Modeling Tumor Growth in the Presence of Anaerobic Bacteria

Joseph Graves1 and James Nolen1,

1Department of Mathematics, Duke University;

Abstract

Mathematical Model: The Tumor Environment

Bacterial, Tumor, and Oxygen Growth over Time

We formed a mathematical model of the tumor environment when introduced to Clostridium bacteria. Our model consists of the bacterial concentration, oxygen concentration, tumor size, and bacteria spore concentration over time along a single axis, based on the system of reaction-diffusion equations for modeling the use of macrophages to treat tumor tissue by Owen et. al (2004).

The simulations show that the hypoxic region created by the tumor allows for the spores to germinate and reduce the size of the tumor. The bacteria concentration decreases in this region from the initial value, while increasing outside of the hypoxic region. The spore concentration decreases as time passes, and eventually is depleted assuming no further input, allowing the possibility for the tumor to grow back.

The initial conditions of the model at time t=0.

The model depicts the tumor environment on the one-dimensional interval [0,1], consisting of the bacteria(b(x,t)), tumor(m(x,t)), oxygen levels(c(x,t)), and spore concentration(s(x,t)) as functions of space and time. Additionally, boundary conditions are incorporated as well. The following equations are used to depict the change over time:

Bacteria Concentration

Tumor Mass

Spore Concentration

Oxygen Concentration

- The boundary conditions:
- , for some positive constant k.
- where cmax is the maximum oxygen concentration.

N=80,000,000

- The initial conditions for the bacteria and tumor are Gaussian profiles.
- The initial conditions for the spore and oxygen concentration are positive constants that are set at the beginning of the simulation.
- N is the number of time steps.
- Dotted lines represent the initial conditions while full lines represent the model at time step N.

Introduction

Despite the success of traditional treatments of cancer, there are many situations where such treatments are ineffective. It is important that any new cancer treatments must be able to treat the tumor cells without harming healthy tissue.

In particular, the growth of tumor tissue results in an increased usage of oxygen and nutrients creating hypoxic (low oxygen) regions in the tumor environment. These hypoxic regions allow for the growth of anaerobic bacteria such as Clostridium, and thus allow for the delivery of anti-cancer agents through such bacteria. More importantly, the bacterial spores only germinate near the tumor cells, meaning that the healthy tissue is unharmed.1

The main focus of such experiments consists of the delivery of prodrug-converting enzymes, which is encoded into the plasmid of the Clostridium. The prepared spores are then inserted into the lab animals. Once the spores germinate, a prodrug is administered to the animal, which is then converted by the encoded enzyme into the active drug used for treatment. Tumor regression has been observed; however there have been a variety of drawbacks for different strands of Clostridium, prodrugs, and enzymes.1

As a result of these findings, we propose a model consisting of a series of equations of the changes in bacteria, tumor size, oxygen levels, and spore concentration over a period of time in order to provide insight on how effective the treatment can be in the tumor environment.

N=120,000,000

The growth of the bacteria and spore germination rate are hindered by high levels of oxygen in the system. Furthermore, it is assumed that, despite the bacteria being used to deliver the means for producing the treatment for the tumor, the effective death rate of the tumor due to the amount of the active drug produced is directly proportional to the amount of bacteria present. It is assumed that the oxygen solely depend on the current level of oxygen and tumor size and is unaffected by the presence of bacteria. Additionally, many of the parameters are nondimensionalized in the simulation, with some of the dimensional parameters being based on estimates of parallel variables from the research in Owen et. al (2004).

N=1,000,000

Note how the oxygen levels are zero in the center of the model,

Indicating the hypoxic region created by the tumor.

N=200,00,000

Summary

- The high consumption of oxygen by the tumor creates a hypoxic region in the center of the tumor environment.
- As the simulation runs, the bacteria levels decrease at the center but increase slightly near the boundary, and eventually remain relatively stable as the spore concentration decreases.
- It is possible that the tumor can grow back, as observed in previous findings once the spore levels are effectively zero.1
- For further research, we plan on making a model for the tumor environment in three dimensions along with the addition of an advection velocity vector for the reaction-diffusion equations.

Discretization of the Equations

- In order to simulate the tumor environment over time, it is important to discretize the differential equations. In this case, we used the implicit forward Euler scheme.
- Take a specific point k such that
- Can represent = for some time n.
- (=)=

N=5,000,000

References

- Substitution leads to expression for
- Boundary conditions are established: It is established that the oxygen levels at the boundary prevent spores from germinating, Thus for each time n,

1. N Minton. Clostrida in cancer therapy. Nature reviews. Volume 1, December 2003.

2. M Owen, H Byrne, C Lewis. Mathematical modelling of macrophages. Journal of Theoretical Biology. 226: 377-391. 2004

N=20,000,000