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A well-known mathematical model of radially symmetric tumour growth is revisited in the present work. Under this aim, a cancerous spherical mass lying in a finite concentric nutritive surrounding is considered. The host spherical shell provides the tumor with vital nutrients, receives the debris of the necrotic cancer cells, and also transmits to the tumour the pressure imposed on its exterior boundary. We focus on studying the type of inhomogeneity that the nutrient supply and the pressure field imposed on the host exterior boundary, can exhibit in order for the spherical structure to be supported. It turns out that, if the imposed fields depart from being homogeneous, only a special type of interrelated inhomogeneity between nutrient and pressure can secure the spherical growth. The work includes an analytic derivation of the related boundary value problems based on physical conservation laws and their analytical treatment. Implementations in cases of special physical interest are examined, and also existing homogeneous results from the literature are fully recovered.

Mathematical modelling of cancer tumour growth helps in understanding the mechanisms underlying the phenomenon in many ways. The main idea is that modelling physical hypotheses in mathematical terms, a biological phenomenon, such as tumour growth, is approached by a mathematical problem. Analytic and numerical or hybrid methods, applied to the problem studied, lead to conclusions on the solution of the problem. The conclusions are interpreted in biological terms that are subject to evaluation with respect to experimental evidence. Thus, mathematical modelling may potentially offer new perspective in the research directions of biological procedures.

As cancer research and the related technology improved, an increasing amount of experimental data was produced, which made it hard to interpret. At the same time, mathematical models begun to develop at the aid of understanding the crucial parameters involved in tumour’s evolution [

Both deterministic, numerical and hybrid models have been developed towards the avascular growth study [

In most works, a spherical tumour structure has been used and different forms of the physical parameters involved have been investigated. Heterogeneity with respect to different cell types has been taken into account [

In particular, in this work, we consider a spherical fully developed avascular tumour, meaning that it consists of a necrotic core, a quiescent layer, and a proliferating layer, which are under dynamical growth until the whole structure reaches a steady state. Then, the tumour may stay in a dormant state or it may further develop only by entering in the vascular phase, a case that is not examined in the present study. We further assume that the tumour grows in a finite concentric spherical host environment, modelling the healthy part of the organ in the interior of which the tumour develops. Both tissues, cancerous and healthy, are assumed to be incompressible fluids of different phase. Here, we need to clarify that the tumour’s avascular phase ends at a maximum tumour size, which is significantly smaller than a typical human host organ. Nevertheless, our study aims to reveal rather qualitative aspects than quantitative effects of the tumour growth. One such aspect is concerned with the confinement effects imposed by the growth environment. The effect of the stress, imposed by the surrounding, has been subject of investigation in several works [

We assume that the tumour receives the nutrient (in general oxygen or glycose) from the surrounding spherical layer with inhomogeneous concentration and also it is affected by a tissue oriented nonhomogeneous pressure field. The proof that either no kind of inhomogeneity or a special kind of inhomogeneity that relates the externally supplied nutrient field with the externally imposed pressure field can be supported by such model in order for its concentric spherical structure to be secured is one of the outcomes of the present work. In the first case, it turns out that a spherical tumour can be developed only under homogeneous nutrient supply and homogeneous pressure field imposed, which confirms all previous works in the field. Nevertheless, the model reveals another option that allows nonhomogeneity in a concentric spherical growth, provided that a special relation holds between the nutrient and the pressure fields, with respect to the cancer cells’ motility parameters. In any case, the concentration profile of both the nutrient and the inhibitor turns out to be strongly affected by the host tissue boundary. Similarly, the pressure distribution throughout the cancer tissue and the host tissue is affected by the host tissue boundary, but the evolution of the tumour boundary is independent of it.

Before we proceed, we present briefly the basic assumptions we make within this work following the framework of [

We consider a spherical shell with radius

In Section

Let

Figure

The spherical domains of the tumour growth model.

According to the mass conservation law in each of the regions

Assuming that the nutrient flow is only diffusive, it follows Fick’s law, which means that it is directed to regions of lower nutrient concentration

Applying the divergence theorem in

In particular, considering the physical assumptions made in the Introduction, the consumption rate in each region is defined as

We turn now to the inhibitor concentration. From the mathematical point of view, the inhibitor concentration problem is very similar to the nutrient concentration problem. The difference lies on the interpretation of the parameters and in their particular expressions. As mentioned in the introduction, the inhibitor is produced inside the necrotic core with production rate

Before we proceed, we will examine the significance of the terms with the time derivatives in the partial differential (

Let us consider an elementary test cylinder lying symmetrically across the tumour interface

Finally, we assume continuity conditions of the nutrient concentration on each interface and an external nutrient supply traced on

Similarly, we derive continuity boundary conditions for the inhibitor flow through the tumour’s interfaces and we assume that the inhibitor concentration is also continuous there. Finally, the asymptotic condition

Combining previous assumptions [

Here we recall that both the tumour tissue and the host tissue are assumed to be incompressible fluids with constant density. The mass conservation law implies that

Substituting the partial differential (

The tumour colony and its host surrounding are assumed to be incompressible fluids of different phase. So the Young-Laplace equation is considered on their interface

The aim of this work is to determine the evolution of the tumour’s exterior boundary

We note here that relationship (

The right-hand side of (

Also, the nutrient value

In what follows the boundary value problems for the nutrient concentration field, for the inhibitor concentration, and for the pressure field are solved, using standard spectral analysis of the Laplace operator. The inhomogeneity imposed by the nutrient supply is reflected upon the conditions (

With respect to Section

Before we proceed with the solution of the problem, it is worth mentioning the eigenfunctions we will use, which are the surface spherical harmonics, defined as

Moreover, for notation convenience, we will replace the triple summation symbol by the following

Within this aim, we expand the nutrient concentration in each domain, in spherical harmonics, demanding the field to be continuous at the center of the spherical core. In that way we, obtain

In addition, the spherical expansion of the nutrient supply field

Substituting expression (

Equation (

In the first case, we imply the orthogonality property of

In the second case, we take the mean value of both sides of (

In what follows, the treatment of either case is indifferent. Nevertheless, we will come back to the different approaches in Section

Both (

The spherical expansions of the inhibitor concentration that solve the boundary value problem (

Following similar track of calculations as in Section

The inhibitor concentration is, by definition, a decreasing function of

The pressure field satisfies the boundary value problem (

In the sequel, we collect all the results for the concentration fields and for the pressure field, as traced on

Following the demand that the nutrient trace on the exterior boundary

Substituting (

We note here that the whole model’s treatment is based on the assumption that as the tumour evolves, all its boundaries remain members of the concentric spherical coordinate system, so that all the Laplace and the Poisson partial differential equations can be treated analytically through the separation of variables method. This implies that

By considering (

Then, the corresponding evolution equation for

The arguments stated in Section

Consequently, the tumour’s boundaries evolve as concentric spheres and the evolution (

We stress that the restrictions on the type of inhomogeneity exhibited by the exterior pressure are implied by the assumption (

Equation (

The third-degree nonlinear algebraic equation (

Equations (

Continuing, the rate

In this section, we implement the results obtained in Section

For the purpose of illuminating the inhomogeneous case, we suppose that the pressure’s trace on

The evolution equation is not affected by this inhomogeneity, and it is given by (

In this Section we consider the host nutritive environment to lie within an infinite medium, considered to be an incompressible fluid characterized by physical parameters different from those of both the host tissue and the tumour. Let us suppose that this medium imposes in the host tissue a uniform pressure

In terms of expression (

Though, the exterior pressure should not affect the evolution equation at all, as it is explained in Section

Following a related idea presented in [

The nutrient field in this case is immediately derived from relationships (

The corresponding modifications upon (

Since (

The above results fully recover the corresponding nutrient results obtained in [

In order to let the results obtained in Section

Moreover, if the inhibitor is considered to vanish far away from the tumour and not on the tumour’s boundary and also if

Then, (

The corresponding results for the pressure field when no exterior pressure is imposed on the tumor are found in [

The evolution equation studied in [

In the present work, we have studied a continuous model of avascular tumour growth, under two basic assumptions. Firstly, it evolves maintaining a spherical multilayer structure, lying inside a finite concentric spherical host medium. Secondly, its evolution is regulated by the diffusion of an inhomogeneous nutrient field and of a growth inhibitory factor and by the pressure that results from the cell proliferation and disintegration, as well as from an externally imposed inhomogeneous pressure from the finite surrounding medium. All the structure interfaces evolve in time, but the time scale of the space expansion of the tumour is so much different than the diffusion time scales of the substances in the tumour that the equilibrium assumption is well justified.

Hence, the model is formulated in three boundary value problems that hold true as the tumour evolves and provide the nutrient field, the internally produced inhibitor field and the pressure field throughout the tumour and the host surrounding. The model includes an assumption for the boundaries’ evolution, which is formulated as a nonlinear ordinary differential equation with respect to the tumour’s exterior boundary, and also it includes connection formulae between all the other boundaries with respect to the tumour’s exterior one.

The same formulation and analysis can probably be adapted in different models with alternative interpretations, for example, the nutrient concentration can be considered as a drug substance supplied externally that inhibits the growth, in which case the evolution equation would be respectively modified and so on.

In the present work we focus on the type of inhomogeneity that can be compatible with the specific model that we present. The inhomogeneities that we are focused on concern the data that are independent of the tumour’s development, that is, the nutrient supply and the pressure field imposed by the exterior to the host organ surrounding. It turns out that a concentric spherical multilayer development could be secured under homogeneous nutrient supply and pressure field, or by a nutrient supply and an exterior pressure that exhibit a special connection between them, expressed in (

These considerations are illuminated, by implementing the general results in special cases, in Section

Clearly, it is a drawback of the model presented, that it incorporates a rather ideal concentric spherical structure and the results refer to such an idealistic configuration or simplified assumptions. Alternative evolution approaches for the same spherical structure in avascular tumour development, and also alternative geometrical structure of the development, which is much more applicable to cancer growing in humans, are under our current investigation.