[Paper Review] Control of tumour growth distributions through kinetic methods

Luigi Preziosi, Giuseppe Toscani+1 more|arXiv (Cornell University)|Jun 11, 2020

Mathematical Biology Tumor Growth56 references4 citations

TL;DR

This paper introduces a novel kinetic model of tumor growth that describes tumor size distributions through mesoscopic interactions, enabling precise feedback control therapies. By scaling to a Fokker–Planck equation with variable coefficients, the model shows that control modifies the drift operator, shifting equilibrium distributions from fat-tailed generalized Gamma to slim-tailed forms, significantly reducing the risk of large, uncontrollable tumors.

ABSTRACT

The mathematical modeling of tumor growth has a long history, and has been mathematically formulated in several different ways. Here we tackle the problem in the case of a continuous distribution using mathematical tools from statistical physics. To this extent, we introduce a novel kinetic model of growth which highlights the role of microscopic transitions in determining a variety of equilibrium distributions. At variance with other approaches, the mesoscopic description in terms of elementary interactions allows to design precise microscopic feedback control therapies, able to influence the natural tumor growth and to mitigate the risk factors involved in big sized tumors. We further show that under a suitable scaling both the free and controlled growth models correspond to Fokker--Planck type equations for the growth distribution with variable coefficients of diffusion and drift, whose steady solutions in the free case are given by a class of generalized Gamma densities which can be characterized by fat tails. In this scaling the feedback control produces an explicit modification of the drift operator, which is shown to strongly modify the emerging distribution for the tumor size. In particular, the size distributions in presence of therapies manifest slim tails in all growth models, which corresponds to a marked mitigation of the risk factors. Numerical results confirming the theoretical analysis are also presented.

Motivation & Objective

• To develop a mesoscopic kinetic model of tumor growth based on cellular-level interactions, moving beyond deterministic ODEs.
• To address the limitations of traditional ODE-based growth models, which struggle with biological uncertainty and lack direct links between cellular therapies and macroscopic outcomes.
• To design microscopic feedback control strategies that explicitly influence tumor size distribution by modifying transition rates in the kinetic framework.
• To analyze the resulting Fokker–Planck equations under scaling, identifying how control alters drift and diffusion coefficients to shape equilibrium distributions.
• To demonstrate that controlled tumor growth leads to slim-tailed distributions, indicating reduced risk of large, therapeutically challenging tumors.

Proposed method

• Formulate a Boltzmann-type kinetic model where tumor size evolution results from elementary growth and death transitions influenced by environmental cues and stochastic fluctuations.
• Define transition functions that encode microscopic growth dynamics, allowing derivation of mesoscopic evolution equations for the distribution function.
• Apply a hydrodynamic scaling limit to derive a Fokker–Planck-type equation with variable diffusion and drift coefficients, capturing the stochastic growth process.
• Introduce additive and multiplicative control mechanisms on the drift term to model therapeutic interventions, with explicit time-dependent modifications to parameters α(t) and xL(t).
• Characterize steady-state solutions of the controlled and uncontrolled models as generalized Gamma distributions, with fat tails in the free case and slim tails under control.
• Use numerical simulations to validate theoretical predictions, showing that control effectively shifts the distribution away from large tumor sizes.

Experimental results

Research questions

• RQ1 How can a kinetic model of tumor growth be constructed to reflect microscopic cellular interactions and environmental fluctuations?
• RQ2 What is the impact of feedback control on the steady-state distribution of tumor sizes, particularly in terms of tail behavior?
• RQ3 How does the scaling limit of the kinetic model lead to a Fokker–Planck equation with variable coefficients, and what does this imply for tumor size dynamics?
• RQ4 In what way does control modify the drift operator in the Fokker–Planck equation, and how does this affect the emergence of equilibrium distributions?
• RQ5 Can the controlled model produce distributions with slim tails, indicating a reduced risk of large tumors compared to the uncontrolled case?

Key findings

• The uncontrolled growth model yields a steady-state distribution that is a generalized Gamma distribution with fat tails, indicating a higher probability of large tumor sizes.
• Under the proposed control strategy, the equilibrium distribution shifts to one with slim tails, significantly reducing the likelihood of tumors reaching large, dangerous sizes.
• The control mechanism modifies the drift operator in the Fokker–Planck equation, which is shown to be the key driver in reshaping the tumor size distribution.
• The scaling limit of the kinetic model results in a Fokker–Planck equation with variable diffusion and drift coefficients, enabling a rigorous connection between microscopic interactions and macroscopic dynamics.
• Numerical results confirm that the controlled model leads to a marked mitigation of risk factors associated with large tumors, as evidenced by the change in tail behavior.
• The model provides a principled framework to link cellular-level therapeutic actions to macroscopic outcomes, offering a path to optimize treatment schedules through explicit control of distributional properties.