Thermostatted Kinetic Theory Structures in Biophysics: Generalizations and Perspectives

Abstract

The mathematical modeling of multicellular systems is an important branch of biophysics, which focuses on how the system properties emerge from the elementary interaction between the constituent elements. Recently, mathematical structures have been proposed within the thermostatted kinetic theory for the modeling of complex living systems and have been profitably employed for the modeling of various complex biological systems at the cellular scale. This paper deals with a class of generalized thermostatted kinetic theory frameworks that can stand in as background paradigms for the derivation of specific models in biophysics. Specifically, the fundamental homogeneous thermostatted kinetic theory structures of the recent literature are recovered and generalized in order to take into consideration further phenomena in biology. The generalizations concern the conservative, the nonconservative, and the mutative interactions between the inner system and the outer environment. In order to sustain the strength of the new structures, some specific models of the literature are reset into the style of the new frameworks of the thermostatted kinetic theory. The selected models deal with breast cancer, genetic mutations, immune system response, and skin fibrosis. Future research directions from the theoretical and modeling viewpoints are discussed in the whole paper and are mainly devoted to the well-posedness in the Hadamard sense of the related initial boundary value problems, to the spatial–velocity dynamics and to the derivation of macroscopic-scale dynamics.

1. Introduction

The mathematical modeling of living systems has gained remarkable interest in this century, and great effort has been dedicated to the development of general structures able to be employed for the derivation of specific models [1,2,3,4,5]. Scholars have been so attracted by the complexity of living systems to the point that the construction of a theory was invoked [6]. If, on the one hand, the proposition of a suitable general theory appears a long-cherished dream, on the other hand, many general structures have been offered to the pertinent literature.

A mathematical structure is usually defined as a set of algebraic or differential equations formulated by means of parameters and functions. Once all of the functions and parameters are established, a specific model is said to be derived. The mathematical structure or framework is then itself a general paradigm or theory. It is understandable that many different theories can be proposed and adapted to a class of systems. In this context, the complexity of the system needs to be taken into consideration, thus reducing the applicability of a theory [7,8,9]. Bearing all the above in mind, in the development of a theory for complex living systems, the common properties shared by the complex systems need to be identified; this is an important issue in the definition of complexity itself, which is a hard question that has prevented a universal definition of complexity [10]. However, some common features have been identified [11]—the number of the elements composing the system, heterogeneity, self-organization, the unpredictable result of the interactions, and the emergence of collective behaviors, among others.

The mathematical structures that can be found in the literature were derived by following some existing theories employed for the inert matter, such as continuum mechanics [12,13,14], kinetic theory [6,15], and nonequilibrium physics [16,17], but also by means of ordinary and/or partial differential equations [18,19]. Hybrid frameworks coming from physics, information theory, and agent-based structures have also been suggested [20,21,22]. However, using only one of these frameworks is neither sufficient to model all of the phenomena occurring in a system nor able to model all types of systems. Moreover, a phenomenon needs to be modeled by looking at it on a specific scale or by considering different scales (multiscale problem); see, among others, the following review papers [23,24,25].

This paper focuses on the modeling of complex biological systems. From the modeling viewpoint, three main representation scales have been identified in biology [26]: Microscopic, mesoscopic, and macroscopic scales, which, roughly speaking, correspond to gene/protein, cell, and tissue/organ scales, respectively. On the one hand, the biological scales are well-established; on the other hand, the mathematical structures are not strictly related to them. Indeed, the same structure can be employed for modeling at each scale, e.g., in the case of an ordinary differential equations-based model each equation can model the density of the gene, cells, or tissues. However, a specific structure can be more appropriate at a specific scale in order to bound the number of parameters. Accordingly, the following convention of modeling choice might be made: microscopic phenomena are modeled by employing agent-based models or ordinary differential equations; mesoscopic phenomena are modeled by generalized kinetic theory and statistical mechanics; macroscopic phenomena are modeled by continuum mechanics or, more generally, partial differential equations. By making further assumptions concerning the system—e.g., homogeneous conditions—macroscopic phenomena can be modeled by ordinary differential equations.

This paper focuses on complex multicellular systems modeled at a cellular scale (kinetic) by making use of the methods of biophysics. In contrast to molecular biophysics [27], which regards biological systems as single entities (a macromolecule, cell, or tissue), systems biophysics [28] is interested in how the system features emerge from the interactions between the constituent elements.

In this context, the methods of thermostatted kinetic theory have been introduced. A review paper [29] has settled the basis of this theory for the modeling of complex biological systems. This paper is devoted to the recent developments of the thermostatted kinetic theory for active particles. Specifically, various mathematical frameworks recently proposed in the literature are generalized in order to model further phenomena occurring at the cellular scale.

The thermostatted kinetic theory for biological systems is based on the new system biology approach proposed in [30], where the whole system is divided into different subsystems composed by cells expressing the same functional state or strategy. The microscopic state of the cells, assumed homogeneous with respect to the mechanical variables (space and velocity), consists of a continuous variable called activity. The time evolution of each functional subsystem is statistical by means of a distribution function. Interactions are responsible for the activity modification or proliferation/destruction/mutation of cells. The interactions are modeled by introducing interactions rates, transition probability functions, and proliferation rates, which depend on the functional state of cells. The interactions are stochastic and can follow the rules of game theory [31]. The mathematical structures, obtained by equating the different flows of cells into the volume of the microscopic states, consist of nonlinear integro-differential equations coupled with suitable initial and/or boundary conditions. Moreover, the nonequilibrium conditions of a biological system are modeled by introducing external force fields coupled with a Gaussian-type thermostat in order to ensure control of the total energy of the system and the reaching of nonequilibrium stationary states, see papers [32,33,34,35,36] and the references therein.

It is worth describing that differently from agent-based modeling, the thermostatted kinetic theory proposes mathematical frameworks that allow us to perform an asymptotic analysis. Moreover, the particles are grouped in a low number of functional subsystems, making it less expensive from a computational viewpoint.

The generalized mathematical structures of the present paper also take into consideration microscopic external actions modeled as outer functional subsystems [37]. Among the generalizations, the introduction of a weighted function that defines an interaction domain [38] is presented in each framework. Generalizations of the nonconservative interactions (proliferation and mutation) refer in particular to the new structure with multivariate activity. Finally, a unified homogeneous framework is proposed as a general paradigm for the definition of specific models.

It is worth stressing that the systems under consideration are assumed homogeneous with respect to the space and velocity variables. However, the introduction of spatial and velocity dynamics are discussed in the last section of the paper.

This paper, after discussing the most important thermostatted structures of kinetic theory, focuses on their applications, which consist of specific models of the literature that can be rewritten within the thermostatted kinetic theory for active particles. Three models are selected from the field of cancer-immune system competition [39] and fibrosis diseases [40].

The present paper is organized into ten sections, including this introduction. The contents are divided as follows:

-

Section 2 deals with the first proposed framework of the thermostatted kinetic theory. Specifically, the system, assumed homogeneous, is composed by only one functional subsystem subjected to conservative interactions. The main macroscopic quantities are defined and the main functions defining the conservative interaction operator are presented at a tutorial level. The first generalization is discussed within the section that consists in the introduction of a function that weights the interactions. The mathematical results existing in the literature and concerning the related Cauchy/Dirichlet problems are mentioned.

-

Section 3 is devoted to the generalizations when the system is composed by different functional subsystems. The mathematical structure consists of a system of nonlinear integro-differential equations with quadratic nonlinearities. As in the previous section, the weighted function is introduced and the related mathematical analysis is discussed.

-

Section 4 is concerned with the nonconservative interactions, namely, the proliferation and destruction of cells because of the interactions, which are generalized with respect to the literature, by introducing the weighted function and by generalizing the interaction operators. Moreover, the natural birth/death processes are presented for the first time and add to the thermostatted kinetic theory framework.

-

Section 5 highlights the role of mutations. Mutations occur because of mutual interactions; in particular, two different types of mutative interactions are defined, generalizing the existing interaction operators. The related mathematical analysis is mentioned by discussing the main difficulties in the context of the existence of a global solution.

-

Section 6 introduces the interactions with the outer environment at the microscopic scale. The outer environment is modeled by introducing known distribution functions. In particular, for the first time, the definition of the nonconservative and the mutative interactions are developed as consequences of interactions with the external system.

-

Section 7 presents a recent development of the thermostatted kinetic theory that is based on the definition of a vectorial activity variable. A multivariate activity variable is a generalization recently proposed for the conservative interactions. In the present paper, for the first time, nonconservative and mutative interactions are set in the multivariable activity framework.

-

Section 8 focuses on the applications. Three existing models presented in the literature are revisited in the style of the thermostatted kinetic theory. Specifically, the first model concerns breast cancer and the competition with some cells of the immune system. The second application refers to the onset of genetic mutations. Finally, the third application is related to a specific skin fibrosis, called keloid—in particular, the role of therapeutic actions is taken into account.

-

Section 9 concludes the paper with a critical analysis on the proposed frameworks and with future research perspectives from the theoretical and modeling viewpoints. Specifically, the role of the space and velocity variables are mentioned as well as the derivation of macroscopic scale models by asymptotic analysis methods.

According to the above description of contents, the paper aims to address three issues: The first issue is theoretical and relevant to the mathematical frameworks with generalizations of the thermostatted kinetic theory (Section 2, Section 3, Section 4, Section 5, Section 6 and Section 7); the second issue is to show the applicability of the frameworks by selecting some specific models of the literature (Section 8); the third issue refers to the known open problems, as thermostatted kinetic theory can be considered a fruitful, interdisciplinary research domain (Section 9).

2. Conservative-Interactions Thermostatted Kinetic Theory

This section deals with the first mathematical framework of a generalized kinetic theory that was proposed for the modeling of complex living systems composed by a large number of cells. According to the theory, the microscopic state of cells is composed of a real scalar variable u called activity that allows us to characterize the heterogeneity of cells. In particular, this variable is employed for establishing a specific property of the cells. In the biophysics system under consideration, the role of the space and velocity variable is neglected. The dynamics of the system is thus assumed independent on the space and velocity dynamics (homogeneous system).

Bearing all above in mind, let S be a system of the biophysics domain. The time evolution of the system is described by introducing a distribution function $f=f(t,u)$, where $t\in [0,+\infty [$ and $u\in D_u\subseteq \mathbb{R}$. The macroscopic density of system is defined as follows:

$${\mathbb{E}}_0\left[f\right]\left(t\right)={\int }_{D_u}f(t,u)\mathrm{d}u.$$

(1)

Moreover, the linear-activity and the energy-activity macroscopic quantities are defined as follows:

$${\mathbb{E}}_1\left[f\right]\left(t\right)={\int }_{D_u}uf(t,u)\mathrm{d}u,{\mathbb{E}}_2\left[f\right]\left(t\right)={\int }_{D_u}{u}^{2}f(t,u)\mathrm{d}u.$$

(2)

The cells are able to interact with each other, and the result of these interactions allows the modification of the magnitude of the activity variable. In order to quantify these interactions, the following positive functions need to be defined:

(A1) 

The interaction/encounter rate $\eta =\eta (u_{*},u^{*})$ between a cell with activity $u_{*}$ and a cell with activity $u^{*}$.

(A2) 

The function $\mathcal{A}=\mathcal{A}(u_{*},u^{*},u)$, called the table of interactions, which models the probability that the cell with activity $u_{*}$, interacting with the cells with activity $u^{*}$, reaches activity u. The probability function $\mathcal{A}$ verifies the following identity:

$${\displaystyle {\int }_{D_u}\mathcal{A}(u_{*},u^{*},u)du=1,\forall u_{*},u^{*}\in D_u.}$$

According to the above description, the interactions among the cells are modeled by introducing a conservative operator $\mathcal{C}=\mathcal{C}\left[f\right](t,u)$, which takes into account the gain $\mathcal{G}=\mathcal{G}\left[f\right](t,u)$ and the loss $\mathcal{L}=\mathcal{L}\left[f\right](t,u)$ of cells with microscopic state u. The conservative operator reads

$$\begin{array}{ccc}\hfill \mathcal{C}\left[f\right]& =& \mathcal{G}\left[f\right]-\mathcal{L}\left[f\right]={\int }_{D_u\times D_u}\eta (u_{*},u^{*})\mathcal{A}(u_{*},u^{*},u)f(t,u_{*})f(t,u^{*})d{u}_{*}d{u}^{*}\hfill \\ & & -f(t,u){\int }_{D_u}\eta (u,u^{*})f(t,u^{*})d{u}^{*}.\hfill \end{array}$$

(3)

From the notation point of view, the function $\mathcal{A}(u_{*},u^{*},u)$ can be rewritten as $\mathcal{A}(u_{*}\to u|u^{*})$. The first notation will be employed in order to avoid a wrong read of the variables.

The evolution equation of the system, which is obtained by balancing the time derivative of f to the flow of interactions into the elementary volume of the microscopic states, writes

$${\partial }_{t}f(t,u)=\mathcal{C}\left[f\right](t,u).$$

(4)

Remark 1. 

The kinetic framework (4) is conservative, namely, the interactions do not modify the density of cells. Indeed, it is easy to see that

$${\int }_{D_u}\mathcal{C}\left[f\right](t,u)du=0,$$

and then

$${\int }_{D_u}{\partial }_{t}f(t,u)du=0.$$

According to the kinetic framework (4), each cell $u_{*}$ is able to interact with all cells $u^{*}$. However, it can be necessary to weight the intensity of the interactions by introducing a function $w=w(u_{*},u^{*})$ such that

$${\int }_{D_u}w(u_{*},u^{*})d{u}^{*}=1,\forall u_{*}\in D_u.$$

The kinetic equation with weighted interactions thus reads

$$\begin{array}{ccc}\hfill {\partial }_{t}f(t,u)& =& {\mathcal{C}}_w\left[f\right](t,u)={\int }_{D_u\times D_u}\eta (u_{*},u^{*})w(u_{*},u^{*})\mathcal{A}(u_{*},u^{*},u)f(t,u_{*})f(t,u^{*})d{u}_{*}d{u}^{*}\hfill \\ & & -f(t,u){\int }_{D_u}\eta (u,u^{*})w(u,u^{*})f(t,u^{*})d{u}^{*}.\hfill \end{array}$$

(5)

As the reader can easily see, the weighted kinetic theory framework (5) is conservative as well.

The kinetic theory framework (4) and the weighted kinetic theory framework (5) need to be coupled to an initial condition $f^0\left(u\right)$. The statement of the Cauchy problem reads

$$\left\{\begin{array}{cc}{\displaystyle {\partial }_{t}f(t,u)={\mathcal{C}}_w\left[f\right](t,u)}\hfill & (t,u)\in [0,+\infty [\times D_u\hfill \\ f(0,u)=f^0\left(u\right)\hfill & u\in D_u\hfill \end{array}\right.$$

(6)

The Cauchy problem (6) can be considered as a general paradigm for the derivation of homogeneous specific models in biophysics.

A biophysics system usually operates out-of-equilibrium. The introduction of an external force field $F=F\left(u\right):D_u\to {\mathbb{R}}^{+}$ coupled to a damping term, which ensures the existence of nonequilibrium stationary states, was proposed in [29]. Specifically, by assuming the conservation of the activity–energy ${\mathbb{E}}_2\left[f\right]$, a thermostat operator was derived, where $f(t,u)=0$ for $u\in \partial D_u$ (the boundary of $D_u$). The thermostatted kinetic theory framework thus reads

$${\partial }_{t}f(t,u)+T_F\left[f\right](t,u)={\mathcal{C}}_w\left[f\right](t,u),$$

(7)

where

$$T_F\left[f\right](t,u)={\partial }_u\left(\left(F\left(u\right)-{\displaystyle \frac{u}{{\mathbb{E}}_2\left[f\right]}}{\int }_{D_u}uF\left(u\right)f(t,u)du\right)f(t,u)\right),$$

is called the thermostat operator.

The conservative property of the thermostatted kinetic theory framework (7) was proved under the following further assumptions on the function $\mathcal{A}$:

(A3) 

$\mathcal{A}$ is an even function with respect to u.

(A4) 

$\mathcal{A}$ is such that

$${\int }_{D_u}{u}^2\mathcal{A}(u,u_{*},u^{*})du=u_{*}^2\forall u_{*},u^{*}\in D_u.$$

The initial boundary value problem now writes

$$\left\{\begin{array}{cc}{\displaystyle {\partial }_{t}f(t,u)+T_F\left[f\right](t,u)={\mathcal{C}}_w\left[f\right](t,u)}\hfill & (t,u)\in [0,+\infty [\times D_u\hfill \\ f(0,u)=f^0\left(u\right)\hfill & u\in D_u\hfill \\ f(t,u)=0\hfill & (t,u)\in [0,+\infty [\times \partial D_u\hfill \end{array}\right.$$

(8)

The mathematical analysis of the previous kinetic theory frameworks was addressed to the existence and uniqueness of the solution and to the existence and uniqueness of the nonequilibrium stationary state when $w=1$, see [41]. The well-posedness in the Hadamard sense was proved in [42,43]. Consequently, the development of specific models was allowed and the construction of numerical solutions was performed. The mathematical analysis of the weighted thermostatted kinetic theory problem (8) can be considered as a further research direction.

3. Multiple-Subsystems Thermostatted Kinetic Theory

This section is devoted to the modeling of a biophysics system composed by a large number of cells expressing different strategies. According to the previous section, the activity variable u allows the decomposition of the whole system into subsystems, called functional subsystems, composed by cells expressing the same functional state (activity), see Figure 1 and Figure 2. Specifically, assume that the system is composed by $n\in \mathbb{N}$ functional subsystems. Let $f_i=f_i(t,u)$ be the distribution function of the ith functional subsystem, for $i\in \{1,2,\cdots ,n\}$. Accordingly, the local and global macroscopic quantities can be defined by generalizing the macroscopic quantities of the previous section as follows:

•

The local macroscopic quantities write

$${\mathbb{E}}_j\left[f_i\right]\left(t\right)={\int }_{D_u}{u}^j{f}_i(t,u)\mathrm{d}u,j\in \{0,1,2\}.$$

•

The global macroscopic quantities write

$${\mathbb{E}}_j\left[\mathbf{f}\right]\left(t\right)=\sum _{i=1}^n{\int }_{D_u}{u}^j{f}_i(t,u)\mathrm{d}u,j\in \{0,1,2\},$$

where $\mathbf{f}=\mathbf{f}(t,u)=(f_1(t,u),f_2(t,u),\cdots ,f_n(t,u))$.

The conservative interactions among the cells can now be defined between the cells of the different functional subsystems. Accordingly, let $i,j\in \{1,2,\cdots ,n\}$; the following functions need to be defined, generalizing the ones of the previous section:

(B1) 

The interaction/encounter rate ${\eta }_{ij}={\eta }_{ij}(u_{*},u^{*})$ between a cell of the functional subsystem $f_i$ with activity $u_{*}$ and a cell of the functional subsystem $f_j$ with activity $u^{*}$.

(B2) 

The function ${\mathcal{A}}_{ij}={\mathcal{A}}_{ij}(u_{*},u^{*},u)$, which models the probability that the cell of the functional subsystem $f_i$ with activity $u_{*}$, interacting with the cell of the functional subsystem $f_j$ with activity $u^{*}$, reaches the activity u. As in the previous section, the function ${\mathcal{A}}_{ij}$ has the structure of a probability function; then,

$${\displaystyle {\int }_{D_u}{\mathcal{A}}_{ij}(u_{*},u^{*},u)du=1,\forall u_{*},u^{*}\in D_u.}$$

(B3) 

The function $w_{ij}=w_{ij}(u_{*},u^{*})$, which models the intensity of the interactions and is such that

$${\int }_{D_u}{w}_{ij}(u_{*},u^{*})d{u}^{*}=1,\forall u_{*}\in D_u.$$

(B4) 

The external action $F_i=F_i\left(u\right)$, which acts on the ith functional subsystem.

(B5) 

The thermostat operator $T_{F_i}\left[f_i\right](t,u)$ allows the conservation of the global activity-energy ${\mathbb{E}}_2\left[\mathbf{f}\right]\left(t\right)$:

$$T_{F_i}\left[f_i\right]={\partial }_u\left(\left(F_i\left(u\right)-{\displaystyle \frac{u}{{\mathbb{E}}_2\left[\mathbf{f}\right]}}\sum _{j=1}^n{\int }_{D_u}uF\left(u\right)f_j(t,u)\mathrm{d}u\right)f_i(t,u)\right).$$

Bearing all above, the conservative operator now writes

$$\begin{array}{ccc}\hfill {\mathcal{C}}_{w,i}\left[\mathbf{f}\right](t,u)& =& \sum _{j=1}^n{\int }_{D_u\times D_u}{\eta }_{ij}(u_{*},u^{*})w_{ij}(u_{*},u^{*}){\mathcal{A}}_{ij}(u_{*},u^{*},u)f_i(t,u_{*})f_j(t,u^{*})d{u}_{*}d{u}^{*}\hfill \\ & & -f_i(t,u)\sum _{j=1}^n{\int }_{D_u}{\eta }_{ij}(u,u^{*})w_{ij}(u,u^{*})f_j(t,u^{*})d{u}^{*}.\hfill \end{array}$$

(9)

The multiple-subsystems thermostatted kinetic theory framework thus consists of a system of n integro-partial differential equations with quadratic nonlinearities coupled to n initial conditions and n homogeneous boundary conditions. The initial boundary value problem reads

$$\left\{\begin{array}{cc}{\displaystyle {\partial }_t{f}_1(t,u)+T_{F_1}\left[f_1\right](t,u)={\mathcal{C}}_{w,1}\left[\mathbf{f}\right](t,u)}\hfill & (t,u)\in [0,+\infty [\times D_u\hfill \\ {\displaystyle {\partial }_t{f}_2(t,u)+T_{F_2}\left[f_2\right](t,u)={\mathcal{C}}_{w,2}\left[\mathbf{f}\right](t,u)}\hfill & (t,u)\in [0,+\infty [\times D_u\hfill \\ ⋮\hfill \\ {\displaystyle {\partial }_t{f}_n(t,u)+T_{F_n}\left[f_n\right](t,u)={\mathcal{C}}_{w,n}\left[\mathbf{f}\right](t,u)}\hfill & (t,u)\in [0,+\infty [\times D_u\hfill \\ f_1(0,u)=f_1^0\left(u\right),f_2(0,u)=f_2^0\left(u\right),\cdots ,f_n(0,u)=f_n^0\left(u\right),\hfill & u\in D_u\hfill \\ f_1(t,u)=0,f_2(t,u)=0,\cdots ,f_n(t,u)=0\hfill & (t,u)\in [0,+\infty [\times \partial D_u\hfill \end{array}\right.$$

(10)

where ${\mathbf{f}}^0\left(u\right)=(f_1^0\left(u\right),f_2^0\left(u\right),\cdots ,f_n^0\left(u\right))$ denotes a suitable initial condition vector.

Remark 2. 

In the multiple functional subsystem case, some of cell functional subsystems can be assumed homogeneous with respect to the activity variable, namely, $f_i(t,u)=f_i\left(t\right)$. Accordingly, the evolution equation of these cell functional subsystems is an ordinary differential equation and the whole framework consists of a system of mixed evolution equations: ordinary and partial differential equations.

The mathematical analysis addressed to the existence and uniqueness of the solution of (10) can be found in [44] when $w_{ij}=1$. The well-posedness in the Hadamard sense can be obtained by generalizing the results of [42,43,45] in the case of functional subsystems, which are homogeneous with respect to the activity variable. As in the previous section, the mathematical analysis of the problem (10) can be considered as a future research perspective.

4. Nonconservative-Interactions Thermostatted Kinetic Theory

This section is concerned with biophysics systems where cell proliferation has an important role in the time evolution of the system. Specifically, cells are able to proliferate because of interactions with other types of cells; in this context, the destruction of cells is also possible. The natural birth and death of cells can also be taken into account. These events modify the density of the cell functional subsystems, thus introducing interactions that are called nonconservative.

Let $i,j\in \{1,2,\cdots ,n\}$. In order to model the proliferation/destruction events, the following functions need to be defined (see the Figure 3):

(C1) 

The function ${\mu }_{ij}^{+}={\mu }_{ij}^{+}\left(u_{*},u^{*},u\right)$ that represents the proliferation rate of the cells with activity u of the functional subsystem $f_i$ after the interaction between the cells with the activity $u_{*}$ of the functional subsystem $f_i$ and the cells with the activity $u^{*}$ of the functional subsystem $f_j$.

(C2) 

The function ${\mu }_{ij}^{-}={\mu }_{ij}^{-}\left(u_{*},u^{*},u\right)$ represents the destruction rate of the cells with activity u of the functional subsystem $f_i$ after the interaction between the cells with the activity $u_{*}$ of the functional subsystem $f_i$ and the cells with the activity $u^{*}$ of the functional subsystem $f_j$.

Bearing all above in mind, let ${\mu }_{ij}\left(u_{*},u^{*},u\right)={\mu }_{ij}^{+}\left(u_{*},u^{*},u\right)-{\mu }_{ij}^{-}\left(u_{*},u^{*},u\right)$ be the net proliferation/destruction rate; then, the mathematical operator $P{D}_{w,i}\left[\mathbf{f}\right]=P{D}_{w,i}\left[\mathbf{f}\right](t,u)$ that models the nonconservative interactions is split as follows:

$$P{D}_{w,i}\left[\mathbf{f}\right](t,u)=P_{w,i}\left[\mathbf{f}\right](t,u)-D_{w,i}\left[\mathbf{f}\right](t,u)$$

where $P_{w,i}\left[\mathbf{f}\right](t,u)$ and $D_{w,i}\left[\mathbf{f}\right](t,u)$ denote the proliferation and destruction operators, respectively, which write

$$P_{w,i}\left[\mathbf{f}\right](t,u)=\sum _{j=1}^n{\int }_{D_u\times D_u}{\eta }_{ij}\left(u_{*},u^{*}\right)w_{ij}(u_{*},u^{*}){\mu }_{ij}^{+}\left(u_{*},u^{*},u\right)f_i(t,u_{*})f_j(t,u^{*})\mathrm{d}{u}_{*}\mathrm{d}{u}^{*},$$

$$D_{w,i}\left[\mathbf{f}\right](t,u)=\sum _{j=1}^n{\int }_{D_u\times D_u}{\eta }_{ij}\left(u_{*},u^{*}\right)w_{ij}(u_{*},u^{*}){\mu }_{ij}^{-}\left(u_{*},u^{*},u\right)f_i(t,u_{*})f_j(t,u^{*})\mathrm{d}{u}_{*}\mathrm{d}{u}^{*},$$

and then

$$P{D}_{w,i}\left[\mathbf{f}\right](t,u)=\sum _{j=1}^n{\int }_{D_u\times D_u}{\eta }_{ij}\left(u_{*},u^{*}\right)w_{ij}(u_{*},u^{*}){\mu }_{ij}\left(u_{*},u^{*},u\right)f_i(t,u_{*})f_j(t,u^{*})\mathrm{d}{u}_{*}\mathrm{d}{u}^{*}.$$

Remark 3. 

The proliferation/destruction operator defined above is a generalization of the classical proliferation/destruction term in population dynamics, which in this context writes

$$P{D}_{w,i}\left[\mathbf{f}\right](t,u)=f_i(t,u)\sum _{j=1}^n{\int }_{D_u\times D_u}{\eta }_{ij}\left(u,u^{*}\right)w_{ij}(u,u^{*}){\mu }_{ij}\left(u,u^{*}\right)f_j(t,u^{*})d{u}^{*}.$$

The latter can be recovered by choosing, for instance, ${\mu }_{ij}\left(u_{*},u^{*},u\right)={\mu }_{ij}(u_{*},u^{*})\delta (u-u_{*})$.

The natural birth/death processes can be modeled by introducing the birth rate $\alpha =\alpha \left(t\right)$ and the death rate $\beta =\beta \left(t\right)$, respectively, which are assumed to be known functions depending on the time variable. Let $\gamma \left(t\right)=\alpha \left(t\right)-\beta \left(t\right)$ be the net natural birth/death rate of cells of the ith functional subsystem; then, the operator $N\left[f_i\right](t,u)=\gamma \left(t\right)f_i(t,u)$ is related to the natural birth/death processes.

In what follows, the mathematical operator modeling the nonconservative interactions (proliferation/destruction and natural birth/death) will be denoted as follows:

$${\mathcal{PDN}}_{w,i}\left[\mathbf{f}\right](t,u)=P{D}_{w,i}\left[\mathbf{f}\right](t,u)+N\left[f_i\right](t,u).$$

The nonconservative-interactions thermostatted kinetic theory framework, coupled with a suitable initial condition vector ${\mathbf{f}}^0\left(u\right)=(f_1^0\left(u\right),f_2^0\left(u\right),\cdots ,f_n^0\left(u\right))$ and homogeneous boundary conditions, is the following initial boundary value problem:

$$\left\{\begin{array}{cc}{\displaystyle {\partial }_t{f}_1(t,u)+T_{F_1}\left[f_1\right](t,u)={\mathcal{C}}_{w,1}\left[\mathbf{f}\right](t,u)+{\mathcal{PDN}}_{w,1}\left[\mathbf{f}\right](t,u)}\hfill & (t,u)\in [0,+\infty [\times D_u\hfill \\ {\displaystyle {\partial }_t{f}_2(t,u)+T_{F_2}\left[f_2\right](t,u)={\mathcal{C}}_{w,2}\left[\mathbf{f}\right](t,u)+{\mathcal{PDN}}_{w,2}\left[\mathbf{f}\right](t,u)}\hfill & (t,u)\in [0,+\infty [\times D_u\hfill \\ ⋮\hfill \\ {\displaystyle {\partial }_t{f}_n(t,u)+T_{F_n}\left[f_n\right](t,u)={\mathcal{C}}_{w,n}\left[\mathbf{f}\right](t,u)+{\mathcal{PDN}}_{w,n}\left[\mathbf{f}\right](t,u)}\hfill & (t,u)\in [0,+\infty [\times D_u\hfill \\ f_1(0,u)=f_1^0\left(u\right),f_2(0,u)=f_2^0\left(u\right),\cdots ,f_n(0,u)=f_n^0\left(u\right),\hfill & u\in D_u\hfill \\ f_1(t,u)=0,f_2(t,u)=0,\cdots ,f_n(t,u)=0\hfill & (t,u)\in [0,+\infty [\times \partial D_u\hfill \end{array}\right.$$

(11)

where the meaning of the other operators can be recovered by the previous sections.

The well-posedness in the Hadamard sense of the problem (11) presents many difficulties due to the nonconservation of the density of the system. Indeed, the local existence and uniqueness of the solution can be easily proved by employing the same proof for the frameworks discussed in the previous sections; however, the global existence and uniqueness of the solution is an open problem and can be obtained by establishing the conservation of a macroscopic quantity related to the system.

It is worth stressing that the thermostat operator was derived by assuming the conservation of the energy-activity when the nonconservative interactions are missing. In this context, the role of the thermostat operator is assumed to control the activity-energy. However, a generalized thermostat operator can be derived when the nonconservative interactions occur, thus allowing the conservation of a macroscopic quantity and consequently ensuring the global existence and uniqueness of the solution. These are important research perspectives that will be discussed again in the last section of the paper.

5. Mutative-Interactions Thermostatted Kinetic Theory

This section aims at presenting a mathematical operator that takes into account the mutative events occurring in the evolution of cells. Indeed, a cell can mutate because of errors in the DNA or because of interactions with other cells. Accordingly, new cell subsystems can be created. Mutations can thus be modeled as transitions among the functional subsystems. Specifically, the following two different events can be envisaged:

•

A mutation can occur if a cell of the ith functional subsystem just changes its structure because of the interactions between the cells of hth and kth functional subsystems, respectively;

•

A mutation can be a nonconservative event if new cells into a new functional subsystem originate because of interactions between the other functional subsystems.

Let $i,j,h,k\in \{1,2,\cdots ,n\}$. In order to model the mutative events, the following functions need to be defined:

(D1) 

The function ${\mathcal{T}}_{hk}^i\left(u_{*},u^{*},u,h\to i\right)$ that represents the probability of transition into the state u of the functional subsystem $f_i$.

(D2) 

The function ${\phi }_{hk}^i\left(u_{*},u^{*},u\right)$ that represents the net mutative rate of the u-cells of the functional subsystem $f_i$ because of the interaction between the $u_{*}$-cells of the functional subsystem $f_h$ and the $u^{*}$-cells of the functional subsystem $f_k$.

Bearing all above in mind, two different mutative operators can be defined; the mutative operator that models the transition of cells of the hth functional subsystem into the ith functional subsystem:

$$\begin{array}{ccc}\hfill {\mathcal{M}}_{w,i}^1\left[\mathbf{f}\right]& =& \sum _{h=1}^n\sum _{k=1}^n{\int }_{D_u\times D_u}{\eta }_{hk}(u_{*},u^{*})w_{hk}(u_{*},u^{*}){\mathcal{T}}_{hk}^i\left(u_{*},u^{*},u,h\to i\right)f_h(t,u_{*})f_k(t,u^{*})d{u}_{*}d{u}^{*}\hfill \\ & & -f_i(t,u)\sum _{k=1}^n{\int }_{D_u}{\eta }_{ik}(u,u^{*})w_{ik}(u,u^{*})f_k(t,u^{*})d{u}^{*},\hfill \end{array}$$

(12)

and the following mutative operator with the mutation rate

$${\mathcal{M}}_{w,i}^2\left[\mathbf{f}\right]=\sum _{h=1}^n\sum _{k=1}^n{\int }_{D_u\times D_u}{\eta }_{hk}(u_{*},u^{*})w_{hk}(u_{*},u^{*}){\phi }_{hk}^{i\ne h}(u_{*},u^{*},u)f_h(t,u_{*})f_k(t,u^{*})d{u}_{*}d{u}^{*}.$$

(13)

Let ${\mathcal{M}}_{w,i}\left[\mathbf{f}\right]={\mathcal{M}}_{w,i}^1\left[\mathbf{f}\right](t,u)+{\mathcal{M}}_{w,i}^2\left[\mathbf{f}\right](t,u)$. The mutative-interactions thermostatted kinetic theory framework, coupled with a suitable initial condition vector

$${\mathbf{f}}^0\left(u\right)=(f_1^0\left(u\right),f_2^0\left(u\right),\cdots ,f_n^0\left(u\right))$$

and homogeneous boundary conditions, is the following initial boundary value problem:

$$\left\{\begin{array}{cc}{\displaystyle {\partial }_t{f}_1(t,u)+T_{F_1}\left[f_1\right](t,u)={\mathcal{C}}_{w,1}\left[\mathbf{f}\right](t,u)+{\mathcal{PDN}}_{w,1}\left[\mathbf{f}\right](t,u)+{\mathcal{M}}_{w,1}\left[\mathbf{f}\right]}\hfill & (t,u)\in [0,+\infty [\times D_u\hfill \\ {\displaystyle {\partial }_t{f}_2(t,u)+T_{F_2}\left[f_2\right](t,u)={\mathcal{C}}_{w,2}\left[\mathbf{f}\right](t,u)+{\mathcal{PDN}}_{w,2}\left[\mathbf{f}\right](t,u)+{\mathcal{M}}_{w,2}\left[\mathbf{f}\right]}\hfill & (t,u)\in [0,+\infty [\times D_u\hfill \\ ⋮\hfill \\ {\displaystyle {\partial }_t{f}_n(t,u)+T_{F_n}\left[f_n\right](t,u)={\mathcal{C}}_{w,n}\left[\mathbf{f}\right](t,u)+{\mathcal{PDN}}_{w,n}\left[\mathbf{f}\right](t,u)+{\mathcal{M}}_{w,n}\left[\mathbf{f}\right]}\hfill & (t,u)\in [0,+\infty [\times D_u\hfill \\ f_1(0,u)=f_1^0\left(u\right),f_2(0,u)=f_2^0\left(u\right),\cdots ,f_n(0,u)=f_n^0\left(u\right),\hfill & u\in D_u\hfill \\ f_1(t,u)=0,f_2(t,u)=0,\cdots ,f_n(t,u)=0\hfill & (t,u)\in [0,+\infty [\times \partial D_u\hfill \end{array}\right.$$

(14)

where the meaning of the other operators can be recovered by the previous sections.

The well-posedness in the Hadamard sense of the thermostatted kinetic theory framework (14) is an open problem and presents the same difficulties discussed in the previous section.

6. Microscopic-External-Actions Thermostatted Kinetic Theory

The mathematical frameworks discussed in the previous sections deal with biophysics systems subjected to external actions at the macroscopic scale. The outer environment is thus represented as a whole by introducing an external action $F_i$ acting on the functional subsystem $f_i$. However, the outer environment can also be modeled at the same scale of the functional subsystems, thus introducing outer functional subsystems that interact with the (inner) system; these external actions or agents can represent, for instance, the action of a vaccine during the competition between the tumor-immune system cells. In this context, the system is called open.

Bearing all the above in mind, assume firstly that the system is composed by only one functional subsystem and subjected to conservative interactions ${\mathcal{C}}_w\left[f\right]$. Thus, the external action at the microscopic scale is an outer/external functional subsystem whose particle microscopic state is denoted by $v\in D_u$ (the same domain of the activity variable u) and the related distribution function $e=e(t,v)$, $v\in D_u$, is a known function of its arguments.

The active particles of the external agent interact with the cells of the inner-system and, consequently, the magnitude of the activity variable u is modified by the external agent as well.

In order to model the inner–outer interactions, the following functions need to be introduced:

• ${\eta }^e={\eta }^e(u_{*},v^{*})$ represents the inner–outer encounter rate between the cells $u_{*}$ of the inner-system and the active particle $v^{*}$ of the external agent.
• $\mathcal{E}=\mathcal{E}(u_{*},v^{*},u)$ denotes the inner–outer transition probability function, which models the probability that the cell $u_{*}$ of the inner-system acquires the microscopic state u after an interaction with the active particle $v^{*}$ of the external agent; as the probability function, the function $\mathcal{E}$ verifies the following condition:

  $${\int }_{D_u}\mathcal{E}(u_{*},v^{*},u)du=1,\forall u_{*},v^{*}\in D_u.$$

  (15)
• $w^e=w^e(u_{*},v^{*})$ weights the interactions between the inner–outer system and is such that

  $${\int }_{D_u}{w}^e(u_{*},v^{*})d{v}^{*}=1,\forall u_{*}\in D_u.$$

Accordingly, the new mathematical operator ${\mathcal{IO}}_w$ modeling the weighted inner–outer conservative interactions reads

$$\begin{array}{ccc}\hfill {\mathcal{IO}}_w[f,e](t,u)& =& {\int }_{D_u\times D_u}{\eta }^e(u_{*},v^{*})w^e(u_{*},v^{*})\mathcal{E}(u_{*},v^{*},u)f(t,u_{*})e(t,v^{*})d{u}_{*}d{v}^{*}\hfill \\ & & -f(t,u){\int }_{D_u}{\eta }^e(u,v^{*})w^e(u,v^{*})e(t,v^{*})d{v}^{*}.\hfill \end{array}$$

(16)

The thermostatted evolution equation of f, always obtained by equating the inlet and outlet flows into the elementary volume $[u,u+du]$ of the space of the microscopic states, reads

$${\partial }_{t}f(t,u)+{\partial }_u\left(\left(F\left(u\right)-{\displaystyle \frac{u}{{\mathbb{E}}_2\left[f\right]}}{\int }_{D_u}uF\left(u\right)f(t,u)du\right)f(t,u)\right)={\mathcal{C}}_w\left[f\right](t,u)+{\mathcal{IO}}_w[f,e](t,u),$$

(17)

where the expression of the operator ${\mathcal{C}}_w\left[f\right](t,u)$ is analogous to that of Section 2.

The case of a biophysics system composed of n interacting cell functional subsystems is technical. It is assumed that the ith cell functional subsystem, for $i\in \{1,2,\cdots ,n\}$, interacts with a known kth external agent, for $k\in \{1,2,\cdots ,m\}$. Let

$$\mathbf{f}=\mathbf{f}(t,u)=\{f_1(t,u),f_2(t,u),\cdots ,f_n(t,u)\},\mathbf{e}=\mathbf{e}(t,v)=\{e_1(t,v),e_2(t,v),\cdots ,e_m(t,v)\},$$

be the vectors whose components are the n distribution functions of the cell functional subsystems and the m distribution functions associated to the external agents, respectively.

Bearing all the above in mind, the microscopic-external-actions thermostatted kinetic theory framework consists of the following evolution equation for each functional subsystem:

$${\partial }_t{f}_i(t,u)+{\partial }_u\left(\left(F_i\left(u\right)-{\displaystyle \frac{u}{{\mathbb{E}}_2\left[f\right]}}\sum _{j=1}^n{\int }_{D_u}u{F}_j\left(u\right)f_j(t,u)du\right)f_i(t,u)\right)={\mathcal{C}}_{w,i}\left[\mathbf{f}\right](t,u)+{\mathcal{IO}}_{w,i}[\mathbf{f},\mathbf{e}](t,u),$$

(18)

where the operator ${\mathcal{C}}_{w,i}\left[\mathbf{f}\right](t,u)$ is given by (9) and

$$\begin{array}{ccc}\hfill {\mathcal{IO}}_{w,i}[\mathbf{f},\mathbf{e}](t,u)& =& \sum _{k=1}^m{\int }_{D_u\times D_u}{\eta }_{ik}^e(u_{*},v^{*})w_{ik}^e(u_{*},v^{*}){\mathcal{E}}_{ik}(u_{*},v^{*},u)f_i(t,u_{*})e_k(t,v^{*})d{u}_{*}d{v}^{*}\hfill \\ & & -f_i(t,u)\sum _{k=1}^m{\int }_{D_u}{\eta }_{ik}^e(u,v^{*})w_{ik}^e(u,v^{*})e_k(t,v^{*})d{v}^{*}.\hfill \end{array}$$

(19)

The functions in the operator ${\mathcal{IO}}_{w,i}[\mathbf{f},\mathbf{e}](t,u)$ have the following meaning:

(E1) 

${\eta }_{ik}^e(u_{*},v^{*})$ denotes the inner–outer encounter rate between the cells with state $u_{*}$ of the ith functional subsystem and the active particles with state $v^{*}$ of the kth external agent.

(E2) 

${\mathcal{E}}_{ik}(u_{*},v^{*},u)$ represents the inner–outer transition probability that cells with state $u_{*}$ of the ith functional subsystem fall into the state u after interactions with the active particles with state $v^{*}$ of the kth external agent; moreover,

$${\int }_{D_u}{\mathcal{E}}_{ik}(u_{*},v^{*},u)du=1,\forall u_{*},v^{*}\in D_u.$$

(20)

(E3) 

$w_{ik}^e=w_{ik}^e(u_{*},v^{*})$ weights the interactions between the ith functional subsystem and the kth external agent; moreover,

$${\int }_{D_u}{w}_{ik}^e(u_{*},v^{*})d{v}^{*}=1,\forall u_{*}\in D_u.$$

Remark 4. 

The kth external action $e_k(t,v)$ can be factorized as follows:

$$e_k(t,v)={ϵ}_k\left(t\right){\phi }_k\left(v\right),v\in D_u,$$

(21)

where the function ${ϵ}_k={ϵ}_k\left(t\right)$ models the temporal intensity of the kth external agent on the ith functional system and ${\phi }_k\left(v\right)$ denotes the probability function associated to the variable v (defined over the same domain $D_u$ of the variable u).

Generalizations of the above mathematical framework refer to the definition of nonconservative and mutative interactions due to the external actions. Specifically, the following new operators yield

$${\mathcal{PDE}}_{w,i}\left[\mathbf{f},\mathbf{e}\right](t,u)=\sum _{j=1}^m{\int }_{D_u\times D_u}{\eta }_{ij}^e\left(u_{*},v^{*}\right)w_{ij}^e(u_{*},v^{*}){\mu }_{ij}^e\left(u_{*},v^{*},u\right)f_i(t,u_{*})e_j(t,v^{*})\mathrm{d}{u}_{*}\mathrm{d}{v}^{*},$$

which allows the modeling of the proliferation/destruction of cells of the ith functional subsystem because of the interactions with the microscopic external agents, and

$${\mathcal{ME}}_{w,i}[\mathbf{f},\mathbf{e}](t,u)=\sum _{h=1}^n\sum _{k=1}^m{\int }_{D_u\times D_u}{\eta }_{hk}^e(u_{*},v^{*})w_{hk}^e(u_{*},v^{*}){\phi }_{hk}^{e,i\ne h}(u_{*},v^{*},u)f_h(t,u_{*})e_k(t,v^{*})d{u}_{*}d{v}^{*},$$

which allows modeling of the mutations of cells into the ith functional subsystem because of the interactions with the microscopic external agents; the meaning of each function can be recovered by generalizing the functions of the previous sections.

From the mathematical analysis viewpoint, the well-posedness in the Hadamard sense of the related initial boundary value problem is missing and can be considered as an important research perspective.

7. Multivariate-Activity Thermostatted Kinetic Theory

This section deals with a biophysics system where the microscopic state of the cells is supposed to be defined by a multivariable activity. The introduction of a multivariate activity variable is an important generalization that allows us to better manage the complexity of the whole system and, at the same time, avoids many approximation assumptions. In particular, in biology, the role of a vectorial activity variable permits better distinguishing the cells that are grouped into the same functional subsystem. The heterogeneity of cells is then taken into account and more specified.

Accordingly, the microscopic state of the cells consists of a vector activity variable $\mathbf{u}=(u_1,u_2,\cdots ,u_p)$, where $u_j\in D_{u_j}\subseteq \mathbb{R}$, for $j\in \{1,2,\cdots ,p\}$. This generalization was firstly introduced in [46] for a system subjected only to conservative binary interactions.

Let $D_{\mathbf{u}}=D_{u_1}\times D_{u_2}\times \cdots \times D_{u_p}$, the distribution function of the ith functional subsystem, for $i\in \{1,2,\cdots ,n\}$, is denoted by $f_i(t,\mathbf{u}):[0,+\infty [\times D_{\mathbf{u}}\to {\mathbb{R}}^{+}$, whereas $\mathbf{f}(t,\mathbf{u})=\left(f_1(t,\mathbf{u}),f_2(t,\mathbf{u}),\cdots ,f_n(t,\mathbf{u}))\right):[0,+\infty [\times D_{\mathbf{u}}\to {\left({\mathbb{R}}^{+}\right)}^n$ denotes the distribution function of the whole system. The (local) density and the activity-energy of the ith functional subsystem now write

$${\mathbb{E}}_0\left[f_i\right]\left(t\right)={\int }_{D_{\mathbf{u}}}{f}_i(t,\mathbf{u})d\mathbf{u},{\mathbb{E}}_2\left[f_i\right]\left(t\right):=\sum _{h=1}^p{\int }_{D_{\mathbf{u}}}{u}_h^2{f}_i(t,\mathbf{u})d\mathbf{u}.$$

(22)

The (global) density and the activity energy of the whole system read

$${\mathbb{E}}_0\left[\mathbf{f}\right]\left(t\right)=\sum _{i=1}^n{\int }_{D_{\mathbf{u}}}{f}_i(t,\mathbf{u})d\mathbf{u},{\mathbb{E}}_2\left[\mathbf{f}\right]\left(t\right):=\sum _{i=1}^n\sum _{h=1}^p{\int }_{D_{\mathbf{u}}}{u}_h^2{f}_i(t,\mathbf{u})d\mathbf{u}.$$

(23)

The macroscopic external force field acting on the whole system is

$$\mathbf{F}\left(\mathbf{u}\right)=(F_1\left(\mathbf{u}\right),F_2\left(\mathbf{u}\right),\cdots ,F_n\left(\mathbf{u}\right)):D_{\mathbf{u}}\to {\left({\mathbb{R}}^{+}\right)}^n,$$

where $F_i\left(\mathbf{u}\right)$ denotes the ith component of the external force field acting on the ith function subsystem. The thermostat term was derived by imposing the conservation of the global activity-energy ${\mathbb{E}}_2\left[\mathbf{f}\right]$.

Bearing all above in mind, the multivariate-activity thermostatted kinetic framework for the functional subsystem $f_i(t,\mathbf{u})$, for $i\in \{1,2,\cdots ,n\}$, writes as follows:

$${\partial }_t{f}_i(t,\mathbf{u})+\sum _{j=1}^p{\partial }_{u_j}\left(\left(F_i\left(\mathbf{u}\right)-{\displaystyle \frac{u_j}{{\mathbb{E}}_2\left[\mathbf{f}\right]}}\sum _{h=1}^p\sum _{k=1}^n{\int }_{D_{\mathbf{u}}}{u}_h{F}_k\left(\mathbf{u}\right)f_k(t,\mathbf{u})d\mathbf{u}\right)f_i(t,\mathbf{u})\right)={\mathcal{C}}_{w,i}\left[\mathbf{f}\right](t,\mathbf{u}),$$

(24)

where ${\mathcal{C}}_{w,i}\left[\mathbf{f}\right](t,\mathbf{u})$ now writes

$$\begin{array}{ccc}\hfill {\mathcal{C}}_{w,i}\left[\mathbf{f}\right]& =& \sum _{h=1}^p\sum _{j=1}^n{\int }_{D_{u_h}\times D_{u_h}}{\eta }_{ij}({\left({\mathbf{u}}_{*}\right)}_h,{\left({\mathbf{u}}^{*}\right)}_h)w_{ij}({\left({\mathbf{u}}_{*}\right)}_h,{\left({\mathbf{u}}^{*}\right)}_h){\mathcal{A}}_{ij}\left({\left({\mathbf{u}}_{*}\right)}_h,{\left({\mathbf{u}}^{*}\right)}_h,{\left(\mathbf{u}\right)}_h\right)\hfill \\ & & \times f_i(t,{\left({\mathbf{u}}_{*}\right)}_h)f_j(t,{\left({\mathbf{u}}^{*}\right)}_h)d{\left({\mathbf{u}}_{*}\right)}_{h}d{\left({\mathbf{u}}^{*}\right)}_h\hfill \\ & & -f_i(t,\mathbf{u})\sum _{h=1}^p\sum _{j=1}^n{\int }_{D_{u_h}}{\eta }_{ij}({\left(\mathbf{u}\right)}_h,{\left({\mathbf{u}}^{*}\right)}_h)w_{ij}({\left(\mathbf{u}\right)}_h,{\left({\mathbf{u}}^{*}\right)}_h)f_j(t,{\left({\mathbf{u}}^{*}\right)}_h)d{\left({\mathbf{u}}^{*}\right)}_h,\hfill \end{array}$$

(25)

where

$${\left({\mathbf{u}}_{*}\right)}_h=\left({\left(u_{*}\right)}_1,{\left(u_{*}\right)}_2,\cdots ,{\left(u_{*}\right)}_h,\cdots ,{\left(u_{*}\right)}_p\right),$$

$${\left({\mathbf{u}}^{*}\right)}_h=\left({\left(u^{*}\right)}_1,{\left(u^{*}\right)}_2,\cdots ,{\left(u^{*}\right)}_h\cdots ,{\left(u^{*}\right)}_p\right),$$

and ${\left(\mathbf{u}\right)}_h=(u_1,u_2,\cdots ,u_h,\cdots ,u_p)$. The meaning of each function can be recovered from the previous sections.

Differently from [46], the conservative operator (25) is generalized in order to relax the binary-interaction assumption by introducing the weight function.

A further generalization of the multivariate-activity thermostatted kinetic theory with respect to the existing literature is the introduction, for the first time to the best of the author’s knowledge, of the nonconservative and mutative interactions. Specifically, the nonconservative operator $P{D}_{w,i}\left[\mathbf{f}\right]=P{D}_{w,i}\left[\mathbf{f}\right](t,\mathbf{u})$ and the mutative operator ${\mathcal{M}}_{w,i}^2\left[\mathbf{f}\right]={\mathcal{M}}_{w,i}^2\left[\mathbf{f}\right](t,\mathbf{u})$ can be rewritten as follows:

$$\begin{array}{ccc}\hfill P{D}_{w,i}\left[\mathbf{f}\right]& =& \sum _{h=1}^p\sum _{j=1}^n{\int }_{D_{u_h}\times D_{u_h}}{\eta }_{ij}\left({\left({\mathbf{u}}_{*}\right)}_h,{\left({\mathbf{u}}^{*}\right)}_h\right)w_{ij}({\left({\mathbf{u}}_{*}\right)}_h,{\left({\mathbf{u}}^{*}\right)}_h){\mu }_{ij}\left({\left({\mathbf{u}}_{*}\right)}_h,{\left({\mathbf{u}}^{*}\right)}_h,{\left(\mathbf{u}\right)}_h\right)\hfill \\ & & \times f_i(t,{\left({\mathbf{u}}_{*}\right)}_h)f_j(t,{\left({\mathbf{u}}^{*}\right)}_h)\mathrm{d}{\left({\mathbf{u}}_{*}\right)}_h\mathrm{d}{\left({\mathbf{u}}^{*}\right)}_h,\hfill \end{array}$$

(26)

$$\begin{array}{ccc}\hfill {\mathcal{M}}_{w,i}^2\left[\mathbf{f}\right]& =& \sum _{l=1}^p\sum _{h=1}^n\sum _{k=1}^n{\int }_{D_{u_l}\times D_{u_l}}{\eta }_{hk}({\left({\mathbf{u}}_{*}\right)}_l,{\left({\mathbf{u}}^{*}\right)}_l)w_{hk}({\left({\mathbf{u}}_{*}\right)}_l,{\left({\mathbf{u}}^{*}\right)}_l){\phi }_{hk}^{i\ne h}({\left({\mathbf{u}}_{*}\right)}_l,{\left({\mathbf{u}}^{*}\right)}_l,{\left(\mathbf{u}\right)}_l)\hfill \\ & & \times f_h(t,{\left({\mathbf{u}}_{*}\right)}_l)f_k(t,{\left({\mathbf{u}}^{*}\right)}_l)d{\left({\mathbf{u}}_{*}\right)}_{l}d{\left({\mathbf{u}}^{*}\right)}_l.\hfill \end{array}$$

(27)

The generalization can be also developed for the open systems under the action of a microscopic external action with multivariable activity $\mathbf{v}$, namely, $\mathbf{e}(t,\mathbf{v})$. Accordingly,

$$\begin{array}{ccc}\hfill {\mathcal{PDE}}_{w,i}\left[\mathbf{f},\mathbf{e}\right]& =& \sum _{h=1}^p\sum _{j=1}^m{\int }_{D_{u_h}\times D_{u_h}}{\eta }_{ij}^e\left({\left({\mathbf{u}}_{*}\right)}_h,{\left({\mathbf{v}}^{*}\right)}_h\right)w_{ij}^e({\left({\mathbf{u}}_{*}\right)}_h,{\left({\mathbf{v}}^{*}\right)}_h){\mu }_{ij}^e\left({\left({\mathbf{u}}_{*}\right)}_h,{\left({\mathbf{v}}^{*}\right)}_h,{\left(\mathbf{u}\right)}_h\right)\hfill \\ & & f_i(t,{\left({\mathbf{u}}_{*}\right)}_h)e_j(t,{\left({\mathbf{v}}^{*}\right)}_h)\mathrm{d}{\left({\mathbf{u}}_{*}\right)}_h\mathrm{d}{\left({\mathbf{v}}^{*}\right)}_h,\hfill \end{array}$$

(28)

$$\begin{array}{ccc}\hfill {\mathcal{ME}}_{w,i}[\mathbf{f},\mathbf{e}]& =& \sum _{l=1}^p\sum _{h=1}^n\sum _{k=1}^m{\int }_{D_{u_l}\times D_{u_l}}{\eta }_{hk}^e({\left({\mathbf{u}}_{*}\right)}_l,{\left({\mathbf{v}}^{*}\right)}_l)w_{hk}^e({\left({\mathbf{u}}_{*}\right)}_l,{\left({\mathbf{v}}^{*}\right)}_l){\phi }_{hk}^{e,i\ne k}({\left({\mathbf{u}}_{*}\right)}_l,{\left({\mathbf{v}}^{*}\right)}_l,{\left(\mathbf{u}\right)}_l)\hfill \\ & & f_h(t,{\left({\mathbf{u}}_{*}\right)}_l)e_k(t,{\left({\mathbf{v}}^{*}\right)}_l)d{\left({\mathbf{u}}_{*}\right)}_{l}d{\left({\mathbf{v}}^{*}\right)}_l,\hfill \end{array}$$

(29)

where ${\left({\mathbf{v}}_{*}\right)}_h=\left({\left(v_{*}\right)}_1,{\left(v_{*}\right)}_2,\cdots ,{\left(v_{*}\right)}_h,\cdots ,{\left(v_{*}\right)}_p\right)$.

The meaning and the properties of all of the functions can be recovered by the previous sections.

The analysis of the related initial boundary value problems is missing and presents the same difficulties mentioned in the previous sections.

8. Applications: Specific Models

The mathematical frameworks discussed in the present paper concern the modeling of systems coming from biophysics. Specifically, the overview and generalizations of the frameworks of the thermostatted kinetic theory were the core of the previous sections. In particular, the role and the effects of interactions were taken into account by defining mathematical operators, which are based on interactions rates, transitions of the microscopic states, proliferation/destruction, mutations, and external actions at the macroscopic and microscopic scales. The main variable describing the microscopic state of the cells is called activity; mechanical variables, such as space and velocity, were neglected by assuming that the system under consideration is homogeneous with respect to these variables. Each mathematical framework can be considered as a background paradigm for the derivation of specific models in biophysics.

This section is meant to be a survey, and discussions of some specific models of the pertinent literature can be considered as part of the thermostatted kinetic theory for active particles revisited in this paper.

According to the generalized frameworks of this paper, different preliminary steps are at the base of the derivation of a specific model. Indeed, the system under consideration needs to be analyzed in order to select the most important actors and interactions to which a certain number of parameters correspond.

Thus, the management of the complexity of the system is necessary and this step can be pursued by analyzing the phenomenon. This analysis aims at identifying the representation scale of the system (kinetic/cellular in the case of the thermostatted kinetic theory), the homogeneous assumption of the system, the functional subsystems with the meaning of the related activity variable and its continuous assumption, and the main relationship and interactions among the cells. Moreover, the role of the external environment needs to be discussed.

The selection of the most suitable framework is a consequence of the previous steps. Depending on the types of interactions, the different functions defining the interaction operators need to be modeled.

The penultimate step is, thus, the derivation of the evolution equations fulfilled by the distribution functions and the related initial and/or boundary data. In this context, the development of a suitable numerical method is necessary in order to describe the numerical solutions and the macroscopic quantities. Finally, the computational analysis of the model allows us to validate its capability to describe the phenomenon under consideration and the possible unexpected, emerging phenomena. The validation step is obtained by showing the qualitative reproduction of the phenomena of the system and by tuning the parameters of the model with empirical or experimental data. The predictive ability of the model is an important step.

8.1. A Kinetic Model for Breast Cancer-Immune System Competition

A mathematical model for the breast cancer-immune system competition was proposed in [47], whose numerical results are in agreement with oncological data.

Breast cancer is one of the most common cancers in women; steroid sexual hormones have key roles in the initiation and progression of breast cancer. Breast cancer is characterized by a highly inflammatory microenvironment, which is supported by the infiltrating immune cells, cytokines, and growth factors [48].

The model is obtained by considering conservative and nonconservative interactions; however, the action of the thermostat is not taken into account and there are no microscopic external actions. The model can be rewritten within the thermostatted kinetic theory frameworks of the present paper.

The mathematical model [47] describes the breast cancer-immune system competition by analyzing the time evolution of six cell functional subsystems:

(CFS1) 

Cancer cells. The activity variable represents the recognition probability by APC. Low values of u imply dangerous cancer cells and vice versa.

(CFS2) 

T-helper cells. The activity variable models the capability to influence the ability of the other immune system cells (normalized quantity of cytokines [49]).

(CFS3) 

Cytotoxic T-cells. The activity variable refers to the probability to eliminate recognized cancer cells [50].

(CFS4) 

Antigen-presenting cells (APCs).

(CFS5) 

Antigen-loaded APCs.

(CFS6) 

Endothelial cells.

The distribution function $f_i(t,u)$ of each cell functional subsystem is labeled by the indices $i\in \{1,2,3,4,5,6\}$, respectively, and the activity variable $u\in D_u=[0,1]$. The antigen-presenting cells, antigen-loaded APCs, and endothelial cells are also assumed homogeneous with respect to the activity variable; then, $f_i(t,u)=f_i\left(t\right)$, for $i\in \{4,5,6\}$. Moreover, the distribution function of the endothelial cells is assumed to be constant in time.

The interactions are assumed to be binary ($w_{ij}(u_{*},u^{*})=1$, for $i,j\in \{1,2,3,4,5,6\}$), homogeneous in space, and instantaneous (no time delay). Specifically, these are as follows:

•

Natural events. The (constant in time) natural proliferation of T-helper cells and cytotoxic T-cells by assuming that the probability of new T-helper cells and cytotoxic T-cells with small activity is greater than that with high activity [50]; accordingly,

$$N^{+}\left[f_i\right](t,u)={\alpha }_i(1-u),{\alpha }_i>0,i\in \{2,3\}.$$

The (constant in time) natural proliferation of the antigen-presenting cells:

$$N^{+}\left[f_4\right](t,u)={\alpha }_4,{\alpha }_4>0.$$

The natural death of the immune system cells (T-helper cells, cytotoxic T-cells, APCs, and antigen-loaded APCs) are taken into account:

$$N^{-}\left[f_i\right](t,u)=-{\beta }_i{f}_i(t,u),{\beta }_i>0,i\in \{2,3,4,5\}.$$

•

The interaction rates are defined as follows:

$${\eta }_{ij}(u_{*},u^{*})=\left\{\begin{array}{cc}{\displaystyle {\eta }_{16}{u}_{*}}\hfill & \mathrm{if}i=1\mathrm{and}j=6\hfill \\ {\eta }_{32}{(1-u_{*})}^2\hfill & \mathrm{if}i=3\mathrm{and}j=2\hfill \\ {\eta }_{25}{(1-u_{*})}^2\hfill & \mathrm{if}i=2\mathrm{and}j=5\hfill \\ 1\hfill & \mathrm{if}i=1\mathrm{and}j=2,3\hfill \\ 1\hfill & \mathrm{if}i=2,3,5\mathrm{and}j=1\hfill \\ 1\hfill & \mathrm{if}i=4\mathrm{and}j=1\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(30)

•

Proliferative interactions.

-

The cancer cells proliferate after the encounter with the endothelial cells.

-

T-helper cells proliferate because of the interactions with the antigen-loaded APCs.

• The proliferation rates are defined as follows:

$${\mu }_{ij}^{+}(u_{*},u^{*},u)=\left\{\begin{array}{cc}{\displaystyle \frac{{\alpha }_{16}}{{\eta }_{16}{u}_{*}}}\hfill & \mathrm{if}i=1\mathrm{and}j=6\hfill \\ \hfill & \\ {\displaystyle \frac{{\alpha }_{25}2u_{*}(1-u)}{{\eta }_{25}{(1-u_{*})}^2}}\hfill & \mathrm{if}i=2\mathrm{and}j=5\hfill \\ \hfill & \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(31)

•

Destructive interactions.

-

Cancer cells are destroyed by T-helper cells and cytotoxic T-cells and the corresponding destruction rates ${\delta }_{12}$ and ${\delta }_{13}$ are assumed to be proportional to their activity state.

-

The immune system cells (T-helper cells, cytotoxic T-cells, and antigen-loaded APCs) are destroyed by cancer cells with destruction rates ${\delta }_{21},{\delta }_{31},{\delta }_{51}$, respectively.

-

A fraction c of APCs is lost because it becomes loaded with tumor antigen after the interaction with cancer cells.

Bearing all the above in mind, the destruction rates are defined as follows:

$${\mu }_{ij}^{-}(u_{*},u^{*},u)=\left\{\begin{array}{cc}{\displaystyle {\delta }_{1j}{u}^{*}}\hfill & \mathrm{if}i=1\mathrm{and}j=2,3\hfill \\ \hfill & \\ {\displaystyle {\delta }_{i1}}\hfill & \mathrm{if}i=2,3,5\mathrm{and}j=1\hfill \\ \hfill & \\ {\displaystyle c{u}_{*}}\hfill & \mathrm{if}i=4\mathrm{and}j=1\hfill \\ \hfill & \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(32)

•

Mutative interactions.

-

The interactions between the T-helper cells and the antigen-loaded APCs lead to the proliferation of the cytotoxic T-cells and the antigen-presenting cells.

The mutation rates are defined as follows:

$${\phi }_{hk}^i(u_{*},u^{*},u)=\left\{\begin{array}{cc}{\displaystyle \frac{{\phi }_{25}^3{u}_{*}(1-u)}{{\eta }_{25}{(1-u_{*})}^2}}\hfill & \mathrm{if}i=3\mathrm{and}h=2,k=5;\hfill \\ \hfill & \\ {\displaystyle \frac{{\phi }_{25}^4{u}_{*}}{{\eta }_{25}{(1-u_{*})}^2}}\hfill & \mathrm{if}i=4\mathrm{and}h=2,k=5;\hfill \\ \hfill & \\ 0\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(33)

•

Conservative interactions.

-

It is assumed that the activity of the cancer cells is reduced by the encounter with the endothelial cells because the immune system is less active to identify cancer cells, then

$${\mathcal{A}}_{16}(u_{*},u^{*},u)=\left\{\begin{array}{cc}{\displaystyle \frac{2u}{u_{*}}}\hfill & \mathrm{if}u\le u_{*}\hfill \\ \hfill & \\ 0\hfill & \mathrm{if}u>u_{*}\hfill \end{array}\right.$$

(34)

-

The activity of the cytotoxic T-cells increases after the interaction with the T-helper cells.

-

The activity of the T-helper cells increase after the interaction with the antigen-loaded APCs.

Accordingly,

$${\mathcal{A}}_{32}(u_{*},u^{*},u)={\mathcal{A}}_{25}(u_{*},u^{*},u)=\left\{\begin{array}{cc}{\displaystyle \frac{2(u-u_{*})}{{(1-u_{*})}^2}}\hfill & \mathrm{if}u\ge u_{*}\hfill \\ \hfill & \\ 0\hfill & \mathrm{if}u<u_{*}\hfill \end{array}\right.$$

(35)

Bearing all the above in mind, the kinetic theory model [47] rewrites

$$\left\{\begin{array}{c}{\displaystyle {\partial }_t{f}_1={\eta }_{16}{f}_6\left(t\right)\left(2u{\int }_u^1{f}_1(t,u^{*})d{u}^{*}-u^2{f}_1(t,u)\right)+{\alpha }_{16}{f}_6\left(t\right){\int }_0^1{f}_1(t,u^{*})d{u}^{*}}\hfill \\ {\displaystyle -{\delta }_{12}{f}_1(t,u){\int }_0^1{u}^{*}{f}_2(t,u^{*})d{u}^{*}-{\delta }_{13}{f}_1(t,u){\int }_0^1{u}^{*}{f}_3(t,u^{*})d{u}^{*},}\hfill \\ \\ {\displaystyle {\partial }_t{f}_2={\eta }_{25}{f}_5\left(t\right)\left(2{\int }_0^u(u-u_{*})f_2(t,u_{*})d{u}_{*}-{(1-u)}^2{f}_2(t,u)\right)}\hfill \\ {\displaystyle +{\alpha }_{25}(1-u)f_5\left(t\right){\int }_0^1{u}^{*}{f}_2(t,u^{*})d{u}^{*}}\hfill \\ {\displaystyle -{\delta }_{21}{f}_2(t,u){\int }_0^1{f}_1(t,u^{*})d{u}^{*}-{\beta }_2{f}_2(t,u)+{\alpha }_2(1-u),}\hfill \\ \\ {\displaystyle {\partial }_t{f}_3={\eta }_{32}\left({\int }_0^1{f}_2(t,u^{*})d{u}^{*}\right)\left(2{\int }_0^u(u-u_{*})f_3(t,u_{*})d{u}_{*}-{(1-u)}^2{f}_3(t,u)\right)}\hfill \\ {\displaystyle +{\phi }_{25}^3(1-u)f_5\left(t\right){\int }_0^1{u}^{*}{f}_2(t,u^{*})d{u}^{*}}\hfill \\ {\displaystyle -{\delta }_{31}{f}_3(t,u){\int }_0^1{f}_1(t,u^{*})d{u}^{*}-{\beta }_3{f}_3(t,u)+{\alpha }_3(1-u),}\hfill \\ \\ {\displaystyle d{f}_4={\phi }_{25}^4{f}_5\left(t\right){\int }_0^1{u}^{*}{f}_2(t,u^{*})d{u}^{*}-c{f}_4\left(t\right){\int }_0^1{u}^{*}{f}_1(t,u^{*})d{u}^{*}-{\beta }_4{f}_4\left(t\right)+{\alpha }_4,}\hfill \\ \\ {\displaystyle d{f}_5=c{f}_4\left(t\right){\int }_0^1{u}^{*}{f}_1(t,u^{*})d{u}^{*}-{\delta }_{51}{f}_5\left(t\right){\int }_0^1{f}_1(t,u^{*})d{u}^{*}-{\beta }_5{f}_5\left(t\right),}\hfill \\ \end{array}\right.$$

(36)

coupled to suitable initial conditions.

The kinetic model was numerically analyzed in order to find the parameter values such that the solutions are in agreement with the results of the mammographical examinations.

It is worth mentioning that similar kinetic models can be found in [51,52] for the mathematical modeling of the humoral response of the immune system to cancer evolution. Further developments were presented in [53,54]. Recently, a deeper analysis of the recognition and learning process of the immune response to antigens was performed in the thermostatted model published in [55]. The effects of autoimmune diseases were modeled within the kinetic framework in [56,57]; the roles of virus infections were taken into account in [58]. The innate immune response to bacterial infections has also been modeled in [59,60]. Another application that can be rewritten within the thermostatted frameworks proposed in this paper can be found in [61].

8.2. A Kinetic Model for Genetic Mutations and Immune System Competition

A mathematical model based on the kinetic theory approach was proposed in [62] for the modeling of genetic mutations under the action of the immune system. Specifically, in the context of cancer progression, the role of mutations is essential [63,64]. According to the phenomenological analysis in view of the modeling approach, it is assumed that normal cells undergo a mutation because of DNA corruptions, becoming pre-neoplastic and neoplastic (cancer) cells. The system of cells is assumed to be homogeneous with respect to the space and velocity variables.

The kinetic model [62] is derived by considering conservative and nonconservative/mutative interactions, but macroscopic/microscopic external actions coupled with the thermostat are not taken into consideration. The kinetic model is based on the time evolution of the following four cell functional subsystems:

(CFS1) 

Normal cells. The activity variable is related to the differentiation.

(CFS2) 

Pre-neoplastic cells. The activity variable models the progression towards malignancy.

(CFS3) 

Neoplastic cells. The activity variable models the progression towards malignancy.

(CFS4) 

Immune system cells. The activity variable models the activation level of the cells.

The distribution function $f_i(t,u)$ of each cell functional subsystem is labeled by the indices $i\in \{1,2,3,4\}$, respectively, and the activity variable $u\in D_u=[0,+\infty [$.

The density of the normal cell functional subsystem is assumed to be constant (homeostasis).

The encounter rate is assumed constant and, in particular, ${\eta }_{ij}=1$ for all $i,j\in \{2,3,4\}$.

The physiological birth and death of cells are not taken into account in the model.

The interactions are assumed binary, homogeneous in space, and instantaneous (no time delay). Specifically, they are described as follows:

•

Conservative interactions.

The conservative interactions modify only the evolution of the pre-neoplastic and neoplastic cell functional subsystems. The activity variable increases with a certain rate because of interactions with the normal cell functional subsystem. In particular, the progression of neoplastic cells is assumed to be greater than the progression of pre-neoplastic cells.

Let $ϵ<1$ be a scale parameter; the transition probability functions are assumed to be defined as follows:

$$\begin{array}{ccc}\hfill {\mathcal{A}}_{21}& =& \Delta (u-(u_{*}+ϵ\alpha )),\hfill \end{array}$$

(37)

$$\begin{array}{ccc}\hfill {\mathcal{A}}_{31}& =& \Delta (u-(u_{*}+\alpha )),\hfill \end{array}$$

(38)

where $\Delta$ denotes the delta Dirac function.

•

Nonconservative interactions.

Proliferation and destruction of cells refer to the pre-neoplastic cell functional subsystem, the neoplastic cell functional subsystem, and the immune system. In particular, the proliferation rate of a neoplastic cell is assumed to be greater than the proliferation rate of a pre-neoplastic cell. Accordingly, the following apply:

-

Proliferation of pre-neoplastic cells (${\mu }_{21}^{+}=ϵ{\beta }_a$) and neoplastic cells (${\mu }_{31}^{+}={\beta }_a$) occurs due to encounters with normal cells.

-

Destruction of pre-neoplastic cells (${\mu }_{24}^{-}=-{\delta }_a$) and neoplastic cells (${\mu }_{34}^{-}=-ϵ{\delta }_a$) is due to the encounters with immune system cells.

-

Proliferation of the immune system cells (${\mu }_{42,43}^{+}=ϵ{\beta }_I$) occurs because of encounters with pre-neoplastic cells and neoplastic cells;

-

Destruction of immune system cells (${\mu }_{43}^{-}=-{\delta }_s{u}^{*}$) is due to encounters with neoplastic cells.

•

Mutative interactions.

Mutations concern normal cells that can undergo a mutation into pre-neoplastic cells (${\phi }_{11}^2={\gamma }_a\delta (u-u_{*})$), which can further undergo a mutation and become neoplastic cells (${\phi }_{21}^3={\gamma }_a\delta (u-u_{*})$). It is assumed that the microscopic state does not change during the mutation.

Bearing all above in mind, the kinetic theory model reads

$$\left\{\begin{array}{c}{\displaystyle {\partial }_t{f}_1=0,}\hfill \\ \\ {\displaystyle {\partial }_t{f}_2=\left(\epsilon {\beta }_a-{\delta }_a{\int }_0^{\infty }{f}_4(t,u^{*})d{u}^{*}-{\gamma }_a-1\right)f_2(t,u)+{\gamma }_a{f}_1(t,u)+f_2(t,u-ϵ\alpha ),}\hfill \\ \\ {\displaystyle {\partial }_t{f}_3=\left({\beta }_a-\epsilon {\delta }_a{\int }_0^{\infty }{f}_4(t,u^{*})d{u}^{*}-1\right)f_3(t,u)+{\gamma }_a{f}_2(t,u)+f_3(t,u-\alpha ),}\hfill \\ \\ {\displaystyle {\partial }_t{f}_4=\left({\beta }_I{\int }_0^{\infty }{f}_2(t,u^{*})d{u}^{*}+{\beta }_I{\int }_0^{\infty }{f}_3(t,u^{*})d{u}^{*}-{\delta }_s{\int }_0^{\infty }{u}^{*}{f}_3(t,u^{*})d{u}^{*}\right)f_4(t,u).}\hfill \end{array}\right.$$

(39)

A specific model requires linking the system to suitable initial conditions. The initial distributions are such that $f_1$ and $f_4$ are different from zero; the other initial distributions are equal to zero.

Numerical simulations refer to the evolution of the density and distribution functions of each cell functional subsystem. The predictive ability of the model was numerically investigated in [62] by employing a sensitive analysis on some of the parameters. The onset of mutated cells, their heterogeneity, and the role of the immune system were investigated and shown by letting the parameters ${\gamma }_a$ and ${\beta }_I$ vary. Moreover, the kinetic model is able to reproduce the two different behaviors:

-

The high heterogeneity of the neoplastic cells and/or low activation of the immune system allow the inhibition of immune system cells and the formation of tumors;

-

The low heterogeneity of the neoplastic cells and/or high activation of the immune system cells allows the depletion of neoplastic cells.

8.3. A Thermostatted Kinetic Theory Model for a Skin Fibrosis

A more recent application of the thermostatted kinetic theory frameworks discussed in the previous sections was the modeling of therapeutic actions in order to avoid the formation of a specific skin fibrosis, called keloid, and the development of further malignancy, see [65,66,67]. According to the phenomenological analysis in view of the modeling approach, keloid is a skin fibrosis that occurs because of mistakes during the normal wound healing process [68,69,70]. The fibroblast cells undergo a mutation because of a virus [71] and genetic susceptibilities [72,73], and acquire a greater proliferation rate [74].

The thermostatted kinetic model is based on the definition of the following five cell functional subsystems:

(CFS1) 

Fibroblast cells. The activity variable represents the proliferation ability of these cells.

(CFS2) 

Virus. The activity variable represents its aggressiveness level.

(CFS3) 

Keloid-fibroblasts. The activity variable represents the proliferation ability of these cells.

(CFS4) 

Malignant cells. The activity variable represents the malignancy level of these cells.

(CFS5) 

The immune system cells. The activity variable represents the degree of activation for the immune system.

The distribution function $f_i(t,u)$ of each cell functional subsystem is labeled by the indices $i\in \{1,2,3,4,5\}$, respectively, and the activity variable $u\in D_u=[0,+\infty [$.

The encounter rate is set ${\eta }_{ij}=1$, for all $i,j\in \{2,3,4\}$.

The physiological birth and death of cells are not taken into account in the model.

The interactions are assumed binary, homogeneous in space, and instantaneous (no time delay). Specifically, they are described as follows:

•

Conservative interactions.

The conservative interactions imply the increase in the magnitude of the activity variable with a different rate.

Let $ϵ<1$ be a scale parameter; the transition probability functions are defined as follows:

$$\begin{array}{ccc}\hfill {\mathcal{A}}_{21}& =& \Delta (u-(u_{*}+ϵ\alpha )),\hfill \end{array}$$

(40)

$$\begin{array}{ccc}\hfill {\mathcal{A}}_{32}& =& \Delta (u-(u_{*}+\alpha )),\hfill \end{array}$$

(41)

$$\begin{array}{ccc}\hfill {\mathcal{A}}_{42}& =& \Delta (u-(u_{*}+{ϵ}^2\alpha )),\hfill \end{array}$$

(42)

where $\Delta$ denotes the delta Dirac function.

•

Nonconservative interactions.

Proliferation and destruction of cells refer to each cell functional subsystem. In particular, the following apply:

-

The proliferation of the fibroblast cells occurs because of the interaction with each other (${\mu }_{11}^{+}={ϵ}^2\beta$) and are destroyed by the virus (${\mu }_{ij}^{-}=ϵ\delta$);

-

The virus proliferation depends on the encounter with the fibroblasts (${\mu }_{21}^{+}=ϵ\beta$) and immune system cells (${\mu }_{25}^{+}=ϵ\beta$), and is depleted by the immune system (${\mu }_{25}^{-}=\delta$);

-

The keloid-fibroblast proliferation depends on the interaction with fibroblasts (${\mu }_{31}^{+}=\beta$) and viruses (${\mu }_{32}^{+}=\beta$) and are destroyed by the virus (${\mu }_{32}^{-}={ϵ}^2\delta$) and by the immune system (${\mu }_{35}^{-}={ϵ}^2\beta$);

-

The proliferation of cancer cells occurs because of the interaction with the virus (${\mu }_{42}^{+}=ϵ\beta$), and they are depleted by the immune system (${\mu }_{45}^{-}=\delta$);

-

The proliferation and destruction of the immune system cells occur because of the interaction with the virus (${\mu }_{52}^{+}={\beta }_i$, ${\mu }_{52}^{-}={\delta }_i{u}^{*}$), the keloid-fibroblasts (${\mu }_{53}^{+}={ϵ}^2{\beta }_i$, ${\mu }_{53}^{-}={ϵ}^2{\delta }_i{u}^{*}$), and the cancer cells (${\mu }_{54}^{+}={\beta }_i$, ${\mu }_{54}^{-}={\delta }_i{u}^{*}$).

•

Mutative interactions.

-

The fibroblast can undergo mutation into keloid-fibroblasts because of the encounters with each other (${\phi }_{11}^3=ϵ\gamma \delta (u-u_{*})$) and with the virus (${\mu }_{12}^3=\gamma \delta (u-u_{*})$);

-

The keloid-fibroblasts can undergo mutation into cancer cells because of the encounters with the virus (${\mu }_{32}^4=\lambda \delta (u-u_{*})$).

•

Macroscopic external actions coupled to the thermostat.

The therapeutic actions, introduced in [67], are mimicked by introducing the macroscopic external forces ${\mathcal{F}}_1$, ${\mathcal{F}}_2$, and ${\mathcal{F}}_3$, see [75], to the virus, keloid-fibroblasts, and malignant cell functional subsystems, respectively.

Bearing all the above in mind, the following thermostatted kinetic theory model yields [67]:

$$\left\{\begin{array}{cc}{\displaystyle {\partial }_t{f}_1=\epsilon \left(\epsilon \beta {\int }_0^{\infty }{f}_1(t,u)du-\delta {\int }_0^{\infty }{f}_2(t,u)du\right)f_1(t,u),}\hfill & \\ \hfill & \\ {\displaystyle {\partial }_t{f}_2+{\partial }_u\left({\mathcal{F}}_2\left(1-u{\int }_{D_u}u\sum _{j=1}^5{f}_j(t,u)du\right)f_i(t,u)\right)=(\epsilon \beta {\int }_0^{\infty }[f_1(t,u)+f_5(t,u)]du}\hfill & \\ \hfill & \\ {\displaystyle -\delta {\int }_0^{\infty }{f}_5(t,u)du)f_2(t,u)-f_2(t,u){\int }_0^{\infty }{f}_1(t,u)du+f_2(t,u-ϵ\alpha ){\int }_0^{\infty }{f}_1(t,u)du,}\hfill & \\ \hfill & \\ {\displaystyle {\partial }_t{f}_3+{\partial }_u\left({\mathcal{F}}_3\left(1-u{\int }_{D_u}u\sum _{j=1}^5{f}_j(t,u)du\right)f_i(t,u)\right)=(\beta {\int }_0^{\infty }{f}_1(t,u)du-(1-\beta +{\epsilon }^2\delta )\times }\hfill & \\ \hfill & \\ {\displaystyle {\int }_0^{\infty }{f}_2(t,u)du)f_3(t,u)-{\epsilon }^2\delta f_3(t,u){\int }_0^{\infty }{f}_5(t,u)du+f_3(t,u-\alpha ){\int }_0^{\infty }{f}_2(t,u)du}\hfill & \\ \hfill & \\ {\displaystyle +\gamma \left(ϵ{\int }_0^{\infty }{f}_1(t,u)du+{\int }_0^{\infty }{f}_2(t,u)du\right)f_1(t,u),}\hfill & \\ \hfill & \\ {\displaystyle {\partial }_t{f}_4+{\partial }_u\left({\mathcal{F}}_4\left(1-u{\int }_{D_u}u\sum _{j=1}^5{f}_j(t,u)du\right)f_i(t,u)\right)=((\epsilon \beta -1){\int }_0^{\infty }{f}_2(t,u)du}\hfill & \\ \\ {\displaystyle -\delta {\int }_0^{\infty }{f}_5(t,u)du)f_4(t,u)+\lambda f_3(t,u){\int }_0^{\infty }{f}_2(t,u)du+f_4(t,u-{\epsilon }^2\alpha ){\int }_0^{\infty }{f}_2(t,u)du,}\hfill & \\ \hfill & \\ {\displaystyle {\partial }_t{f}_5={\beta }_i\left({\int }_0^{\infty }[f_2(t,u)+f_4(t,u)]du+{\epsilon }^2{\int }_0^{\infty }{f}_3(t,u)du\right)f_5(t,u)}\hfill & \\ \hfill & \\ {\displaystyle -{\delta }_i\left({\int }_0^{\infty }u[f_2(t,u)+f_4(t,u)]du+{\epsilon }^2{\int }_0^{\infty }u{f}_3(t,u)du\right)f_5(t,u).}\hfill \end{array}\right.$$

(43)

which needs to be linked to the related initial and homogeneous boundary conditions.

Numerical simulations were performed in the absence of therapeutic actions—namely, ${\mathcal{F}}_2={\mathcal{F}}_3={\mathcal{F}}_4=0$—showing that the model is able to catch different scenarios, such as the following: The birth of keloid-fibroblasts without or with cancer cells, high and low heterogeneity of keloid-fibroblasts and malignant cells, and elimination of malignant cells but rebirth of keloid-fibroblasts because of the genetic susceptibilities. Moreover, the computational analysis has shown the existence of nontrivial bifurcations. The computational analysis developed in the presence of therapeutic actions has shown that the action against the virus strongly depends on the heterogeneity of the virus, and even if the action is able to remove the virus, the genetic susceptibilities do not avoid keloid formation. The action against keloid formation has shown that depending on the heterogeneity of the keloid-fibroblasts, the magnitude of the action needs to be increased and, in some cases, only surgery action is possible. Finally, the action against the malignant cells has shown that boosting of the immune system has an important role.

The reader interested in mathematical models for keloid formation based on other modeling approaches is referred to a recent review paper: [76].

9. Conclusions and Research Perspectives

The goal of the present paper is to summarize the main mathematical frameworks of the thermostatted kinetic theory that have been proposed in the last two decades for the modeling of homogeneous complex systems in the biophysics domain. The presentation was at the tutorial level, starting from the simple structure of a functional subsystem with conservative interactions within a scalar activity variable to the case of multiple functional subsystems with conservative, nonconservative, and mutative interactions, with macroscopic and microscopic external actions coupled to a Gaussian-type thermostat within a vectorial activity variable. Thus, the goal was to achieve the most general framework of the thermostatted kinetic theory that acts as a fundamental background paradigm for the derivation of specific homogeneous models in biophysics. However, the paper was not limited to a survey of the existing frameworks, considering that some further developments have been discussed. Specifically, the binary interaction assumption was relaxed by introducing a weighted function w that depends on the activity variable of the interacting cells (intensity). This function allows us to define an activity interaction domain of the cell $u_{*}$, which contains only the cells $u^{*}$ that are able to interact with the cell $u_{*}$. Moreover, two different types of mutative interactions have been introduced: proliferation of the cells of a functional subsystem due to the interactions between the cells of two other functional subsystems, and the onset of new cells of a functional subsystem due to the interactions with the cells of another functional subsystems. Finally, the nonconservative interactions were enlarged by considering the proliferation/destruction and mutation of the cells of a functional subsystem because of microscopic external actions. The nonconservative interactions were also introduced in the vectorial activity variable case. In the context of thermostatted kinetic theory models, to the best of the author’s knowledge, this is the first paper where these dynamics have been defined.

The mathematical analysis, addressed to the existence and uniqueness of the solution of the corresponding initial/boundary value problem, was mentioned casewise. It is worth mentioning that the well-posedness in the Hadamard sense of a mathematical problem (Cauchy, Dirichlet, Neumann, and Robin) is an essential step in the modeling theory because it permits the development of numerical schemes for obtaining the numerical solutions and performing the numerical simulations of the emerging behaviors.

It is worth stressing that the thermostatted kinetic theory frameworks presented in the paper do not exhaust the conceivable ones. Indeed, further thermostatted frameworks can be found in the pertinent literature—for instance, in the inert matter case, the thermostatted Kac and Boltzmann equations [77,78,79]. The selected mathematical structures of the present paper come from the generalized kinetic theory frameworks that have been proposed for the modeling of complex living systems. Moreover, additional states can be required, thus broadening the dependence of the functions defined into the different interaction operators on the distribution functions and their moments; in this context, frameworks with nonlinear interactions have been suggested [80]. This type of dependence has not been revisited in this review paper and will be discussed in a deeper fashion in due course.

The mathematical frameworks and the specific models discussed in the present paper were selected as representative of a preliminary, even if general, mathematical theory for out-of-equilibrium complex living systems [29]. Three specific models have been revisited in the style of the thermostatted kinetic theory. The selected models come from the biophysics domain and, specifically, in the context of cancer-immune system competition and wound healing disease. The selected scale is that of cells, which represent the active particles of the system; however, the microscopic scale was taken into account by considering the activity variable, which allows the definition of the interactions among the cells. Indeed, modeling of the proliferation, destruction, and mutation of cells is related to the molecular scale.

The specific models reviewed in the present paper are not the only models derived within the thermostatted kinetic theory that the reader can find in the literature. The interested reader is referred to, among others, the following papers: [81,82]. Indeed, one of the aims of this paper is to select three representative models that cover the main interactions defined within the thermostatted kinetic theory frameworks (conservative, nonconservative, and mutative interactions) and the possibility to introduce a macroscopic external action coupled to the thermostat. By examining the three models found in Section 8, the reader should, on the one hand, obtain a large view of how the thermostatted methods work and, on the other hand, be able to derive a specific model. As the reader should have understood, the main role of the activity variable, sometimes called the functional state, is to differentiate the cell of the same functional subsystem (cell heterogeneity). More generally, the activity variable allows to distinguish the inner matter from the active matter [83]. The meaning of the activity variable depends on the functional subsystem that can be also assumed independent on the functional state of its cells. The structure of the activity variable is not necessarily common to every subsystem but takes values in the same subdomain.

In the thermostatted kinetic theory frameworks revisited in this paper, the activity variable is assumed to attain values in a continuous subdomain of the real numbers; accordingly, the related frameworks are called continuous and consist of a system of integro-differential equations with quadratic nonlinearity. On the contrary, when the activity variable assumes only discrete (integer) values, the discrete frameworks consist of a system of ordinary differential equations. The reader interested in the discrete or hybrid (continuous–discrete) frameworks of the thermostatted kinetic is referred to the following papers: [84,85,86].

The microscopic state of the cells was assumed to be homogeneous with respect to the space and velocity variables. The latter variables do not have significant meaning in the systems considered in this paper. Thus, the spatial and velocity dynamics are neglected and, consequently, the interactions are assumed to occur only because of chemical signals such as autocrine signaling, paracrine signaling, endocrine signaling, and direct signaling across gap junctions depending on the distance that the signal travels to reach the target cell [87].

From the perspective point of view, there are different issues that can be pursued from the theoretical and applied outlooks. Firstly, many of the mathematical problems based on the frameworks discussed in the paper need to be investigated from the analytical point-of-view in order to ensure the well-posedness of the problem in the Hadamard sense (existence, uniqueness, and continuous dependence on the initial conditions of the solution). The main difficulty is concerned with the nonconservative frameworks where the density of the system cannot be controlled (blow-up of the solution, namely, the solution tends to infinity for some finite time). If, on the one hand, the local well-posedness of the problems can be easily obtained by generalizing the proof of the conservative interaction case; on the other hand, the global well-posedness of the problem can be ensured by assuming a priori the conservation of some quantities. This is a very hard problem, and the well-posedness of a model could be ensured casewise.

The research directions from the modeling standpoint are various and at different levels. Firstly, the mathematical models of Section 8 could be improved by further specializing the phenomena under investigation. For instance, the role of the immune system could be more specialized in the second and third models where the immune system is modeled as a whole. Concerning the first and the second models of Section 8, the role of external actions coupled to the thermostat could be inserted in order to mimic a therapeutic action. In the third model, further mutations beyond viral-induced mutations in fibroblast connections with cancer cells can be explored. Moreover, for all three models, the different rates could also be improved in order to better take into account the medium where the interactions occur. Finally, the evolution equations for the density and the linear momentum could be derived by employing an asymptotic analysis. However, an important research perspective, considering the possibility to introduce phenomena that require cell motility, is the introduction of the space and velocity dynamics. Some preliminary frameworks and models have been proposed in the pertinent literature where the space dynamics is just guided by the classical linear transport term and the velocity variable changes according to some jump processes [88,89], as firstly suggested in [90]. However, it was assumed that the interactions do not modify the space and velocity dynamics. It is clear that further developments are required in order to construct a more robust spatial-velocity thermostatted kinetic theory framework. Further research directions can be addressed to the derivation of the following thermostatted kinetic theory frameworks:

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Those where the interactions are weighted by taking into consideration some empirical/experimental data by employing optimization problems (inverse problems [91]).

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Those that permit the conservation of more than one moment. A preliminary thermostatted framework was discussed in [85] in the discrete activity case.

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Those that are based on fractional derivatives, which is today a fertile research domain that has gained much attention in the last ten years—see, among others, papers [92,93,94].

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Those that include the role of time delays.

Finally, the most important research perspective is the derivation of a multiscale approach that allows modeling a system from the lowest scale to the highest scale. The following three main scales need to be considered: microscopic (gene level), mesoscopic (cellular level), and macroscopic (tissue level) scales. The thermostatted kinetic framework is in between the microscopic scale (interactions) and mesoscopic scale (kinetic equations). For the dynamics at the macroscopic scale, asymptotic methods of the kinetic theory of gases have been employed in the inert matter case [79,95,96,97,98]. Similar methods have been proposed for the thermostatted frameworks, see [99,100,101]. These methods were derived for the specific space–velocity dynamic discussed above.

The thermostatted kinetic theory thus opens the doors to important research perspectives, most of them requiring an interdisciplinary interplay of mathematicians, physicists, and computer scientists on the one hand and biologists, immunologists, and physicians on the other hand.