# On the Controllability of a System Modeling Cell Dynamics Related to Leukemia

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## Abstract

**:**

## 1. Introduction

#### 1.1. Biological Background

#### 1.2. Mathematical Model and Approach

## 2. Controllability of a Fixed Point Equation

#### 2.1. A General Controllability Principle

**Proposition**

**1.**

**Proof.**

#### 2.2. Stability of the General Control Problem

**Proposition**

**2.**

- (a)
- For each solution $\phantom{\rule{4pt}{0ex}}\left(w,\lambda \right)\in \Sigma $ and $\phantom{\rule{4pt}{0ex}}\overline{\epsilon}>0,$ there exists $\phantom{\rule{4pt}{0ex}}\delta =\delta \left(\overline{\epsilon},w,\lambda \right)>0$ such that$$\left(\overline{w},\overline{\lambda}\right)\in \Sigma ,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{d}_{\mathrm{\Lambda}}\left(\overline{\lambda},\lambda \right)\le \delta \phantom{\rule{4pt}{0ex}}\u27f9\phantom{\rule{4pt}{0ex}}{d}_{W}\left(\overline{w},w\right)\le \overline{\epsilon};$$
- (b)
- For each solution $\phantom{\rule{4pt}{0ex}}\left(w,\lambda \right)\in \Sigma \cap D$ and $\phantom{\rule{4pt}{0ex}}\epsilon \in (0,{\epsilon}_{0}),$ there exists $\phantom{\rule{4pt}{0ex}}\overline{\epsilon}=\overline{\epsilon}\left(\epsilon ,w,\lambda \right)>0$ such that$$\left(\overline{w},\overline{\lambda}\right)\in \Sigma ,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{d}_{W}\left(\overline{w},w\right)\le \overline{\epsilon}\phantom{\rule{4pt}{0ex}}\u27f9\phantom{\rule{4pt}{0ex}}\left(\overline{w},\overline{\lambda}\right)\in {D}_{\epsilon}.$$Then, the control problem (2) is $\mathcal{D}$-stable.

**Proof.**

## 3. First Control Problem for the Normal–Leukemic System

#### 3.1. Solving of the Control Problem

**objective condition**of the type

**objective conditions**can be considered instead (see Section 3.4).

**interpretation**is as follows: the patient begins the treatment with the initial ratio $ln{y}_{0}/ln{x}_{0}\phantom{\rule{4pt}{0ex}}$, and during the treatment, it is expected that the ratio $lny\left(t\right)/lnx\left(t\right)\phantom{\rule{4pt}{0ex}}$ is $\phantom{\rule{4pt}{0ex}}k\left(t\right)$-times smaller than the initial ratio, and at the end T of the treatment time interval, it reaches a prescribed safe level. Of course, the control variable $\lambda $ by which the cell death rates $\phantom{\rule{4pt}{0ex}}c$ and $\phantom{\rule{4pt}{0ex}}C$ are increased in order to guarantee the desired patient evolution depends on the drug dose.

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

- (a)
- (control regularity) In case $\phantom{\rule{4pt}{0ex}}r\in {C}^{1}\left[0,T\right],$ one has $\phantom{\rule{4pt}{0ex}}\lambda \in C\left[0,T\right].$
- (b)
- (continuous dependence) If $\phantom{\rule{4pt}{0ex}}{u}_{k}\to u$ and $\phantom{\rule{4pt}{0ex}}{v}_{k}\to v$ as $\phantom{\rule{4pt}{0ex}}k\to \infty ,$ uniformly on $\left[0,T\right]$ and $\phantom{\rule{4pt}{0ex}}{\lambda}_{k}\left(t\right):=dH\left({u}_{k},{v}_{k}\right)/dt,$ then $\phantom{\rule{4pt}{0ex}}{\lambda}_{k}\to \lambda $ as $\phantom{\rule{4pt}{0ex}}k\to \infty $ in $\phantom{\rule{4pt}{0ex}}{L}^{\infty}\left(0,T\right).$

#### 3.2. Stability of the Control Problem

**objective condition**$\phantom{\rule{4pt}{0ex}}v\left(t\right)/u\left(t\right)=r\left(t\right).$ The stability issue that we consider is the following: how close to the exact drug prescription given by function $\lambda \left(t\right)$ should be the administrated treatment, let it be $\phantom{\rule{4pt}{0ex}}\overline{\lambda}\left(t\right),$ in order that the final result $\phantom{\rule{4pt}{0ex}}\overline{v}\left(T\right)/\overline{u}\left(T\right)=:\overline{r}\left(T\right)$ differs from the desired one $\phantom{\rule{4pt}{0ex}}r\left(T\right)$ by at most $\phantom{\rule{4pt}{0ex}}\epsilon ?$ More exactly, for a given $\phantom{\rule{4pt}{0ex}}\epsilon >0,$ find $\phantom{\rule{4pt}{0ex}}\delta >0$ such that

**Theorem**

**2.**

**Proof.**

**Remark**

**2.**

#### 3.3. Numerical Simulations

#### 3.4. A Different Objective Condition

**objective conditions**can be considered for the first control problem. One of these, with immediate practical importance, consists of following a predefined path for the ratio $y\left(t\right)/x\left(t\right)$ instead of $lny\left(t\right)/lnx\left(t\right)$ from (7). Hence, given a function $r\left(t\right)$ with ${k}_{0}\frac{{y}_{0}}{{x}_{0}}\le r\left(t\right)\le \frac{{y}_{0}}{{x}_{0}}\phantom{\rule{4pt}{0ex}}\left(t\in \left[0,T\right]\right),$ $r\left(0\right)={y}_{0}/{x}_{0}$ and $r\left(T\right)={k}_{0}{y}_{0}/{x}_{0},$ we look for $\lambda \left(t\right)$ such that problem (4) has a solution $\left(x,y\right)$ such that

## 4. Second Control Problem for the Normal–Leukemic System

#### 4.1. Solving of the Control Problem

**objective condition**

**Theorem**

**3.**

**Proof.**

**Remark**

**3.**

#### 4.2. Stability of the Control Problem

**Theorem**

**4.**

**Proof.**

**Remark**

**4.**

## 5. Discussion

**Model applicability:**In the developmental hierarchy of bone marrow, whether normal or leukemic, there are four cell types (stem cells, progenitors, differentiated, and terminally differentiated). Our model aggregates over the properties of all cell types in the hierarchy to model a unique cell type for the normal phenotype and another unique cell type for the leukemic phenotype. Our abstract cells have regulatory negative feedback on the growth found at the stem cell level $1/(1+{b}_{1}x+{b}_{2}y)$ and $1/\left(1+Bx+By\right),$ respectively), the high growth rates of progenitor cells (large a and A), and the differentiated and terminally differentiated bulk mass effect, which eventually accounts for the disease dynamics.

**First control problem:**We divided up the treatment protocol into two phases. The first is a burn-in phase, from time $t=0$ to ${T}_{0},$ in which the drug infusion only freezes the cellular status quo, keeping the malignant/normal cells ratio constant in time (at pretreatment value). In practice, this phase should be very short, as it exposes the patient to additional drug toxicity. It could be useful in clinical trial settings to assess the side effects of chemotherapy separately from the effects of the disease progression or to avoid tumor lysis syndrome at the onset of an induction cycle.

**Second control problem:**We obtained a similar exponential relation for the stability of the second control problem (Theorem 4), where the drug dose lambda is constant in time. The caveat here is that factor is $\phantom{\rule{4pt}{0ex}}{e}^{-\left(a+\lambda A\right)T}$ instead of $\phantom{\rule{4pt}{0ex}}{e}^{-\left(a+A\right)T}.$ This implies that, with faster leukemic cell production rates A and a subsequent need for higher drug levels $\lambda ,$ to achieve the same end result, the control is exponentially harder to obtain.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The graph of the function $r\left(t\right)$ and the graph of the control function $\lambda \left(t\right)$ in Case 1 (chronic phase) of the simulations.

**Figure 2.**The approximate solutions of the system of two nonlinear Volterra integral equations, given by (10), obtained in Case 1 (chronic phase).

**Figure 3.**The graph of the function $r\left(t\right)$ and the graph of the control function $\lambda \left(t\right)$ in Case 2 (accelerated–acute phase) of simulations.

**Figure 4.**The approximate solutions of the system of two nonlinear Volterra integral equations, given by (10), obtained in Case 2 (accelerated–acute phase).

**Table 1.**Numerical simulations of the system of two nonlinear Volterra integral equations, given by (10), obtained in Case 1 (the chronic phase of the disease).

t | u(t) | v(t) | v(t)/u(t) | $\mathit{\lambda}\left(\mathit{t}\right)$ |
---|---|---|---|---|

0.0 | 15.76142 | 16.11810 | 1.02263 | 1.03667861 |

1.0 | 15.74559 | 16.04691 | 1.01914 | 1.82681929 |

2.0 | 15.69218 | 15.82858 | 1.00869 | 2.72096808 |

3.0 | 15.60090 | 15.46682 | 0.99141 | 3.73208846 |

4.0 | 15.47141 | 14.96789 | 0.96745 | 4.88700566 |

5.0 | 15.30324 | 14.34054 | 0.93709 | 6.21220861 |

6.0 | 15.09671 | 13.59667 | 0.90064 | 7.72168000 |

7.0 | 14.85444 | 12.75238 | 0.85849 | 9.42156274 |

8.0 | 14.58279 | 11.82826 | 0.81111 | 11.31917081 |

9.0 | 14.29297 | 10.84895 | 0.75904 | 13.40181913 |

10.0 | 14.00196 | 9.84183 | 0.70289 | 15.58459339 |

11.0 | 13.73138 | 8.83391 | 0.64334 | 17.68482475 |

12.0 | 13.50188 | 7.84645 | 0.58114 | 19.48688614 |

13.0 | 13.32522 | 6.89067 | 0.51712 | 20.85563950 |

14.0 | 13.20103 | 5.96906 | 0.45217 | 21.77786674 |

15.0 | 13.12077 | 5.08115 | 0.38726 | 22.31403995 |

16.0 | 13.07384 | 4.22854 | 0.32343 | 22.53852452 |

17.0 | 13.05129 | 3.41688 | 0.26180 | 22.51022240 |

18.0 | 13.04694 | 2.65569 | 0.20355 | 22.26679888 |

19.0 | 13.05703 | 1.95760 | 0.14993 | 21.82786481 |

20.0 | 13.07965 | 1.33756 | 0.10226 | 21.19948845 |

**Table 2.**Numerical simulations of the system of two nonlinear Volterra integral equations, given by (10), obtained in Case 2 (the accelerated–acute phase of the disease).

t | u(t) | v(t) | v(t)/u(t) | $\mathit{\lambda}\left(\mathit{t}\right)$ |
---|---|---|---|---|

0.0 | 15.76142 | 16.11810 | 1.02263 | 3.15804 |

1.0 | 15.59321 | 15.89161 | 1.01914 | 6.05741 |

2.0 | 15.27793 | 15.41073 | 1.00869 | 9.93469 |

3.0 | 14.76795 | 14.64103 | 0.99141 | 16.19792 |

4.0 | 13.93492 | 13.48141 | 0.96745 | 29.76677 |

5.0 | 12.57768 | 11.78644 | 0.93709 | 57.07073 |

6.0 | 11.49280 | 10.35085 | 0.90064 | 66.27579 |

7.0 | 11.14817 | 9.57058 | 0.85849 | 66.82106 |

8.0 | 11.05090 | 8.96350 | 0.81111 | 67.63740 |

9.0 | 11.01412 | 8.36017 | 0.75904 | 68.47741 |

10.0 | 10.99174 | 7.72598 | 0.70289 | 69.17105 |

11.0 | 10.97355 | 7.05969 | 0.64334 | 69.69562 |

12.0 | 10.95824 | 6.36825 | 0.58114 | 70.06188 |

13.0 | 10.94651 | 5.66061 | 0.51712 | 70.28064 |

14.0 | 10.93937 | 4.94642 | 0.45217 | 70.35631 |

15.0 | 10.93770 | 4.23573 | 0.38726 | 70.28740 |

16.0 | 10.94224 | 3.53910 | 0.32343 | 70.06809 |

17.0 | 10.95364 | 2.86770 | 0.26180 | 69.68924 |

18.0 | 10.97248 | 2.23344 | 0.20355 | 69.13897 |

19.0 | 10.99933 | 1.64909 | 0.14993 | 68.40266 |

20.0 | 11.03477 | 1.12845 | 0.10226 | 67.46265 |

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**MDPI and ACS Style**

Haplea, I.Ş.; Parajdi, L.G.; Precup, R. On the Controllability of a System Modeling Cell Dynamics Related to Leukemia. *Symmetry* **2021**, *13*, 1867.
https://doi.org/10.3390/sym13101867

**AMA Style**

Haplea IŞ, Parajdi LG, Precup R. On the Controllability of a System Modeling Cell Dynamics Related to Leukemia. *Symmetry*. 2021; 13(10):1867.
https://doi.org/10.3390/sym13101867

**Chicago/Turabian Style**

Haplea, Ioan Ştefan, Lorand Gabriel Parajdi, and Radu Precup. 2021. "On the Controllability of a System Modeling Cell Dynamics Related to Leukemia" *Symmetry* 13, no. 10: 1867.
https://doi.org/10.3390/sym13101867