The Space-Jump Model of the Movement of Tumor Cells and Healthy Cells

and Applied Analysis 3 3. Interaction of Two Cell Populations Now we show how to deduce a system of PDEs which describes the interaction of two cell populations. Here the two considered cell populations are called by u-cell and V-cell. What the variables and functions (E(x, t) and g(E)) mean is as above; moreover, denote the density of u-cell and Vcell populations on position x at time t by u(x, t) and V(x, t), respectively. On the other hand, we write w(x, t) := u(x, t) + V(x, t) to describe the total cell density. There is also another vague function, g(E), which needs to be defined clearly. Given that space limitation influences the movement of cells, the probability of cells moving to position x decreases with how the position is crowded with cells. We choose w(x, t), the total cell density, to express the information of cells on position x, namely, E(x, t) = w(x, t). Hence g(E) = g(w) = 1 − (w/T) shows that the probability of cells moving to position x decreases with the total cell density on position x, where T ≫ w initially and T is a constant. Here the assumption on g(E) follows the paper written by Painter and Sherratt (2003) [10]. After defining those variables, the model of interaction of two cell populations (u-cell and V-cell) can be deduced. According to (7), replacing g(E) by 1 − (w(x, t)/T) ≡ 1 − (w/T), then


Introduction
In the 1980s, the movement of isolated single cells was researched and was modelled by a range of authors (Oster [1]; Oster and Perelson [2]; Bottino and Fauci [3]; and Bottino, et al. [4]).In mathematics and biomedicine, not only of one-cell population but of multiple cell populations, there are many researches on the movement.
A consequential early paper written by Keller and Segel [5] modelled a partial differential equation to study the biochemical regulation of bacteria movement; their research has been the basis for models of the movement of diversified cell populations, such as slime mould aggregation (Höfer et al. [6]), tumor angiogenesis (Chaplain and Stuart [7]), primitive streak formation (Painter et al. [8]), and wound repair (Pettet et al. [9]).
In the recent years, most of the researches on cell movement focused on the interaction of multiple cell populations, precise cell behavior, and the development of the mathematics modelling.In this study we follow the contour of two-cell interaction developed by Painter and Sherratt [10].The modelling of interaction of tumor-and healthycell populations was developed with the concept of randomwalk (space-jump).Assuming the movement is according to space limitation and the diffusion coefficients of two cell populations are the same, we develop a system of partial differential equations (PDEs).Through some calculations, the system of PDEs is simplified to a system of ordinary differential equations (o.d.es.).Analyzing the system of o.d.es., it is obtained that the number of two cell populations per unit area in a unit amount of time is finite no matter when; namely, the density of each cell population does not blow up.
To model the motion of biological organisms, there are three major concepts which would be used: In this work we adopt space-jump concept to establish our model and from it we show how a PDE of cell movement could be deduced.Then we use the same concept and expand the PDE which has been deduced to reason a system of PDEs describing the interaction of two cell population.

Movement of One-Cell Population
We will deduce an equation of cell movement on a lattice from the space-jump concept; moreover, we translate that equation into a PDE of cell movement through changing variables.First, we list the functions and variables that will be used in this content and call the considering cell population by -cell as follows: (  , ) ≡   number of -cell at site   at time  per unit area in a unit amount of time (the density of cell at site   at time ), (  , ) ≡   the information of -cell at site   at time , ( +1 ) the probability of -cell moving from   to  +1 (to right), ( −1 ) the probability of -cell moving from   to  −1 (to left).
Moreover, the meaning of ( +1 ) is that the probability of the cell moving to the target would be influenced by the information of the cell's jumping target.
For example, we choose that the cell density on position  +1 at time  is the information of cells on  +1 at ; then the probability of cells moving from   to  +1 would be influenced by  +1 , which is the density of cell population on position  +1 at time .Reasonably, a decreasing function ( +1 ) with respect to  +1 implies that a lower probability results from the more crowded target.
Supposing that cells move continuously in time on a lattice (discrete space), a PDE of -cell movement would be modelled.
In the lattice space, the -cells' movement at time  can be modelled as We explain our idea as shown in Figure 1.
Figure 1 shows the movement of cells; the function on the figure is the moving probability.The changing of the cell density at site   at time  is equal to that of the -cell number jumping from site  −1 and site  +1 minus the cell number jumping to site  −1 and site  +1 .  / means the changing of -cell density at site   and time .The function (  )  ( −1 , )( −1 , )+(  )  ( +1 , )( +1 , ) is the increase of -cell density at site   at time  with cells moving from site  −1 and site  +1 to site   , where   (  , ) is the jumping (diffusion) coefficient of -cell at site   at time .And −  (  , )(  , )(( −1 ) + ( +1 )) is the decrease of -cell density at site   at time  with cells moving to site  −1 and site  +1 from site   .Thus, (1) is obtained.
The model of -cell movement in continuous space can be deduce from (1) in a lattice space through changing variables.
The -cell movement can be modelled as where   is a diffusion coefficient and (, ) ≡  is the information of -cell on position  at time .

Interaction of Two Cell Populations
Now we show how to deduce a system of PDEs which describes the interaction of two cell populations.Here the two considered cell populations are called by -cell and V-cell.
What the variables and functions ((, ) and ()) mean is as above; moreover, denote the density of -cell and Vcell populations on position  at time  by (, ) and V(, ), respectively.On the other hand, we write (, ) := (, ) + V(, ) to describe the total cell density.There is also another vague function, (), which needs to be defined clearly.
Given that space limitation influences the movement of cells, the probability of cells moving to position  decreases with how the position is crowded with cells.We choose (, ), the total cell density, to express the information of cells on position , namely, (, ) = (, ).Hence () = () = 1 − (/) shows that the probability of cells moving to position  decreases with the total cell density on position , where  ≫  initially and  is a constant.Here the assumption on () follows the paper written by Painter and Sherratt (2003) [10].
After defining those variables, the model of interaction of two cell populations (-cell and V-cell) can be deduced.According to (7), replacing () by 1 − ((, )/) ≡ 1 − (/), then where   is a constant.Similarly, the same processes are applied to V. We obtain the following equation: Consequently, we get the interaction of two cell populations.Following space limitation, the interaction of two cell populations can be modelled as where   and  V are diffusion coefficients with respect to cell and V-cell (  and  V are constants), respectively.Furthermore, through changing variables, with the consequence that Rewriting system (10) as the system of P.D.Es (10) can be simplified as where   and  ] are diffusion coefficients.Now, the interaction of -cell and V-cell has been modelled.Model ( 14) will be used frequently in the following context, and some properties of two cell populations can be deduced from analyzing model (14).We show the analyzing procedures and some results in the next section.

The Behavior and the Meaning of
](, ) = ]() as  → 0 We have got the system of PDEs ( 14) which shows the interaction of two cell populations.In this section, model (14) will be transformed to a system of o.d.es.and then analyzed to obtain some properties of ](, ) = ]() as  approaches to zero and infinite; furthermore, the properties of (, ) = () will be deduced from the properties of ]() and (), where () is () + ]().
Our purpose is to obtain a simpler form of ( 14) in order to analyze the model conveniently.Supposing that cell and V-cell have the same diffusion coefficient (  is equal to  ] ),  denotes the diffusion coefficients   and  ] .Through changing variables, the system of PDEs ( 14) could be transformed to a system of o.d.es.(14) can be shown as a system of o.d.es.as follows:

Lemma 1. Given two cell populations with the same diffusion coefficient, the system of PDEs
where  = /√,  ≡   =  ] .
In that case, the simpler form (model ( 15)) will be analyzed in the following subsections in order to obtain some properties of ]().
Before deducing that ](, ) = ]() is bounded for  in [0, ] ( is very small), we must know the behavior of total cells.

Lemma 2. The movement of total cells (𝑢-cell and V-cell) can be modelled as a classical diffusion equation 𝜔
Proof.Adding the two equations in the system (15), we obtain Imposing () upon (18), equation ( 18) could be rewritten as follows: In consequence, where  0 =  0 /√ 0 , for some site  0 at initial time  0 .According to above assumptions, (, ) ≡ () = () + ]() and () = ()/ and ]() = V()/, () can be restored to (()/) + (V()/), where  is a constant.In that case, equation ( 20) can be transformed into the form and then written as where  is /√ and  is a constant.The last equation shows the behavior of total cells; moreover, that is the classical representation of the solution of the fundamental diffusion equation.
In order to simplify the representation of the following equations, we let The following theorem would show that ]() and ]() are bounded on [0, ] for some small .
(a) the space-jump process in which the individual jumps between sites on a lattice, (b) the velocity-jump process in which discontinuous changes in the speed or direction of an individual are generated by a Poisson process, (c) the flux motion in which the movement of cells are treated as the flux motion.

Figure 1 :
Figure 1: The movement of cells.