# Optimal CAR T-cell Immunotherapy Strategies for a Leukemia Treatment Model

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## Abstract

**:**

## 1. Introduction

- With what frequency and at what intervals should chimeric leukocytes be injected and in what concentrations?
- How to take a medicine that suppresses the “surge” of the immune system? If between injections the patient seems to be resting, is it necessary to simultaneously stop giving the immunosuppressant, or are these two independent procedures?

## 2. Model and Optimal Control Problem

**Assumption**

**1.**

**Lemma**

**1.**

## 3. Pontryagin Maximum Principle

- ${\psi}_{*}\left(t\right)$ is a nontrivial solution of the adjoint system:$$\left\{\begin{array}{cc}\hfill {{\psi}_{1}^{*}}^{\prime}\left(t\right)& =-{H}_{x}^{\prime}({x}_{*}\left(t\right),{y}_{*}\left(t\right),{z}_{*}\left(t\right),{w}_{*}\left(t\right),{u}_{*}\left(t\right),{v}_{*}\left(t\right),{\psi}_{1}^{*}\left(t\right),{\psi}_{2}^{*}\left(t\right),{\psi}_{3}^{*}\left(t\right),{\psi}_{4}^{*}\left(t\right))\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =-(r-{\alpha}_{1}{y}_{*}\left(t\right)-{\beta}_{1}{z}_{*}\left(t\right)+m{u}_{*}\left(t\right)){\psi}_{1}^{*}\left(t\right)+{\alpha}_{2}{y}_{*}\left(t\right){\psi}_{2}^{*}\left(t\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}+{\alpha}_{3}{z}_{*}\left(t\right){\psi}_{3}^{*}\left(t\right)-n(1-{v}_{*}\left(t\right)){w}_{*}\left(t\right){\psi}_{4}^{*}\left(t\right),\hfill \\ \hfill {{\psi}_{2}^{*}}^{\prime}\left(t\right)& =-{H}_{y}^{\prime}({x}_{*}\left(t\right),{y}_{*}\left(t\right),{z}_{*}\left(t\right),{w}_{*}\left(t\right),{u}_{*}\left(t\right),{v}_{*}\left(t\right),{\psi}_{1}^{*}\left(t\right),{\psi}_{2}^{*}\left(t\right),{\psi}_{3}^{*}\left(t\right),{\psi}_{4}^{*}\left(t\right))\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =-(p-\eta {y}_{*}\left(t\right)-{\alpha}_{2}{x}_{*}\left(t\right)-{\beta}_{2}{z}_{*}\left(t\right)){\psi}_{2}^{*}\left(t\right)+{\alpha}_{1}{x}_{*}\left(t\right){\psi}_{1}^{*}\left(t\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}+\eta {y}_{*}\left(t\right){\psi}_{2}^{*}\left(t\right)+{\beta}_{3}{z}_{*}\left(t\right){\psi}_{3}^{*}\left(t\right)+1,\hfill \\ \hfill {{\psi}_{3}^{*}}^{\prime}\left(t\right)& =-{H}_{z}^{\prime}({x}_{*}\left(t\right),{y}_{*}\left(t\right),{z}_{*}\left(t\right),{w}_{*}\left(t\right),{u}_{*}\left(t\right),{v}_{*}\left(t\right),{\psi}_{1}^{*}\left(t\right),{\psi}_{2}^{*}\left(t\right),{\psi}_{3}^{*}\left(t\right),{\psi}_{4}^{*}\left(t\right))\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =-(q-\gamma {z}_{*}\left(t\right)-{\alpha}_{3}{x}_{*}\left(t\right)-{\beta}_{3}{y}_{*}\left(t\right)){\psi}_{3}^{*}\left(t\right)+{\beta}_{1}{x}_{*}\left(t\right){\psi}_{1}^{*}\left(t\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}+{\beta}_{2}{y}_{*}\left(t\right){\psi}_{2}^{*}\left(t\right)+\gamma {z}_{*}\left(t\right){\psi}_{3}^{*}\left(t\right)-\mu ,\hfill \\ \hfill {{\psi}_{4}^{*}}^{\prime}\left(t\right)& =-{H}_{w}^{\prime}({x}_{*}\left(t\right),{y}_{*}\left(t\right),{z}_{*}\left(t\right),{w}_{*}\left(t\right),{u}_{*}\left(t\right),{v}_{*}\left(t\right),{\psi}_{1}^{*}\left(t\right),{\psi}_{2}^{*}\left(t\right),{\psi}_{3}^{*}\left(t\right),{\psi}_{4}^{*}\left(t\right))\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =(\nu -n(1-{v}_{*}\left(t\right)){x}_{*}\left(t\right)){\psi}_{4}^{*}\left(t\right)+\delta ,\hfill \\ \hfill {\psi}_{1}^{*}\left(T\right)& =-{I}_{x\left(T\right)}^{\prime}=0,\phantom{\rule{0.277778em}{0ex}}{\psi}_{2}^{*}\left(T\right)=-{I}_{y\left(T\right)}^{\prime}=-1,\phantom{\rule{0.277778em}{0ex}}{\psi}_{3}^{*}\left(T\right)=-{I}_{z\left(T\right)}^{\prime}=\mu ,\hfill \\ \hfill {\psi}_{4}^{*}\left(T\right)& =-{I}_{w\left(T\right)}^{\prime}=-\delta ;\hfill \end{array}\right.$$
- the controls ${u}_{*}\left(t\right)$ and ${v}_{*}\left(t\right)$ maximize the Hamiltonian$$H({x}_{*}\left(t\right),{y}_{*}\left(t\right),{z}_{*}\left(t\right),{w}_{*}\left(t\right),u,v,{\psi}_{1}^{*}\left(t\right),{\psi}_{2}^{*}\left(t\right),{\psi}_{3}^{*}\left(t\right),{\psi}_{4}^{*}\left(t\right))$$$${u}_{*}\left(t\right)=\left\{\begin{array}{cc}{u}_{max}\hfill & ,\phantom{\rule{0.277778em}{0ex}}\mathrm{if}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}m{x}_{*}\left(t\right){\psi}_{1}^{*}\left(t\right)>0,\hfill \\ \mathrm{any}\phantom{\rule{0.277778em}{0ex}}u\in [0,{u}_{max}]\hfill & ,\phantom{\rule{0.277778em}{0ex}}\mathrm{if}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}m{x}_{*}\left(t\right){\psi}_{1}^{*}\left(t\right)=0,\hfill \\ 0\hfill & ,\phantom{\rule{0.277778em}{0ex}}\mathrm{if}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}m{x}_{*}\left(t\right){\psi}_{1}^{*}\left(t\right)<0,\hfill \end{array}\right.$$$${v}_{*}\left(t\right)=\left\{\begin{array}{cc}1\hfill & ,\phantom{\rule{0.277778em}{0ex}}\mathrm{if}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}-n{w}_{*}\left(t\right){x}_{*}\left(t\right){\psi}_{4}^{*}\left(t\right)>0,\hfill \\ \mathrm{any}\phantom{\rule{0.277778em}{0ex}}v\in [0,1]\hfill & ,\phantom{\rule{0.277778em}{0ex}}\mathrm{if}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}-n{w}_{*}\left(t\right){x}_{*}\left(t\right){\psi}_{4}^{*}\left(t\right)=0,\hfill \\ 0\hfill & ,\phantom{\rule{0.277778em}{0ex}}\mathrm{if}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}-n{w}_{*}\left(t\right){x}_{*}\left(t\right){\psi}_{4}^{*}\left(t\right)<0.\hfill \end{array}\right.$$

## 4. Properties of the Switching Functions and Optimal Controls

**Lemma**

**2.**

**Proof of Lemma**

**2.**

**Theorem**

**1.**

**Remark**

**1.**

**Lemma**

**3.**

**Proof of Lemma**

**3.**

**Theorem**

**2.**

## 5. Reducing the Matrix of System to the Upper Triangular Form

## 6. Splitting Quadratic System

## 7. Analysis of Quadratic Subsystems

## 8. Estimating the Number of Zeros of Switching Function $\mathit{L}\left(\mathit{t}\right)$

- if ${\alpha}_{2}{y}_{*}\left(T\right)-{\alpha}_{3}\mu {z}_{*}\left(T\right)\ge 0$, then$$L\left(t\right)\left\{\begin{array}{ccc}>0,\hfill & \mathrm{if}\hfill & 0\le t<{\tau}_{1}^{*},\hfill \\ =0,\hfill & \mathrm{if}\hfill & t={\tau}_{1}^{*},\hfill \\ <0,\hfill & \mathrm{if}\hfill & {\tau}_{1}^{*}<t<{\tau}_{2}^{*},\hfill \\ =0,\hfill & \mathrm{if}\hfill & t={\tau}_{2}^{*},\hfill \\ >0,\hfill & \mathrm{if}\hfill & {\tau}_{2}^{*}<t<T,\hfill \\ =0,\hfill & \mathrm{if}\hfill & t=T;\hfill \end{array}\right.$$
- if ${\alpha}_{2}{y}_{*}\left(T\right)-{\alpha}_{3}\mu {z}_{*}\left(T\right)<0$, then$$L\left(t\right)\left\{\begin{array}{ccc}<0,\hfill & \mathrm{if}\hfill & 0\le t<{\tau}_{1}^{*},\hfill \\ =0,\hfill & \mathrm{if}\hfill & t={\tau}_{1}^{*},\hfill \\ >0,\hfill & \mathrm{if}\hfill & {\tau}_{1}^{*}<t<{\tau}_{2}^{*},\hfill \\ =0,\hfill & \mathrm{if}\hfill & t={\tau}_{2}^{*},\hfill \\ <0,\hfill & \mathrm{if}\hfill & {\tau}_{2}^{*}<t<T,\hfill \\ =0,\hfill & \mathrm{if}\hfill & t=T,\hfill \end{array}\right.$$

## 9. Types of Optimal Control ${\mathit{u}}_{*}\left(\mathit{t}\right)$

- if ${\alpha}_{2}{y}_{*}\left(T\right)-{\alpha}_{3}\mu {z}_{*}\left(T\right)\ge 0$, then$${u}_{*}\left(t\right)=\left\{\begin{array}{ccc}{u}_{max},\hfill & \mathrm{if}\hfill & 0\le t\le {\tau}_{1}^{*},\hfill \\ 0,\hfill & \mathrm{if}\hfill & {\tau}_{1}^{*}<t\le {\tau}_{2}^{*},\hfill \\ {u}_{max},\hfill & \mathrm{if}\hfill & {\tau}_{2}^{*}<t\le T;\hfill \end{array}\right.$$
- if ${\alpha}_{2}{y}_{*}\left(T\right)-{\alpha}_{3}\mu {z}_{*}\left(T\right)<0$, then$${u}_{*}\left(t\right)=\left\{\begin{array}{ccc}0,\hfill & \mathrm{if}\hfill & 0\le t\le {\tau}_{1}^{*},\hfill \\ {u}_{max},\hfill & \mathrm{if}\hfill & {\tau}_{1}^{*}<t\le {\tau}_{2}^{*},\hfill \\ 0,\hfill & \mathrm{if}\hfill & {\tau}_{2}^{*}<t\le T,\hfill \end{array}\right.$$

## 10. Numerical Results

**Case 1.**The inequalities (3)–(6), or, what is the same, the inequalities (54) and (55) are satisfied.

## 11. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The orange area is the set of parameters that satisfy inequalities (54).

**Figure 2.**The orange area is the set of parameters that satisfy inequalities (56).

**Figure 3.**The optimal solutions ${x}_{*}\left(t\right)$, ${y}_{*}\left(t\right)$, ${z}_{*}\left(t\right)$ and optimal control ${u}_{*}\left(t\right)$ for the values of the parameters $r=0.6,p=1.1,q=1.0,\eta =0.9,\gamma =1.1,{\alpha}_{1}=1.0,{\alpha}_{2}=0.8,{\alpha}_{3}=0.9,{\beta}_{1}=0.9,$${\beta}_{2}=0.9,$ ${\beta}_{3}=1.1,m=0.2,\mu =100,{u}_{max}=1,T=30$ and the initial conditions ${x}_{0}=0.55,{y}_{0}=0.8,{z}_{0}=0.6$.

**Figure 4.**The optimal solutions ${x}_{*}\left(t\right)$, ${y}_{*}\left(t\right)$, ${z}_{*}\left(t\right)$ and optimal control ${u}_{*}\left(t\right)$ for the values of the parameters $r=0.6,p=1.1,q=1.0,\eta =1.2,\gamma =0.8,{\alpha}_{1}=1.2,{\alpha}_{2}=0.9,{\alpha}_{3}=0.9,{\beta}_{1}=0.6,$${\beta}_{2}=0.75,$ ${\beta}_{3}=1.3,m=0.4,\mu =100,{u}_{max}=1,T=30$ and the initial conditions ${x}_{0}=0.6,{y}_{0}=0.6,{z}_{0}=0.6$.

**Figure 5.**The optimal solutions ${x}_{*}\left(t\right)$, ${y}_{*}\left(t\right)$, ${z}_{*}\left(t\right)$ and optimal control ${u}_{*}\left(t\right)$ for the values of the parameters $r=0.6,p=1.1,q=1.0,\eta =0.9,\gamma =1.0,{\alpha}_{1}=1.0,{\alpha}_{2}=0.8,{\alpha}_{3}=0.8,{\beta}_{1}=0.9,$${\beta}_{2}=0.9,$ ${\beta}_{3}=1.0,m=0.5,\mu =50,{u}_{max}=1,T=30$ and the initial conditions ${x}_{0}=0.6,{y}_{0}=0.8,{z}_{0}=0.8$.

**Figure 6.**The optimal solutions ${x}_{*}\left(t\right)$, ${y}_{*}\left(t\right)$, ${z}_{*}\left(t\right)$ and optimal control ${u}_{*}\left(t\right)$ for the values of the parameters $r=0.6,p=1.0,q=1.1,\eta =0.8,\gamma =1.0,{\alpha}_{1}=1.1,{\alpha}_{2}=0.8,{\alpha}_{3}=0.7,{\beta}_{1}=0.9,$${\beta}_{2}=0.8,$ ${\beta}_{3}=1.0,m=0.3,\mu =1,{u}_{max}=1,T=30$ and the initial conditions ${x}_{0}=0.6,{y}_{0}=0.7,{z}_{0}=0.7$.

**Figure 7.**The optimal solutions ${x}_{*}\left(t\right)$, ${y}_{*}\left(t\right)$, ${z}_{*}\left(t\right)$ and optimal control ${u}_{*}\left(t\right)$ for the values of the parameters $r=0.6,p=1.0,q=1.1,\eta =0.8,\gamma =1.0,{\alpha}_{1}=1.1,{\alpha}_{2}=0.8,{\alpha}_{3}=0.1,{\beta}_{1}=0.9,$${\beta}_{2}=0.8,$ ${\beta}_{3}=1.0,m=0.3,\mu =1,{u}_{max}=1,T=30$ and the initial conditions ${x}_{0}=0.7,{y}_{0}=0.7,{z}_{0}=0.1$.

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Khailov, E.; Grigorieva, E.; Klimenkova, A. Optimal CAR T-cell Immunotherapy Strategies for a Leukemia Treatment Model. *Games* **2020**, *11*, 53.
https://doi.org/10.3390/g11040053

**AMA Style**

Khailov E, Grigorieva E, Klimenkova A. Optimal CAR T-cell Immunotherapy Strategies for a Leukemia Treatment Model. *Games*. 2020; 11(4):53.
https://doi.org/10.3390/g11040053

**Chicago/Turabian Style**

Khailov, Evgenii, Ellina Grigorieva, and Anna Klimenkova. 2020. "Optimal CAR T-cell Immunotherapy Strategies for a Leukemia Treatment Model" *Games* 11, no. 4: 53.
https://doi.org/10.3390/g11040053