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December  2018, 15(6): 1435-1463. doi: 10.3934/mbe.2018066

## Enhancement of chemotherapy using oncolytic virotherapy: Mathematical and optimal control analysis

 1 Department of Mathematics and Applied Mathematics, University of Pretoria, Private Bag X 20, Hatfield, Pretoria 0028, South Africa 2 Department of Mathematics, The College of Saint Rose, Albany, New York, USA 3 Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal 4 Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK

Received  March 27, 2018 Accepted  July 10, 2018 Published  September 2018

Oncolytic virotherapy has been emerging as a promising novel cancer treatment which may be further combined with the existing therapeutic modalities to enhance their effects. To investigate how virotherapy could enhance chemotherapy, we propose an ODE based mathematical model describing the interactions between tumour cells, the immune response, and a treatment combination with chemotherapy and oncolytic viruses. Stability analysis of the model with constant chemotherapy treatment rates shows that without any form of treatment, a tumour would grow to its maximum size. It also demonstrates that chemotherapy alone is capable of clearing tumour cells provided that the drug efficacy is greater than the intrinsic tumour growth rate. Furthermore, virotherapy alone may not be able to clear tumour cells from body tissue but would rather enhance chemotherapy if viruses with high viral potency are used. To assess the combined effect of virotherapy and chemotherapy we use the forward sensitivity index to perform a sensitivity analysis, with respect to chemotherapy key parameters, of the virus basic reproductive number and the tumour endemic equilibrium. The results from this sensitivity analysis indicate the existence of a critical dose of chemotherapy above which no further significant reduction in the tumour population can be observed. Numerical simulations show that a successful combinational therapy of the chemotherapeutic drugs and viruses depends mostly on the virus burst size, infection rate, and the amount of drugs supplied. Optimal control analysis was performed, by means of the Pontryagin's maximum principle, to further refine predictions of the model with constant treatment rates by accounting for the treatment costs and sides effects. Results from this analysis suggest that the optimal drug and virus combination correspond to half their maximum tolerated doses. This is in agreement with the results from stability and sensitivity analyses.

Citation: Joseph Malinzi, Rachid Ouifki, Amina Eladdadi, Delfim F. M. Torres, K. A. Jane White. Enhancement of chemotherapy using oncolytic virotherapy: Mathematical and optimal control analysis. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1435-1463. doi: 10.3934/mbe.2018066
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Visual representation of the elasticity indices of $R_0$ with respect to model parameters. The bar graph shows that the virus burst size, infection and decay rates $b$, $\beta$ & $\gamma$ have the highest elasticity indices with virus decay being negatively correlated to $R_0$
Plots of the elasticity indices, $e_q$, and $\Gamma^{U^*+ I^*}_q$ against the drug dosage $q$. Both Figures (a) and (b) depict that increasing the amount of drug infused reduces viral multiplication thus reducing the sensitivity indices. The figures further suggest that values of $q$ from 40 to 100 mg/l have minimal negative impact on viral replication
(a) Plots of uninfected cell density with constant drug infusion and for different virus burst sizes. The plot shows that an increase in the virus burst size reduces the tumour density. (b)Plots of uninfected cell density with constant drug infusion and for different drug infusion rates. The plot shows that an increase in the drug infusion rate reduces the tumour density by a relatively small magnitude
Plots of the virus and infected tumour densities for different virus burst size. The plots show that both densities increase with increasing virus burst size
(a) Plots of the virus density for different virus infection rate. The plots show the virus density reduces with increasing virus infection rates. (b) Plots of infected tumour density for different infection rates. The figure shows that the infected tumour density increases as the infection rate increases
Total tumour density with optimal control. The tumour density is reduced in a very short time period
Optimal control variable variables $u_1$: The external supply of viruses and $u_2$: Drug dosage. The Figures depict 500 virions as the optimal number of viruses and the optimal drug dosage to be 50 m/g per
The model variables
 Variable Description Units $U(t)$ Uninfected tumour density cells per mm$^3$ $I(t)$ Virus infected tumour cell density cells per mm$^3$ $V(t)$ Free virus particles virions per mm$^3$ $E_v(t)$ Virus specific immune response cells per mm$^3$ $E_T(t)$ Tumour specific immune response cells per mm$^3$ $C(t)$ Drug concentration grams per millilitre (g/ml)
 Variable Description Units $U(t)$ Uninfected tumour density cells per mm$^3$ $I(t)$ Virus infected tumour cell density cells per mm$^3$ $V(t)$ Free virus particles virions per mm$^3$ $E_v(t)$ Virus specific immune response cells per mm$^3$ $E_T(t)$ Tumour specific immune response cells per mm$^3$ $C(t)$ Drug concentration grams per millilitre (g/ml)
The model parameters, their description and base values
 Symbol Description Value & units Ref. $K$ Tumour carrying capacity $10^6$ cells per mm$^3$ per day [4] $\alpha$ Tumour growth rate $0.206$ cells per mm$^3$ per day [4] $\beta$ Infection rate of tumour cells $0.001-0.1$ cells per mm$^3$ per day [4] $\delta$ Infected tumour cells death $0.5115$ day$^{-1}$ [4] $\gamma$ Rate of virus decay $0.01$ day$^{-1}$ [4] $b$ Virus burst size $0-1000$ virions per mm$^3$ per day [11] $\psi$ Rate drug decay $4.17$ millilitres per mm$^3$ per day [39] $\delta_U$ Lysis rate of $U$ by the drug $50$ cells per mm$^3$ per day [39] $\delta_I$ Lysis rate of $I$ by the drug $60$ cells per mm$^3$ per day [39] $\phi$ $E_V$ production rate $0.7$ day$^{-1}$ [9] $\beta_T$ $E_T$ production rate $0.5$ cells per mm$^3$ per day [19,27] $\delta_v,\delta_T$ immune decay rates $0.01$ day$^{-1}$ [19,27] $K_u,K_c,\kappa$ Michaelis--Menten constants $10^5$ cells per mm$^3$ per day [25] $\nu_U$ Lysis rate of $U$ by $E_T$ $0.08$ cells per mm$^3$ per day est $\nu_I$ Lysis rate of $I$ by $E_T$ $0.1$ cells per mm$^3$ per day est $\tau$ Lysis rate of $I$ by $E_V$ $0.2$ cells per mm$^3$ per day est
 Symbol Description Value & units Ref. $K$ Tumour carrying capacity $10^6$ cells per mm$^3$ per day [4] $\alpha$ Tumour growth rate $0.206$ cells per mm$^3$ per day [4] $\beta$ Infection rate of tumour cells $0.001-0.1$ cells per mm$^3$ per day [4] $\delta$ Infected tumour cells death $0.5115$ day$^{-1}$ [4] $\gamma$ Rate of virus decay $0.01$ day$^{-1}$ [4] $b$ Virus burst size $0-1000$ virions per mm$^3$ per day [11] $\psi$ Rate drug decay $4.17$ millilitres per mm$^3$ per day [39] $\delta_U$ Lysis rate of $U$ by the drug $50$ cells per mm$^3$ per day [39] $\delta_I$ Lysis rate of $I$ by the drug $60$ cells per mm$^3$ per day [39] $\phi$ $E_V$ production rate $0.7$ day$^{-1}$ [9] $\beta_T$ $E_T$ production rate $0.5$ cells per mm$^3$ per day [19,27] $\delta_v,\delta_T$ immune decay rates $0.01$ day$^{-1}$ [19,27] $K_u,K_c,\kappa$ Michaelis--Menten constants $10^5$ cells per mm$^3$ per day [25] $\nu_U$ Lysis rate of $U$ by $E_T$ $0.08$ cells per mm$^3$ per day est $\nu_I$ Lysis rate of $I$ by $E_T$ $0.1$ cells per mm$^3$ per day est $\tau$ Lysis rate of $I$ by $E_V$ $0.2$ cells per mm$^3$ per day est
Sensitivity and elasticity indices of $R_0$ with respect to model parameters
 Parameter Sensitivity index Elasticity index $b$ $S_b = 9.88$ $e_b = 1$ $\beta$ $S_{\beta} = 988.4$ $e_{\beta}=1$ $\gamma$ $S_{\gamma} = - 9884$ $e_{\gamma}=-1$ $\delta$ $S_{\delta} = 2.2325$ $e_{\delta}= 0.011553$ $\alpha$ $S_{\alpha} = 2.4372$ $e_{\alpha} = 5.1 \times 10^{-12}$ $\delta_U$ $S_{\delta_U} = -1.004 \times 10^{-7}$ $e_{\delta_U} = -5.1 \times 10^{-12}$ $\delta_I$ $S_{\delta_I} = -90.32$ $e_{\delta_I} =- 0.011553$ $K_c$ $S_{K_c}= 0.2048x$ $e_{K_c}= 0.0000414$ $K_u$ $S_{K_u}=-1.01\times 10^{-6}$ $e_{K_u} = -2.05 \times 10^{-10}$ $\psi$ $S_{\psi} = 0.004$ $e_{\psi}= 0.000414$ $q$ $S_{q} = -0.008$ $e_{q} = - 0.000414$
 Parameter Sensitivity index Elasticity index $b$ $S_b = 9.88$ $e_b = 1$ $\beta$ $S_{\beta} = 988.4$ $e_{\beta}=1$ $\gamma$ $S_{\gamma} = - 9884$ $e_{\gamma}=-1$ $\delta$ $S_{\delta} = 2.2325$ $e_{\delta}= 0.011553$ $\alpha$ $S_{\alpha} = 2.4372$ $e_{\alpha} = 5.1 \times 10^{-12}$ $\delta_U$ $S_{\delta_U} = -1.004 \times 10^{-7}$ $e_{\delta_U} = -5.1 \times 10^{-12}$ $\delta_I$ $S_{\delta_I} = -90.32$ $e_{\delta_I} =- 0.011553$ $K_c$ $S_{K_c}= 0.2048x$ $e_{K_c}= 0.0000414$ $K_u$ $S_{K_u}=-1.01\times 10^{-6}$ $e_{K_u} = -2.05 \times 10^{-10}$ $\psi$ $S_{\psi} = 0.004$ $e_{\psi}= 0.000414$ $q$ $S_{q} = -0.008$ $e_{q} = - 0.000414$
Selected sensitivity indices of the total tumour equilibria, $\Gamma ^{U+I} _p$, in response to drug, $q$ with the corresponding value of $R_0$
 q (mg/l) 5 10 15 35 50 100 $\Gamma ^{U^*+I^*} _p$ $-8.3\times 10^{-5}$ $-2.5\times 10^{-5}$ $-1\times 10^{-5}$ $-6.1\times 10^{-5}$ $-9.6\times 10^{-6}$ $-2.4\times 10^{-6}$ $R_0$ $51.0476$ $51.0473$ $51.0470$ $51.0459$ $51.0450$ $51.0421$
 q (mg/l) 5 10 15 35 50 100 $\Gamma ^{U^*+I^*} _p$ $-8.3\times 10^{-5}$ $-2.5\times 10^{-5}$ $-1\times 10^{-5}$ $-6.1\times 10^{-5}$ $-9.6\times 10^{-6}$ $-2.4\times 10^{-6}$ $R_0$ $51.0476$ $51.0473$ $51.0470$ $51.0459$ $51.0450$ $51.0421$
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