Control of tumor growth distributions through kinetic methods

Highlights

• Novel probabilistic approach for the evolution of the distribution of the tumor size.

• Large sized tumors are linked to emerging fat tailed distributions.

• Therapeutical protocols designed through control strategies in the growth factor.

• The distribution of the tumor size in presence of therapies exhibits slim tails.

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.

Introduction

Since the early years of cancer research one of the basic questions addressed by scientists aimed at the identification of the growth law followed by tumors. The natural related purpose was the need of using it to model the effect of cancer treatment and optimize therapy. The easiest but still most used way to do that is to model growth by an ODE, usually of first order.According to the right hand side, they are named after Malthus (i.e., the exponential growth law), Verhulst (i.e., logistic growth law), Gompertz, Richards, von Bertalanffy, West, and so on. In particular, West et al. (2001) gave a new insight to von Bertalanffy’s growth model starting from an original viewpoint. The parameters of the model are then optimized to fit the available experimental data in absence and presence of therapies. Due to the need of fitting the same data, they mostly give rise to a similar sigmoidal behaviour characterized by an asymptotic tendency to an equilibrium related to the presence of a carrying capacity. The literature on the subject is huge. So, for more information we refer to the recent review papers (Gerlee, 2013, Rodriguez-Brenes et al., 2013, Sarapata and de Pillis, 2014) and volumes (Schättler and Ledzewicz, 2015, Wodarz and Komarova, 2014). The process of parameter identification is affected by many sources of uncertainty stemming out at different and independent levels of observation. To name a few, the first one consists in the fact that the evaluation of the number of cells in a tumor is obtained using only partial information, e.g., approximating the tumor as an ellipsoid on the basis of the maximum and the minimum dimension measured ex-vivo (the middle axis of the ellipsoid is then approximated as the mean of the measurements above), or obtained by two-dimensional in vivo images assuming that the observed section is the one containing the longest and shortest axis of the ellipsoid. The second one regards the presence within the same body of many metastasis of different sizes growing in different environmental conditions. The third regards the fact that in a cohort of individuals, from nude mice used in experiments up to humans, the evolution is not the same because in each host the response of the body is different.

So, in spite of the apparent simplicity of the question, at present there is no general consensus on the type of growth law that is better to be used to fit data, with stochasticity playing a role that is often overwhelming with respect to the difference among the evolutions predicted by the different models. In addition, the relation between the therapeutic action operating at the cellular level and the macroscopic parameter, e.g. the carrying capacity, is not immediate. On the other hand, regardless of the exact fitting of the growth law, as stated for instance in Langer et al. (1980), one of the therapeutic goals in oncology is to control tumor growth and to reduce the probabilities of having tumors growing to sizes that are too large to be physiologically or therapeutically controllable, or that are harmful to the human body.

In order to accomplish this task, rather than modelling the tumor with a stochastic adaptation of the ODE growth models, we present here a novel kinetic approach, which aims to describe the growth of tumor cells in terms of the evolution of a distribution function whose temporal variation is the result of transitions occurring at the cellular level that lead to an increase or decrease in tumor size, related to growth and death processes. The mathematical description proposed here is based on a Boltzmann-type model where the elementary variations describing the number of cancer cells are determined by a transition function which takes environmental cues and random fluctuations into account. Under a suitable limit procedure, different choices of parameters in the transition probability will characterize the equilibrium distribution.

The notion of growth in random environment has been formulated before in the framework of stochastic birth and death processes by several Authors (see, for instance, Nobile and Ricciardi, 1980, Prajneshu, 1980, Tan, 1986 and references therein) to take into account of environmental fluctuations. In this framework, a stochastic model of tumor growth was introduced by Albano and Giorno (2006). Application of tools from nonlinear statistical physics to describe biological phenomena involving a huge number of entities represents one of the major challenges in contemporary mathematical modeling (Adam and Bellomo, 1997, Bellomo, 2008, Bellomo and Delitala, 2008, Bellomo et al., 2008, Perthame, 2007). A consistent part of these applications makes a substantial use of methods borrowed from kinetic theory of rarefied gases, which, starting from a mesoscopic description of microscopic cells interactions, leads to construct master equations of Boltzmann type, usually referred to as kinetic equations, able to drive the system towards universal statistical profiles. An important example of emergent behavior is concerned with the building of tumors by cancer cells and their migration through the tissues (Bellomo and Delitala, 2008, Grizzi and Chiriva-Internati, 2006, Hatzikirou et al., 2010, Moreira and Deutsch, 2002). Another example to consider in this context is the classical Luria–Delbrück mutation problem treated by statistical methods (Kashdan and Pareschi, 2012, Toscani, 2013). In multicellular organisms, these two examples are closely related, since the connection between mutagenesis and carcinogenesis is broadly accepted (see, for instance, Frank, 2007, Frank, 2002, Kendall, 1960). Furthermore, the Luria–Delbrück distribution plays an important role in the study of cancer, because tumor progression depends on how heritable changes (mutations) accumulate in cell lineages. While the basic entities in these examples differ from the physical particles in that they already have an intermediate complexity, for some specific phenomena, like the statistical growth of mutated cells, one can reasonably assume that the statistical behavior of the system is mainly related to the peculiar way entities interact and not to their internal complex structure. From our point of view, the most important output of using a kinetic model is to have an equilibrium distribution stemming from stochastic interactions occurring at the microscopic level, i.e. the cellular level. In particular, the distribution function will give the probability of having tumours of size bigger than a given alerting size. Most importantly, it will be shown that, in different regimes of parameters of the general transition law, the emerging equilibrium distribution of the Boltzmann-type model shows a radically heterogeneous behavior in terms of the decay of the tails. In details, transitions laws that in a suitable limit are related to logistic-type growths are associated to a generalized Gamma density function which is characterized by slim tail, i.e. by exponential decay. On the other hand, transitions laws that in a suitable limit are related to von Bertalanffy-type growths are associated to Amoroso-type distributions that are rather characterized by fat tail, i.e. by polynomial decay. The border case between the two distributions leads to lognormal-type equilibria which exhibits slim tail, but with a possible dramatic increase of higher moments. From a statistical physics point of view, it is worth to remark that in the context of tumor growth the dynamics leading to fat-tailed distributions imply the formation of big sized tumors with high probability. Therefore, the distributions with fat tails can be associated to an increased risk for the human body. For this reason, once characterized the emerging distributions of the mentioned growth dynamics, we concentrate on implementable therapeutical control strategies so that fat tails can be transformed in thin tails which means mitigating the risk of having big tumors. The control is determined analytically for any growth law.The control of emerging phenomena described by kinetic models or mean field theories is relatively recent (Albi et al., 2015, G. Albi et al., 2017, Albi et al., 2014, Bensoussan et al., 2013, Degond et al., 2017, Fornasier et al., 2014). In particular, the proposed approach can be derived from a model predictive control (MPC) strategy which is based on determining the control by optimising a given cost functional over a finite time horizon which recedes as time evolves (Camacho and Bordons Alba, 2007, Sontag, 1998). Assuming that the the minimisation horizon coincides with the duration of a single transition, we obtain a feedback solution to the control problem that can be implemented efficiently in the Boltzmann-type kinetic model to observe its aggregate effects. It is well known that MPC leads typically to suboptimal controls. Nevertheless, performance bounds are computable to guarantee the consistency of the MPC approximation in a kinetic framework (Grüne, 2009, Herty and Zanella, 2017).

In more detail, the paper is organized as follows. The kinetic model for tumor growth is presented in Section 2 where we introduce elementary variations of the number of cancer cells depending on a transition function determining the deterministic variations of the tumors’ size, and on random fluctuations. In suitable regimes we will obtain a classification of equilibrium distributions corresponding to the introduced growth models, some of them exhibiting fat tails. The controlled model is presented in Section 3 and the emerging slim tailed distributions are computed for two possible therapeutical strategies. Finally, we summarise the highlights of the work and draw some conclusions. In Appendix A-B we present a brief review of microscopic and mean-field models for tumour growth.