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Density evolution by the low-field limit of kinetic frameworks with thermostat and mutations
Accepted Manuscript Short communication Density evolution by the low-field limit of kinetic frameworks with thermostat and mutations Carlo Bianca, Annie Lemarchand PII: DOI: Reference:
S1007-5704(14)00215-9 http://dx.doi.org/10.1016/j.cnsns.2014.05.009 CNSNS 3192
To appear in:
Communications in Nonlinear Science and Numerical Simulation
Please cite this article as: Bianca, C., Lemarchand, A., Density evolution by the low-field limit of kinetic frameworks with thermostat and mutations, Communications in Nonlinear Science and Numerical Simulation (2014), doi: http:// dx.doi.org/10.1016/j.cnsns.2014.05.009
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Density evolution by the low-field limit of kinetic frameworks with thermostat and mutations Carlo Bianca1,2 , Annie Lemarchand1,2 1 Sorbonne Universit´es, UPMC Univ Paris 06, UMR 7600, Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee, 4, place Jussieu, case courrier 121, 75252 Paris cedex 05, France
2
CNRS, UMR 7600 LPTMC, France
Abstract This paper is concerned with an asymptotic limit of a thermostatted kinetic framework which can be proposed for the modeling of complex biological and chemical systems where proliferative/destructive/mutative interactions and random velocity-jump processes occur. Specifically the macroscopic equations fulfilled by the local and global densities of the system are obtained by performing an asymptotic limit of a low-field rescaling of a kinetic framework with a thermostat and mutative interactions. Keywords: Kinetic theory, low-field scaling, integro-differential equation, nonlinearity, diffusion
1
Introduction
Recently in [1] a new thermostatted kinetic approach has been proposed for the modeling of complex biological and chemical systems in which mutations occur. According to this approach, the whole system is composed of active particles that express a specific strategy (activity) grouped, depending on their function, in subsystems called functional subsystems. The system is then statistically described by employing distribution functions and stochastic interactions. In order to take into account further phenomena in biology, this paper aims at developing a complete framework that takes into account also space and velocity phenomena occuring in biological and chemical systems. In particular in this paper a stochastic velocity perturbation of the spatially homogeneous dynamics is proposed: the velocity-jump process, where the direction of particle velocity fluctuates continuously. In this random process the turning event is usually governed by a Poisson process, see among others, papers [2, 3, 4]. Similar random processes of the velocity of the particles have been also considered in [5] to simulate multiple collision dynamics. Finally in order to have a macroscopic description of the system, a low-field scaling is introduced. The asymptotic limit of the rescaled thermostatted kinetic framework is performed under technical, but general, assumptions on the operators that describe the different types of interactions that change the microscopic state of particles, 1
proliferation, inhibition, mutation. The derived macroscopic equation, which is of diffusion type, is of interest for example in cancer modeling, for styding the diffusion and growth processes of malignant cells at the tissue scale [6, 7, 8, 9]. The method developed in this paper is thus of interest in the multiscale analysis of tumor formation that aims at obtaining a complete description of the tumor growth at tissue scale by linking the information that are achieved at a lower scale, i.e. cellular, genetic, molecular, see [10, 11] and therein references. Moreover the asymptotic method proposed in the present paper can be used to study the physics of social systems, see the review paper [12], as well as to understand the statistical mechanics of evolutionary and coevolutionary games [13, 14]. As already mentioned, the present paper is devoted to the low-field limit analysis only. However the interested reader in a deeper understanding of the asymptotic analysis by the low-field and high-field limits is referred, among others, to papers [15, 16, 17, 18, 20, 19] and the references cited therein. The contents of the present paper is outlined as follows: After this introduction, Section 2 deals with a technical generalization of the thermostatted mathematical framework developed in [1] by introducing the space structure and the velocity-jump process. Section 3 is concerned with the low-field limit of the generalized thermostatted kinetic framework introduced in Section 2. Specifically Section 3 is presented through three sequential subsections. In detail: Section 3.1 is concerned with the rescaled thermostatted framework by the low-field scaling and preliminary results; the main result of the present paper regarding the derivation of the macroscopic equation fulfilled by the local and global densities of the system is outlined in Section 3.2; Section 3.3 is devoted to the derivation of a specific model that allows to illustrate the method step by step. Finally a critical analysis and future research directions are highlighted in Section 4.
2
The thermostatted kinetic framework
This section aims at introducing the thermostatted kinetic model that will be the underlying framework for the analysis developed in the present paper. Specifically we generalize the mathematical framework recently proposed in [1] by introducing a space-velocity structure and a velocity-jump process and we perform an asymptotic analysis by the low-field limit. Let x ∈ Dx ⊂ R3 be the position, v ∈ Dv ⊂ R3 the velocity and u ∈ Du ⊂ R the activity of a particle of the system under consideration assumed to be composed of n ∈ N different subsystems (functional subsystems) of active particles, see [1]. The time evolution of the ith functional subsystem is described by the distribution function fi = fi (t, x, v, u) and the time evolution of the whole system is depicted by the vector f = (f1 , f2 , . . . , fn ). The system evolves because of the following interactions among the particles of the functional subsystem. Conservative interactions. Particles interact with each other and the results of these kinds of interactions is the modification of the magnitude of the activity
2
variable according to the following operator Ji [f ] = Ji [f ](t, x, v, u): Z n X Ji [f ] = ηij Aij (u∗ , u∗ , u)fi (t, x, v, u∗ ) fj (t, x, v, u∗ ) du∗ du∗ j=1D ×D u u
−fi (t, x, v, u)
n Z X j=1
ηij fj (t, x, v, u∗ ) du∗ ,
(2.1)
Du
where ηij models the probability that a particle of the ith functional subsystem with state (x, v, u∗ ) interacts with a particle of the jth functional subsystem with state (x, v, u∗ ); Aij = A(u∗ , u∗ , u) is the probability density that the particles of the ith functional subsystem with state (x, v, u∗ ) interacting with the particles of the jth functional subsystem with state (x, v, u∗ ) reach the activity (x, v, u). The function Aij is such that kAij kL1 (Du ,dv) = 1, ∀ u∗ , u∗ ∈ Du . Proliferative/destructive interactions. Interactions among the particles result in a change in the number of particles (birth-death process) and are modeled by the following operator N [f ] = N [f ](t, x, v, u): n Z X Ni [f ] = fi (t, x, v, u) ηij (u∗ , u∗ ) µij (u∗ , u∗ ) fj (t, x, v, u∗ ) du∗ . (2.2) j=1
Du
where µij denotes the net proliferation/destruction rate. Mutative interactions. Particles can modify their shape and type of activity becoming a particle of another functional subsystem (mutation). These kinds of interactions are modeled by the following operator Mi [f ] = Mi [f ](t, x, v, u): Z n X n X Mi [f ] = ηhk (u∗ , u∗ ) ϕihk (u∗ , u∗ , u) fh (t, x, v, u∗ ) (2.3) ×
h=1 k=1D ×D u u fk (t, x, v, u∗ ) du∗
du∗ .
(2.4)
ϕihk
denotes the net mutation rate into the ith functional subsystem, due where to interactions that occur with rate ηhk between the particles of the hth functional subsystem, with state (x, v, u∗ ), and the particles of the kth functional subsystem, with state (x, v, u∗ ). Velocity-jump process. This process, which models the particle that moves with constant speed describing a straight line whose direction changes continuously, is modeled by the following operator Vi [fi ] ≡ Vi [fi ](t, x, v, u): Z Vi [fi ] = Ti (v∗ , v)fi (t, x, v∗ , u) − Ti (v, v∗ )fi (t, x, v, u) dv∗ , (2.5) Dv
where Ti (v , v) is the turning kernel which gives the probability that the velocity v∗ ∈ Dv jumps into the velocity v ∈ Dv (if a jump occurs). The set v ∈ Dv is assumed to be bounded and spherically symmetric with respect to (0, 0, 0). ∗
Considering the balance of the inlet and outlet flows into the elementary volume of the space of the microscopic states, the underlying thermostatted kinetic framework for the distribution function fi thus reads: (∂t + v · ∇x ) fi + TFi [f ] = Ji [f ] + Ni [f ] + Mi [f ] + ν Vi [fi ], 3
(2.6)
where • v · ∇x fi is the transport operator; • ν is the turning rate or turning frequency of the velocity jump, hence 1/ν is the mean run time; • TFi [f ] is the thermostat term that has the role of controlling the second moment of local activity: Z u2 f˜(t, x, v, u) du. (2.7) E2 [f ](t, x, v) = Du
This term, according to [21], reads: TFi [f ] := ∂u
Fi (u) 1 − u
Z
!
u f˜(t, x, v, u) dx dv du fi (t, x, v, u) ,
Ω
(2.8)
where Fi (u) : Du → R is a known external force field that moves the system far from equilibrium, Ω = Dx × Dv × Du and f˜(t, x, v, u) =
n X
fj (t, x, v, u).
j=1
Remark 2.1 In the thermostatted kinetic framework (2.6) the space and velocity variables are not modified by the interactions. In this context, the local density %[fi ](t, x, u) of the ith functional subsystem defined at time t in the position x and activity u, reads: Z fi (t, x, v, u) dv, (2.9) %i := %[fi ](t, x, u) = Dv
and the global density %[f ](t, x, u) of the whole system reads: Z % := %[f ](t, x, u) = f˜(t, x, v, u) dv,
(2.10)
Dv
The main aim of this paper is to obtain the macroscopic equation of %[fi ] and %[f ].
3
The low-field limit
The derivation of the macroscopic equation for the global density %[f ] strongly depends on the properties of the turning operator Vi [fi ]. Indeed the asymptotic analysis will be performed by considering the following preliminary results. Lemma 3.1 ([22]) Let the following assumptions be satisfied: (A1 ) kVi [fi ]kL1 (Dv ,dv) = kv Vi [fi ]kL1 (Dv ,dv) = 0, ∀ x ∈ Dx , u ∈ Du .
4
(A2 ) If there exists a bounded equilibrium velocity distribution Gi (v) : Dv → R+ , independent of t and x, such that: Ti (v∗ , v) Gi (v) = Ti (v, v∗ )Gi (v∗ ).
(3.1)
and kv Gi (v)kL1 (Dv ,dv) = 0, kGi (v)kL1 (Dv ,dv) = 0, ∀ x ∈ Dx , u ∈ Du . (A3 ) The kernel Ti (v, v∗ ) is bounded, and there exists a constant σi > 0 such that Ti (v, v∗ ) ≥ σi Gi (v), ∀(v, v∗ ) ∈ Dv × Dv . (3.2) Then (i) For fi ∈ L2 (Dv , dv/Gi ), the integral equation Vi [gi ] = fi has a unique solution gi ∈ L2 Dv , dv/Gi such that kgi (v)kL1 (Dv ) = 0
if and only if
kfi (v)kL1 (Dv ) = 0.
(ii) The integral operator Vi defines a self-adjoint Fredholm operator on the space L2 Dv , dv/Gi .
(iii) The equation Vi [fi ] = v Gi (v) has a unique solution given by fi (v) = χi (v). (iv) The null-space of Vi is spanned by a unique normalized and nonnegative function G(v): Ker(Vi ) = Span{Gi }. The Hilbert space L2 (Dv , dv) endowed with the usual scalar product Z f (v) g(v) dv, f, g ∈ L2 (Dv , dv), hf, gi = Dv
will be used in the sequel and the average of the function ϕ with respect to variable v will be denoted by Z hϕi := hϕ, 1i = ϕ(v) dv. Dv
Moreover the following Kronecker delta will be used: 1 if i = j δij = 0 if i 6= j 3.1
The rescaled thermostatted kinetic framework
This subsection deals with the rescaled thermostatted kinetic framework whose solutions are assumed to be bounded in a space of functions where all needed convergence results will be true. The following low-field scaling is considered: t ` (3.3) , x, v, u, Fi , ` ≥ 1, (t, x, v, u, Fi ) → 5
and considering the meaning and the relationship that exists among the rates, the following rates are chosen: η = r ,
µ = q ,
ϕ = m ,
ν=
1 , s
(3.4)
where m, q, r, s ≥ 1. Therefore the rescaled distribution function reads: t fi (t, x, v, u) = fi , x, v, u . Let f = (f1 , f2 , . . . , fn ) be the vector of the rescaled distribution functions, then the rescaled thermostatted framework (2.6) is as follows: ˜ i [f ]+ ∂t fi +v·∇x fi +` TFi [f ] = r J˜i [f ]+r+q N˜i [f ]+r+m M
1 ˜ Vi [fi ], (3.5) s
˜ i [f ], V˜i [f ], and TF [f ] are the rescaled conservative, where J˜i [f ], N˜i [f ], M i i nonconservative, mutating, turning and thermostat operators, respectively, that read: n Z X ˜ Ji [f ] = Aij (u∗ , u∗ , u)fi (t, x, v, u∗ ) fj (t, x, v, u∗ ) du∗ du∗ j=1
−
Du ×Du
n X
fi (t, x, v, u)
= fi (t, x, v, u)
n Z X j=1
˜ i [f ] M
=
n X n X
fj (t, x, v, u∗ ) du∗ ,
(3.6)
Du
j=1
˜i [f ] N
Z
Z
fj (t, x, v, u∗ ) du∗ ,
(3.7)
Du
fh (t, x, v, u∗ ) fk (t, x, v, u∗ ) du∗ du∗ ,
(3.8)
h=1 k=1D ×D u u
V˜i [fi ] TFi [f ]
Z
Ti (v∗ , v)fi (t, x, v∗ , u) − Ti (v, v∗ )fi (t, x, v, u) dv∗ , (3.9) Dv ! Z ˜ = ∂u Fi (u) 1 − u u f dx dv du fi (t, x, v, u) (3.10) =
Ω
The following lemmas hold true. Lemma 3.2 ([23]) Let fi (t, x, v, u) be a sequence of solutions of the rescaled thermostatted kinetic equation (3.5). Assume that the turning operator Vi satisfies the assumptions (A1 -A2 -A3 ) and, when → 0, the following statements hold true: fi −→ fi a.e. in [0, ∞) × Dx × Dv × Du , V˜i [fi ] −→ V˜i [fi ]
(3.11) (3.12)
Then the asymptotic limit fi of the sequence fi (modulo the extraction of a subsequence) admits the following factorization: fi (t, x, v, u) = %i (t, x, u) Gi (v),
(3.13)
where %i is the local macroscopic density (2.9) of the ith functional subsystem. 6
In order to calculate the asymptotic limit of the rescaled thermostatted framework (3.5), we need to perform the limit, when goes to zero, of the term 1 hv · ∇x f i.
(3.14)
Lemma 3.3 ([23]) Let fi (t, x, v, u) be a sequence of solutions of the scaled thermostatted kinetic equation (3.5). Assume that the turning operator Vi satisfies the assumptions (A1 -A2 -A3 ) and, when → 0, the following statements hold true: fi −→ V˜i [fi ] −→
fi a.e. in [0, ∞) × Dx × Dv × Du , V˜i [fi ],
(3.15) (3.16)
then
1
v · ∇x fi −−−→ δs,1 divx (χi (v) ⊗ v) ∇x %i . →0 where χi (v) is the only solution of the equation V˜i [fi ] = v Gi (v). 3.2
(3.17)
The macroscopic equation
This subsection is devoted to the derivation of the macroscopic equation which describes the evolution of the global macroscopic density %. The analysis is based on the following assumptions. Let fi (t, x, v, u) be a sequence of solutions of the rescaled thermostatted kinetic equation (3.5). (A4 ) The following quantities hfi i, hvfi i, hv ⊗ vfi i, converge, in the sense of distributions on R∗+ × Dx × Du , to the corresponding quantities hfi i, hvfi i, hv ⊗ vfi i, (A5 ) The following quantities D E D E D E
˜i [f ] , M ˜ i [f ] , TF [f ] , J˜i [f ] , N i
converge, in the sense of distributions on R∗+ × Dx × Du , to the corresponding quantities E
E D E D D ˜ i [f ] , TF [f ] , ˜i [f ] , M J˜i [f ] , N i
(A6 ) The following quantities D E D E D E
˜i [f ] , vM ˜ i [f ] , vTF [f ] vJ˜i [f ] , vN i
converge, in the sense of distributions on R∗+ × Dx × Du , to the corresponding quantities D E D E D E
˜i [f ] , vM ˜ i [f ] , vTF [f ] vJ˜i [f ] , vN i 7
The main result of this paper follows. Theorem 3.1 Let fi (t, x, v, u) be a sequence of solutions of the rescaled thermostatted kinetic equation (3.5). Assume that the turning operator Vi satisfies the assumptions (A1 -A2 -A3 ) and, when → 0, the following statements hold true: fi −→ V˜i [fi ] −→
fi a.e. in [0, ∞) × Dx × Dv × Du , V˜i [fi ]
(3.18) (3.19)
If the assumptions (A4 -A5 -A6 ) hold true and every formally small term in vanishes, then the local macroscopic density %i of the system is the weak solution of the following equation ∂t %i + δ`,1 ∂u Fi (u)(1 − uA[%](t))%i = δs,1 divx (D%i · ∇x %i ) + δr,1 Hi [%] + δr+q,1 Si [%] + δr+m,1 L[%] (3.20) where % = (%1 , %2 , . . . , %n ) and • A[%](t) is the following operator: A[%](t) =
n Z X j=1
u %j (t, x, u) dx du =
Dx ×Du
Z
u %(t, x, u) dx du,
Dx ×Du
(3.21)
• D%i is the following tensor: D%i = −
Z
v ⊗ χi (v) dv,
(3.22)
Dv
• Hi [%](t) is the following operator: Hi [%]
=
n X
hGi (v), Gj (v)i ×
j=1
Z
Aij (u∗ , u∗ , u) %i (t, x, u∗ ) %j (t, x, u∗ ) du∗ du∗
Du ×Du
− %i (t, x, u)
n X
hGi (v), Gj (v)i
Z
%j (t, x, u∗ ) du∗ .
(3.23)
Du
j=1
• Si [%](t) is the following operator: Si [%] = %i (t, x, u)
n X
hGi (v), Gj (v)i
Z
%j (t, x, u∗ ) du∗ .
(3.24)
Du
j=1
• L[%](t) is the following operator: L[%] =
n X n X
h=1 k=1
hGh (v), Gk (v)i
Z
Du
8
%h (t, x, u∗ ) du∗
!
Z
∗
%k (t, x, u ) du
Du
(3.25)
∗
!
.
Proof. By taking the average of the equation (3.5) with respect to v, using the assumption A1 and dividing by , one obtains:
where
1
˜ ]. ∂t fi + v · ∇x fi + `−1 TFi [f ] = Z[f
(3.26)
D D D E E E ˜ ] = r−1 J[f ˜ ] + r+q−1 N ˜ [f ] + r+m−1 M ˜ [f ] Z[f
(3.27)
Assume that ` > 1, r > 1, r + q > 1, and r + m > 1. When → 0, the term ˜ ] goes to zero, and by using Lemma 3.3 we have the limit (3.17). Z[f By using Lemma 3.2, the asymptotic limit of the thermostat term reads: * !+ Z
`−1 `−1 u f˜ dx dv du fi TFi [f ] = ∂u Fi (u) 1 − u Ω
−−−→ δl,1 →0
=
*
∂u Fi (u) 1 − u
n Z X j=1
Ω
u %j Gj dx dv du %i Gi
δl,1 ∂u Fi (u) 1 − u A[%](t) %i .
+
(3.28)
Then, it is an easy task to show that r−1 hJ˜i [f ]i
−−−→
δr,1 Hi [%]
(3.29)
˜i [f ]i r+q−1 hN
−−−→
δr+q,1 Si [%]
(3.30)
˜ i [f ]i r+m−1 hM
−−−→
δr+m,1 L[%]
(3.31)
→0
→0
→0
where Hi [%], Si [%] and L[%] are the operators defined in Eqs. (3.23), (3.24), and (3.25) respectively. Therefore the proof is concluded. Bearing the above theorem in mind we have the following result. Theorem 3.2 Let fi (t, x, v, u) be a sequence of solutions of the rescaled thermostatted kinetic equation (3.5). Assume that the turning operator Vi satisfies the assumptions (A1 -A2 -A3 ) and, when → 0, the following statements hold true: fi −→ V˜i [fi ] −→
fi a.e. in [0, ∞) × Dx × Dv × Du , ˜ Vi [fi ]
(3.32) (3.33)
If the assumptions (A4 -A5 -A6 ) hold true and every formally small term in vanishes, then the global macroscopic density % of the system is the weak solution of the following equation ∂t % + δ`,1 ∂u (1 − uA[%](t)) αF [%] = δs,1 divx (D% · ∇x %) + δr,1 H[%] ˜ + δr+q,1 S[%] + δr+m,1 L[%] (3.34) where A[%](t) is given by (3.21), 9
• αF [%] = αF [%](t, x, u) is the following operator: αF [%](t) =
n X
Fj (u)%j (t, x, u),
(3.35)
j=1
• D% · ∇ x % =
n X
D%j · ∇x %j , where D%j is given by (3.22);
j=1
• H[%](t) =
n X
Hj [%](t), where Hi [%](t) is given by (3.23);
j=1
• S[%](t) =
n X
Sj [%](t), where Hi [%](t) is given by (3.24);
j=1
˜ • L[%](t) = nL[%](t), where L[%](t) is given by (3.25). 3.3
A specific model
This section aims at showing the asymptotic method by means of an example in biology and specifically a model for the study of cancer-immune system competition. Accordingly: the first step is the definition of the functional subsystems that are the main actors involved in the evolution of the system under modeling; the second step is the definition of the microscopic interactions; the third and final step is the derivation of the macroscopic equations that can be pursued following the statement of the theorems 3.1 and 3.2. 1. Functional subsystems and activity variable. It is assumed that the cancer-immune system competition involves the following three interacting functional subsystems and their mutual interactions may also generate mutations: - Normal cells Nc distributed according to f1 (t, x, v, u). The activity variable u represents the mutation ability. A normal cell may undergo a mutation as consequence of DNA corruptions and mutate in cancer cells. - Cancer cells Cc distributed according to f2 (t, x, v, u). The activity variable u represents the progression ability. - Immune system cells ISc distributed according to f3 (t, x, v, u). The activity variable u is a magnitude of the activation. 2. Microscopic interactions. The encounter rate is assumed to be constant for all interacting pairs: ηij = η for all i, j ∈ {1, 2, 3}. The probability density Aij is assumed to be defined by a delta Dirac function (deterministic output mij (u∗ , u∗ ) of a pair interaction) depending on the microscopic state of the interacting pairs: Aij (u∗ , u∗ , u) = δ(u − mij (u∗ , u∗ )). (3.36) Finally we assume that Du = [0, +∞), Dv is the 3-sphere of radius R > 0 and we let Dx and Fi (u) arbitrary.
10
• Conservative Interactions. It is assumed that Cc is the only subsystem subject to conservative interactions, therefore J1 [f ](t, u) = J3 [f ](t, u) = 0. The conservative term J2 [f ](t, u) is derived under the assumption that the Cc have a tendency to increase their microscopic state with a certain rate, regulated by Nc. Accordingly we define ( u∗ + ψ if j = 1 and i = 2, ∗ (3.37) mij (u∗ , u ) = u∗ otherwise. where α is a positive parameter related to the ability of Cc to reach high states of activity and 0 < ψ < 1 a scale factor. Therefore the conservative term reads:
Ji [f ](t, x, v, u) =
(
η ρ[f1 ](t, x, u) [f2 (t, x, v, u − ψ α) − f2 (t, x, v, u)] 0
if i = 2, otherwise.
• Proliferative/Destructive Interactions. Each subsystem may proliferate (without mutations) when it encounters another subsystem. Specifically: P.1 Nc proliferate when encounter each other; P.2 Cc proliferate when encounter Nc and ISc; P.3 ISc proliferate when encounter Cc. Accordingly, the proliferation rate reads: β if j = 1 and i = 1, ψβ if j ∈ {1, 3} and i = 2, p ∗ µij (u, u ) = β if j = 2 and i = 3, I 0 otherwise,
(3.38)
where β > 0 is the proliferation rate of Nc and βI is the proliferation rate of the ISc. The proliferative term thus reads: η βf1 (t, x, v, u) ρ[f1 ](t, x, u) η ψβf2 (t, x, v, u) [ρ[f1 ](t, x, u) + ρ[f3 ](t, x, u)] Pi [f ](t, x, v, u) = η βI f3 (t, x, v, u) ρ[f2 ](t, x, u)
if i = 1, if i = 2, if i = 3.
The cells of each subsystem may be depleted when they encounter cells of another subsystem. Specifically: D.1 Nc are depleted by Cc; D.2 Cc are depleted by ISc; D.3 ISc are depleted by Cc.
11
Accordingly, we have: ψδ δ d ∗ µij (u, u ) = δI 0
if j = 2 and i = 1, if j = 3 and i = 2, if j = 2 and i = 3, otherwise,
(3.39)
where δ is the destruction rate of Cc by ISc and δI the destruction rate of the ISc by Mc. Thus the destructive term reads: η ψδf1 (t, x, v, u) ρ[f2 ](t, x, u) η δ f2 (t, x, v, u) ρ[f5 ](t, x, u) Di [f ](t, x, v, u) = η δI f5 (t, x, v, u) ρ[f2 ](t, x, u)
if i = 1, if i = 2, if i = 3.
According to Section 2, the net proliferation/destruction rate reads: µij (u∗ , u∗ ) = µpij (u∗ , u∗ ) − µdij (u∗ , u∗ ), and then Ni [f ](t, x, v, u) = Pi [f ](t, x, v, u) − Di [f ](t, x, v, u).
• Mutative Interactions. It is assumed that may occur genetic mutations. Specifically: T.1 Nc may mutate in Cc when interact with Nc and Cc. In particular, we assume that the microscopic state of the entities does not change during the mutation. Accordingly, we have: if h = 1, k = 1, and i = 2, ψγ δ(u − u∗ ) ∗ i γ δ(u − u∗ ) if h = 1, k = 2, and i = 2, ϕhk (u∗ , u , u) = (3.40) 0 otherwise,
where γ is the mutation rate of Nc when they encounter Cc. Thus the mutative term reads:
Ti [f ](t, x, v, u) =
η γ f1 (t, x, v, u) [ψ ρ[f1 ](t, x, u) + ρ[f2 ](t, x, u)] 0
if i = 2, otherwise.
Velocity-jump process. Since the jump process is described by the kernel Ti , we can assume without restrictions that Ti (v∗ , v) = γi Gi (v) =
12
γi , |Dv |
0 < γi < σi ,
where we have assumed Gi (v) = 1/|Dv |. Therefore the turning operator reads: Z Vi [fi ](t, x, v, u) = Ti (v∗ , v)fi (t, x, v∗ , u) − Ti (v, v∗ )fi (t, x, v, u) dv∗ , Dv = γi ρi (t, x, u)Gi (v) − fi (t, x, v, u) . Bearing all above in mind, the thermostatted kinetic model thus reads: ! Z Z f2 du f1 f1 du − ψδ (∂t + v · ∇x ) f1 + TF1 [f ] = η β Du Du Z (∂t + v · ∇x ) f2 + TF2 [f ] = η [f2 (t, x, v, u − ψ α) − f2 (t, x, v, u)] f1 du+ Du ! Z Z ψβ f5 du f2 η (f + f ) du − δ 1 3 Du Du ! Z Z f2 du − δI f2 du(t, x, u) f5 (∂t + v · ∇x ) f3 + TF3 [f ] = η βI Du
Du
3. The macroscopic equation. According to Lemma 3.1, the function χi (v) reads: γi ρi − v Gi (v)(γi ρi − v) = , χi (v) = γi γi |Dv | and according to Theorem 3.1, the diffusion tensor Dρi defined in (3.22) reads: Z Z R2 1 1 v ⊗ v dv = [v ⊗ vGi (v)] dv = I, (3.41) Dρi = γi Dv γi |Dv | Dv 3γi where I is the identity matrix and, as it is known, the following formula holds Z R2 δij . vi vj dv = |Dv | 3 Dv By straightforward computations we have: divx (Dρi · ∇x ρi ) =
R2 ∆x ρi , 3γi
and hGi (v), Gi (v)i =
1 . |Dv |
Let Ξ[ρi , Fi , γi ](t, x, u) be the following operator: R2 Ξl,s [ρi , Fi , γi ](t, x, u) = ∂t ρi + δl,1 ∂u (Fi (u) 1 − uA[%](t))ρi − δs,1 ∆x ρi , 3γi
and Γr,m [%](t, x, u) the following one:
! ! Z Z 3 3 δr+m,1 X X ρk (t, x, u) du . ρh (t, x, u) du Γr,m [%](t, x, u) = |Dv | Du Du h=1 k=1
13
Then the macroscopic model (3.20) related to the evolution of the local densities of the system thus reads: ρ1 (t, x, u) Ξl,s [ρ1 , F1 , γ1 ] = (δr+q,1 − δr,1 ) ρ˜(t, x, u) + Γr,m [%](t, x, u) |Dv | Z ρ1 (t, x, u) du+ Ξl,s [ρ2 , F2 , γ2 ] = δr,1 (ρ2 (t, x, u − ψα) − ρ2 (t, x, u)) Du
ρ2 (t, x, u) ρ˜(t, x, u) + Γr,m [%](t, x, u) (δr+q,1 − δr,1 ) |Dv | ρ3 (t, x, u) ρ˜(t, x, u) + Γr,m [%](t, x, u) Ξl,s [ρ3 , F3 , γ3 ] = (δr+q,1 − δr,1 ) |Dv |
where
ρ˜(t, x, u) =
3 Z X j=1
ρj (t, x, u) du.
Du
The same arguments hold true for the macroscopic model given by (3.34) related to the evolution of the global density of the system.
4
Critical Analysis and Perspective
Thermostatted kinetic frameworks have been recently proposed in [24] for the modeling of complex systems in physics and life sciences. As shown in [24], these models are able to model complex systems at the mesoscopic scale that in the case of biological systems corresponds to the cellular scale. However a complete multiscale approach that allows to obtain a full description of the complex dynamics occurring in many systems of the real-world is missing. Multiscale approach means to obtain a link among mathematical/computational models developed at different scales, e.g. atomic-molecular-mesoscopic-macroscopic. The pertinent literature in this field needs the development of a formal method, possibly a mathematical theory. This is a very hard problem and a new challenge, of the last and new century, for the researchers in applied mathematicians and, in general, in the applied sciences. The goal of the present paper is the development of a mathematical method that allows to link the dynamics at the mesoscopic (cellular) scale with the dynamics at the macroscopic (tissue) scale. Specifically the paper has been concerned with the generalization of the asymptotic method proposed in [23] to a new thermostatted kinetic framework, which includes mutative events that are typical of biological and chemical systems and the velocity-jump process. Under suitable assumptions, the macroscopic equations for the local and global densities have been derived and the main steps of the method have been summarized by considering a specific model for cancer-immune system competition. It is worth stressing that the method proposed in the present paper has been developed by considering a general complex system modeled according to a thermostatted kinetic framework that can acts as a general paradigm for the derivation of specific models in biology and chemistry. We are aware that much research activity is needed in order to obtain a complete multiscale approach. Nevertheless the method proposed in this paper can be considered as an important contribution in the pertinent literature especially because, to the best of our knowledgement, this is the first time that the 14
asymptotic method is developed for the thermostatted kinetic framework with mutations. It is worth precising that the asymptotic limit developed in the present paper refers to a low-field scaling of the thermostatted kinetic framework. Further research directions include the possibility to consider the derivation of macroscopic equations by high-field scalings or mixed scalings. Finally the asymptotic method can be further generalized to thermostatted kinetic frameworks which include particles refuge, see [25, 26]. This is a work in progress and the results will be presented in due course.
Acknowledgement The authors were partially supported by l’Agence Nationale de la Recherche (ANR T-KiNeT Project).
References [1] C. Bianca, How do mutative events modify moments evolution in thermostatted kinetic models? Commun Nonlinear Sci Numer Simulat 19 (2014) 2155-2159. [2] T. Hillen and H. Othmer, The diffusion limit of transport equations derived from velocity jump processes, SIAM J. Appl. Math. 61 (2000) 751-775. [3] T. Hillen, Hyperbolic models for chemosensitive movement, Math. Models Methods Appl. Sci. 12 (2002) 1007-1034. [4] H. G. Othmer, S. R. Dunbar, W. Alt, Models of dispersal in biological systems, J. Math. Biol. 26 (1988) 263-298. [5] R. Kapral, Multiparticle collision dynamics: Simulation of complex systems on mesoscales, Adv. Chem. Phys. 140 (2008) 89-146. [6] L. Loeb, Tissue growth and tumor growth, Cancer Res 2 (1917) 135-150. [7] J.E. Hobbs, A. Zakarija, D.L. Cundiff, J.A. Doll, E. Hymen, M. Cornwell, S.E. Crawford, N. Liu, M. Signaevsky, G.A. Soff, Alternatively spliced human tissue factor promotes tumor growth and angiogenesis in a pancreatic cancer tumor model, Thromb Res. 120 (2007) S13-S21. [8] T. Aikawa, J. Gunn, S.M. Spong, S.J. Klaus, M. Korc, Connective tissue growth factor-specific antibody attenuates tumor growth, metastasis, and angiogenesis in an orthotopic mouse model of pancreatic cancer, Mol Cancer Ther. 5 (2006) 1108-1116. [9] J.M. Butler, H. Kobayashi, S. Rafii, Instructive role of the vascular niche in promoting tumour growth and tissue repair by angiocrine factors, Nature Reviews Cancer 10 (2010) 138-146. [10] C. L. Jorcyk, M. Kolev, K. Tawara, B. Zubik-Kowal, Experimental versus numerical data for breast cancer progression, Nonlinear Analysis: Real World Applications 13 (2012) 78-84. [11] C. Bianca and M. Pennisi, The triplex vaccine effects in mammary carcinoma: A nonlinear model in tune with simtriplex, Nonlinear Analysis: Real World Applications 13 (2012) 1913-1940.
15
[12] C. Castellano, S. Fortunato and V. Loreto, Statistical physics of social dynamics, Reviews of Modern Physics, 81 (2009) 591-646. [13] M. Perc, J. G´ omez-Garde˜ nes, A. Szolnoki, L.M. Flor´ıa and Y. Moreno, Evolutionary dynamics of group interactions on structured populations: a review, J. R. Soc. Interface 10 (2013) 20120997. [14] M. Perc, A. Szolnoki, Coevolutionary games - A mini review, BioSystems 99 (2010) 109-125. [15] C. Bardos, F. Golse, C.D. Levermore, Sur les limites asymptotiques de la th´eorie cin´etique conduisant ` a la dynamique des fluides incompressibles, C.R. Acad. Sci. Paris S´er. I 309 (1989) 727-732. [16] C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations I, Formal derivations, J. Statist. Phys. 21 (1992) 531-555. [17] P. L. Lions and N. Masmoudi, From Boltzmann equation to the equation of incompressible fluid mechanics - I, Arch. Rational. Mech. Anal. 158 (2001) 173193. [18] C. Dogbe, Fluid dynamic limits for gas mixture. I. Formal derivations, Math. Models Methods Appl. Sci., 18 (2008) 1633-1672. [19] N. Bellomo, A. Bellouquid, J. Nieto, J. Soler, On the asymptotic theory from microscopic to macroscopic growing tissue models: An overview with perspectives, Math. Models Methods Appl. Sci. 21 (2012) 1-37. [20] P. Degond and B. Wennberg, Mass and energy balance laws derived from highfield limits of thermostatted Boltzmann equations, Commun Math Sci. 5 (2007) 355-382. [21] G. P. Morris, C.P. Dettmann, Thermostats: analysis and application, Chaos 8 (1998) 321-336. [22] H. G Othmer, T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations, SIAM J Appl Math 62 (2002) 1222-1250. [23] C. Bianca, C. Dogbe, Kinetic models coupled with Gaussian thermostats: Macroscopic frameworks, Nonlinearity, accepted (2014). [24] C. Bianca, Thermostatted kinetic equations as models for complex systems in physics and life sciences, Physics of Life Reviews 9 (2012) 359-399. [25] C. Bianca, Modeling complex systems with particles refuge by thermostatted kinetic theory methods, Abstract and Applied Analysis 2014 (2014) 152174. [26] C. Bianca, C. Dogbe, L. Guerrini, A thermostatted kinetic framework with particle refuge for the modeling of tumors hiding, Applied Mathematics and Information Sciences 8 (2014) 469-473.
16
Highlights: A system of integro-differential equations with quadratic nonlinearity; Modeling of nonequilibrium complex systems with mutations; Complex system decomposed into a finite number of functional subsystems; Low-field scaling and asymptotic limit; Macroscopic equations of local and global densities.
S1007-5704(14)00215-9 http://dx.doi.org/10.1016/j.cnsns.2014.05.009 CNSNS 3192
To appear in:
Communications in Nonlinear Science and Numerical Simulation
Please cite this article as: Bianca, C., Lemarchand, A., Density evolution by the low-field limit of kinetic frameworks with thermostat and mutations, Communications in Nonlinear Science and Numerical Simulation (2014), doi: http:// dx.doi.org/10.1016/j.cnsns.2014.05.009
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Density evolution by the low-field limit of kinetic frameworks with thermostat and mutations Carlo Bianca1,2 , Annie Lemarchand1,2 1 Sorbonne Universit´es, UPMC Univ Paris 06, UMR 7600, Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee, 4, place Jussieu, case courrier 121, 75252 Paris cedex 05, France
2
CNRS, UMR 7600 LPTMC, France
Abstract This paper is concerned with an asymptotic limit of a thermostatted kinetic framework which can be proposed for the modeling of complex biological and chemical systems where proliferative/destructive/mutative interactions and random velocity-jump processes occur. Specifically the macroscopic equations fulfilled by the local and global densities of the system are obtained by performing an asymptotic limit of a low-field rescaling of a kinetic framework with a thermostat and mutative interactions. Keywords: Kinetic theory, low-field scaling, integro-differential equation, nonlinearity, diffusion
1
Introduction
Recently in [1] a new thermostatted kinetic approach has been proposed for the modeling of complex biological and chemical systems in which mutations occur. According to this approach, the whole system is composed of active particles that express a specific strategy (activity) grouped, depending on their function, in subsystems called functional subsystems. The system is then statistically described by employing distribution functions and stochastic interactions. In order to take into account further phenomena in biology, this paper aims at developing a complete framework that takes into account also space and velocity phenomena occuring in biological and chemical systems. In particular in this paper a stochastic velocity perturbation of the spatially homogeneous dynamics is proposed: the velocity-jump process, where the direction of particle velocity fluctuates continuously. In this random process the turning event is usually governed by a Poisson process, see among others, papers [2, 3, 4]. Similar random processes of the velocity of the particles have been also considered in [5] to simulate multiple collision dynamics. Finally in order to have a macroscopic description of the system, a low-field scaling is introduced. The asymptotic limit of the rescaled thermostatted kinetic framework is performed under technical, but general, assumptions on the operators that describe the different types of interactions that change the microscopic state of particles, 1
proliferation, inhibition, mutation. The derived macroscopic equation, which is of diffusion type, is of interest for example in cancer modeling, for styding the diffusion and growth processes of malignant cells at the tissue scale [6, 7, 8, 9]. The method developed in this paper is thus of interest in the multiscale analysis of tumor formation that aims at obtaining a complete description of the tumor growth at tissue scale by linking the information that are achieved at a lower scale, i.e. cellular, genetic, molecular, see [10, 11] and therein references. Moreover the asymptotic method proposed in the present paper can be used to study the physics of social systems, see the review paper [12], as well as to understand the statistical mechanics of evolutionary and coevolutionary games [13, 14]. As already mentioned, the present paper is devoted to the low-field limit analysis only. However the interested reader in a deeper understanding of the asymptotic analysis by the low-field and high-field limits is referred, among others, to papers [15, 16, 17, 18, 20, 19] and the references cited therein. The contents of the present paper is outlined as follows: After this introduction, Section 2 deals with a technical generalization of the thermostatted mathematical framework developed in [1] by introducing the space structure and the velocity-jump process. Section 3 is concerned with the low-field limit of the generalized thermostatted kinetic framework introduced in Section 2. Specifically Section 3 is presented through three sequential subsections. In detail: Section 3.1 is concerned with the rescaled thermostatted framework by the low-field scaling and preliminary results; the main result of the present paper regarding the derivation of the macroscopic equation fulfilled by the local and global densities of the system is outlined in Section 3.2; Section 3.3 is devoted to the derivation of a specific model that allows to illustrate the method step by step. Finally a critical analysis and future research directions are highlighted in Section 4.
2
The thermostatted kinetic framework
This section aims at introducing the thermostatted kinetic model that will be the underlying framework for the analysis developed in the present paper. Specifically we generalize the mathematical framework recently proposed in [1] by introducing a space-velocity structure and a velocity-jump process and we perform an asymptotic analysis by the low-field limit. Let x ∈ Dx ⊂ R3 be the position, v ∈ Dv ⊂ R3 the velocity and u ∈ Du ⊂ R the activity of a particle of the system under consideration assumed to be composed of n ∈ N different subsystems (functional subsystems) of active particles, see [1]. The time evolution of the ith functional subsystem is described by the distribution function fi = fi (t, x, v, u) and the time evolution of the whole system is depicted by the vector f = (f1 , f2 , . . . , fn ). The system evolves because of the following interactions among the particles of the functional subsystem. Conservative interactions. Particles interact with each other and the results of these kinds of interactions is the modification of the magnitude of the activity
2
variable according to the following operator Ji [f ] = Ji [f ](t, x, v, u): Z n X Ji [f ] = ηij Aij (u∗ , u∗ , u)fi (t, x, v, u∗ ) fj (t, x, v, u∗ ) du∗ du∗ j=1D ×D u u
−fi (t, x, v, u)
n Z X j=1
ηij fj (t, x, v, u∗ ) du∗ ,
(2.1)
Du
where ηij models the probability that a particle of the ith functional subsystem with state (x, v, u∗ ) interacts with a particle of the jth functional subsystem with state (x, v, u∗ ); Aij = A(u∗ , u∗ , u) is the probability density that the particles of the ith functional subsystem with state (x, v, u∗ ) interacting with the particles of the jth functional subsystem with state (x, v, u∗ ) reach the activity (x, v, u). The function Aij is such that kAij kL1 (Du ,dv) = 1, ∀ u∗ , u∗ ∈ Du . Proliferative/destructive interactions. Interactions among the particles result in a change in the number of particles (birth-death process) and are modeled by the following operator N [f ] = N [f ](t, x, v, u): n Z X Ni [f ] = fi (t, x, v, u) ηij (u∗ , u∗ ) µij (u∗ , u∗ ) fj (t, x, v, u∗ ) du∗ . (2.2) j=1
Du
where µij denotes the net proliferation/destruction rate. Mutative interactions. Particles can modify their shape and type of activity becoming a particle of another functional subsystem (mutation). These kinds of interactions are modeled by the following operator Mi [f ] = Mi [f ](t, x, v, u): Z n X n X Mi [f ] = ηhk (u∗ , u∗ ) ϕihk (u∗ , u∗ , u) fh (t, x, v, u∗ ) (2.3) ×
h=1 k=1D ×D u u fk (t, x, v, u∗ ) du∗
du∗ .
(2.4)
ϕihk
denotes the net mutation rate into the ith functional subsystem, due where to interactions that occur with rate ηhk between the particles of the hth functional subsystem, with state (x, v, u∗ ), and the particles of the kth functional subsystem, with state (x, v, u∗ ). Velocity-jump process. This process, which models the particle that moves with constant speed describing a straight line whose direction changes continuously, is modeled by the following operator Vi [fi ] ≡ Vi [fi ](t, x, v, u): Z Vi [fi ] = Ti (v∗ , v)fi (t, x, v∗ , u) − Ti (v, v∗ )fi (t, x, v, u) dv∗ , (2.5) Dv
where Ti (v , v) is the turning kernel which gives the probability that the velocity v∗ ∈ Dv jumps into the velocity v ∈ Dv (if a jump occurs). The set v ∈ Dv is assumed to be bounded and spherically symmetric with respect to (0, 0, 0). ∗
Considering the balance of the inlet and outlet flows into the elementary volume of the space of the microscopic states, the underlying thermostatted kinetic framework for the distribution function fi thus reads: (∂t + v · ∇x ) fi + TFi [f ] = Ji [f ] + Ni [f ] + Mi [f ] + ν Vi [fi ], 3
(2.6)
where • v · ∇x fi is the transport operator; • ν is the turning rate or turning frequency of the velocity jump, hence 1/ν is the mean run time; • TFi [f ] is the thermostat term that has the role of controlling the second moment of local activity: Z u2 f˜(t, x, v, u) du. (2.7) E2 [f ](t, x, v) = Du
This term, according to [21], reads: TFi [f ] := ∂u
Fi (u) 1 − u
Z
!
u f˜(t, x, v, u) dx dv du fi (t, x, v, u) ,
Ω
(2.8)
where Fi (u) : Du → R is a known external force field that moves the system far from equilibrium, Ω = Dx × Dv × Du and f˜(t, x, v, u) =
n X
fj (t, x, v, u).
j=1
Remark 2.1 In the thermostatted kinetic framework (2.6) the space and velocity variables are not modified by the interactions. In this context, the local density %[fi ](t, x, u) of the ith functional subsystem defined at time t in the position x and activity u, reads: Z fi (t, x, v, u) dv, (2.9) %i := %[fi ](t, x, u) = Dv
and the global density %[f ](t, x, u) of the whole system reads: Z % := %[f ](t, x, u) = f˜(t, x, v, u) dv,
(2.10)
Dv
The main aim of this paper is to obtain the macroscopic equation of %[fi ] and %[f ].
3
The low-field limit
The derivation of the macroscopic equation for the global density %[f ] strongly depends on the properties of the turning operator Vi [fi ]. Indeed the asymptotic analysis will be performed by considering the following preliminary results. Lemma 3.1 ([22]) Let the following assumptions be satisfied: (A1 ) kVi [fi ]kL1 (Dv ,dv) = kv Vi [fi ]kL1 (Dv ,dv) = 0, ∀ x ∈ Dx , u ∈ Du .
4
(A2 ) If there exists a bounded equilibrium velocity distribution Gi (v) : Dv → R+ , independent of t and x, such that: Ti (v∗ , v) Gi (v) = Ti (v, v∗ )Gi (v∗ ).
(3.1)
and kv Gi (v)kL1 (Dv ,dv) = 0, kGi (v)kL1 (Dv ,dv) = 0, ∀ x ∈ Dx , u ∈ Du . (A3 ) The kernel Ti (v, v∗ ) is bounded, and there exists a constant σi > 0 such that Ti (v, v∗ ) ≥ σi Gi (v), ∀(v, v∗ ) ∈ Dv × Dv . (3.2) Then (i) For fi ∈ L2 (Dv , dv/Gi ), the integral equation Vi [gi ] = fi has a unique solution gi ∈ L2 Dv , dv/Gi such that kgi (v)kL1 (Dv ) = 0
if and only if
kfi (v)kL1 (Dv ) = 0.
(ii) The integral operator Vi defines a self-adjoint Fredholm operator on the space L2 Dv , dv/Gi .
(iii) The equation Vi [fi ] = v Gi (v) has a unique solution given by fi (v) = χi (v). (iv) The null-space of Vi is spanned by a unique normalized and nonnegative function G(v): Ker(Vi ) = Span{Gi }. The Hilbert space L2 (Dv , dv) endowed with the usual scalar product Z f (v) g(v) dv, f, g ∈ L2 (Dv , dv), hf, gi = Dv
will be used in the sequel and the average of the function ϕ with respect to variable v will be denoted by Z hϕi := hϕ, 1i = ϕ(v) dv. Dv
Moreover the following Kronecker delta will be used: 1 if i = j δij = 0 if i 6= j 3.1
The rescaled thermostatted kinetic framework
This subsection deals with the rescaled thermostatted kinetic framework whose solutions are assumed to be bounded in a space of functions where all needed convergence results will be true. The following low-field scaling is considered: t ` (3.3) , x, v, u, Fi , ` ≥ 1, (t, x, v, u, Fi ) → 5
and considering the meaning and the relationship that exists among the rates, the following rates are chosen: η = r ,
µ = q ,
ϕ = m ,
ν=
1 , s
(3.4)
where m, q, r, s ≥ 1. Therefore the rescaled distribution function reads: t fi (t, x, v, u) = fi , x, v, u . Let f = (f1 , f2 , . . . , fn ) be the vector of the rescaled distribution functions, then the rescaled thermostatted framework (2.6) is as follows: ˜ i [f ]+ ∂t fi +v·∇x fi +` TFi [f ] = r J˜i [f ]+r+q N˜i [f ]+r+m M
1 ˜ Vi [fi ], (3.5) s
˜ i [f ], V˜i [f ], and TF [f ] are the rescaled conservative, where J˜i [f ], N˜i [f ], M i i nonconservative, mutating, turning and thermostat operators, respectively, that read: n Z X ˜ Ji [f ] = Aij (u∗ , u∗ , u)fi (t, x, v, u∗ ) fj (t, x, v, u∗ ) du∗ du∗ j=1
−
Du ×Du
n X
fi (t, x, v, u)
= fi (t, x, v, u)
n Z X j=1
˜ i [f ] M
=
n X n X
fj (t, x, v, u∗ ) du∗ ,
(3.6)
Du
j=1
˜i [f ] N
Z
Z
fj (t, x, v, u∗ ) du∗ ,
(3.7)
Du
fh (t, x, v, u∗ ) fk (t, x, v, u∗ ) du∗ du∗ ,
(3.8)
h=1 k=1D ×D u u
V˜i [fi ] TFi [f ]
Z
Ti (v∗ , v)fi (t, x, v∗ , u) − Ti (v, v∗ )fi (t, x, v, u) dv∗ , (3.9) Dv ! Z ˜ = ∂u Fi (u) 1 − u u f dx dv du fi (t, x, v, u) (3.10) =
Ω
The following lemmas hold true. Lemma 3.2 ([23]) Let fi (t, x, v, u) be a sequence of solutions of the rescaled thermostatted kinetic equation (3.5). Assume that the turning operator Vi satisfies the assumptions (A1 -A2 -A3 ) and, when → 0, the following statements hold true: fi −→ fi a.e. in [0, ∞) × Dx × Dv × Du , V˜i [fi ] −→ V˜i [fi ]
(3.11) (3.12)
Then the asymptotic limit fi of the sequence fi (modulo the extraction of a subsequence) admits the following factorization: fi (t, x, v, u) = %i (t, x, u) Gi (v),
(3.13)
where %i is the local macroscopic density (2.9) of the ith functional subsystem. 6
In order to calculate the asymptotic limit of the rescaled thermostatted framework (3.5), we need to perform the limit, when goes to zero, of the term 1 hv · ∇x f i.
(3.14)
Lemma 3.3 ([23]) Let fi (t, x, v, u) be a sequence of solutions of the scaled thermostatted kinetic equation (3.5). Assume that the turning operator Vi satisfies the assumptions (A1 -A2 -A3 ) and, when → 0, the following statements hold true: fi −→ V˜i [fi ] −→
fi a.e. in [0, ∞) × Dx × Dv × Du , V˜i [fi ],
(3.15) (3.16)
then
1
v · ∇x fi −−−→ δs,1 divx (χi (v) ⊗ v) ∇x %i . →0 where χi (v) is the only solution of the equation V˜i [fi ] = v Gi (v). 3.2
(3.17)
The macroscopic equation
This subsection is devoted to the derivation of the macroscopic equation which describes the evolution of the global macroscopic density %. The analysis is based on the following assumptions. Let fi (t, x, v, u) be a sequence of solutions of the rescaled thermostatted kinetic equation (3.5). (A4 ) The following quantities hfi i, hvfi i, hv ⊗ vfi i, converge, in the sense of distributions on R∗+ × Dx × Du , to the corresponding quantities hfi i, hvfi i, hv ⊗ vfi i, (A5 ) The following quantities D E D E D E
˜i [f ] , M ˜ i [f ] , TF [f ] , J˜i [f ] , N i
converge, in the sense of distributions on R∗+ × Dx × Du , to the corresponding quantities E
E D E D D ˜ i [f ] , TF [f ] , ˜i [f ] , M J˜i [f ] , N i
(A6 ) The following quantities D E D E D E
˜i [f ] , vM ˜ i [f ] , vTF [f ] vJ˜i [f ] , vN i
converge, in the sense of distributions on R∗+ × Dx × Du , to the corresponding quantities D E D E D E
˜i [f ] , vM ˜ i [f ] , vTF [f ] vJ˜i [f ] , vN i 7
The main result of this paper follows. Theorem 3.1 Let fi (t, x, v, u) be a sequence of solutions of the rescaled thermostatted kinetic equation (3.5). Assume that the turning operator Vi satisfies the assumptions (A1 -A2 -A3 ) and, when → 0, the following statements hold true: fi −→ V˜i [fi ] −→
fi a.e. in [0, ∞) × Dx × Dv × Du , V˜i [fi ]
(3.18) (3.19)
If the assumptions (A4 -A5 -A6 ) hold true and every formally small term in vanishes, then the local macroscopic density %i of the system is the weak solution of the following equation ∂t %i + δ`,1 ∂u Fi (u)(1 − uA[%](t))%i = δs,1 divx (D%i · ∇x %i ) + δr,1 Hi [%] + δr+q,1 Si [%] + δr+m,1 L[%] (3.20) where % = (%1 , %2 , . . . , %n ) and • A[%](t) is the following operator: A[%](t) =
n Z X j=1
u %j (t, x, u) dx du =
Dx ×Du
Z
u %(t, x, u) dx du,
Dx ×Du
(3.21)
• D%i is the following tensor: D%i = −
Z
v ⊗ χi (v) dv,
(3.22)
Dv
• Hi [%](t) is the following operator: Hi [%]
=
n X
hGi (v), Gj (v)i ×
j=1
Z
Aij (u∗ , u∗ , u) %i (t, x, u∗ ) %j (t, x, u∗ ) du∗ du∗
Du ×Du
− %i (t, x, u)
n X
hGi (v), Gj (v)i
Z
%j (t, x, u∗ ) du∗ .
(3.23)
Du
j=1
• Si [%](t) is the following operator: Si [%] = %i (t, x, u)
n X
hGi (v), Gj (v)i
Z
%j (t, x, u∗ ) du∗ .
(3.24)
Du
j=1
• L[%](t) is the following operator: L[%] =
n X n X
h=1 k=1
hGh (v), Gk (v)i
Z
Du
8
%h (t, x, u∗ ) du∗
!
Z
∗
%k (t, x, u ) du
Du
(3.25)
∗
!
.
Proof. By taking the average of the equation (3.5) with respect to v, using the assumption A1 and dividing by , one obtains:
where
1
˜ ]. ∂t fi + v · ∇x fi + `−1 TFi [f ] = Z[f
(3.26)
D D D E E E ˜ ] = r−1 J[f ˜ ] + r+q−1 N ˜ [f ] + r+m−1 M ˜ [f ] Z[f
(3.27)
Assume that ` > 1, r > 1, r + q > 1, and r + m > 1. When → 0, the term ˜ ] goes to zero, and by using Lemma 3.3 we have the limit (3.17). Z[f By using Lemma 3.2, the asymptotic limit of the thermostat term reads: * !+ Z
`−1 `−1 u f˜ dx dv du fi TFi [f ] = ∂u Fi (u) 1 − u Ω
−−−→ δl,1 →0
=
*
∂u Fi (u) 1 − u
n Z X j=1
Ω
u %j Gj dx dv du %i Gi
δl,1 ∂u Fi (u) 1 − u A[%](t) %i .
+
(3.28)
Then, it is an easy task to show that r−1 hJ˜i [f ]i
−−−→
δr,1 Hi [%]
(3.29)
˜i [f ]i r+q−1 hN
−−−→
δr+q,1 Si [%]
(3.30)
˜ i [f ]i r+m−1 hM
−−−→
δr+m,1 L[%]
(3.31)
→0
→0
→0
where Hi [%], Si [%] and L[%] are the operators defined in Eqs. (3.23), (3.24), and (3.25) respectively. Therefore the proof is concluded. Bearing the above theorem in mind we have the following result. Theorem 3.2 Let fi (t, x, v, u) be a sequence of solutions of the rescaled thermostatted kinetic equation (3.5). Assume that the turning operator Vi satisfies the assumptions (A1 -A2 -A3 ) and, when → 0, the following statements hold true: fi −→ V˜i [fi ] −→
fi a.e. in [0, ∞) × Dx × Dv × Du , ˜ Vi [fi ]
(3.32) (3.33)
If the assumptions (A4 -A5 -A6 ) hold true and every formally small term in vanishes, then the global macroscopic density % of the system is the weak solution of the following equation ∂t % + δ`,1 ∂u (1 − uA[%](t)) αF [%] = δs,1 divx (D% · ∇x %) + δr,1 H[%] ˜ + δr+q,1 S[%] + δr+m,1 L[%] (3.34) where A[%](t) is given by (3.21), 9
• αF [%] = αF [%](t, x, u) is the following operator: αF [%](t) =
n X
Fj (u)%j (t, x, u),
(3.35)
j=1
• D% · ∇ x % =
n X
D%j · ∇x %j , where D%j is given by (3.22);
j=1
• H[%](t) =
n X
Hj [%](t), where Hi [%](t) is given by (3.23);
j=1
• S[%](t) =
n X
Sj [%](t), where Hi [%](t) is given by (3.24);
j=1
˜ • L[%](t) = nL[%](t), where L[%](t) is given by (3.25). 3.3
A specific model
This section aims at showing the asymptotic method by means of an example in biology and specifically a model for the study of cancer-immune system competition. Accordingly: the first step is the definition of the functional subsystems that are the main actors involved in the evolution of the system under modeling; the second step is the definition of the microscopic interactions; the third and final step is the derivation of the macroscopic equations that can be pursued following the statement of the theorems 3.1 and 3.2. 1. Functional subsystems and activity variable. It is assumed that the cancer-immune system competition involves the following three interacting functional subsystems and their mutual interactions may also generate mutations: - Normal cells Nc distributed according to f1 (t, x, v, u). The activity variable u represents the mutation ability. A normal cell may undergo a mutation as consequence of DNA corruptions and mutate in cancer cells. - Cancer cells Cc distributed according to f2 (t, x, v, u). The activity variable u represents the progression ability. - Immune system cells ISc distributed according to f3 (t, x, v, u). The activity variable u is a magnitude of the activation. 2. Microscopic interactions. The encounter rate is assumed to be constant for all interacting pairs: ηij = η for all i, j ∈ {1, 2, 3}. The probability density Aij is assumed to be defined by a delta Dirac function (deterministic output mij (u∗ , u∗ ) of a pair interaction) depending on the microscopic state of the interacting pairs: Aij (u∗ , u∗ , u) = δ(u − mij (u∗ , u∗ )). (3.36) Finally we assume that Du = [0, +∞), Dv is the 3-sphere of radius R > 0 and we let Dx and Fi (u) arbitrary.
10
• Conservative Interactions. It is assumed that Cc is the only subsystem subject to conservative interactions, therefore J1 [f ](t, u) = J3 [f ](t, u) = 0. The conservative term J2 [f ](t, u) is derived under the assumption that the Cc have a tendency to increase their microscopic state with a certain rate, regulated by Nc. Accordingly we define ( u∗ + ψ if j = 1 and i = 2, ∗ (3.37) mij (u∗ , u ) = u∗ otherwise. where α is a positive parameter related to the ability of Cc to reach high states of activity and 0 < ψ < 1 a scale factor. Therefore the conservative term reads:
Ji [f ](t, x, v, u) =
(
η ρ[f1 ](t, x, u) [f2 (t, x, v, u − ψ α) − f2 (t, x, v, u)] 0
if i = 2, otherwise.
• Proliferative/Destructive Interactions. Each subsystem may proliferate (without mutations) when it encounters another subsystem. Specifically: P.1 Nc proliferate when encounter each other; P.2 Cc proliferate when encounter Nc and ISc; P.3 ISc proliferate when encounter Cc. Accordingly, the proliferation rate reads: β if j = 1 and i = 1, ψβ if j ∈ {1, 3} and i = 2, p ∗ µij (u, u ) = β if j = 2 and i = 3, I 0 otherwise,
(3.38)
where β > 0 is the proliferation rate of Nc and βI is the proliferation rate of the ISc. The proliferative term thus reads: η βf1 (t, x, v, u) ρ[f1 ](t, x, u) η ψβf2 (t, x, v, u) [ρ[f1 ](t, x, u) + ρ[f3 ](t, x, u)] Pi [f ](t, x, v, u) = η βI f3 (t, x, v, u) ρ[f2 ](t, x, u)
if i = 1, if i = 2, if i = 3.
The cells of each subsystem may be depleted when they encounter cells of another subsystem. Specifically: D.1 Nc are depleted by Cc; D.2 Cc are depleted by ISc; D.3 ISc are depleted by Cc.
11
Accordingly, we have: ψδ δ d ∗ µij (u, u ) = δI 0
if j = 2 and i = 1, if j = 3 and i = 2, if j = 2 and i = 3, otherwise,
(3.39)
where δ is the destruction rate of Cc by ISc and δI the destruction rate of the ISc by Mc. Thus the destructive term reads: η ψδf1 (t, x, v, u) ρ[f2 ](t, x, u) η δ f2 (t, x, v, u) ρ[f5 ](t, x, u) Di [f ](t, x, v, u) = η δI f5 (t, x, v, u) ρ[f2 ](t, x, u)
if i = 1, if i = 2, if i = 3.
According to Section 2, the net proliferation/destruction rate reads: µij (u∗ , u∗ ) = µpij (u∗ , u∗ ) − µdij (u∗ , u∗ ), and then Ni [f ](t, x, v, u) = Pi [f ](t, x, v, u) − Di [f ](t, x, v, u).
• Mutative Interactions. It is assumed that may occur genetic mutations. Specifically: T.1 Nc may mutate in Cc when interact with Nc and Cc. In particular, we assume that the microscopic state of the entities does not change during the mutation. Accordingly, we have: if h = 1, k = 1, and i = 2, ψγ δ(u − u∗ ) ∗ i γ δ(u − u∗ ) if h = 1, k = 2, and i = 2, ϕhk (u∗ , u , u) = (3.40) 0 otherwise,
where γ is the mutation rate of Nc when they encounter Cc. Thus the mutative term reads:
Ti [f ](t, x, v, u) =
η γ f1 (t, x, v, u) [ψ ρ[f1 ](t, x, u) + ρ[f2 ](t, x, u)] 0
if i = 2, otherwise.
Velocity-jump process. Since the jump process is described by the kernel Ti , we can assume without restrictions that Ti (v∗ , v) = γi Gi (v) =
12
γi , |Dv |
0 < γi < σi ,
where we have assumed Gi (v) = 1/|Dv |. Therefore the turning operator reads: Z Vi [fi ](t, x, v, u) = Ti (v∗ , v)fi (t, x, v∗ , u) − Ti (v, v∗ )fi (t, x, v, u) dv∗ , Dv = γi ρi (t, x, u)Gi (v) − fi (t, x, v, u) . Bearing all above in mind, the thermostatted kinetic model thus reads: ! Z Z f2 du f1 f1 du − ψδ (∂t + v · ∇x ) f1 + TF1 [f ] = η β Du Du Z (∂t + v · ∇x ) f2 + TF2 [f ] = η [f2 (t, x, v, u − ψ α) − f2 (t, x, v, u)] f1 du+ Du ! Z Z ψβ f5 du f2 η (f + f ) du − δ 1 3 Du Du ! Z Z f2 du − δI f2 du(t, x, u) f5 (∂t + v · ∇x ) f3 + TF3 [f ] = η βI Du
Du
3. The macroscopic equation. According to Lemma 3.1, the function χi (v) reads: γi ρi − v Gi (v)(γi ρi − v) = , χi (v) = γi γi |Dv | and according to Theorem 3.1, the diffusion tensor Dρi defined in (3.22) reads: Z Z R2 1 1 v ⊗ v dv = [v ⊗ vGi (v)] dv = I, (3.41) Dρi = γi Dv γi |Dv | Dv 3γi where I is the identity matrix and, as it is known, the following formula holds Z R2 δij . vi vj dv = |Dv | 3 Dv By straightforward computations we have: divx (Dρi · ∇x ρi ) =
R2 ∆x ρi , 3γi
and hGi (v), Gi (v)i =
1 . |Dv |
Let Ξ[ρi , Fi , γi ](t, x, u) be the following operator: R2 Ξl,s [ρi , Fi , γi ](t, x, u) = ∂t ρi + δl,1 ∂u (Fi (u) 1 − uA[%](t))ρi − δs,1 ∆x ρi , 3γi
and Γr,m [%](t, x, u) the following one:
! ! Z Z 3 3 δr+m,1 X X ρk (t, x, u) du . ρh (t, x, u) du Γr,m [%](t, x, u) = |Dv | Du Du h=1 k=1
13
Then the macroscopic model (3.20) related to the evolution of the local densities of the system thus reads: ρ1 (t, x, u) Ξl,s [ρ1 , F1 , γ1 ] = (δr+q,1 − δr,1 ) ρ˜(t, x, u) + Γr,m [%](t, x, u) |Dv | Z ρ1 (t, x, u) du+ Ξl,s [ρ2 , F2 , γ2 ] = δr,1 (ρ2 (t, x, u − ψα) − ρ2 (t, x, u)) Du
ρ2 (t, x, u) ρ˜(t, x, u) + Γr,m [%](t, x, u) (δr+q,1 − δr,1 ) |Dv | ρ3 (t, x, u) ρ˜(t, x, u) + Γr,m [%](t, x, u) Ξl,s [ρ3 , F3 , γ3 ] = (δr+q,1 − δr,1 ) |Dv |
where
ρ˜(t, x, u) =
3 Z X j=1
ρj (t, x, u) du.
Du
The same arguments hold true for the macroscopic model given by (3.34) related to the evolution of the global density of the system.
4
Critical Analysis and Perspective
Thermostatted kinetic frameworks have been recently proposed in [24] for the modeling of complex systems in physics and life sciences. As shown in [24], these models are able to model complex systems at the mesoscopic scale that in the case of biological systems corresponds to the cellular scale. However a complete multiscale approach that allows to obtain a full description of the complex dynamics occurring in many systems of the real-world is missing. Multiscale approach means to obtain a link among mathematical/computational models developed at different scales, e.g. atomic-molecular-mesoscopic-macroscopic. The pertinent literature in this field needs the development of a formal method, possibly a mathematical theory. This is a very hard problem and a new challenge, of the last and new century, for the researchers in applied mathematicians and, in general, in the applied sciences. The goal of the present paper is the development of a mathematical method that allows to link the dynamics at the mesoscopic (cellular) scale with the dynamics at the macroscopic (tissue) scale. Specifically the paper has been concerned with the generalization of the asymptotic method proposed in [23] to a new thermostatted kinetic framework, which includes mutative events that are typical of biological and chemical systems and the velocity-jump process. Under suitable assumptions, the macroscopic equations for the local and global densities have been derived and the main steps of the method have been summarized by considering a specific model for cancer-immune system competition. It is worth stressing that the method proposed in the present paper has been developed by considering a general complex system modeled according to a thermostatted kinetic framework that can acts as a general paradigm for the derivation of specific models in biology and chemistry. We are aware that much research activity is needed in order to obtain a complete multiscale approach. Nevertheless the method proposed in this paper can be considered as an important contribution in the pertinent literature especially because, to the best of our knowledgement, this is the first time that the 14
asymptotic method is developed for the thermostatted kinetic framework with mutations. It is worth precising that the asymptotic limit developed in the present paper refers to a low-field scaling of the thermostatted kinetic framework. Further research directions include the possibility to consider the derivation of macroscopic equations by high-field scalings or mixed scalings. Finally the asymptotic method can be further generalized to thermostatted kinetic frameworks which include particles refuge, see [25, 26]. This is a work in progress and the results will be presented in due course.
Acknowledgement The authors were partially supported by l’Agence Nationale de la Recherche (ANR T-KiNeT Project).
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Highlights: A system of integro-differential equations with quadratic nonlinearity; Modeling of nonequilibrium complex systems with mutations; Complex system decomposed into a finite number of functional subsystems; Low-field scaling and asymptotic limit; Macroscopic equations of local and global densities.