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There are many works (i.e. [1]) aiming to find out numerically how positive feedback affects the formation of invadopodia and invasion of cancer cells; however, studies on the cancer cell invasion model with free boundary are fairly rare. In this paper, we study modified cancer cell invasion model with free boundary, where, free boundary stands for cancer cell membrane, so that we can more precisely describe the positive feedback affects. Firstly, we simplized the model by means of characteristic curve and semi-groups’ property, and obtained the Stefan-like problem by introducing Gaussian Kernel and Green function. Secondly, based on the classical Stefan problem, we derived the integral solution of simplified model, which could lead us a further step to find the solution of modified cancer cell invasion model.

As well known, cancer disease is one of the leading causes of death worldwide. Many natural and man-made factors (for example, smoking, car exhaust fumes, ultraviolet rays, air pollution and radiations, etc.) are the main risks for cancer disease. Metastasis is a leading death process, in which two processes are crucial from the viewpoint of cancer therapy. The first one is angiogenesis, nucleation of new blood vessels, which can provide enough nutrients to further development of tumor cells. The other one is tissue invasion. After the vascular growth of the tumor, cells become more aggressive that it can invade into the extracellular matrix, even into blood vessels, to complete the metastasis. Tissue invasion is a process in which cells can migrate and establish a new colony in new organs.

There are many studies about angiogenesis inhibition, because cancer cells have a certain size and cannot grow further without nutrients from blood vessels. Signaling molecule VEGF (vascular endothelial growth factor) can be secreted by cancer cells and can bind the normal endothelial cells to form new blood vessels. Scientists found inhibitors, such as bevacizumab, to block VEGF [

To reduce the ability of invasion is also one way to prevent metastasis. Cancer cells can spread by degrading ECMs. ECMs are degraded by the assembly of the actin cytoskeleton in invadopodia―the invasive feet of cancer cells. In fact, MMPs (matrix metalloproteinase), the family of ECM degrading enzymes [

Mathematical medicine and biology have become one of the popular topics in the study of modern applied mathematics. Where, cancer cell invasion models have received much attention in recent years [

To the best of our knowledge, studies on the cancer cell invasion model with free boundary are fairly rare. For that reasons, in this paper, we study a modified cancer cell invasion model with free boundary problem. The method used in this paper is motivated by Stefan problem.

In order to obtain cancer cell invasion model with free moving boundary, we need to consider the biological background of the problem. For the reader’s convenience, we will introduce the process of invadopodia formation. Invadopodia are the invasive feet (as shown in

ECM fragments are decreased by the reorganization of actin cytoskeleton indirectly [

We improved the model considered in [

where,

where,

Free boundary

where,

Since the membrane pushed by the F-actin which is reorganized by signals from cell receptors, therefore, the velocity of the membrane depends on the gradient of signals which cause F-actin polymerization. Hence, boundary velocity defined as follows:

Finally, we derived the modified cancer cell invasion model with cell boundary described as (1-5), our main purpose is to generalize this model into Stefan problem, then analytically discuss its solution.

The organization of this paper is as follows. In section 2 we present some basic definitions, assumptions and related properties, such as characteristic curve of the problem, Greens functions etc., to simplify the problem. In section 3, the main results, related theorem and its proof, of our paper was stated. Finally, the detailed calculation from (48) to (49) and (50) is given in Appendix A, B, C.

The previous section, we stated the biological background and modified cancer cell invasion model with free moving boundary (1 - 5). In this section, we will introduce some basic definitions and related preliminaries, such as Gaussian Kernel, Green function and derivation of Stefan problem etc., which would be useful in proving main results and solutions of Stefan problem (27) in section 3.

In this paper, for simplicity, we take

where,

Take

where

Consider the following equation of c,

One can write

Then we easily have the solution

where

By (8), it follows that

By

When

Now we take

and finally we can get

By using an argument similar to the above, from the equation

we have the solution

Then signals

Finally we have,

with

The Equation ((10) with (11)) is our key problem for the solution of (1-5). If we can get the solution of sigma from (10) and (11), then we can easily find the solution

First, consider initial condition problem

Now, we multiply

If

where,

However,

Thus, we can say that

Next, consider parameters

If we consider

Note that

In order to find the solution of (14), we need to introduce new function as follows

Clearly, we can see that

and

Since,

by (18), we can easily prove (23) and (24). Using the third property of

where,

Now, we consider free moving boundary

where,

Similar to (13), we have the following system

where,

Define a domain D as (

Take

Furthermore, from (24), we know

in D. Combining (29) and (30), we have

According to Green’s formula, the left side of the Equation (31) becomes

Finally, we have

since,

When

In this section we will state the main results, for convenience we will divide this section into two parts. In the first part, we will give three propositions and one theorem. Where, Proposition 1 and Proposition 2 are useful in proving Proposition 3, and Proposition 3 proves Theorem 1. Theorem 1 represents the solution of

In order to prove Theorem 1, we need to prove following three propositions.

Proposition 1. Suppose that,

holds.

Proof. We consider the left side of (34),

Since,

hence,

This completes the proof of Proposition 1.

Proposition 2. Suppose that,

holds.

Proof. We consider the left side of (35),

where,

Since

Next, from the continuity of

therefore,

Similarly we can get

Therefore, (35) follows.

Proposition 3. Suppose that,

Proof. From the assumption of Proposition 3, we have

By Proposition 1 and Proposition 2, we can prove Proposition 3. This completes the proof.

Theorem 1. Suppose that

Proof. Proposition 3 gives the calculation of the first term of form (33). Regarding to the second term of (33), we are using the third property of

for

since

The above results show that the left side of Equation (32) satisfies

Then, we have

For the right side of (39), it is clear that point

and

Moreover, note

for

Equation (41) is useful to prove Theorem 1, which can be written as,

This completes the proof of Theorem 1.

In the above result (37) in Theorem 1, all variables are known except

where,

From the definition of

then,

If we use above property, (45) becomes,

Next, from (23), we have

When

(See appendix for specific information).

When

Integrate both sides of (43) from 0 to t, we have

therefore, we have

Refer appendix for specific information about how (49) and (51) are followed by (48) as

Actin filaments are cytoskeleton in cytoplasm, which can drive cell deformation, migration [

Colin et al [

We have more interesting topics which deserve further investigations, such as numerical simulations of the integral equations (48, 50 - 52) and how we can get the solution of the original modified model (1 - 4) based on integral solution.

This work was supported by the National Natural Science Foundation of China (Grant Nos.11401509) and the Natural Science Foundation of Xinjiang University (Starting Fund for Doctors, Grant No. BS130102).

All authors of this article declare: there is no conflict of interests regarding the publication of this article.

Mahemuti, R., Muhammadhaji, A. and Suzuki, T. (2017) Research on the Solution of Cell Invasion Model with Free Boundary. Open Journal of Applied Sciences, 7, 242-261. https://doi.org/10.4236/ojapps.2017.76021

This appendix provides specific information about how we get (49) from (48).

a) Confirmation of

where,

Proof. From the definition of

Then, from the definition of

where,

For further calculations, we introduce inequality

where,

then, by the definition of Function-limit, we have for any

then,

Therefore, applying (54) we can get

Take

Hence, we have

which implies that

and continuous for all

On the other hand,

Similarly, we can get

b) confirmation of

Proof. Set

if

From the definition of

Take

Since,

Now, prove the continuity of

holds for all

where

for all

Therefore (58) holds, which implies

c) Conformation of

where,

Proof. From the definition of

where,

Suppose,

hence,

when

When

next, from (59) and (60), we can get

since,

From the proof of Appendix B, one can know

Finally, we get

where,

Next, we can get

since,

is continuous on

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