Computational and Mathematical Methods in Medicine

Volume 2019, Article ID 7576591, 16 pages

https://doi.org/10.1155/2019/7576591

## Mathematical Analysis of a Mathematical Model of Chemovirotherapy: Effect of Drug Infusion Method

Department of Mathematics, University of Eswatini, Private Bag 4, Kwaluseni, Eswatini

Correspondence should be addressed to Joseph Malinzi; az.ca.smia@iznilamhpesoj

Received 17 November 2018; Revised 14 January 2019; Accepted 28 January 2019; Published 11 March 2019

Guest Editor: Madjid Soltani

Copyright © 2019 Joseph Malinzi. 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

A mathematical model for the treatment of cancer using chemovirotherapy is developed with the aim of determining the efficacy of three drug infusion methods: constant, single bolus, and periodic treatments. The model is in the form of ODEs and is further extended into DDEs to account for delays as a result of the infection of tumor cells by the virus and chemotherapeutic drug responses. Analysis of the model is carried out for each of the three drug infusion methods. Analytic solutions are determined where possible and stability analysis of both steady state solutions for the ODEs and DDEs is presented. The results indicate that constant and periodic drug infusion methods are more efficient compared to a single bolus injection. Numerical simulations show that with a large virus burst size, irrespective of the drug infusion method, chemovirotherapy is highly effective compared to either treatments. The simulations further show that both delays increase the period within which a tumor can be cleared from body tissue.

#### 1. Introduction

Tumors possess mechanisms that suppress antitumor activity such as ligands that block natural killer cells and cytotoxic tumor infiltrating cell functions [1]. Greatly because of this, successful cancer treatment often requires a combination of treatment regimens.

Nearly all traditional monotherapies, including chemotherapy, surgery, and radiation therapy are not a definite cure for cancer and are highly toxic [2]. Chemotherapy for example, which is the most commonly used regimen, involves the use of medical drugs to lyse cancer cells. These chemotherapeutic drugs circulate in the body and kill rapidly multiplying cells nonselectively, which ultimately results into the destruction of both healthy and cancerous cells [2, 3]. Chemotherapy can thus be toxic to a patient with adverse side effects and can also damage their immune system [2].

Presently, combination cancer treatment is a centerpiece of cancer therapy [4]. The amalgamation of anticancer drugs increases efficacy compared to single-drug treatments. Further, anticancer drug combination provides therapeutic benefits such as reducing tumor growth, arresting mitotically active cells, reducing the population of cancer stem cells, and inducing apoptosis [4]. Despite the fact that combination therapy might as well be toxic if one of the agents used is chemotherapeutic, the toxicity is lesser because different pathways would be targeted [4]. Moreover with the use of combination therapy, the toxicity on normal cells can be prevented while concurrently producing cytotoxic effects on cancer cells [4, 5].

In the recent past, virotherapy, a less toxic treatment has been identified as a possible cancer remedy [6–11]. Virotherapy involves the use of oncolytic viruses that infect, multiply, and directly lyse cancer cells with less or no toxicity [9]. Their tumor specific properties allow for viral binding, entry, and replication [12]. Oncolytic viruses can greatly enhance the cytotoxic mechanisms of chemotherapeutic drugs [13]. Further, chemotherapeutic drugs lyse fast multiplying cells and, in general, virus infected tumor cells quickly replicate [14].

Chemovirotherapy is a combination treatment strategy that involves the use of oncolytic viruses and chemotherapeutic drugs. Recent experimental and mathematical studies have shown that chemovirotherapy is a plausible cancer treatment and leads to enhanced therapeutic effects not achievable when either therapies are independently used [12, 13, 15–20]. Nguyen et al. [12] gave an account of the mechanisms through which drugs can successfully be used in a combination with oncolytic viruses. They however note that the success of this combination depends on several factors including the type of oncolytic virus- (OV-) drug combination used, the timing, frequency, dosage, and cancer type targeted. To date, the best method of OV drug delivery is debatable [21, 22].

The main goal of this study is to, thus, consider and compare the efficacy of three drug infusion methods, use mathematical analysis to predict the outcome of OV-drugs combination treatment and determine the effect of drug response and virus infection delays. To this end, we construct a mathematical model in the form of ODEs which we later extend to DDEs to include the virus infection and drug response delays. The model constructed combines elements from existing mathematical models [10, 11, 19, 20, 23–30]. Tian [10] presented a mathematical model that incorporates burst size for oncolytic virotherapy. His study showed that virotherapy is highly effective provided that viruses with large burst sizes are used. Malinzi et al. [19] constructed a spatiotemporal mathematical model to investigate the outcome of chemovirotherapy. Their study suggested that combining chemotherapeutic drugs with oncolytic viruses is more efficient than using either treatments alone. A similar study by Malinzi et al. [20] indicates that chemotherapy alone is capable of clearing tumor cells from body tissue if the drug efficacy is greater than the tumor growth rate. Nevertheless, the study contends that oncolytic viruses highly enhance chemotherapy in lysing tumor cells. The study further postulates that half the maximum tolerated doses of chemotherapy and virotherapy optimize chemovirotherapy, thus answering a very pertinent question in combination cancer therapy.

The article is organised as follows: Section 2 presents a comprehensive description of the both the ODE and DDE models and the underlying assumptions made in constructing them. In Section 3, the model without delay is analysed. First, without any form of treatment, then with either treatments (that is, with chemo only and virotherapy alone) and with both treatments. The delay model is then analysed in Section 4 and numerical experiments for both the ODE and DDE models are carried out in Section 5. Finally, before concluding in Section 7, a comparison of this study with related works is done is Section 6.

#### 2. Model Description

##### 2.1. Model without Delay

Time-dependent cell concentrations of uninfected tumor cells , infected tumor cells , a free virus population , and a chemotherapeutic drug in an avascular tumor localization are considered. The uninfected tumor grows logistically at an intrinsic rate *α* per day, and the total tumor carrying capacity is *K* cells in a tumor nodule. The infected tumor cells die off at a rate *δ* per day. Virus multiplication in the tumor is represented by the function , where *β* is the virus replication rate measured per day per cells or viruses. The response of the drug to the uninfected and infected tumor is, respectively, modelled by the functions and where and are induced lysis rates caused by the chemotherapeutic drug measured per day per cell. Virus lifespan is taken to be and its production is considered to be where *b* is the virus burst size, measured in number of viruses per day per cell, and *δ* is the infected tumor cells’ death rate measured per day. Chemotherapeutic drug infusion into the body is modelled with a function and that the drug gets depleted from body tissue at a rate *λ* per day.

Drug infusion into the body is simulated using (a) a constant rate , (b) an exponential , and (c) a sinusoidal function , where *q* is the rate of drug infusion. The constant *a* determines the exponential drug decay and period for the sinusoidal infusion. Constant drug infusion may relate to a situation where a patient is put on an intravenous injection or a protracted venous infusion and the drug is constantly pumped into the body [31, 32]. The exponential drug infusion may relate a situation where a cancer patient is given a single bolus and the drug exponentially decays in the body tissue. This form of infusion is not common although it is now used for some drugs, for example, a single dose of carboplatin can be given to patients with testicular germ cell tumors and breast cancer ([33, 34]). The third scenario is possible when a cancer patient makes several visits to a health facility and is given injections or anticancer drugs periodically [35, 36].

The assumptions above lead to the following system of nonlinear first-order differential equations (also similarly derived in [11, 19, 20]):subject to initial concentrations

##### 2.2. Delay Model

The model is further extended to account for delays as a result of the infection of tumor cells by the virus and responses of the chemotherapeutic drug. In fact, the viruses need time to develop suitable responses when they meet the uninfected tumor cells (e.g., [37]). The drug does not instantaneously kill the cells (e.g., [38, 39]). By denoting the virus and chemotherapeutic response delays as and , respectively, model (1)

#### 3. Mathematical Analysis of Model without Delay

In this section, the model without delay (1) is analysed. The variables in system (1) are first rescaled by setting , , , , and . Taking , the parameters are renamed to become

For simplicity, we drop the bars and equation (1) becomes

, , and , respectively, are the constant, exponential, and sinusoidal infusion functions. For this model to be biologically meaningful, its solutions should be positive and bounded because they represent concentrations. Well-posedness theorems of model (5) are stated and proved in Appendix A.

##### 3.1. Model Solutions

To investigate the efficacy of each treatment and their combination, we first study the dynamics of the system without treatment. Without any form of treatment, model (5) is reduced to only one equation:whose solution isimplying that the tumor logistically grows to its maximum fractional size. Next, the model (5) is analysed, with chemotherapy, with virotherapy, and then with both treatments incorporated. We obtain, where possible analytical and time invariant solutions which predict the long term dynamics of the model equation (1).

Without virotherapy (), the system (5) is transformed towith and . The second equation in equation (8) is a first-order linear ordinary differential equation which can easily be solved to givewhere is a constant of integration. The solution to the first equation in equation (8) depends on the infusion function . For a fixed infusion function *ϕ*,

From the solution of in equation (10),

Biologically, it can be inferred that with a constant drug infusion and without virotherapy, the tumor is not completely cleared and a certain proportion of the drug remains in body tissue. The tumor clearance depends on the drug induced lysis of the tumor and the drug infusion rate which should be maximized and the tumor growth and drug decay rate which should be minimized.

For ,where .

From equation (12),where is a fractional tumor cell concentration between 0 and 1. This suggests that with a single dosage infusion of the chemotherapeutic drug with exponential decay and without virotherapy, the tumor cannot be cleared from body tissue. The drug is also completely depleted from the body.

When is substituted in equation (8), the resulting differential equations are solved to givewhere

This suggests that with time, some drug concentration remains in the body tissue.

Theorem 1. *The system (8), with constant infusion, has no periodic solutions for positive and .*

*Proof. *Using Dulac’s criterion ([40]):

Suppose and is continuously differentiable on a simply connected domain . If there exists a real valued function such that has one sign in , then there are no closed orbits in . Using Dulac’s criterion, it is sufficient to show thatConsider

Theorem 2. *The system (8) has at least two steady states for each of the drug infusion functions: *(1)*For the constant drug infusion function , there are two steady states of equation (8): which is locally asymptotically stable provided that and which is locally asymptotically stable provided that ; otherwise, it is unstable.*(2)*For the exponential drug infusion , equation (8) has two steady states: which is unstable and which is locally asymptotically stable.*(3)*For the sinusoidal infusion function, there are four steady states of equation (8): , , and which are unstable and which is locally asymptotically stable if and .*

*Proof. *(1)It is easy to show that when equation (8) is equated to zero, one obtains two steady states. The characteristic polynomial of the Jacobian matrix for equation (8) evaluated at iswhose roots *λ* can only be negative if . The characteristic polynomial evaluated at iswhose roots are negative provided that .(2)By letting , equation (8) is turned into an autonomous system: The eigenvalues of the Jacobian matrix for equation (20) evaluated at which are and 0 and at which are and are all negative.(3)Similarly, by letting , equation (8) becomes the autonomous system:The eigenvalues of the Jacobian matrix for equation (20) evaluated at are and 0 and the eigenvalues evaluated at are and 0. For the third steady state to exist, , and the eigenvalues evaluated at this state are , implying that for it to be locally asymptotically stable, ; yet for this to happen, the steady state will not exist. The eigenvalues evaluated at are implying that this steady state is locally asymptotically stable if and .

Theorems 1 and 2 show that there are no periodic solutions in the dynamics of equation (8) and with a constant drug infusion, the tumor can be eliminated from body tissue by chemotherapy provided that the combination of the chemotherapeutic drug-induced lysis of the tumor and the drug infusion is greater than the combination of the intrinsic tumor growth rate and the drug deactivation rate. The tumor can also be wiped out with a periodic drug infusion provided that the combination of the tumor-induced lysis by the drug and the dosage is greater than the intrinsic tumor growth rate and drug decay rate. With the exponential infusion method, the tumor is not removed from body tissue and may grow to its maximum size.

Without chemotherapy, equation (5) is reduced toThe analytical solutions to system (22) are not easy to obtain. The derivatives of equation (22) are therefore equated to zero to obtain time invariant solutions and investigate their stability by linearizing equation (22) about the steady states.

*Theorem 3. (1)If , the system (12) has two steady states: a tumor free cell state which is unstable and an infection tumor free state which is locally asymptotically stable.(2)If , the system (22) has three steady states: the tumor free state and the infected free state which are unstable and a tumor dormant state:which is locally asymptotically stable if and where are coefficients of the characteristic equation.*

*Proof. *(1)The characteristic equation evaluated at isfrom which , and , thus rendering it unstable. The characteristic equation evaluated at isfrom which and which are all negative since .(2)The characteristic polynomial evaluated at the tumor dormant state is whereand are the coordinates of the tumor dormant state. Using Routh–Hurwitz stability criterion, this state will only be locally asymptotically stable if and .

*Since the infected tumor-free state is undesirable, the reverse of the condition is necessary for tumor eradication from body tissue. In other words, , that is, the product of the virus replication rate and their burst size should be greater than the sum of the burst size and virus replication rate. We also notice from equation (23) that*

*It is therefore evident that high virus replication rate and burst size lead to lower tumor cell concentrations. The steady-state solutions of equation (23) involve many parameters, thereby giving rise to large expressions in the conditions for its stability. It is therefore a difficult undertaking to infer biological implications from these conditions. Nevertheless, it can be observed that virotherapy may only succeed in eliminating cancer from body tissue when the virus deactivation rate is very small or even zero and the virus replication rate very high.*

*Next, the model with both treatments is analysed. For a constant drug infusion rate ϕ, the system (5) has three steady states;(i)Tumor-free steady state:*

*Here, the tumor and viruses are cleared from body tissue by the coupled treatment and a fraction of the chemotherapeutic drug remains in body tissue. The eigenvalues of the Jacobian matrix evaluated at this state areimplying that this desirable state is locally asymptotically stable if From this condition, in order to clear a tumor, the combination of the rate at which the drug kills the uninfected tumor cells and the drug infusion must be higher than the tumor growth rate and deactivation of the drug from body tissue.(ii)Infected tumor-free state: In this state, the whole tumor is not cleared as a fraction of uninfected tumor cells remain and all the infected ones are cleared by the treatment combination. Using the parameter values in Table 1, the eigenvalues of the Jacobian matrix evaluated at the infected tumor-free state are 0.403, and , implying that the infected tumor-free state is unstable.(iii)Tumor dormant state:where*