In this short review article, two atherosclerosis models are presented, one as a scalar equation and the other one as a system of two equations. They are given in terms of reaction-diffusion equations in an infinite strip with nonlinear boundary conditions. The existence of traveling wave solutions is studied for these models. The monostable and bistable cases are introduced and analyzed.
This is a short review article about atherosclerosis models and existence of traveling wave solutions for such models. We present both the monostable and bistable case.
Reaction-diffusion problems with nonlinear boundary conditions arise in various applications. In physiology, such problems describe in particular the development of atherosclerosis and other inflammatory diseases . In this context, nonlinear boundary conditions show the influx of white blood cells from blood flow into the tissue where the inflammation occurs. Among other possible applications, let us indicate molecular transport through biological membrane where some molecules can amplify their own transport opening membrane channels, as it is the case, for example, with calcium induced calcium release.
Atherosclerosis begins as an inflammation in blood vessels walls (intima). The inflammatory response of the organism leads to the recruitment of monocytes. Trapped in the intima, they differentiate into macrophages and foam cells, leading to the production of inflammatory cytokines and further recruitment of white blood cells. This self-accelerating process, strongly influenced by low-density lipoproteins (cholesterol), results in a dramatic increase of the width of blood vessels walls, formation of an atherosclerotic plaque and, possibly, of its rupture.
We present in Section 2 below two reaction-diffusion models for atherosclerosis. In Section 3, we discuss the existence of waves for a scalar equation in the bistable case. The method employed here is the Leray-Schauder method. The monostable case is not discussed here for a single equation, but directly for a system of two reaction-diffusion equations. Section 4 is devoted to this aspect. We end the paper with a section of conclusions and future work.
The results of the present paper are mainly included in the papers [1,2,3,4,5]. Paper  presents some blood flow simulations in atherosclerotic vascular networks. Basic theory and results about travelling waves and different methods can be found in [7,8,9,10].
2. Atherosclerosis Models
Papers [1,4] propose a mathematical model for atherosclerosis initiation and development. The model is given by a reaction-diffusion system in a strip with nonlinear boundary conditions which describe the recruitment of the monocytes M as a function g of the concentration A of inflammatory cytokines. So the PDE model is the following:
in the two-dimensional strip with the boundary conditions
and the initial conditions
Here M is the concentration of the white blood cells (monocytes, macrophages, and foam cells) inside the intima (blood vessels walls), A is the concentration of the pro- and anti- inflammatory cytokines, is the rate of production of the cytokines which depends on their concentration and on the concentration of the blood cells. Terms and describe the natural death of the blood cells and of the cytokines, respectively, while the term b represents a constant source of the activator in the intima (a ground level of cytokines in the intima). It can be the oxidized LDL coming from the blood. All constants b are positive.
Functions f and g are supposed smooth enough and satisfying the following conditions:
A stationary solution of systems (2.1) and (2.2) is So is a constant level of cytokines in the intima such that the corresponding concentration M of the monocytes is zero and they are not recruited through the boundary.
A simplified model that takes into account only the concentration of the white blood cells is given by the scalar parabolic PDE ()
in the infinite strip with the nonlinear boundary conditions:
It is a mathematical model of atherosclerosis and other inflammatory diseases. Ω corresponds to the blood vessel wall (intima) where the disease develops, v is the concentration of white blood cells in the tissue. The nonlinear boundary condition describes the cell influx through the boundary. The influx depends on the cell concentration in the tissue.
3. Reaction-Diffusion Waves for One Equation in the Bistable Case
3.1. Formulation of the Problem
We begin with the study of the simplified model, that is the reaction-diffusion equation
with nonlinear boundary conditions:
in the infinite strip . This is the problem studied in [2,3].
We study the existence of traveling wave solutions of this problem. They are solutions of the form . Then function u satisfies the equation
with the boundary conditions
Here c is an unknown constant, the wave speed, that should be found together with function Assume that f and g are of class .
We look for solutions of problems (3.3) and (3.4) with the limits
where are some functions which satisfy the problem in the cross section (i.e., independent of x):
Let us introduce the bistable and the monostable cases as in . To do this, consider the equation linearized about solutions and the corresponding eigenvalue problems:
If both problems have all eigenvalues in the left-half plane, then we call it the bistable case. If one of these problems has all eigenvalues in the left-half plane and the other one has some eigenvalues in the right-half plane, then it is the monostable case. We will present here the bistable case  for a single equation. The monostable case will be analyzed directly for systems (see Section 4). It is a more difficult and more complete study than for a single equation. The study of systems in the bistable case we have in preparation.
As a particular situation , we can assume that are constant (do not depend on y) and that
We also assume that f and g have a single zero between and (), such that This is also a bistable case and has been analyzed in the paper .
Investigation of problems (3.3) and (3.4) relies on the properties of the corresponding operators. It will be shown that in the bistable case, the linearized operator satisfies the Fredholm property. Moreover for the nonlinear operator we can introduce a topological degree. These tools allow us to use the Leray-Schauder method based on the topological degree and a priori estimates of solutions.
3.2. Solutions in the Cross-Section
3.2.1. General case
We start with the problem in the cross-section (independent on x)
in the interval . Suppose here that functions f and g are continuous together with their first derivatives. One can reduce the second-order equation to the system of two first-order equations
and then to the equation
We can solve this equation analytically. Consider for simplicity only monotone solutions w and denote , . In the case of decreasing solutions , and the boundary conditions become
Under the assumption that
We obtain the length of the interval as a function of the maximal value of solution:
Depending on the functions f and g, solution can exist, it can be unique or non-unique, or it may not exist. The case of increasing solutions can be studied in a similar way.
3.2.2. Constant solutions
In the next sections, when we study the wave existence, we will consider problems which depend on parameters. So we discuss here problem (3.7) where and δ is a positive parameter. Suppose that and are continuous together with their first derivatives and
for some and , and that these functions have a single zero in the interval ,
Lemma 3.1. () Let functions f and g satisfy conditions (3.8) and (3.9). Then there exists such that problem (3.7) with has only constant solutions for any and any positive δ.
Now we discuss the stability of these solutions. Denote a zero of the function g by . Then the corresponding eigenvalue problem writes
If , then the principal eigenvalue of this problem is positive. This is the case for . If (ex. ), then the eigenvalues are negative.
3.3. Properties of the Operators
3.3.1. Fredholm property of the linearized operator
A linear operator (between Banach spaces) has the Fredholm property if is closed, L has a finite dimensional kernel and the codimension of is finite.
Consider the operator corresponding to problem (3.3), (3.4) and linearized about a solution :
The operator acts from the space into the space . Consider the limiting operators
and the corresponding equations
Let E be a Banach space with the norm and be a partition of unity. Then is the space of functions u for which the expression
This is the norm in the space . The result below has been proved under stronger hypotheses in . We present here the most general case, which is due to .
Theorem 3.2. () If condition (3.8) is satisfied and (or more generally, if we are in the bistable case), then the operator acting from into or from into satisfies the Fredholm property.
3.3.2. Properness and topological degree of the nonlinear operator
Consider the nonlinear operator in the domain Ω
and the boundary operator
Let , where for for . Set
We consider the operator acting in weighted spaces,
with the weight function . The norm in the weighted space:
Theorem 3.3. () In the bistable case, P is proper in the weighted spaces and a topological degree can be defined.
3.4. A Priori Estimates
A priori estimates of solutions are obtained below only for monotone solutions. In order to apply the Leray-Schauder method, separation of monotone solutions from non-monotone solutions is proved. This means the norm of their difference is uniformly bounded from below by a positive number. It follows from a subjacent result (see Lemma 3.5). This permits one to construct the domain in the function space, which contains all monotone solutions and does not contain non-monotone solutions. The degree is found with respect to this domain. This remark is the key point of the proof of Theorem 3.12 below.
3.4.1. Auxiliary results
Consider the problem
Lemma 3.4. () Let be a solution of problems (3.10) and (3.11) such that for all . Then the last inequality is strict.
Lemma 3.5. () Let be a sequence of solutions of problems (3.10) and (3.11) such that in , where is a solution monotonically decreasing with respect to x. Then for all n sufficiently large , .
3.4.2. Wave speed
Lemma 3.6. () Suppose that is a solution of problem (3.6) in the cross section of the domain, and . Assume, next, that the corresponding eigenvalue problem
has some eigenvalues in the right-half plane. If a monotone with respect to x function satisfies the problem
Lemma 3.7. () In the conditions of the previous lemma, if
instead of , then .
Lemma 3.8. () If problems (3.10)–(3.12) has a solution w, then the value of the speed admits the estimate , where the constant M depends only on
3.4.3. Estimates of solutions
Consider the problem
where f and g depend on the parameter .
Lemma 3.9. () Suppose that the above problem admits a solution that satisfies with some positive constant M, and
where K is a positive constant. Then the Hölder norm , of the solution u is bounded by a constant which depends only on K, M and c.
Theorem 3.10. () If there exists a solution of the above problem for some , then the norm is bounded independently of the solution .
Under stronger hypotheses, this result has been also proved in .
3.5. Existence of the Traveling Wave Solutions
Consider the model problem for
Assume that , , and there is a single zero of f in the interval , .
Lemma 3.11. ([2,3]) There exists a unique monotone in x solution of problems (3.17)–(3.19) up to translation in space.
Consider next the operators corresponding to problems (3.14)–(3.16) :
Suppose that for . Then the equation
has a unique solution The topological degree of this operator with respect to a small neighborhood of the solution, equal 1.
Assume now that are bounded and continuous, together with their derivatives of order 3 with respect to w and of the second order with respect to τ.
have all eigenvalues in the left-half plane for any . Suppose that for any other solution , the eigenvalue problem
has some eigenvalues in the right-half plane. If the problem in
has a unique solution monotone with respect to x for , then it also has a unique monotone solution for any .
The proof of this result is based on the remark from the beginning of Section 3.4 about the separation of all monotone solutions from nonmonotone solutions.
Theorem 3.13. () Let and be some constants and the following conditions be satisfied: (1). , , (2). , for some , and there are no other zeros of these functions in this interval.
Then for all sufficiently small, the problem
considered in has a unique solution monotone with respect to x.
Theorem 3.14. ([2,5]) Let the function satisfy conditions of the previous theorem. Then for all positive L, the problem in
has a unique solution monotone with respect to
Theorem 3.15. () Suppose that and for some there exists a monotone solution of the problem
Then for all ϵ sufficiently small, the problem
considered in the domain has a unique solution monotone with respect to x. Here are solutions of the problem
as uniformly in y.
These results show that atherosclerosis develops as a reaction-diffusion wave.
4. Reaction-Diffusion Waves for Systems
We approach in this section the atherosclerosis models (2.1) and (2.2) in the monostable case. The bistable case is not yet studied and it is the topic of a future work. Recall that here M is the concentration of the white blood cells inside the intima, while A is the concentration of the inflammatory cytokines.
Suppose that the stationary solution is unstable and that there exists a stable stationary solution such that
One studies the existence of waves with the limits at and at Assume that there are no other stationary solutions such that and This means that is the smallest solution above
The wave solutions have the form where the constant c is the wave speed. Then verifies the system
in with the boundary conditions
One looks for the solutions with the property
Theorem 4.1. () Problems (4.1)—(4.3) admits a solution if and only if c is greater or equal than a specific constant value The solution is strictly monotone with respect to
This result shows the wave propagation if and only if In this case, atherosclerosis develops as a reaction-diffusion wave.
5. Conclusions and Future Work
In the present paper, we have put together some results about traveling wave solutions for reaction-diffusion equations with nonlinear boundary solutions arising in atherosclerosis models. In such a way the reader can find easily these results and compare the behavior of solution in different cases. In Section 4 we have presented the existence of such solutions in the monostable case for system (2.1). Since the bistable case is more complicated and infers a more elaborated mathematical tools, we have preferred to study it first for a single equation of the form (3.1) (Section 3). The bistable case for systems will be the goal of a future work.
We can study this even in a more general reaction-diffusion system, namely consider
with the nonlinear boundary conditions
where D is a diagonal matrix with positive diagonal elements.
We are looking for solutions of this system under the form They satisfy the problem
We are going to employ the Leray - Schauder method to prove the existence of reaction-diffusion waves for this system.
This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS—UEFISCDI, project number PN-II-ID-PCE-2011-3-0563, contract no. 343/5.10.2011 “Models from medicine and biology: mathematical and numerical insights”.
The aim of this paper is to assemble some results about the existence of travelling wave solutions for reaction-diffusion equations with nonlinear boundary solutions arising in atherosclerosis models. In such a way the reader can compare the behavior of solution in different cases. All the results of this review paper have been published before.
Conflicts of Interest
The author declares no conflict of interest.
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