Abstract
We establish the interaction model of two cell populations following the concept of the random-walk, and assume the cell movement is constrained by space limitation primarily. Furthermore, we analyze the model to obtain the behavior of two cell populations as time is closed to initial state and far into the future.
1. 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 healthy-cell populations was developed with the concept of random-walk (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:(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.
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.
2. 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 , the probability of -cell moving from to (to right), the probability of -cell moving from to (to left).
Moreover, the meaning of 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 at time is the information of cells on at ; then the probability of cells moving from to would be influenced by , which is the density of cell population on position at time . Reasonably, a decreasing function with respect to 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 and site minus the -cell number jumping to site and site . means the changing of -cell density at site and time . The function is the increase of -cell density at site at time with cells moving from site and site to site , where is the jumping (diffusion) coefficient of -cell at site at time . And is the decrease of -cell density at site at time with cells moving to site and site 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. Let , . , , , and ; hence, (1) becomes
For a continuum flow we consider that the jumping coefficient is a constant. Denote and by Taylor’s series
In consequence, ; similarly,
Consequently, and then we get Therefore, we consider (1) as the following.
The -cell movement can be modelled as where is a diffusion coefficient and is the information of -cell on position at time .
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 -cell and -cell. What the variables and functions and mean is as above; moreover, denote the density of -cell and -cell populations on position at time by and , respectively. On the other hand, we write 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 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 -cell) can be deduced. According to (7), replacing by , then where is a constant. Similarly, the same processes are applied to . 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 are diffusion coefficients with respect to -cell and -cell ( and 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 -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.
4. The Behavior and the Meaning of as
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 -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.
Lemma 1. Given two cell populations with the same diffusion coefficient, the system of PDEs (14) can be shown as a system of o.d.es. as follows: where , .
Proof. According to the system of PDEs (14), we could obtain
Let and , with the consequence that
The system of PDEs (16) can be written as model (15).
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 ( is very small), we must know the behavior of total cells.
Lemma 2. The movement of total cells (-cell and -cell) can be modelled as a classical diffusion equation .
Proof. Adding the two equations in the system (15), we obtain
Imposing upon (18), equation (18) could be rewritten as follows:
In consequence,
where , for some site at initial time .
According to above assumptions, and and , can be restored to , 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.
After describing the behavior of total cells, following (15), we replace by in the equation
Hence,
Given that , the equation (24) is simplified as
where is as (20), with the consequence that
Lemma 3. Equation (26) can be transformed to where
Proof. Assuming that , equation (26) is transformed as follows:
where and . Hence we denote as .
Therefore,
Hence, , where .
In order to simplify the representation of the following equations, we let The following theorem would show that and are bounded on for some small .
Before we make the following theorem complete, the substantiation of the next lemma must be finished.
Theorem 4. The solution of is bounded where is a constant and is closed to zero as if the solution of is bounded as .
Proof. Assume ; the solution of is given by
where and are constants.
We say that is the solution of and is the solution of . Then we have
Let , , and
The equation can be written as
Let be a fundamental solution matrix of . Then
where is the super norm and , .
By Granwall's inequality and for all in , then
for all in .
Hence, the solution of is bounded as .
Theorem 5. is bounded on for some small ; moreover, is bounded on .
Proof. Supposing that is closed to as and , then when .
According to the above assumptions, we have
For , , can be estimated immediately.
Thus the equation can be written as
for all .
Because the solution of is bounded as , the solution of is also bounded as . Consequently, , the solution of for all , is bounded on for some small , saying that and is a constant.
Hence,
where for some ; moreover, since , for all in and for some . In consequence, is bounded by where is a constant and on for some .
It is verified that is bounded by where is a constant and on , where is and is very small. Furthermore, we restore to , where is a positive constant. expresses that time approximates infinite. Therefore, Theorem 5 indicates that the density of -cell population approximates finite number as time approaches infinite. Through writing as , it could be deduced immediately that the density of -cell population is finite no matter how long time passes.
5. The Behavior and the Meaning of as
Near (namely, approaches zero), the boundedness of has been shown. Hence, we obtain that the density of -cell and -cell populations would not blow up when time approached infinity. In this section, through justifying that is bounded by first, we will show that is also bounded when approaches .
Theorem 6. The solution of , got by Lemma 3, is bounded by as approaches , where .
Proof. Supposing approaches , there is a such that approaches as . As tends to infinity, could be rewritten as the following asymptotic form:
Consider
and let be a solution of (43). Immediately,
is obtained. Assume , where , , and are constants; then
Consequently, and ; then . Hence, we get , where .
Now let be another solution of (43). Assume that , , with the consequence that . We get
Moreover, is convergent since , as . Therefore, the solution of is
and then
where is a constant and is defined as .
After substantiating that is bounded by as approaches , where , it is not difficult to verify that is also bounded as approaches .
Theorem 7. is bounded when approaches .
Proof. Given , in above Theorem 6, we have transformed
to through changing to , and
In consequence,
where and . Hence, is bounded by
as .
Restoring to , according to Theorem 7, we know that is bounded by as approaches ; namely, approaches initial time. In consequence, it is obtained immediately that the density of -cell population which is denoted by tends to a finite number as -cell population has begun moving for a fleeting time. Furthermore, the density of -cell population would also approximate a finite number for the same time.
If it is possible, we hope the solutions of (10) could be obtained by using our methods that were analytically used in [11–18] or numerically used in [19, 20].
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
Thanks are due to Professor Long-Yi Tsai, Professor Tai-Ping Liu, Professor Ton Yang, and Professor Shih-Shien Yu for their continuous encouragement and discussions over this work, to Metta Education, Grand Hall, and Auria Solarfor for their financial assistance, and to the referee for his interest and helpful comments on this paper.