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Kinetic and mean-field modeling of muscular dystrophies

Tommaso Lorenzi, Horacio Tettamanti, Mattia Zanella

TL;DR

This work develops a kinetic framework for the distributions of four key cell populations in muscular dystrophy, then derives a mean-field (Fokker-Planck) limit and a macroscopic description for cohort-level means and variances. By specifying detailed microscopic rules for normal/damaged cells and immune populations, the authors obtain explicit evolution equations for the means, a nonclosed variance system in the kinetic setting, and a closed variance system in the mean-field limit. They show that the mean-field solution exhibits inverse-Gamma quasi-equilibria with tractable expressions for shapes, scales, and long-time means, and prove convergence to equilibrium in the $\dot H_{-p}$ norm. Numerical simulations using DSMC and structure-preserving discretizations validate the kinetic-to-mean-field limit and illustrate convergence to the predicted equilibria, offering a principled framework to integrate heterogeneous patient data and potentially assess therapeutic interventions.

Abstract

We present a new class of models for assessing the cell dynamics characterising muscular dystrophies. The proposed approach comprises a system of integro-differential equations for the statistical distributions, over a large patient cohort, of the densities of muscle fibers and immune cells implicated in muscle inflammation, degeneration, and regeneration, which underpin disease development. Considering an appropriately scaled version of this model, we formally derive, as the corresponding mean-field limit, a system of Fokker-Planck equations, from which we subsequently derive, as a macroscopic model counterpart, a system of differential equations for the mean densities of muscle and immune cells in the cohort of patients and the related variances. Then, we study long-time asymptotics for the mean-field model by determining the quasi-equilibrium cell distribution functions, which are in the form of probability density functions of inverse Gamma distributions, and proving the long-time convergence to such quasi-equilibrium distributions. The analytical results obtained are illustrated by means of a sample of results of numerical simulations. The modeling approach presented here has the potential to offer new insights into the balance between degeneration and regeneration mechanisms in the progression of muscular dystrophies, and provides a basis for future extensions, including the modeling of therapeutic interventions.

Kinetic and mean-field modeling of muscular dystrophies

TL;DR

This work develops a kinetic framework for the distributions of four key cell populations in muscular dystrophy, then derives a mean-field (Fokker-Planck) limit and a macroscopic description for cohort-level means and variances. By specifying detailed microscopic rules for normal/damaged cells and immune populations, the authors obtain explicit evolution equations for the means, a nonclosed variance system in the kinetic setting, and a closed variance system in the mean-field limit. They show that the mean-field solution exhibits inverse-Gamma quasi-equilibria with tractable expressions for shapes, scales, and long-time means, and prove convergence to equilibrium in the norm. Numerical simulations using DSMC and structure-preserving discretizations validate the kinetic-to-mean-field limit and illustrate convergence to the predicted equilibria, offering a principled framework to integrate heterogeneous patient data and potentially assess therapeutic interventions.

Abstract

We present a new class of models for assessing the cell dynamics characterising muscular dystrophies. The proposed approach comprises a system of integro-differential equations for the statistical distributions, over a large patient cohort, of the densities of muscle fibers and immune cells implicated in muscle inflammation, degeneration, and regeneration, which underpin disease development. Considering an appropriately scaled version of this model, we formally derive, as the corresponding mean-field limit, a system of Fokker-Planck equations, from which we subsequently derive, as a macroscopic model counterpart, a system of differential equations for the mean densities of muscle and immune cells in the cohort of patients and the related variances. Then, we study long-time asymptotics for the mean-field model by determining the quasi-equilibrium cell distribution functions, which are in the form of probability density functions of inverse Gamma distributions, and proving the long-time convergence to such quasi-equilibrium distributions. The analytical results obtained are illustrated by means of a sample of results of numerical simulations. The modeling approach presented here has the potential to offer new insights into the balance between degeneration and regeneration mechanisms in the progression of muscular dystrophies, and provides a basis for future extensions, including the modeling of therapeutic interventions.

Paper Structure

This paper contains 19 sections, 1 theorem, 107 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Kinetic and mean-field models
  3. Microscopic rules underlying the kinetic model
  4. Microscopic rules underlying the dynamics of normal and damaged cells.
  5. Microscopic rules underlying the dynamics of cytotoxic T lymphocytes and macrophages.
  6. The kinetic model
  7. Evolution of key observable macroscopic quantities for the kinetic model
  8. Evolution equations for $m_J(t)$, $J \in \{N,D,M,C\}$.
  9. Evolution equations for $V_J(t)$, $J \in \{N,D,M,C\}$.
  10. Corresponding mean-field model
  11. Evolution of key observable macroscopic quantities for the mean-field model
  12. Long-time asymptotics for the mean-field model
  13. Characterisation of the quasi-equilibrium distribution
  14. Long-time convergence to the equilibrium distribution
  15. Numerical tests
  16. ...and 4 more sections

Key Result

Theorem 3.1

Let $f_J(0,x)$, $J \in \{N,D,M,C\}$, be probability density functions on $\mathbb R_+$ such that conditions eq:unitmassIC, eq:boundedmeanIC, eq:ICNpmD, and eq:boundedvarIC hold and such that $\| f_J(0,x) - f_J^\infty(x)\|_{\dot H_{-p}}<\infty$ for $\frac{1}{2}<p<1$, with $f_J^\infty(x)$ given by eq:

Figures (6)

  • Figure 1: Evolution of key observable macroscopic quantities for the mean-field model.Top row. Plots of the components of the numerical solution of the system \ref{['eq:mean_K_NDMC']} (left panel) and the system \ref{['eq:var_MF_NDMC']} (right panel) for $t \in [0,100]$. Center and bottom rows. Corresponding trajectories of the numerical solutions in the phase planes $(m_N, m_D)$ (center, left panel), $(m_M, m_C)$ (center, right panel), $(V_N, V_D)$ (bottom, left panel), and $(V_M, V_C)$ (bottom, right panel). The red dots highlight the points corresponding to the components of the equilibria \ref{['eq:MNDMCinfty']} and \ref{['eq:VNDMCinfty']}. Numerical simulations were carried out under initial conditions of components $m_N(0) = 9$, $m_D(0)= 1$, $m_M(0) = 0.1$, $m_C(0) = 0.5$, and $V_J(0) = 0.1$ for all $J \in \{N,D,M,C\}$, and the parameter values $\beta_N = 0.2$, $\beta_D = 0.1$, $\beta_M = 0.2$, $\beta_C = 0.1$, $\gamma_M = \gamma_C = 0.05$, and $\sigma_J^2 = 0.01$ for all $J \in \{N,D,M,C\}$, which were chosen with exploratory aim and are to be intended as being dimensionless.
  • Figure 2: Consistency between the kinetic model and its mean-field limit. Solid, coloured lines highlight the dynamics of the means, $m_J(t)$ defined via \ref{['eq:mJ']}, of the distribution functions, $f_J(x,t)$, with $J \in \{N,D,M,C\}$, of the kinetic model \ref{['eq:system']}, under the time scaling \ref{['eq:timescalingeps']} and the parameter scaling \ref{['eq:paramscalingeps']}, for the values of the scaling parameter $\epsilon$ provided in the legends. Dashed, black lines highlight the dynamics of the means of the distribution functions of the mean-field model \ref{['eq:MFN']}-\ref{['eq:MFoBCs']}. Numerical simulations were carried out under initial conditions of components defined via \ref{['eq:init_f']} and the parameter values listed in Table \ref{['Table:params']}.
  • Figure 3: Consistency between the kinetic model and its mean-field limit. Solid, coloured lines highlight the dynamics of the variances, $V_J(t)$ defined via \ref{['eq:VJ']}, of the distribution functions, $f_J(x,t)$, with $J \in \{N,D,M,C\}$, of the kinetic model \ref{['eq:system']}, under the time scaling \ref{['eq:timescalingeps']} and the parameter scaling \ref{['eq:paramscalingeps']}, for the values of the scaling parameter $\epsilon$ provided in the legends. Dashed, black lines highlight the dynamics of the variances of the distribution functions of the mean-field model \ref{['eq:MFN']}-\ref{['eq:MFoBCs']}. Numerical simulations were carried out under initial conditions of components defined via \ref{['eq:init_f']} and the parameter values listed in Table \ref{['Table:params']}.
  • Figure 4: Convergence to equilibrium of the mean-field model. Plots of the distribution functions, $f_J(x,t)$, with $J \in \{N,D,M,C\}$, of the mean-field model \ref{['eq:MFN']}-\ref{['eq:MFoBCs']} for $t \in \{0, 1, 8, 75\}$ (lines without markers) and the corresponding equilibrium distributions defined via \ref{['eq:finftyJ']} (lines with circle markers). Numerical simulations were carried out under initial conditions of components defined via \ref{['eq:init_f']} and the parameter values listed in Table \ref{['Table:params']}.
  • Figure 5: Convergence to equilibrium of the mean-field model. Plots of $\mathcal{E}^p(f_J,f_J^q)$ defined via \ref{['def:mathcalE']}, for $p \in (\frac{1}{2},1)$, i.e. $p \in \{ \frac{5}{8}, \frac{3}{4}, \frac{7}{8}\}$, with $J \in \{N,D,M,C\}$. Numerical simulations were carried out under initial conditions of components defined via \ref{['eq:init_f']} and the parameter values listed in Table \ref{['Table:params']}.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Theorem 3.1
  • proof