# A Hybrid Computation Model to Describe the Progression of Multiple Myeloma and Its Intra-Clonal Heterogeneity

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## Abstract

**:**

## 1. Introduction

## 2. Hybrid Model of MM Development

#### 2.1. Cell Motion

#### 2.2. Extracellular Regulation

#### 2.3. Intracellular Regulation

#### Myeloma Cells Division and Mutations

## 3. Results

## 4. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Numerical Implementation

#### Appendix A.1. Equations for Cell Motion and Intracellular Regulation

#### Appendix A.2. Reaction-Diffusion Equations for Extracellular Cytokines Concentration

## Appendix B. Parameter Values

**Table A1.**Values of intracellular regulation parameters for myeloma cells. δ is an arbitrary length unit.

Parameter | Value | Unit |
---|---|---|

Myeloma cell cycle length | 26 | h |

Cell cycle variation | 13 | h |

space variable and step | 1 | δ |

time variable | 1 | min |

time step | 0.01 | min |

${\beta}_{1}$ | 0.001 | min${}^{-1}$ |

${\alpha}_{2}$ | 0.03 | min${}^{-1}$ |

${\beta}_{2}$ | 0.002 | min${}^{-1}$ |

${\alpha}_{3}$ | 0.001 | min${}^{-1}$ |

${\beta}_{3}$ | 0.01 | min${}^{-1}$ |

${\gamma}_{3}$ | 0.00083 | min${}^{-1}$ |

Parameter | Value | Unit |
---|---|---|

D | $0.5\times {10}^{-5}$ | ${\delta}^{2}$·min${}^{-1}$ |

W | 0.0003 | molecules·${\delta}^{-2}$·min${}^{-1}$ |

λ | 0.1 | NU |

σ | $1\times {10}^{-7}$ | min${}^{-1}$ |

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**Figure 1.**(

**a**) The intracellular regulation of myeloma cells as described in the model. Bone marrow stromal cells (BMSCs) secrete the cytokines insulin-like growth factor 1 (IGF-1), interleukin-6 (IL-6), and chemokine stromal cell-derived factor 1 (SDF-1) which are necessary to the survival, homing and proliferation of myeloma cells. Via their respective receptors, IGF-1 and IL-6 activate the RAS/ERK pathway which promotes the cell proliferation. They inhibit apoptosis through the phosphatidylinositol-3 kinase/protein kinase B/Forkhead in rabdomyosarcoma (Akt/FKHR) pathway. The cell migrates and homes to BMSCs through the SDF-1/CXCR4 axis. The IRF4 mutation, which has been associated with concomitant RAS mutations, promotes survival and proliferation. BMSCs, which are much larger cells than the myeloma cells, are shown in reduced in size in this figure as well as the receptors of both IGF-1 and IL-6; (

**b**) The parallel evolution pattern of multiple myeloma clones resulting in intra-clonal heterogeneity. More aggressive clones result from a more aggressive N-RAS mutation in clone 2 or the acquisition of IRF4 mutation in addition to the less aggressive K-RAS in clone 4. Each clone is shown by its corresponding color in the model.

**Figure 2.**(

**a**) The activation rate of RAS protein as a function of the genotype function z; (

**b**) The ERK threshold for division as a function of the genotype function z, it decreases due to the IRF4 mutation found in the clone ${c}_{4}$.

**Figure 3.**The sequential steps of myeloma cells homing to a BMSC. Myeloma cells are represented by the cells with the smaller radii and the BMSC is the large cell in the middle. Each clone of multiple myeloma (MM) cells is denoted by a specific color. The red color gradient represents the summed concentrations of the cytokines SDF-1, IL-6, and IGF-1 shown with white (0) to red (1). (

**a**) After their infiltration, the myeloma cells move towards the BMSC; (

**b**) The surviving cells surround the BMSC; (

**c**) The myeloma cells divide and form the tumor niche; (

**d**) The tumor expands and more aggressive subclones start emerging.

**Figure 4.**(

**a**) The total concentration of the IL-6 and IGF-1 cytokines over time in log scale; (

**b**) The total number of malignant cells over time.

**Figure 5.**Snapshots of a simulation showing the competition between clones. (

**a**) The tumor consists mainly of the clone ${c}_{1}$ (shown in yellow) with few cells in the intermediate state (in purple) between clones and the emergence of the clone ${c}_{2}$ (in cyan) in the sides. The size of the tumor remains limited because clone ${c}_{1}$ cells need relatively high concentration of I to survive; (

**b**) Compared to the clone ${c}_{1}$, the clone ${c}_{2}$ cells expand and survive in areas with lower concentration of cytokines; (

**c**) The clone ${c}_{2}$ cells surround the tumor and crowd out the cells of the clone ${c}_{1}$ leading to the reduction of their population. The clones ${c}_{3}$ (in blue) and ${c}_{4}$ (in magneta); (

**d**) The subclone ${c}_{4}$ is as aggressive as the subclone ${c}_{2}$ due to the additional IRF4 mutation and it manages to coexist with it in the remote areas with fewer cytokines.

**Figure 7.**Gaussian kernel density plots indicating the frequency of cells acquiring each specific mutation at the different stages of the simuation. (

**a**) The initial population composed of cells belonging to the subclone ${c}_{1}$; (

**b**) ${c}_{2}$ is the most predominant subclone after 16 hours of the beginning of the simulation; (

**c**) The K-RAS subclones ${c}_{3}$ and ${c}_{4}$ emerge and start expanding; (

**d**) The clones ${c}_{2}$ and ${c}_{4}$ represent the majority of the population because they are equally aggressive.

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Bouchnita, A.; Belmaati, F.-E.; Aboulaich, R.; Koury, M.J.; Volpert, V. A Hybrid Computation Model to Describe the Progression of Multiple Myeloma and Its Intra-Clonal Heterogeneity. *Computation* **2017**, *5*, 16.
https://doi.org/10.3390/computation5010016

**AMA Style**

Bouchnita A, Belmaati F-E, Aboulaich R, Koury MJ, Volpert V. A Hybrid Computation Model to Describe the Progression of Multiple Myeloma and Its Intra-Clonal Heterogeneity. *Computation*. 2017; 5(1):16.
https://doi.org/10.3390/computation5010016

**Chicago/Turabian Style**

Bouchnita, Anass, Fatima-Ezzahra Belmaati, Rajae Aboulaich, Mark J. Koury, and Vitaly Volpert. 2017. "A Hybrid Computation Model to Describe the Progression of Multiple Myeloma and Its Intra-Clonal Heterogeneity" *Computation* 5, no. 1: 16.
https://doi.org/10.3390/computation5010016