# Optimal Combinations of Chemotherapy and Radiotherapy in Low-Grade Gliomas: A Mathematical Approach

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Patient Population

#### 2.2. Image Acquisition and Radiological Measurements of Tumor Size

#### 2.3. Mathematical Model

#### 2.4. Parameter Estimation

#### 2.4.1. TMZ Concentration in the Brain

#### 2.4.2. Parameter Fitting

#### 2.5. RT-Only Virtual Experiments

#### 2.6. In-Silico Patients

#### 2.7. In-Silico Trial Comparing Different Treatment Modalities/Schedules

- I.
- Control group without treatment;
- II.
- RT (1.8 Gy)+TMZ (concurrently 75 mg/m${}^{2}$/day and post-radiation 150–200 mg/m${}^{2}$/ day) according to RTOG 0424. For simplicity, we assumed a fixed-dose equal to 150 mg/m${}^{2}$/day in all TMZ cycles;
- III.
- RT+TMZ matching the doses of RT (1.8 Gy) to those of TMZ (150 mg/m${}^{2}$/day) in the first five days of each cycle (of length 28 days).

## 3. Results

#### 3.1. Validation of the Mathematical Model

#### 3.1.1. The Mathematical Model Given by Equation (1) Described the Tumor Longitudinal Dynamics in Patients

#### 3.1.2. Delaying Radiation Therapy Did Not Have an Effect on Overall Survival

#### 3.2. Protracted Radiotherapy Schemes Lead to Increased Tumor Growth

#### 3.3. Concurrent Cycles of RT and TMZ Produce the Longest-Lasting Therapeutic Effect

#### 3.4. In-Silico Simulation of Clinical Trials

## 4. Discussion and Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

LGG | Low-Grade Glioma |

O | Oligodendroglioma |

A | Astrocytoma |

OA | Oligoastrocytoma |

RT | Radiation Therapy |

TMZ | Temozolomide |

CS | Cycle Size |

PCV | Procarbazine, Lomustine and Vincristine |

MRI | Magnetic Resonance Imaging |

OSG | Overall Survival Gain |

NA | Not Available |

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**Figure 1.**Diagram illustrating the model interactions. Proliferative cells $P\left(t\right)$ grow at rate $\rho $, some enter a quiescent state $Q\left(t\right)$ at rate ${\beta}_{1}$ and vice versa at rate ${\beta}_{2}$. Radiation damages a fraction $1-{S}_{p}\left(d\right)$ of $P\left(t\right)$ cells to produce a subpopulation of lethally damaged cells $D\left(t\right)$ that die at rate ${\beta}_{3}$, and also stimulates a fraction $1-{S}_{q}\left(d\right)$ of $Q\left(t\right)$ to proliferate. Chemotherapy $C\left(t\right)$ causes some $P\left(t\right)$ cells to undergo delayed death (as radiation does) at rate ${\alpha}_{1}$ and immediate death at rate ${\alpha}_{2}$.

**Figure 2.**(

**a**,

**b**,

**e**,

**f**) Examples of longitudinal volumetric dynamics (green circles) with their respective 18 % error bars (green vertical lines) and best fits obtained using Equation (1) (solid purple line). Data shown correspond to patients $10,36,40$ and 66 (green circles). (

**c**,

**d**,

**g**,

**h**) Time evolution of $100P\left(t\right)/\left(P\right(t)+Q(t\left)\right)$ interpreted as virtual (solid purple line) and initial (red circle) Ki-67 labeling index. The vertical dashed lines indicate the onset and end times of the treatments received by each patient (either RT, QT or both).

**Figure 3.**(

**a**–

**c**) Evolution of $P\left(t\right),Q\left(t\right)$ and $D\left(t\right)$ for patient 36. The sum of each of these cell populations corresponds to the total tumor volume.

**Figure 4.**(

**a**–

**h**) Delaying radiotherapy does not have an effect on overall survival. RT was simulated to start one year later (gray solid line) than when actually received by the patient (solid purple line), except for patient 107, in whose case it was simulated to occur one year earlier. The horizontal black dashed line represents the fatal volume and the green circles, with their respective error bars, the longitudinal volumetric data.

**Figure 5.**(

**a**–

**h**) Protracted radiotherapy schemes lead to increased tumor growth in silico. Simulations of longitudinal tumor growth dynamics of patients receiving radiation therapy fractionations spaced at 5 days (black solid line), 2 days (red solid line) and the conventional scheme (purple solid line). The vertical red dashed line shows the treatment start time and the green circles the longitudinal volumetric data.

**Figure 6.**Same simulations as in Figure 5 for three different sets of parameters taking into account the 18% error in the real tumor volumes. (

**a**) ${V}_{0}=43.56,\phantom{\rule{4pt}{0ex}}\rho =0.091,\phantom{\rule{4pt}{0ex}}{\beta}_{1}=1.08,\phantom{\rule{4pt}{0ex}}{\beta}_{2}=0.03,\phantom{\rule{4pt}{0ex}}{\beta}_{3}=0.0076,\phantom{\rule{4pt}{0ex}}{S}_{q}=0.861$ and ${S}_{p}=0.019$. (

**b**) ${V}_{0}=44.86,\phantom{\rule{4pt}{0ex}}\rho =0.094,\phantom{\rule{4pt}{0ex}}{\beta}_{1}=1.04,\phantom{\rule{4pt}{0ex}}{\beta}_{2}=0.029,{\beta}_{3}=0.0077,\phantom{\rule{4pt}{0ex}}{S}_{q}=0.866$ and ${S}_{p}=0.003$ and (

**c**) ${V}_{0}=44.99,\phantom{\rule{4pt}{0ex}}\rho =0.101,\phantom{\rule{4pt}{0ex}}{\beta}_{1}=1,\phantom{\rule{4pt}{0ex}}{\beta}_{2}=0.027,\phantom{\rule{4pt}{0ex}}{\beta}_{3}=0.0081,\phantom{\rule{4pt}{0ex}}{S}_{q}=0.869$ and ${S}_{p}=0$.

**Figure 7.**Schematic representation of the combined treatment protocols. In the RTOG 0424 protocol, RT was given concurrently with TMZ (75 mg/m${}^{2}$/day) in weeks 1–6 (top of treatment timeline) while in the simultaneous scheme, RT was only administered together with TMZ (150 mg/m${}^{2}$/day) (bottom of treatment timeline).

**Figure 8.**(

**a**–

**d**) Simulated tumor growth curves for patient 36 and for three in-silico patients treated according to RTOG 0424 (purple solid line) and the RT+TMZ simultaneous (red solid line) regimens. Treatment onset is denoted by a vertical green dashed line and the fatal tumor volume by a horizontal black dashed line.

**Figure 9.**Kaplan -Meier survival curves for the control group (black line) and for the groups treated according to RTOG 0424 (red line) and the RT+TMZ simultaneous (green line) schedules. Differences between curves were statistically significant p-value (p) < 0.05.

**Figure 10.**In-silico studies comparing the KM survival curves for virtual patients with a followup period of 15 years treated according to RTOG 0424 vs simultaneous RT+TMZ. The pseudocolor plot shows the p-values for each trial as the number of patients per arm is increased. For a patient population greater than 720 (360 per arm) the probability of finding statistically significant differences between curves is $\ge 95\%$.

**Table 1.**Patients included in the study. Patient 36 was the only one treated with concurrent RT and TMZ. Patient 151 received RT and 11 years later TMZ. The meaning of the abbreviations used in the table can be found at the end of this document.

ID | Age at Diagnosis | Sex | Histology | Treatment | Ki-67 Labeling Index (%) | ${\scriptstyle \#}$RT Sessions | ${\scriptstyle \#}$TMZ Cycles |
---|---|---|---|---|---|---|---|

10 | 48 | F | O | TMZ | 5 | - | 15 |

25 | 47 | M | O | TMZ | 5 | - | 4 |

36 | 40 | M | O | TMZ & RT | 3 | 35 | 13 |

40 | 30 | F | A | RT | 1 | 34 | - |

57 | 54 | M | O | TMZ | 4 | - | 10 |

66 | 28 | M | A | RT | NA | 30 | - |

91 | 34 | M | A | RT | NA | 30 | - |

105 | 33 | M | O | TMZ | 3 | - | 20 |

107 | 34 | M | A | RT | 3 | 30 | - |

108 | 46 | M | O | TMZ | 3 | - | 5 |

121 | 48 | M | A | RT | 3 | 35 | - |

124 | 27 | M | OA | RT | 3 | 35 | - |

151 | 58 | M | O | RT | 3 | 7 | - |

TMZ | 5 | - | 13 | ||||

159 | 46 | M | NA | TMZ | NA | - | 12 |

203 | 51 | M | O | TMZ | 6 | - | 10 |

213 | 58 | F | O | TMZ | 3 | - | 20 |

234 | 44 | M | A | RT | 2 | 30 | - |

Parameter | Description | Value | Sources |
---|---|---|---|

c | Standard dose per day | 150 mg/m${}^{2}$ | [26] |

b | Patient body surface | 1.9 m${}^{2}$ (men) | [41] |

1.6 m${}^{2}$ (women) | [41] | ||

$\gamma $ | Fraction of TMZ reaching | 2.1$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$${10}^{-6}/$mL (men) | [25] |

the brain interstitium | $2.5\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{10}^{-6}/$mL (women) | [42] | |

${t}_{1/2}$ | TMZ half-life clearance time | ≃2 h | [43] |

$\lambda $ | Rate of TMZ decay | 0.3466/h | [25] |

**Table 3.**Parameter values obtained for the best fits of patients to Equation (1). Patient 151 was considered independently for each therapy. The units for the initial conditions and parameters are: for ${P}_{0}$ and ${Q}_{0}$, cm${}^{3}$; for Ki-67${}_{0}$$\to \%$; for $\rho $, ${\beta}_{1}$, ${\beta}_{2}$ and ${\beta}_{3}$, 1/day. Finally, ${\alpha}_{1}$ and ${\alpha}_{2}$ are measured in cm${}^{3}$/$(\mathsf{\mu}$g day). * Estimated values.

ID | #RT Sessions | #TMZ Cycles | ${\mathit{P}}_{0}$ | ${\mathit{Q}}_{0}$ | Ki-67${}_{0}$ | $\mathit{\rho}$ | ${\mathit{\beta}}_{1}$ | ${\mathit{\beta}}_{2}$ | ${\mathit{\beta}}_{3}$ | ${\mathit{S}}_{\mathit{q}}$ | ${\mathit{S}}_{\mathit{p}}$ | ${\mathit{\alpha}}_{1}$ | ${\mathit{\alpha}}_{2}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Radiotherapy + Temozolomide | |||||||||||||

36 | 35 | 13 | 2.7 | 87.6 | 3 | 0.023 | 1.5 | 0.046 | 0.0002 | 0.45 | 0.55 | 2.3 | 3.9 |

Radiotherapy | |||||||||||||

40 | 34 | - | 0.02 | 2.5 | 1 | 0.15 | 1 | 0.008 | 0.0011 | 0.32 | 0 | - | - |

66 | 30 | - | 0.64 | 20.9 | 3 * | 0.06 | 1.5 | 0.04 | 0.0001 | 0.39 | 0 | - | - |

91 | 30 | - | 0.8 | 39.5 | 2 * | 0.033 | 2.1 | 0.04 | 0.0008 | 0.16 | 0.04 | - | - |

107 | 30 | - | 1.3 | 42.8 | 3 | 0.092 | 1.08 | 0.03 | 0.0078 | 0.86 | 0 | - | - |

121 | 35 | - | 0.18 | 6 | 3 | 0.1 | 0.33 | 0.002 | ${\scriptstyle 1\times {10}^{-5}}$ | 0.1 | 0.7 | - | - |

124 | 35 | - | 0.08 | 2.8 | 3 | 0.013 | 0.87 | 0.02 | 0.004 | 0.01 | 0 | - | - |

151 | 7 | - | 0.86 | 27.9 | 3 | 0.01 | 1.3 | 0.04 | 0.01 | 0.94 | 0.09 | - | - |

234 | 30 | - | 1.2 | 6.9 | 2 | 0.032 | 1.6 | 0.04 | 0.0024 | 0.78 | 0 | - | - |

Temozolomide | |||||||||||||

10 | - | 15 | 2.2 | 43.6 | 5 | 0.01 | 3.5 | 0.184 | 0.0008 | - | - | 6.01 | 4.31 |

25 | - | 4 | 1.7 | 32.6 | 5 | 0.028 | 6.42 | 0.33 | 0.004 | - | - | 20 | 20 |

57 | - | 10 | 1.8 | 45.1 | 4 | 0.01 | 0.23 | 0.009 | 0.006 | - | - | 15 | 9.8 |

105 | - | 20 | 4.3 | 141.7 | 3 | 0.035 | 0.7 | 0.02 | 0.0018 | - | - | 2.8 | 6.5 |

108 | - | 5 | 0.4 | 15.2 | 3 | 0.04 | 10.6 | 0.32 | 0.003 | - | - | 45 | 3 |

151 | - | 13 | 1.7 | 33.5 | 5 | 0.012 | 0.53 | 0.02 | 0.0013 | - | - | 14.8 | 1 |

159 | - | 12 | 2 | 66.1 | 3 * | 0.012 | 3.5 | 0.1 | 0.004 | - | - | 11 | 6 |

203 | - | 10 | 1.7 | 27.2 | 6 | 0.059 | 1.6 | 0.098 | 0.021 | - | - | 8.2 | 0.37 |

213 | - | 20 | 0.7 | 23.1 | 3 | 0.011 | 4.7 | 0.14 | 0.0005 | - | - | 15.2 | 15.1 |

**Table 4.**Real (in black, see Table 3) and virtual (in blue) values of the parameters related to therapies and the numbers of TMZ cycles and RT sessions received for the patients included. ${\alpha}_{1}$ and ${\alpha}_{2}$ are measured in cm${}^{3}$/$(\mathsf{\mu}$g day). The final column corresponds to the overall survival gain (OSG) for each patient.

ID | #RT Sessions | #TMZ Cycles | ${\mathit{S}}_{\mathit{q}}$ | ${\mathit{S}}_{\mathit{p}}$ | ${\mathit{\alpha}}_{1}$ | ${\mathit{\alpha}}_{2}$ | OSG (Years) |
---|---|---|---|---|---|---|---|

10 | 30 | 15 | 0.28 | 0.47 | 6.01 | 4.31 | >10 |

25 | 30 | 4 | 0.1 | 0.65 | 20 | 20 | −1.2 |

36 | 35 | 13 | 0.45 | 0.55 | 2.3 | 3.9 | >10 |

40 | 34 | 12 | 0.32 | 0 | 34.6 | 14.9 | >10 |

57 | 30 | 10 | 0.68 | 0.69 | 15 | 9.8 | 8 |

66 | 30 | 12 | 0.39 | 0 | 30.3 | 1 | 8.7 |

91 | 30 | 12 | 0.16 | 0.04 | 7.7 | 18.3 | >10 |

105 | 30 | 20 | 0.51 | 0 | 2.8 | 6.5 | 4.7 |

107 | 30 | 12 | 0.86 | 0 | 30.3 | 3.7 | 0.4 |

108 | 30 | 5 | 0.2 | 0.48 | 45 | 3 | 2.9 |

121 | 35 | 12 | 0.1 | 0.7 | 10.3 | 10 | >10 |

124 | 35 | 12 | 0.01 | 0 | 14.1 | 13.7 | >10 |

151 | 7 | 12 | 0.94 | 0.09 | 5.5 | 1.4 | 0.26 |

151 | 30 | 13 | 0.83 | 0.24 | 14.8 | 1 | 1.9 |

159 | 30 | 12 | 0.18 | 0.17 | 11 | 6 | >10 |

203 | 30 | 10 | 0.6 | 0.27 | 8.2 | 0.37 | 1.4 |

213 | 30 | 20 | 0.36 | 0.41 | 15.2 | 15.1 | >10 |

234 | 30 | 12 | 0.78 | 0 | 2.5 | 15.6 | 3.1 |

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**MDPI and ACS Style**

Ayala-Hernández, L.E.; Gallegos, A.; Schucht, P.; Murek, M.; Pérez-Romasanta, L.; Belmonte-Beitia, J.; Pérez-García, V.M. Optimal Combinations of Chemotherapy and Radiotherapy in Low-Grade Gliomas: A Mathematical Approach. *J. Pers. Med.* **2021**, *11*, 1036.
https://doi.org/10.3390/jpm11101036

**AMA Style**

Ayala-Hernández LE, Gallegos A, Schucht P, Murek M, Pérez-Romasanta L, Belmonte-Beitia J, Pérez-García VM. Optimal Combinations of Chemotherapy and Radiotherapy in Low-Grade Gliomas: A Mathematical Approach. *Journal of Personalized Medicine*. 2021; 11(10):1036.
https://doi.org/10.3390/jpm11101036

**Chicago/Turabian Style**

Ayala-Hernández, Luis E., Armando Gallegos, Philippe Schucht, Michael Murek, Luis Pérez-Romasanta, Juan Belmonte-Beitia, and Víctor M. Pérez-García. 2021. "Optimal Combinations of Chemotherapy and Radiotherapy in Low-Grade Gliomas: A Mathematical Approach" *Journal of Personalized Medicine* 11, no. 10: 1036.
https://doi.org/10.3390/jpm11101036