor growth.

The growth of tumors has also been attributed to the dynamics of the cellular immune system within the human host. Two principal components of this immune system include the natural killer cells and cytotoxic

Over the years, there has been a lot of development in mathematical modeling of cancer. However, the mechanisms that are involved in the interactions of tumor cells with the immune system are still not clear. This paper attempts to make a new contribution in this direction by developing a coupled mathematical model that incorporates tumor dynamics and interactions between the dendritic cells, natural killer cells, and

In this work, we will consider a model that consists of four main cell populations including tumor cells (

For developing the model for each of the cell populations, a standard approach is to begin with applying conservation of mass with diffusion and activation. This would often yield the following equation for the dynamics of the various types of cells:

Here, the functions

Also, note that, in all the models, we will consider the effect of a chemotherapy drug (dynamics described later) kill term through

While equation (

We begin with modeling tumor cells

To model natural killer (NK) cells, we will assume that these cells have a constant source

Next, the growth of NK cells will be impacted by two different interactions, namely, the interaction between NK cells and tumor cells [

Note that we have also included a natural death of NK cells through

Dendritic cells play an important role in the immune system response and in controlling tumor growth. Also known as antigen-presenting cells, they update and present antigens to ^{+} T cells. There is, however, experimental evidence that dendritic cells play an important role in modeling tumor immunotherapy [

To study the dynamics of dendritic cells, we will assume ^{+} T cells, and

Among many factors that impact the growth of tumor cells, it is well known that

In our model, to describe the dynamics of the

It has also been seen that

Here,

In this study, we incorporate a variety of external intervention treatment options including tumor-infiltrating lymphocyte (TIL) injections as well as chemotherapy and immunotherapy drugs. TIL drug intervention may be thought of as an immunotherapy approach in which the

To include the chemotherapy and immunotherapy drugs, we describe the dynamics of the respective concentrations in the blood stream as follows:

The drug intervention terms in these equations reflect the amount of chemotherapy and immunotherapy drug given over time. Note that we assume that the chemotherapy and immunotherapy drugs will be eliminated from the body over time at a rate proportional to its concentration, and these are given by

From Sections

Figure

Network of the dynamics for system (

In this section, we employ mathematical analysis to identify conditions that can help eliminate tumor cells. Also, we will determine conditions for when tumor-free equilibrium is unstable and the tumor grows without bound.

We now consider the system of (

At an equilibrium point, we have

Since we assume there is constant recruitment through source terms

For tumor-free equilibrium, at an equilibrium point, we have

Similarly, setting

Substituting (

Solving the quadratic equation yields

Hence, we have 2 tumor-free equilibrium given by

For these to have biological meaning, we need

These conditions suggest critical values for the death rate and the source term for NK cells to be

The Jacobian matrix for linearization of system (

Evaluating these terms at the general tumor-free equilibrium point gives

To solve for the eigenvalues

It may be noted that

Solving (

From (

Using (

This implies that the eigenvalues

In order for the tumor-free equilibrium to be stable, we require

In this section, we will consider the system of (

^{+} T cell dendritic cells kill rate estimated as

^{+} T cell estimated as

^{+} T cells estimated as

^{+} T cells estimated as

For the first computation, we will assume there is no additional recruitment terms for

Dynamics of (a) tumor, (b) NK, (c) dendritic, and (d)

Next, we consider the effect of one of the terms in system (

Dynamics of (a) tumor, (b) NK, (c) dendritic, and (d)

Along with

Dynamics of (a) tumor, (b) NK, and (c)

Next, to study the effect of TIL drug intervention term only for the

Dynamics of tumor cells as the immunotherapy TIL drug intervention term

Next, we consider the effect of the chemotherapy drug only that is introduced through the term

Dynamics of tumor cells as immunotherapy drug intervention with constant

We want to point out that we also performed the study on the influence of immunotherapy drug intervention

Next, we turn our attention to the effect of the nonlinear term introduced as an inactivation term, which describes the regulation and suppression of

Influence of nonlinearity on dynamics of (a) tumor and (b)

In summary, our computations seem to suggest that a combination of immunotherapy through TIL drug intervention

Figure

Influence of no drug intervention, independent drug interventions, and combined drug interventions.

In this section, we focus on estimating some parameters used in system (

The purpose of parameter estimation is to identify values of parameters for given experimental data. In this work, we will demonstrate how to check the reliability of a mathematical model to estimate parameters optimally. For this, we consider a discrete dataset for tumor dynamics corresponding to the values of

Experimental data for tumor cells for

Parameter estimation description.

Next, we make a guess for values of

We then set up an error expression

Using same initial conditions with poor guesses for

Prediction of the dynamics of tumor cells through estimated values for

In this work, we developed a mathematical model that incorporated the dynamics of four coupled cell populations including tumor cells, natural killer cells, dendritic cells, and cytotoxic

While this paper investigates a system of ordinary differential equations, one must computationally study the corresponding partial differential equations (PDES) presented as we developed the model along with fluid equations that help the drugs to move towards the cancer cells. This will require the use of sophisticated numerical methods like the finite element methods to solve the associated system of PDEs. This will be the focus of a forthcoming paper.

Also, we hope to extend our work to apply the models and validate them against actual experimental or laboratory data along with applying machine learning type algorithms to predict behaviors of the growth of the tumor cells. These predictions can help develop control mechanisms such as drug therapy. This will also be a focus of our future work.

All data supporting the results reported have sources provided in the manuscript through references or generated during the study.

The authors declare that they have no conflicts of interest.

^{+}T cells-mediate eradication of established tumors by peritumoral injection of CpG-containing oligodeoxynucleotides

^{+}T cells and DCs in lymph nodes