In order to include some effects of geometry of the tumor domain, we choose to work in a cylindrical polar coordinate system with axial symmetry. While tumors are likely more spherical than cylindrical, because in our simulation software (COMSOL) the axisymmetric polar system is already built-in, this choice makes the computations more straightforward.

We thus take the tumor domain Ω to be a disk of radius R (polar coordinates system with 1 − D axi-symmetry as defined by COMSOL Multi-Physics system). Within that disk, we somewhat arbitrarily divide the domain into three subregions: Core, Central, and Peripheral, with the core region nearest the symmetry axis (r = 0), and the peripheral region near to the outer boundary (r = R), and the central region in between those two. To summarize the numerical results later, we compute the mass of the tumor within those three regions to see how the combination of chemotaxis and therapeutics redistributes and potentially eliminates the cancer cells in the tumor.

The dependent variables in our model consist of: C(r,t), the cancer cell concentration; A(r,t), the chemo-attractant concentration; and T(r,t), the therapeutic agent concentration. Here, r is the radial variable in polar coordinates and t is time.

We start with the partial differential equation (PDE) that describes the evolution of the cancer cell concentration C(r,t). Essentially it is a conservation equation of the form:

$\frac{\partial C}{\partial t}+\nabla \cdot J=S$

in which $J$ is the flux of cancer cells and S is their net rate of production within the region. The flux $J=vC-D_c\nabla C$ includes an advective term characterized by velocity v and a diffusive term characterized by diffusivity $D_c$. In the absence of any bulk flow, the advection velocity is purely due to chemotaxis and would be given by $v={\chi }_a\nabla A$ in the presence of chemo-attractant with concentration A. The source term S includes expressions for growth rate of cancer cells, as well as their removal or death rates (with negative signs). More specifically, the PDE for cancer cell concentration becomes:

$\frac{\partial C}{\partial t}=D_c{\nabla }^{2}C-{\chi }_a\nabla \cdot \left(C\nabla A\right)+\alpha C\left(1-\frac{C}{K}\right)-\beta C-\delta CT$ (1)

The individual terms in this equation can be interpreted as follows. The term $\partial C/\partial t$ is the time rate-of-change of cancer cell concentration resulting from all the transport and growth and removal processes on the right-hand side. The term $D_c{\nabla }^{2}C$ represents the transport of cancer cells by diffusion, with $D_c$ being the diffusion coefficient due to random motility of cancer cells within the tumor microenvironment. Cancer cells exhibit random movement due to interactions with the extracellular matrix and other cells. The term ${\chi }_a\nabla \cdot \left(C\nabla A\right)$ represents transport due to chemotaxis towards higher concentrations of chemo-attractants. Here, ${\chi }_a$ is the chemotactic sensitivity coefficient, A is the chemoattractant concentration, and $\nabla A$ is its gradient. Cancer cells are known to migrate towards higher concentrations of certain chemicals that promote their growth and survival such as growth factors or nutrients.

The next set of terms describe the growth and death rates of cancer cells, comprising the source term in the original conservation equation. The term $\alpha C\left(1-C/K\right)$ is an assumed logistic growth rate for cancer cells; $\alpha$ is the proliferation rate, and K is the carrying capacity of the cancer microenvironment. Cancer cell proliferation follows a logistic growth pattern where growth slows down as the population approaches the carrying capacity of the environment. Proliferation of cancer cells can be limited by the finite availability of space and nutrients. The term $\beta C$ is the intrinsic (natural) cancer cells death rate, where $\beta$ is the death rate coefficient. Cancer cells are subject to finite lifetime. Finally, the term $\delta CT$ is a removal rate that represents the death of cancer cells due to the presence of the therapeutic agent, $\delta$ being the proportionality coefficient of the death rate with therapeutic agent concentration T, such as chemotherapy drugs that kill cancer cells upon contact. Therapeutic agents are designed to specifically target and kill cancer cells, reducing the overall tumor burden.

Clearly Equation (1) shows the coupling between the evolution of cancer cell concentration C with the concentrations of chemo-attractants A and therapeutic agents T. We thus also write transport equations for these two concentrations. Both of these are assumed to be governed by diffusion equations in which we also allow for degradation of the molecules and potentially a distributed source within the tumor domain. The respective PDEs for the chemo-attractant concentration and the therapeutic agent concentration thus read:

$\frac{\partial A}{\partial t}=D_a{\nabla }^{2}A-{\lambda }_{a}A$ (2)

$\frac{\partial T}{\partial t}=D_t{\nabla }^{2}T-{\lambda }_{t}T$ (3)

In these equations the left-hand sides are the rates of change of the respective concentrations and the right-hand sides include diffusive transport terms with diffusivities $D_a$ and $D_t$, and decay rates with rate constants ${\lambda }_a$ and ${\lambda }_t$. Unlike general formulations that include bulk source terms $S_a$ and $S_t$, our model supplies the chemo-attractant and therapeutic agents exclusively through fluxes imposed at the inner boundary of the domain. We therefore omit bulk sources from the governing equations to streamline the presentation.

In summary, Equations (1) - (3) constitute a system of three coupled PDEs that govern the transport and evolution of cancer cell, chemo-attractant and therapeutic agent concentrations within the tumor. The initial and boundary conditions imposed on this system are specified in Section 3 below in the context of the numerical simulations. This system and its associated initial and boundary conditions constitute our complete mathematical model. Each term in the governing equations corresponds to specific physical or biological processes that are crucial for modeling cancer cell behavior in the tumor microenvironment. By incorporating these terms, the model can simulate how cancer cells evolve, proliferate, and respond to treatments and environmental cues, providing a comprehensive framework for understanding and optimizing cancer therapy. In the simulations, we set up the boundary conditions so as to ensure that chemo-attractants draw cancer cells towards the core of the tumor where we expect them to be eradicated by therapeutic agents that are also introduced mainly in that location.

For computational and visualization purposes, we now proceed to non-dimensionalize the mathematical model described in Equations (1) - (3). To that end, we choose as our length scale the radius R of the disk-like domain and for our time scale, the diffusion time of cancer cells over that distance: ${R^2/D}_c$. We thus define a dimensionless radial coordinate $\hat{r}=r/R$ and a dimensionless time $\hat{t}=D_{c}t/R^2$, and we scale all concentrations with the carrying capacity K, taking $\hat{C}=C/K$, $\hat{A}=A/K$, and $\hat{T}=T/K$. Upon dropping the hats for notational clarity, we obtain:

$\frac{\partial C}{\partial t}={\nabla }^{2}C-{\beta }_1\nabla \cdot \left(C\nabla A\right)+{\beta }_{2}C\left(1-C\right)-{\beta }_{3}C-{\beta }_{4}CT$ (4)

$\frac{\partial C}{\partial t}={\beta }_5{\nabla }^{2}A-{\beta }_{6}A$ (5)

$\frac{\partial C}{\partial t}={\beta }_7{\nabla }^{2}T-{\beta }_{8}T$ (6)

Where the eight dimensionless parameters appearing therein are given by

${\beta }_1=\frac{{\chi }_{a}K}{D_c},{\beta }_2=\frac{\alpha R^2}{D_c},{\beta }_3=\frac{\beta R^2}{D_c},{\beta }_4=\frac{\delta R^{2}K}{D_c}$ (7)

${\beta }_5=\frac{D_a}{D_c},{\beta }_6=\frac{\lambda \alpha R^2}{D_c},{\beta }_7=\frac{D_t}{D_c},{\beta }_8=\frac{{\lambda }_T{R}^2}{D_c}$ (8)

Note that we model the introduction of attractants and therapeutic agents as a boundary flux, corresponding to localized delivery methods such as intratumoral injection or infusion devices. Instead of having distributed sources of attractants and therapeutic agents, we introduce them into the tumor through Neumann boundary conditions.

The tumor domain Ω that we have in mind is a sphere with radial symmetry. However, for convenience, we model a disk-shaped tumor in polar coordinates with axisymmetry. This means that the properties of the tumor are invariant under rotation about a central axis, with no dependence on the angular coordinate. Thus all quantities depend only on the radial coordinate r. In COMSOL this geometry is built-in and referred to as “1-D axisymmetric”.

At this point, an explanation of a small subtlety in COMSOL formulation of the so called 1-D axisymmetric geometry is in order: In many finite element (and other numerical) simulations involving cylindrical coordinates, the origin (r = 0) is a singular point. In COMSOL Multiphysics that point is not treated as a standard boundary but rather as an interior point. This modeling strategy avoids the issues associated with applying a boundary condition at a singular point. Instead, the underlying formulation takes advantage of the invariance of the physical laws under rotation about the axis, ensuring that the appropriate conditions are implicitly satisfied at the origin without requiring an explicit boundary condition [26] - [28] . In our case, however, we would like to impose flux-type boundary conditions at the core of the tumor domain. To get around this limitation, we modify our domain from the dimensionless range 0 ≤ r < 1 to a new one: 0.02/R < r < 1, starting away from the axis of symmetry. We can now impose Neumann boundary conditions at the point r = 0.02/R, to specify the fluxes of all three components in our model. In our simulations the numerical value of R was taken to be 2.56 (in centimeters), and the value 0.02 cm corresponds to 200 microns.

Using COMSOL Multiphysics software, we simulated several instances of the model by varying two key parameters: The Drift Coefficient β₁ and the Therapeutic Agent Coefficient β₄ of the cancer cells PDE (4). We expected that these particular parameters would play a pivotal role in determining the qualitative and numerical evolution of the body mass of cancer cells within the tumor microenvironment. Indeed, our overarching hypothesis is that the combined effect of convecting cancer cells towards the core of the tumor domain, coupled with targeted therapy in that same region, would dramatically reduce, if not eliminate, the tumor from the entire microenvironment.

After many trials, we settled on {0, 30, 60, 90} as the set of possible values for the drift coefficients β₁ and {0, 10, 20, 30, 40} as that of the therapeutic coefficient β₄. Against each of the 5 values in the therapeutic coefficient set, we ran four instances that correspond to the four values in the drift coefficients set, resulting in 20 total completed numerical simulations. Other special cases or variants of simulations were also performed for the purpose of further illustrations and validations of the results. For each instance of the above simulations, we used the built-in integration capability of COMSOL to compute the non-dimensional body mass of cancer cells in the three regions of the tumor: The core region (0.02/R < r < 0.52/R) , the central region (0.52/R < r < 0.54/R), and the peripheral region (0.52/R < r < 2.56/R). In polar coordinates, these integrals have the forms:

$\text{mass}=2\pi {\displaystyle {\int }_{ri}^{rj}rC\left(r,t\right)}dr.$

We also computed the total mass in the entire domain separately. We performed this process for each of the 3 stages. The results are tabulated and visualized below. Oncological staging is implemented by assigning differing initial values to the cancer concentration C for each stage.

We performed numerical simulations on the evolution of the three concentration fields $C\left(r,t\right)$, $A\left(r,t\right)$, and $T\left(r,t\right)$ starting with three different initial conditions that were meant to represent three of the stages of cancer at diagnosis. For all three cases, we took the initial conditions on the attractant and therapeutic to be

$A\left(r,0\right)=0,T\left(r,0\right)=0$

which means that they were not initially present in the system. For the cancer cell concentration, however, we took the initial conditions to be one of:

$\text{Stage I}:C\left(r,0\right)=\{\begin{array}{l}0.1\left(0.02/R<r<0.52/R\right)\\ 0\text{elsewhere}\end{array}$

$\text{Stage II}:C\left(r,0\right)=\{\begin{array}{l}1\left(0.02/R<r<0.52/R\right)\\ 0.1\left(0.52/R\le r<2.54/R\right)\\ 0\text{elsewhere}\end{array}$

$\text{Stage III}:C\left(r,0\right)=1\text{everywhere}$

Stage III thus represents a case where the cancer cells have reached the carrying capacity of unity throughout the domain at the point of diagnosis. Stage I starts with the cancer cells at one-tenth of the carrying capacity in a small region near the center. In stage II, we have cancer at the carrying capacity near the center and one-tenth of it in the central region, and zero in the peripheral region. Recall that the numerical value of R is 2.56 in our computations.

As far as the boundary conditions are concerned, in all cases, on the cancer cell concentration C, we impose no-flux boundary conditions at both ends of the interval. For the other two variables A and T, we impose no-flux conditions at the periphery (r = 1), but at the core (r = 0.02/R) we impose an incoming flux into the tumor domain, with a dimensionless value of unity. The boundary conditions are thus:

$\frac{\partial C}{\partial r}=0,\frac{\partial A}{\partial r}=0,\frac{\partial T}{\partial r}=0,\text{at}r=1$

and

$\frac{\partial C}{\partial r}=0,\frac{\partial A}{\partial r}=-1,\frac{\partial T}{\partial r}=-1,\text{at}r=0.02/R$

for all time t > 0.

This modeling choice reflects a localized delivery mechanism, such as direct intratumoral injection into the tumor core or infusion through an implanted device. Intratumoral injection has been investigated in experimental oncology as a means to achieve high local concentrations of therapeutic agents while minimizing systemic exposure [29] [30] . Representing the introduction of chemo-attractants and therapeutics as boundary fluxes thus captures the effect of sustained, spatially focused administration without assuming a uniform distribution throughout the tumor.