A Cellular Automata Model of Oncolytic Virotherapy in Pancreatic Cancer

Abstract

Oncolytic virotherapy is known as a new treatment to employ less virulent viruses to specifically target and damage cancer cells. This work presents a cellular automata model of oncolytic virotherapy with an application to pancreatic cancer. The fundamental biomedical processes (like cell proliferation, mutation, apoptosis) are modeled by the use of probabilistic principles. The migration of injected viruses (as therapy) is modeled by diffusion through the tissue. The resulting diffusion-reaction equation with smoothed point viral sources is discretized by the finite difference method and integrated by the IMEX approach. Furthermore, Monte Carlo simulations are done to quantitatively evaluate the correlations between various input parameters and numerical results. As we expected, our model is able to simulate the pancreatic cancer growth at early stages, which is calibrated with experimental results. In addition, the model can be used to predict and evaluate the therapeutic effect of oncolytic virotherapy.

Keywords: Cancer treatment; Cellular automata; Computational modeling; Monte Carlo simulations; Virotherapy.

Fig. 1

A schematical figure of oncolytic virotherapy. The viruses can specifically infect cancer cells and then replicate themselves until cancer cells rupture. Subsequently, the newborn viruses are released to infect more cancer cells

Fig. 2

Historical milestones in the development of oncolytic virotherapy

Fig. 3

A specified lattice i at position ${\mathbf{x}}_i=[x_i,y_i,z_i]$ with its neighborhood in the 3D simulations

Fig. 4

Consecutive snapshots of cancer progression, where blue color and red color are visualized as epithelial and cancer cells, respectively. The 3D domain $15\times 15\times 15{\text{mm}}^3$ meshes into $30\times 30\times 30$ lattices. As a result, cancer cells occupy the entire computational domain when $t=1400$ h.

Fig. 5

Growth curves of pancreatic tumor under different situations. a A comparison of growth curves of pancreatic tumors with various ${\alpha }_0$ value (see Eq. (6)), where ${\lambda }_{\max }=5\times {10}^{-3}$; b A comparison of numerical results with experimental results referring to (Durrant et al. 2015), where control and gem in the legend denote tumor growth without drug and with gemcitabine drug, respectively. In the simulation with curve 1, ${\lambda }_{\max }$ and ${\alpha }_0$ are equal $1\times {10}^{-3}$ and 0.94, respectively. However, to calibrate the model to curve 2, ${\lambda }_{\max }$ decreases to $5.5\times {10}^{-4}$ and $\alpha$ falls to 0.85

Fig. 6

A comparison of viruses diffusion by using the FDM method with a color bar indicating the concentration of viruses, where red color represents a high concentration of virus, dark blue hints a neglectable viral concentration and other colors denote values in between. a No cancer cells are present and viral infection is not simulated, which means no new proliferating viruses. Therefore, most viruses are mainly concentrated in the center; b In the presence of cancer cells, viral infection ensues, viruses replicate leading to rupture of cancer cells, which then releases the viruses. The viruses are thus found also at distant locations. The isosurface in grey color has a concentration value of slightly less than 100 pfu ${\text{mm}}^3$

Fig. 7

Consecutive slice plots of viral spread. No cancer cells are present and viral infection is not simulated, which means no new proliferating viruses. The slices are taken from the angle of a z-axis top view, which is located in the middle of the computational domain. A color bar indicates the concentration of viruses

Fig. 8

Consecutive slice plots of viral spread. In the presence of cancer cells (cancer cells are not shown for clarity), viral infection ensues, viruses replicate leading to rupture of cancer cells, which then releases the viruses. The slices are taken from the angle of a z-axis top view, which is located in the middle of the computational domain. A color bar indicates the concentration of viruses

Fig. 9

Consecutive snapshots of oncolytic virotherapy. The blue, red and black colors are visualized as epithelial, cancer and infected cancer cells, respectively. In addition, white color means the dead cells or unoccupied lattice points. A small scale of cancerous tissue that returns to normal tissue by cell reproduction or migration under the oncolytic virotherapy after $t=110$ h

Fig. 10

a Changes in viral quantity in the computational domain; b Changes in cancer volume with time

Fig. 11

a Changes in viral quantity as the evolution of time with an injection rate $\gamma$ = $1\times {10}^4$ pfu/h; b Changes in viral quantity as the evolution of time with an injection rate $\gamma$ = $0.5\times {10}^5$ pfu/h; c Changes in viral quantity as the evolution of time with an injection rate $\gamma$ = $1\times {10}^5$ pfu/h

Fig. 12

a Histogram of the total number of remaining particles of the viruses in Monte Carlo simulations on parameters $\gamma$, $\widehat{c}$ and $\eta$; b Histogram of cancer area in Monte Carlo simulations on parameters $\gamma$, $\widehat{c}$ and $\eta$

Fig. 13

a Scatter plot of injection rate $\gamma$ and total particles of the remaining viruses. The corresponding correlation coefficient is $R=0.0665$; b Scatter plot of injection rate of $\gamma$ and the final cancer area with a correlation coefficient $R=-0.0668$

Fig. 14

a Scatter plot of infection threshold $\widehat{c}$ and the total particles of the remaining viruses. The corresponding correlation coefficient is $R=-0.8413$; b Scatter plot of infection threshold $\widehat{c}$ and the final cancer area. The correlation coefficient of infection threshold $\widehat{c}$ and the final cancer area is $R=0.9210$

Fig. 15

a Scatter plot of immune strength and the remaining viral quantity. The correlation coefficient of immune strength and the remaining viral quantity is $R=-0.4320$; b Scatter plot of immune strength and the final cancer area. The correlation coefficient of immune strength and the cancer area is $R=0.2978$