# Spatio-temporal aspects of the interplay of cancer and the immune system

## Abstract

The conventional mean-field kinetic models describing the interplay of cancer and the immune system are temporal and predict exponential growth or elimination of the population of tumour cells provided their number is small and their effect on the immune system is negligible. More complex kinetics are associated with non-linear features of the response of the immune system. The generic model presented in this communication takes into account that the rates of the birth and death of tumour cells inside a tumour spheroid can significantly depend on the radial coordinate due to diffusion limitations in the supply of nutrients and/or transport of the species (cells and proteins) belonging to the immune system. In this case, non-trivial kinetic regimes are shown to be possible even without appreciable perturbation of the immune system.

## Keywords

Cancer Tumour Growth Diffusion Kinetic modelCancer occurs via initiation, tumour growth, and propagation of metastases. During these stages, its development depends on the response of the immune system (reviewed in [1, 2]) and the enhancement of this response can be used as a basis for efficient anticancer therapies [3]. The interplay of cancer and the immune system is complex and our understanding of this interplay remains limited. At the conceptual level, this interplay can be illustrated by using the corresponding kinetic models. Customarily, such models are temporal and operate with the population of cancer cells forming a tumour and populations of the species (cells and proteins) belonging to the immune system (reviewed in [4, 5]; see also recent treatment [6]; for a more general perspective on the kinetic models of cancer, see reviews [7, 8, 9, 10]). In this framework, the growth or elimination of the population of tumour cells is predicted to be described by the first-order equation and to be exponential provided their number is small and their effect on the immune system is negligible, and accordingly the emphasis is shifted towards the complexity related to non-linear features of the behaviour of the immune system. The effect of the immune system on the initial growth of tumour cells was also theoretically analyzed [11] in the contexts of the experiments determining the lifetime risk of cancer (Ref. [12]; briefly reviewed in [13]) and the interaction of tumours via the immune system. The spatio-temporal aspects were, however, not treated there in detail. In some of the recent multivariable models (see, e.g. [14, 15, 16]), these aspects are taken into account to some extent, but the corresponding mathematical analysis is rather cumbersome and the results reported do not allow one to see the physics behind. Herein, I (i) propose a generic spatio-temporal model focused on the initial phase of the growth of a tumour in the regime where its effect on the the populations of species belonging to the immune system is negligible and (ii) show that even in this limit the kinetics are not necessarily reduced to the exponential growth or elimination of the population of cancer cells.

*n*

_{t}), effector cells (

*n*

_{e}) and interleukin-2 (

*n*

_{p}) are as follows [4, 17]:

*k*,

*w*

_{e},

*w*

_{p},

*κ*

_{e}and

*κ*

_{p}are the birth and death rate constants and rates,

*n*

_{∗}is the maximum population of tumour cells,

*γ*is the rate constant of elimination of tumour cells and

*v*

_{1},

*v*

_{2},

*v*

_{3},

*m*

_{1},

*m*

_{2}and

*m*

_{3}are the other rate constants and parameters characterizing the function of the immune system. The analysis of Eqs. 1–3 can be simplified taking into account that on the time scale of the tumour growth the response of the immune system is rapid, and accordingly (2) and (3) can be solved in the steady-state approximation by setting

*d*

*n*

_{e}/

*d*

*t*=

*d*

*n*

_{p}/

*d*

*t*= 0. Describing the initial phase of the growth of a tumour, one can in addition neglect the effect of tumour cells on the immune system. Mathematically, this means that the third term on the right-hand part in Eq. 3 can be neglected, and accordingly one has:

*n*

_{t}/

*n*

_{∗}in the first term and

*n*

_{t}in the denominator of the second term in Eq. 1. Then, using expression (4) for

*n*

_{e}, Eq. 1 can be rewritten as:

*n*

_{t}is the number of cells forming a tumour, and

*W*

_{b}and

*W*

_{d}are the rates of birth and death of these cells. To calculate these rates, the tumour is assumed to be spherical with radius

*R*, and accordingly the population of tumour cells is expressed via its volume as:

*v*is the volume per cell.

*r*≤

*R*(

*r*is the radial coordinate) is represented as:

*D*is the nutrient diffusion coefficient, and

*η*is the rate constant associated with the nutrient consumption by tumour cells. The boundary condition for this equation is:

*c*

_{∘}is the nutrient concentration outside the tumour. Taking into account that on the time scale of the tumour growth the nutrient diffusion inside the tumour is rapid, Eq. 8 can be solved in the steady-state approximation by setting

*∂*

*c*/

*∂*

*t*= 0. The corresponding textbook solution of Eq. 8 is given by:

*λ*≡ (

*D*/

*η*)

^{1/2}.

*c*(

*r*). In this approximation, the total birth rate can be represented as:

*k*is the birth rate constant (

*c*

_{∘}is considered to be included into this rate constant),

*k*

*n*

_{t}is the birth rate calculated assuming the diffusion limitations to be negligible, and

*R*/

*λ*for the convenience of its derivation as it is usually done in the literature focused on the kinetics of catalytic reactions occurring inside grains (the corresponding models of catalytic reactions can be tracked down to the seminal study by Thiele [18]). In the context of tumour growth,

*R*should be expressed via

*n*

_{t}taking relation (7) into account, i.e., Eq. 11 should be rewritten as:

*χ*≡ (4

*π*/3

*v*)

^{1/3}

*λ*.

*c*,

*D*,

*η*and

*λ*by the corresponding parameters,

*c*

_{∗},

*D*

_{∗},

*η*

_{∗}and

*λ*

_{∗}≡ (

*D*

_{∗}/

*η*

_{∗})

^{1/2}. Then, by analogy with Eq. 13, the death rate is represented as:

*γ*is the death rate constant, and

*χ*

_{∗}≡ (4

*π*/3

*v*)

^{1/3}

*λ*

_{∗}.

*k*>

*γ*and

*k*<

*γ*, respectively.

*χ*

*k*>

*χ*

_{∗}

*γ*and

*χ*

*k*<

*χ*

_{∗}

*γ*, respectively. It is of interest that the growth is algebraic rather than exponential.

*χ*

_{∗}/

*χ*is larger or smaller than unity as illustrated graphically in Fig. 1.

Thus, the model clearly shows that, with inclusion of spatial features, non-trivial kinetic regimes of tumour growth may be possible even at relatively small populations of tumour cells in the situations when the effect of tumour cells on the population of the species (cells and proteins) belonging to the immune system is negligible. Physically, this is related to the transition from the kinetically limited birth and death to diffusion-limited birth and death with increasing population of tumour cells. For birth and death, this transition can easily take place at different populations of cells, and it can result in the appearance of a non-trivial stable or unstable steady state.

Finally, I can add that the model under consideration can be extended in different directions. For example, its current version implies that the tumour cells are of one type. In reality, the population of tumour cells is well known to be heterogeneous [19, 20], and this factor can easily be taken into account in the analysis presented. The model can also be reformulated in the terms of chemotherapy, and accordingly it can be used in the latter area as well.

## Notes

### Funding Information

Open access funding provided by Chalmers University of Technology

### Compliance with ethical standards

### **Conflict of interest**

The author declares that he has no conflict of interest.

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