From cellular to tissue scales by asymptotic limits of thermostatted kinetic models

Tumor growth strictly depends on the interactions occurring at the cellular scale. In order to obtain the linking between the dynamics described at tissue and cellular scales, asymptotic methods have been employed, consisting in deriving tissue equations by suitable limits of mesoscopic models. In this paper, the evolution at the cellular scale is described by thermostatted kinetic theory that include conservative, nonconservative (proliferation, destruction and mutations), stochastic terms, and the role of external agents. The dynamics at the tissue scale (cell-density evolution) is obtained by performing a low-field scaling and considering the related convergence of the rescaled framework when the scaling parameter goes to zero.


Introduction
Tumor disease has recently attracted the attention of many scientists, including physicists, applied mathematicians and bioinformatics researchers. The mechanisms which are responsible of tumor growth comprises many phenomena occurring at different scales, among others, molecular, cellular and tissue scales, see [1]. In particular at molecular scale (length scales of the order of nm-mm and time scales of ns-s) the dynamics of populations of proteins, peptides, and lipids are studied [2,3]. The cellular scale (length scales of the order of mm and time scales of min-hour) is concerned with the description of the malignant transformation of normal cells, cell-cell interactions, the heterogeneous tumor environment [4]. The tissue scale (length scales of the order of mm-cm and time scales of day-year) focuses on the dynamics of the gross tumor behavior including morphology, shape, extent of vascularization, and invasion, under different environmental conditions (cells can be treated as a single continuum) [5]. Therefore a complete understanding of tumor formation requires a multiscale approach. In particular the description of transport phenomena are of great importance in the tumor onset.
At the molecular scale, the dynamics of individual molecules is followed, for example by molecular dynamics simulations. Mean-field approaches to phenomena at cellular scale are based on ordinary differential equations (ODE), see the review paper [6]. More refined descriptions rely on generalized kinetic theory theory [7] and agent-based approach [8] which provide the evolution of cell distribution function. Tumor growth at tissue scale is usually described by employing continuum mechanics approach, which allows to obtain conservation laws for spatiotemporally varying densities of different cell types, see book [9] and the references cited therein. However in order to have a complete description of tumor disease and for understanding the different phenomena occurring at the different scales that have as results tumor metastasis, it is fundamental to link the dynamics described by the different approaches developed at the different scales (multiscale approach). The multiscale approach requires the development of mathematical methods, possibly a theory, that transfer information from the lower to the upper scale. A first attempt to the a e-mail: bianca@lps.ens.fr b e-mail: christian.dogbe@unicaen.fr c e-mail: anle@lptmc.jussieu.fr

The kinetic framework at cellular scale
This section is concerned with the mathematical modeling of the tumor growth at cellular scale by thermostatted kinetic theory that acts as a general paradigm for the derivation of specific models. Specifically in the tumor formation a large number of cells (active particles) that interact in nonlinear matter are involved and macroscopic external force fields F i extent an action on the system. The microscopic state of a cell is the triplet (x, v, u), which means that the cell Cells expressing the same function are grouped into a subsystem, called functional subsystem. The evolution of each functional subsystem is depicted by the distribution function . . , n}, and such that, for any fixed time t, the quantity f i (t, x, v, u)dx dv du represents the density of cells in the volume element dx dv du centered at (x, v, u). Let . . , f n (t, x, v, u)) be the vector whose components are the distribution functions of the functional subsystems, Ω = D x × D v × D u be the domain of all possible microscopic states, dΩ = dx dv du the Lebesgue measure on Ω and The evolution equation for the ith functional subsystem is obtained by equating the time derivative of f i to the balance of the inlet and outlet flows in the elementary volume dΩ. Accordingly we have where is the operator that models the gain-loss of cells due to transitions in the activity variable and it reads where: -η ij models the probability that a cell of the i-th functional subsystem with activity u * interacts instantaneously with a cell of the j-th functional subsystem with activity u * ; is the density function modeling the probability that cells of the i-th functional subsystem with activity u * interacting with cells of the j-th functional subsystem with activity u * reach the activity u. In particular A(u * , u * , u) satisfies the following identity: x, v, u) models the velocity-jump process, and it reads where T i (v * , v) is the turning kernel which gives the probability that, if a jump occurs, the velocity v * ∈ D v jumps into the velocity v ∈ D v . The domain D v is assumed to be bounded and spherically symmetric with respect to origin (i.e. v and −v ∈ D v ). In particular ν is the turning rate or turning frequency of the velocity-jump, hence 1/ν is the mean run time.
is the transport term that models the Gaussian thermostat [31,32], and it reads In particular (5) is a damping operator adjusted to control Du

Proliferative/desctructive and mutative events
The framework (2) can be further generalized by introducing the role of nonconservative processes. Specifically, interactions among the cells may generate proliferation/destruction of other cells (birth-death process). This type of interaction is modeled by the following operator: where α ij = η ij μ ij , being μ ij the net proliferative/destructive rate. Because of DNA corruptions, cells can become cells of another functional subsystem. These kinds of interactions are modeled by the following operator where β ihk = η hk ϕ i hk , being ϕ i hk the net mutative rate into the i-th functional subsystem, due to interactions that occur with rate η hk between the cells (x, v, u * ) of the h-th functional subsystem and the cells (x, v, u * ) of the k-th functional subsystem.
Bearing all the above in mind, the thermostatted kinetic framework with proliferative/destructive and mutative interactions reads

Thermostatted kinetic framework for open systems
The previous thermostatted kinetic structures refer to the modeling of tumor growth subjected to external force fields at the macroscopic scale but in the absence of external interactions at the microscopic scale. The modeling of external agents at the microscopic scale is performed by representing the external agents as functional subsystems that have the ability to modify the state u of the system by a particular action related to the variable ω ∈ D u . Assuming that the i-th inner functional subsystem interacts with the r-th external agent, for r ∈ {1, 2, . . . , m}, and denoting by the related distribution function (known function of its arguments), the microscopic external actions are modeled by the following operator where g i = (g i1 , . . . , g ir ) and -η e ir is the inner-outer encounter rate between the r-th external agent, with state ω * , and the cell of the i-th population, with state u * .
-B ir (u * , ω * , u) is the inner-outer transition probability density which describes the probability density that a cell of the i-th population, with state u * , falls into the state u after an interaction with the r-th external agent whose state is ω * . The density B ir satisfies, for all r ∈ {1, 2, . . . , m} and i ∈ {1, 2, . . . , n}, the following condition: Bearing all the above in mind, the thermostatted kinetic framework for open systems with proliferative/destructive and mutative interactions reads The thermostatted framework (11) is the underlying framework for the derivation of the related macroscopic equation by the low-field scaling. In particular solutions of (11) are assumed to be bounded in a space of functions where all needed convergence results will be true.

Macroscopic quantities and turning operator
This section is devoted to the definition of the local macroscopic quantities and the derivation of the main properties of the turning operator related to the velocity-jump process that will be used in the next section.
Bearing the thermostatted framework (11) in mind, the local macroscopic quantities, for i ∈ {1, 2, . . . , n}, can be defined as follows: -The local density [f i ](t, x, u) of the i-th functional subsystem defined at time t in the position x and activity u, reads -The relative mass velocity of particles -The pressure term reads -The local linear activation moment at time t in x reads where Note that, by definition of U, one has Macroscopic gross quantities can be easily obtained by summing in the local quantities with respect to i. The Hilbert space L 2 (D v , dv) endowed with the usual scalar product will be used in the sequel and the average of the function ϕ with respect to variable v will be denoted by Moreover the following Kronecker delta will be used: Preliminary to the low-field limit are the properties of the turning operator V i [f i ], which are summarized in the following lemma, see [33] for a detailed proof. Lemma 1. Assume that There exists a bounded equilibrium velocity distribution G i (v) : D v → R + , independent of t and x, such that and

Then i) (H-Theorem). The entropy equality holds
ii) The null-space of V i is spanned by a unique normalized and nonnegative function F i (v): iii) There exists a constant κ > 0 such that

The macroscopic framework at tissue scale
This section deals with the low-field limit of the thermostatted kinetic framework (11). In particular the macroscopic equation at the tissue scale is derived according to the following low-field scaling with the following rate choice: where the numbers h, m, q, r, s ≥ 1 are chosen according to biological requests. Therefore the distribution function of the i-th functional subsystem and the distribution function of the ir-th external agent are rescaled as follows: . . , f n ) be the vector of the rescaled distribution functions and g i = (g i1 , g i2 , . . . , g ir ), then the rescaled thermostatted framework (11) reads whereJ and T Fi [f ] are the rescaled conservative, nonconservative, mutating, turning, external agents and thermostat operators, respectively, that read The following lemma holds true.
Lemma 2. Let f i (t, x, v, u) be a sequence of solutions of the rescaled thermostatted kinetic equation (23). Assume that the turning operator V i satisfies the assumptions (A 1 -A 2 -A 3 ) and Then the asymptotic limit f i of the sequence f i (modulo the extraction of a subsequence) admits the following factorization: where i is the local macroscopic density (12) of the i-th functional subsystem. Moreover where Proof. The first part of the Lemma is gained by multiplying eq. (23) by ε s , letting ε goes to zero and considering Lemma 1. The limit of the transport term is gained by considering the following equality: and Lemma 1 again.
The main result of this paper follows. Theorem 1. Let f i (t, x, v, u) be a sequence of solutions of the rescaled thermostatted kinetic equation (23). Assume that the turning operator V i satisfies the assumptions (A 1 -A 2 -A 3 ) and Furthermore, assume that The following quantities The following quantities converge, in the sense of distributions on R * + × D x × D u , to the corresponding quantities The following quantities converge, in the sense of distributions on R * + × D x × D u , to the corresponding quantities and every formally small term in vanishes, then the local macroscopic density i of the i-th functional subsystem is the weak solution of the following equation: where = ( 1 , 2 , . . . , n ) and -D i is the following tensor: t, x, u) is the following operator: x, u) is the following operator: , x) is the following operator: -W i [ ](t, x, u) is the following operator: Proof. Assume that > 1, r > 1, q > 1, m > 1, and h > 1. Considering the average of the rescaled thermostatted framework equation (23) with respect to v, using the assumption A 1 and dividing by , one obtains When → 0, the termZ[f ] goes to zero, and by using Lemma 2 we have the limit (34). Then the asymptotic limit of the thermostat term reads Then, it is an easy task to show that  (45), respectively. Therefore the proof is concluded.

A model for tumor-immune-system competition under the action of a vaccine
This section deals with the derivation of a mathematical model for tumor-immune-system competition. Since the aim of this section is to show the main steps of the asymptotic method proposed in the present paper, the phenomenological analysis will be limited to the interactions among the following three functional subsystems: Normal cells Nc, whose distribution function is f 1 (t, x, v, u) and the activity variable u ∈ D u = [0, +∞) represents the mutation ability.
Mammary cancer cells Cc, whose distribution function is f 2 (t, x, v, u) and the activity variable u ∈ D u = [0, +∞) represents the progression towards high state of malignancy.
Immune system cells ISc, whose distribution function is f 3 (t, x, v, u) and the activity variable u represents the activation.
In particular, we assume the existence of one external action at the microscopic scale that acts on the Cc by reducing their malignancy state; we consider the role of the triplex cellular vaccine, whose distribution function is g 21 (t, x, v, ω) and the activity variable ω ∈ D u = [0, +∞) represents the ability to stimulate the immune system, see paper [34] for further details.
We assume that inner-inner and outer-inner encounter rates are constants for all interacting cells. Specifically η ij = η for all i, j ∈ {1, 2, 3}, and η e 21 = η e . The probability density A ij and B 21 are assumed to be defined by a delta Dirac function (deterministic output m ij (u * , u * ) and n 21 (u * , ω * ) of a pair interaction) depending on the microscopic state of the interacting cells: Moreover we assume that D v is the 3-sphere of radius R > 0 and we let D x and F i (u) arbitrary.
Interactions with transitions of activity. We assume that only Cc are subjected to transitions of activity, then J 1 [f ] = J 3 [f ] = 0. In particular Cc have a tendency to increase their malignancy when they interact with Nc. Specifically we have where ψ is a positive parameter related to the ability of Cc to reach high states of malignancy. Accordingly the conservative operator Interactions with the triplex vaccine. The cellular vaccine produces a decrease in the microscopic state of Cc towards lower states of malignancy, then n 21 (u * , ω * ) = u * − ψ 2 , where ψ 2 is a positive parameter. Assuming that g 21 (t, x, v, ω) = ae −bω (exponential decay with respect to ω), a, b > 0, we have Q[f 2 , g 2 ] = a b η e e −bω [f 2 (t, x, v, u) − f 2 (t, x, v, u + ψ)].

Conclusions
In this paper, a mathematical framework at the cellular scale has been proposed for the modeling of tumor-immunesystem competition. The main aim of the paper was the possibility to link the dynamics described at the cellular scale with the dynamics that occurs at macroscopic (tissue) scale. In particular the paper has been focused on the possibility to model the role of external agents, e.g. the environment, a cellular vaccine, that can affect the whole dynamics. The macroscopic equation has been obtained by employing the asymptotic method of the rescaled thermostatted framework under the low-field limit. The main steps of the method have been shown by means of the definition of a specific model. Looking at the macroscopic equation, it is noticeable that conservative and nonconservative interactions at the cellular scale become source terms for the macroscopic modeling of the tumor tissue growth. Indeed, as the tissue equation shows, different macroscopic phenomena and dynamics can appear depending on the relations among the rates at the cellular scale.
It is worth stressing that, even if this paper can be considered as an important contribution to the multiscale approach, and specifically to the problem of linking the dynamics at the cellular scale with the dynamics at the tissue scale, the problem of linking the dynamics at molecular scale with the dynamics at cellular scale remains elusive and can be considered as a research perspective.