Separable Transition Density in the Hybrid Model for Tumor-Immune System Competition

Abstract

A hybrid model, on the competition tumor cells immune system, is studied under suitable hypotheses. The explicit form for the equations is obtained in the case where the density function of transition is expressed as the product of separable functions. A concrete application is given starting from a modified Lotka-Volterra system of equations.

1. Introduction

The competition between tumor cells and the immune system is mainly due to a significant presence of the proliferation and/or destructive events. In particular, cancer cells have the ability of expressing their biological activity to escape from the immune system which, in principle, have to challenge the progressing cells. The biological activity is not generally the same for all cells since it is statically distributed.

Several authors [1–7] have applied the methods of the classical mathematical kinetic theory of gases to study the immune competition with special attention to cancer phenomena. In this approach, one has to take account of statistical averages and stochastic parameters, typical of macromodels.

Other authors [8–15] have proposed mathematical models based on nonlinear differential equations, which generalize the classical Lotka-Volterra equations. These equations, as known, follow from a deterministic approach on a microscale.

In some recent papers [16–20], a hybrid model was proposed which can be considered as an alternative method between the above two approaches, aiming to mix the two scales into a unique set of equations, the hybrid model. In this model, a system of nonlinear ordinary differential equations are coupled with a stochastic parameter generated by the (kinetic) interaction between the tumor cells and the immune system.

This time-depending stochastic parameter was linked [17] to the hiding-learning information process which underlies the cells competition. In particular [17], the hiding-learning dynamics appears between two populations (tumor cells-immune system) in which the first one has an uncontrolled proliferating and hiding ability and the second one has higher destructive ability and the need of learning about the presence of the first population.

In this paper, we study the above hybrid model by assuming a particular form of the stochastic coefficient. There follow interesting results on the model and, moreover, the classical model of Lotka-Volterra modified by the hiding-learning process can be derived as a special case.

2. Modelling the Immune Competition of Complex Systems

Let us consider a system of two interacting and competing populations. Each population is constituted by a large number of individuals called active particles; their microscopic state is called (biological) activity. This activity enables the particle to organize a suitable response with respect to any information process. In absence of prior information, the activity reduces either to a minimal loss of energy or to a random process.

In active particle competitions, the simplest model of binary interaction is based on proliferation-destructive competition. So that, when the first population get aware of the existence of the other challenging population, it starts to proliferate and destroy the competing cells. However, in this process the most important step is the ability of cells to hide themselves and to learn about the activity of the competing population.

In details consider a physical system of two interacting populations each one constituted by a large number of active particles with sizes: 𝑛_𝑖=𝑛_𝑖(𝑡),𝑛_𝑖(𝑡)∶[0,𝑇]⟶ℝ₊;𝑖=1,2. (1)

Particles are homogeneously distributed in space, while each population is characterized by a microscopic state, called activity, denoted by the variable 𝑢. The physical meaning of the microscopic state may differ for each population. We assume that the competition model depends on the activity by a function of the overall distribution: 𝜇=𝜇𝑓_𝑖(𝑡,𝑢),𝜇𝑓_𝑖(𝑡,𝑢)∶ℝ₊⟶ℝ₊. (2)

The description of the overall distribution over the microscopic state within each populations is given by the probability density function:𝑓_𝑖=𝑓_𝑖(),𝑓𝑡,𝑢_𝑖(𝑡,𝑢)∶[0,𝑇]×𝐷_𝑢⟶ℝ₊,𝐷_𝑢⊆ℝ;𝑖=1,2 (3) such that 𝑓_𝑖(𝑡,𝑢)𝑑𝑢 is the probability that the activity 𝑢 of particles of the 𝑖th population, at the time 𝑡, ranges in the interval [𝑢,𝑢+𝑑𝑢].

Moreover, it is∀𝑖,∀𝑡≥0∶0≤𝑓_𝑖(𝑡,𝑢)≤1,_𝐷𝑢𝑓_𝑖(𝑡,𝑢)𝑑𝑢=1. (4)

We will see in Sections 3 and 4 how the microscopic structure influences the macroscopic system.

3. Hybrid Model

We consider, in this section, the competition between two cell populations: the first one with uncontrolled proliferating ability and with hiding ability; the second one with higher destructive ability, but with the need of learning about the presence of the first population. The analysis developed in what follows is referring to a specific case where the second population attempts to learn about the first population which, instead, escapes by modifying its appearance. Specifically, the hybrid evolution equations can be formally written as follows:𝑑𝑛_𝑖𝑑𝑡=𝐺_𝑖𝑛₁,𝑛₂;𝜇𝑓,𝜕𝑓_𝑖𝜕𝑡=𝒜_𝑖𝑓, (5) where (1)𝐺_𝑖, for 𝑖=1,2, is a function of 𝑛={𝑛₁,𝑛₂}, (2)𝜇, acts over 𝑓={𝑓₁,𝑓₂},(3)𝒜_𝑖, for 𝑖=1,2, is a nonlinear operator acting on 𝑓, (4)𝜇[𝑓] is a functional (0≤𝜇≤1) which describes the ability of the second population to identify the first one.

As a consequence, (5) denotes a hybrid system of a deterministic system coupled with a microscopic system statistically described by a kinetic theory approach. In the following, the evolution of density distribution will be taken within the kinetic theory.

The derivation of (5)₂ can be obtained starting from a detailed analysis of microscopic interactions. Specifically, consider binary interactions between a test, or candidate, particle with state 𝑢_∗ belonging to the 𝑖th population, and field particle with state 𝑢^∗ belonging to the 𝑗th population. We assume that microscopic interactions are characterized by the following quantities.(i)The encounter rate, which depends, for each pair of interacting populations on a suitable average of the relative velocity 𝜂_𝑖𝑗, with 𝑖,𝑗=1,2.(ii)The transition density function 𝜑_𝑖𝑗(𝑢_∗,𝑢^∗,𝑢), denotes the probability density that a candidate particle with activity 𝑢_∗ belonging to the 𝑖th population, falls into the state 𝑢∈𝐷_𝑢, of the test particle, after an interaction with a field entity, belonging to the 𝑗th population, with state 𝑢^∗. The probability density 𝜑_𝑖𝑗(𝑢_∗,𝑢^∗,𝑢) fulfills the condition ∀𝑖,𝑗,∀𝑢_∗,𝑢^∗∶_𝐷𝑢𝜑_𝑖𝑗𝑢_∗,𝑢^∗𝜑,𝑢𝑑𝑢=1,_𝑖𝑗𝑢_∗,𝑢^∗,𝑢>0. (6) Then, by using the mathematical approach, developed in [17], it yields the following class of evolution equations: 𝜕𝑓_𝑖()=𝜕𝑡𝑡,𝑢²_{𝑗=1𝐷𝑢×𝐷𝑢}𝜂_𝑖𝑗𝜑_𝑖𝑗𝑢_∗,𝑢^∗,𝑢×𝑓_𝑖𝑡,𝑢_∗𝑓_𝑗𝑡,𝑢^∗𝑑𝑢_∗𝑑𝑢^∗−𝑓_𝑖(𝑡,𝑢)²_{𝑗=1𝐷𝑢}𝜂_𝑖𝑗𝑓_𝑗𝑡,𝑢^∗𝑑𝑢^∗, (7) which can be formally written as (5)₂.

Since our model is based on the hiding-learning dynamics, one has to introduce the functional which takes into account the “distance” between the two distribution so that 𝜇 in (5) is defined as𝜇𝑓_𝑖,𝑓_𝑗(𝑡)=𝜇𝑓_𝑖−𝑓_𝑗(𝑡) (8) with 0≤𝜇𝑓_𝑖,𝑓_𝑗(𝑡)≤1,∀𝑢∈𝐷_𝑢𝜇𝑓∧𝑡∈𝑇,_𝑖,𝑓_𝑗(𝑡)=1⟺𝑓_𝑖=𝑓_𝑗,𝜇𝑓_𝑖,𝑓_𝑗(𝑡)=0⟺𝑓_𝑖=0∨𝑓_𝑗=0, (9) where the maximum learning result is obtained when the second population is able to reproduce the distribution of the first one: 𝑓₁=𝑓₂, while the minimum learning is achieved when one distribution is vanishing.

In some recent papers [5–7, 17], it has been assumed that 𝜇𝑓_𝑖,𝑓_𝑗(𝑡)=𝜇𝑓_𝑖−𝑓_𝑗(𝑡)=1−_𝐷𝑢𝑓₁−𝑓₂²(𝑡,𝑢)𝑑𝑢. (10)

In this case, it is 𝜇=1, when 𝑓₁=𝑓₂, otherwise 𝜇≠1 with 𝜇↓0, depending on the time evolution of the distance between 𝑓₁ and 𝑓₂. There follows that this parameter could have an infinite value range.

Thus, we have 0≤𝜇𝑓(𝑡)≤1⟹0≤_𝐷𝑢𝑓₁−𝑓₂²()[].𝑡,𝑢𝑑𝑢≤1,∀𝑡∈0,𝑇 (11) Notice that 𝜇 is the coupling term which links the macroscopic model (5)₁ to the microscopic model (5)₂.

4. Transition Density Function Based on Separable Functions

In order to find some classes of solutions of (7), we assume that the transition density is the product of separable density functions as𝜑_𝑖𝑗𝑢_∗,𝑢^∗,𝑢=1−𝛿_𝑖𝑗𝜓_𝑖𝑢_∗,𝑢𝜉_𝑗𝑢^∗,𝑢, (12) that is,𝜑₁₁=𝜑₂₂𝜑=0,₁₂𝑢_∗,𝑢^∗,𝑢=𝜓₁𝑢_∗,𝑢𝜉₂𝑢^∗,𝜑,𝑢₂₁𝑢_∗,𝑢^∗,𝑢=𝜓₂𝑢_∗,𝑢𝜉₁𝑢^∗,𝑢, (13) and using (10) one has_𝐷𝑢𝜓_𝑖𝑢_∗,𝑢𝜉_𝑗𝑢^∗,𝜓,𝑢𝑑𝑢=1𝑖≠𝑗_𝑖𝑢_∗,𝑢>0,𝜉_𝑗𝑢^∗,𝑢>0𝑖,𝑗=1,2. (14) By a substitution of the above terms into (7) we get𝜕𝑓₁()=𝜕𝑡𝑡,𝑢_{2𝑗=1𝐷𝑢×𝐷𝑢}𝜂_1𝑗𝜑_1𝑗𝑢_∗,𝑢^∗,𝑢𝑓₁𝑡,𝑢_∗𝑓_𝑗𝑡,𝑢^∗𝑑𝑢_∗𝑑𝑢^∗−𝑓₁(𝑡,𝑢)²_{𝑗=1𝐷𝑢}𝜂_1𝑗𝑓_𝑗𝑡,𝑢^∗𝑑𝑢^∗,𝜕𝑓₂()=𝜕𝑡𝑡,𝑢_{2𝑗=1𝐷𝑢×𝐷𝑢}𝜂_2𝑗𝜑_2𝑗𝑢_∗,𝑢^∗,𝑢𝑓₂𝑡,𝑢_∗𝑓_𝑗𝑡,𝑢^∗𝑑𝑢_∗𝑑𝑢^∗−𝑓₂(𝑡,𝑢)²_{𝑗=1𝐷𝑢}𝜂_2𝑗𝑓_𝑗𝑡,𝑢^∗𝑑𝑢^∗, (15) from where, by taking into account (13), we obtain𝜕𝑓₁()𝜕𝑡𝑡,𝑢=𝜂_12𝐷𝑢𝜓₁𝑢_∗,𝑢𝑓₁𝑡,𝑢_∗𝑑𝑢_∗𝐷𝑢𝜉₂𝑢^∗,𝑢𝑓₂𝑡,𝑢^∗𝑑𝑢^∗−𝑓₁(𝑡,𝑢)𝜂_11𝐷𝑢𝑓₁𝑡,𝑢^∗𝑑𝑢^∗+𝜂_12𝐷𝑢𝑓₂𝑡,𝑢^∗𝑑𝑢^∗,𝜕𝑓₂()𝜕𝑡𝑡,𝑢=𝜂_21𝐷𝑢𝜓₂𝑢_∗,𝑢𝑓₂𝑡,𝑢_∗𝑑𝑢_∗𝐷𝑢𝜉₁𝑢^∗,𝑢𝑓₁𝑡,𝑢^∗𝑑𝑢^∗,−𝑓₂(𝑡,𝑢)𝜂_21𝐷𝑢𝑓₁𝑡,𝑢^∗𝑑𝑢^∗+𝜂_22𝐷𝑢𝑓₂𝑡,𝑢^∗𝑑𝑢^∗. (16) According to (4) and (13), we have the more general system for the transition density based on separable functions 𝜕𝑓₁()𝜕𝑡𝑡,𝑢=𝜂_12𝐷𝑢𝜓₁𝑢_∗,𝑢𝑓₁𝑡,𝑢_∗𝑑𝑢_∗×_𝐷𝑢𝜉₂𝑢^∗,𝑢𝑓₂𝑡,𝑢^∗𝑑𝑢^∗−𝜂₁₁+𝜂₁₂𝑓₁(𝑡,𝑢),𝜕𝑓₂()𝜕𝑡𝑡,𝑢=𝜂_21𝐷𝑢𝜓₂𝑢_∗,𝑢𝑓₂𝑡,𝑢_∗𝑑𝑢_∗×_𝐷𝑢𝜉₁𝑢^∗,𝑢𝑓₁𝑡,𝑢^∗𝑑𝑢^∗−𝜂₂₁+𝜂₂₂𝑓₂(𝑡,𝑢). (17) This system (17) can be solved when the two functions of (14)₂ are given.

As an example, let us solve this system under the following hypotheses: 𝜓₁𝑢_∗,𝑢=𝜓₂𝑢_∗,𝑢=𝛿𝑢−𝑢_∗,𝜉₁𝑢^∗,𝑢=𝜉₂𝑢^∗,𝑢=𝛿𝑢−𝑢^∗, (18) so that 𝜓_𝑖 and 𝜉_𝑗(𝑖,𝑗=1,2) are a Dirac-delta which fulfill (14)_1𝐷𝑢𝛿𝑢−𝑢_∗𝛿𝑢−𝑢^∗𝑑𝑢=𝛿𝑢_∗−𝑢^∗. (19) The system (17), by using (18), becomes 𝜕𝑓₁()𝜕𝑡𝑡,𝑢=𝜂₁₂𝑓₁(𝑡,𝑢)𝑓₂(𝑡,𝑢)−𝜂₁₁+𝜂₁₂𝑓₁(𝑡,𝑢),𝜕𝑓₂()𝜕𝑡𝑡,𝑢=𝜂₂₁𝑓₁(𝑡,𝑢)𝑓₂(𝑡,𝑢)−𝜂₂₁+𝜂₂₂𝑓₂(𝑡,𝑢). (20) Moreover, by assuming that𝜂₁₁=𝜂₂₁=𝜂₁₁=𝜂=₂₂^def𝜂, (21) and putting 𝑓(𝑡,𝑢)=𝑓₁(𝑡,𝑢)−𝑓₂(𝑡,𝑢), (22) from (20), one has𝜕𝑓(𝑡,𝑢)().𝜕𝑡=−2𝜂𝑓𝑡,𝑢 (23) The more general solution of this equation is 𝑓(𝑡,𝑢)=𝑓(0,𝑢)𝑒^−2𝜂𝑡. (24) Assuming that 𝑓()=10,𝑢√𝜋𝑒^−𝑢2, (25) equation (24) becomes𝑓()=1𝑡,𝑢√𝜋𝑒^{−(𝑢2+2𝜂𝑡)}. (26) From (10), by virtue of (22) and (26), we have𝜇(𝑡)=1−_𝐷𝑢1𝜋𝑒^{−2(𝑢2+2𝜂𝑡)}𝑑𝑢. (27) Taking into account that^+∞_−∞1𝜋𝑒^{−2(𝑢2+2𝜂𝑡)}𝑒𝑑𝑢=^−4𝜂𝑡√2𝜋, (28) equation (27) gives 𝜇(𝑡)𝑒=1−^−4𝜂𝑡√.2𝜋 (29)

5. A Simple Application

It is well known that the pioneering Lotka-Volterra's model of two interacting and competing populations (𝑥=prey, 𝑦=predatory) is based on the following differential system:𝑑𝑥𝑑𝑡=𝑎𝑥−𝑏𝑥𝑦,𝑑𝑦𝑑𝑡=𝑐𝑥𝑦−𝑑𝑦, (30) where 𝑎, 𝑏, 𝑐, and 𝑑 are constants.

In this model, the hiding-learning processes are not considered and the interaction and competition of the two populations start immediately. The orbits of the solutions of (30) are circles around the equilibrium point: 𝑥=𝑑/𝑐, 𝑦=𝑎/𝑏 (see Figure 1).

(a)

(b)

Figure 1 

Time evolutions of the orbits of (30) with parameters 𝑎=2, 𝑏=𝑐=1, and 𝑑=3 (a); distribution function 𝜇(𝑡) for 𝜇=1/4 (b).

If the hiding-learning processes occur, by using the results discussed in the previous sections, we propose the following system:𝑑𝑥𝑑𝑡=𝑎𝑥−𝜇𝑏𝑥𝑦,𝑑𝑦𝑑𝑡=𝑐𝑥𝑦−𝑑𝑦, (31) where 𝜇, given by (29), is the functional (stochastic) parameter depending on the distribution of populations (see Figure 2).

(a)

(b)

Figure 2 

Time evolutions of the orbits of (32) with parameters 𝑎=2, 𝑏=𝑐=1, and 𝑑=3 and initial populations 𝑥0=4, 𝑦0=1 (a), 𝑥0=0.1, 𝑦0=9 (b).

The system (31) becomes𝑑𝑥𝑒𝑑𝑡=𝑎𝑥−𝑏1−^−4𝜂𝑡√2𝜋𝑥𝑦,𝑑𝑦𝑑𝑡=𝑐𝑥𝑦−𝑑𝑦. (32) The nonzero equilibrium point is𝑑𝑥=𝑐𝑎,𝑦=𝑏1−𝑒^−4𝜂𝑡/√,2𝜋 (33) where lim_𝑡→∞𝑎𝑏1−𝑒^−4𝜂𝑡/√=𝑎2𝜋𝑏. (34) For 𝜂=1/4, the solutions of the system (32) are shown in Figure 2.

From Figure 2 it can be noticed that 𝑥₀>𝑦₀ so that the hiding-learning process delay the achievement of the circle around the nonzero equilibrium point. If 𝑥₀≪𝑦₀, then the cricle is reached more quickly.

6. Conclusion

In this paper, it has been studied a hybrid system of competition tumor cells versus immune system, within the kinetic model. A stochastic parameters is computed explicitly in the case of special transition density functions. A simple application shows that due to this parameters we obtain some more realistic solutions of the Lotka-Volterra system, where the cicle around the nonzero equilibrium point is shifted in time, thus showing the importance of the stochastic parameters in a correct approach to the analysis of competition models.