Modeling protective anti-tumor immunity via preventative cancer vaccines using a hybrid agent-based and delay differential equation approach

Abstract

A next generation approach to cancer envisions developing preventative vaccinations to stimulate a person's immune cells, particularly cytotoxic T lymphocytes (CTLs), to eliminate incipient tumors before clinical detection. The purpose of our study is to quantitatively assess whether such an approach would be feasible, and if so, how many anti-cancer CTLs would have to be primed against tumor antigen to provide significant protection. To understand the relevant dynamics, we develop a two-compartment model of tumor-immune interactions at the tumor site and the draining lymph node. We model interactions at the tumor site using an agent-based model (ABM) and dynamics in the lymph node using a system of delay differential equations (DDEs). We combine the models into a hybrid ABM-DDE system and investigate dynamics over a wide range of parameters, including cell proliferation rates, tumor antigenicity, CTL recruitment times, and initial memory CTL populations. Our results indicate that an anti-cancer memory CTL pool of 3% or less can successfully eradicate a tumor population over a wide range of model parameters, implying that a vaccination approach is feasible. In addition, sensitivity analysis of our model reveals conditions that will result in rapid tumor destruction, oscillation, and polynomial rather than exponential decline in the tumor population due to tumor geometry.

Figure 1. Time plots of cell populations for a simulation of the ABM-DDE system.

See also ABM plots in Figure 2. Parameters are taken from the base values shown in Table 1. (left) Numerical solution of the system (4). Populations displayed are , immature APCs in the periphery; , mature APCs in the lymph node; memory CTLs in the lymph node; , effector CTLs in the lymph node; and , effector CTLs in the periphery. (right) Plot of tumor cell and CTL populations at tumor site.

Figure 2. Progression of ABM simulation.

See also time plots in Figure 1. Plots show tumor cells (black circles), CTLs that are circulating at the tumor site (gray circles), and CTLs that are engaging tumor cells (white circles). Parameters are taken from the base values shown in Table 1. (Day 14) Tumor grows from one cell to 1,714 cells, triggering a CTL response in the lymph node. CTLs begin circulating in the periphery and occasionally enter the tumor site. (Day 20) Tumor has grown to 6,709 cells. CTLs discover tumor mass, begin to engage tumor cells, and recruit additional CTLs. (Day 22) CTL population overcomes tumor growth causing tumor mass to begin to decrease. Tumor mass is currently 6,880 cells. (Day 36) CTLs continue to engage the tumor, decreasing tumor to 191 cells. All tumor cells are eliminated on day 42.

Figure 3. Time plots showing oscillating tumor cell and CTL populations at the tumor site.

The average CTL recruitment time, , is 24 h. All other parameters are taken from the base values shown in Table 1. The tumor population peaks and declines 12 times. Low points of tumor remissions range from 1 to 105 residual cells. The tumor is eliminated on day 1,600.

Figure 4. Plots showing tumor extinction times (top row) and of maximum tumor populations (bottom row) versus parameters, , , and (columns 1, 2, and 3, respectively).

Black circles and solid lines represent means over 5 simulations. Dotted lines represent minimum and maximum values obtained over the 5 simulations.

Figure 5. Plots showing tumor extinction times (left) and maximum tumor populations (right) versus frequency of memory CTLs at steady state, i.e. .

Black circles and solid lines represent means over 5 simulations. Dotted lines represent minimum and maximum values obtained over the 5 simulations.

Figure 6. Time plots of tumor and CTL populations from a simulation of the ABM-DDE system.

(left) Tumor antigenicity . All other parameters are taken from the base values shown in Table 1. The tumor population is extinct on day 213. (right) Plot of the cube root and natural logarithm of the tumor population from day 190 to extinction. The numerical solution of has a high linear correlation , implying that decays as a cubic function of . (The linear regression is .) On the other hand, the numerical solution of does not exhibit linear behavior, showing that does not decay exponentially.

Figure 7. Time plots of tumor population, , and CTL population, , from (1).

(left) Parameters , , and . All other parameters are taken from the base values in Table 1. The tumor population is extinct, i.e., identically 0, on day 19.04. (right) Time plot of the cube root of the tumor population. The final decline appears nearly linear.

Figure 8. Possible outcomes of cell division.

(a) Space is available, so the cell divides successfully, and a new cell is generated adjacent to the old cell. (b) No space is available, so the cell fails to divide. To simplify diagrams, figures are shown in 2-D, although the model occurs in 3-D. package.

Figure 9. Plots of tumor cells growing from one cell at days 0, 5, and 10.

The average time between cell divisions . To simplify diagrams, figures are shown in 2-D, although the model occurs in 3-D.

Figure 10. When CTLs move during a time step, two scenarios may occur.

(a) No collision – the CTL moves the entire length of , (b) Collision – the CTL moves as far as possible without colliding with another cell.

Figure 11. Possible CTL actions during a time step.

At each time step, CTLs move according to a 3-D Wiener process. A CTL in contact with a cancer cell stops moving and engages the cancer cell. A CTL engaging cancer cell may recruit an additional anti-cancer CTL with probability or kill the cancer cell with probability . When the cancer cell dies, the CTL disengages and accelerates up to the maximum rate. Although not shown, all CTLs may die with probability .

Figure 12. Model of CTL migration between the region of interest of radius and the CTL cloud of thickness .

In this example, two CTLs already exist in the region of interest, i.e., 1 and 2. At the beginning of the time step, new CTLs are randomly generated in the CTL cloud, i.e., 3 and 4. CTL motion is calculated for the next time step, and at the end of the time step, CTLs beyond the region of interest are eliminated, i.e., 2 and 4, while CTLs in the region of interest are retained, i.e., 1 and 3.

Figure 13. Model of dynamics in the lymph node.

(1) Tumor cells produce antigen at the tumor site. (2) APCs pick up tumor antigen, mature, and migrate to the lymph node. (3) Mature antigen-bearing APCs present antigen to memory CTLs causing them to activate and enter the division program of divisions. (4) Effector CTLs that have completed the division program continue to divide upon further interaction with mature, antigen-bearing APCs. (5) Effector CTLs continually migrate to the periphery. Although not indicated, each cell in the diagram also has a natural death rate.