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In an environment that is neither static nor in equilibrium, but is dynamic and changing, the kinetics of the reactions that cause the growth of a tumor, which depend on the state of the evolving environment, cannot be parametrized in terms of constant rates. We propose a simple model for describing the growth on an untreated tumor in such environments, which is characterized by a minimal number of parameters and is generalizable to include the effects of various types of therapies. In the simplest version that we consider here, it consists of a linear equation with a time-dependent growth rate, which we interpret as the coupling of the system with a dynamic environment. A complete solution is given in terms of the integral of the growth rate. The essential features of the general solution are illustrated with a few examples, and comparison is made with the models that have been proposed to describe recent data.

Many applications of the mathematical modeling of tumor growth as a function of time have been based on the Gompertz equation [

However, some of the models consist on generalizing another model by modifying a parameter in a way that seems to be motivated mostly by the fact of being able to fit certain data. For example, the so-called Generalized Logistic model is the result of modifying the logistic equation by inserting a term that contains an arbitrary power

On the other hand, a few models of tumor growth that have been recently proposed, are based on assumptions and considerations that take into account basic physical principles, such as energy conservation, scaling, a fluctuating growth rate [

The main objective of this article is to present a basic model of growth of an untreated tumor in an environment that is not static, nor in equilibrium, but is dynamic and changing. In such a case the kinetics of the reactions that cause the growth of the tumor, which depend on the state of the evolving environment, cannot be parametrized in terms of constant rates. The model borrows some ideas from those mentioned above, contains a minimal number of parameters, and can be generalized to include other effects, such as those of therapies. In its simplest version, which we consider in some detail here, it consists of a linear first-order differential equation with a time-dependent growth rate, which we interpret as being due to the coupling of the system, the tumor in our case, with the dynamic environment. As argued in [

As we show, the solution, which can be expressed in terms of an integral of the growth rate, reproduces the usual S-shaped form that vanishes at

The rest of this paper is organized as follows. In Section 2 we write down the model that we propose, we discuss the framework that motivates it, the assumptions and idealizations involved, and the interpretation of the parameters that appear in it. In Section 3 the general solution to the equation is displayed, and it is illustrated by considering various specific examples that can be of practical use. In particular, here we confront the model with the data presented in [

Our starting point is the equation that describes the growth of a population that is sustained by an environment which is not necessarily static. In situations in which the population function grows up to a certain asymptotic limit

In situations in which the environment is static, e.g., large enough that it is not affected by the population,

The problem of the time evolution of the population of a given specie appears in many physical contexts. Two basic principles that guide the development of a population are the Master Equation and the Principle of Detailed Balance. The master equation takes the form

where W depends on f itself and the other variables that describe the rest of the system with which the population can interact. W is decomposed into a series of terms, each of which represents the contribution due to a particular process that causes the population to change. The principle of detailed balance states that there is a precise relation between the so-called direct process and its inverse.

For example, let us consider a process in which only one cell participates and let us denote such process in symbols by

where

where

Similar equations also describe the kinetic approach to equilibrium of systems that are put in contact with a reservoir. In such cases, which are governed by physical kinetic equations, the Principle of Detailed Balance can be applied to establish a relationship between

The procedure outlined above for the case of single cell processes could be generalized to more complicated ones. For example, consider the processes in which two cells participate, which we denote in symbols by

Because the direct process involves two cells, its rate is proportional to

where

As a typical rule in those contexts in which these equations have already been applied, the processes in which more than two members participate are rare and not important. Therefore, we are tempted to state that the master equation

is a good starting point for further exploration of these ideas in the present context as well.

We are interested in the cases in which the environment can change, due to external influences or by its interaction with the population itself. For us this means that the parameters

In the present paper, we will restrict ourselves to the linear term only, as written in Equation (1). The assumption behind this approximation is that the process in which the cells participate in pairs are rare compared to those in which only one cell participates. Should this linear approximation prove to be inadequate, it could indicate that the pair interactions are important and the quadratic terms in Equation (8) should be taken into account. Overall, we believe that this approach provides a framework for carrying a systematic analysis, based on incremental approximations, on a firm footing and in an organized fashion.

When

where

We consider the case in which the environment can change due to external influences, and therefore

where G satisfies

with

A suitable Green function for Equation (1) is

where

satisfying the condition

and therefore the solution for

Needless to say, if

A particularly simple form of the solution is obtained when

where we have used

In order to illustrate some general features of the solution, we will consider below various specific cases.

Let us assume that

where a is positive constant and n is a positive integer. First, from Equation (15),

where we have defined

for simplicity of the notation. The solution obtained from Equation (17) is then

In

As before, we assume that

where are

where

In

In order to make contact with experimental data, here we confront the model we have presented with the numerical results obtained with the model considered in [

The model of [

where

where

Therefore, K determines the saturation limit, while

In order to fit their data, the authors of [

In

Which has the solution as given in Equation (21), with

For the plot in

Obviously, the function

By inspection, Equation (23) can be generalized in an obvious way to the case in which

in t. Although the models considered above, with

We have presented a model for the growth of an untreated tumor in an environment that is dynamic and not in equilibrium. In such environments, the equations that describe the kinetics of the reactions that cause the growth of a tumor, which depend on the state of the evolving environment, cannot be parametrized in terms of constant rates. The model we propose to study these cases is based on ideas borrowed from models used in some physical contexts, together with plausible assumptions and idealizations that pertain to the application at hand here. In the simplest version, the model consists of a linear first-order differential equation with a time-dependent growth rate, which we interpret as being due to the coupling of the system, the tumor in our case, with the dynamic environment. As shown in Section 3, for that case a complete solution can be readily obtained in terms of an integral of the growth rate function. The solutions were given explicitly for a few sample cases, and they exhibit the known characteristic features of tumor growth. Moreover, the model was compared with the mathematical models that have been used to describe recent data, and it was shown that it can fit the data equally well, but without the unphysical features that those models have.

The approach that we have followed is fruitful in several ways. Firstly, the model contains a minimal number of parameters, which have a well-defined meaning, and are in principle determined by the interactions that govern the underlying microscopic mechanism. Secondly, this model could in turn shed light on such mechanisms, thereby providing a basis for pursuing this line of work. Thirdly, the model can be extended beyond the linear approximation that we have used if the terms that we have neglected are believed to be important in a particular system and the quadratic terms in Equation (8) should be taken into account. While in this paper we have restricted ourselves to treat the growth of an untreated tumor, similar ideas can be applied to include the effects of therapy.