Limits...
Towards a Mathematical Formalism for Semi-stochastic Cell-Level Computational Modeling of Tumor Initiation.

Vermolen FJ, Meijden RP, Es Mv, Gefen A, Weihs D - Ann Biomed Eng (2015)

Bottom Line: The model is based on a cell-based formalism, where each cell is represented as a circle or sphere in two-and three dimensional simulations, respectively.Fundamental biological processes such as random walk, haptotaxis/chemotaxis, contact mechanics, cell proliferation and death, as well as secretion of chemokines are taken into account.The developed formalism is based on the representation of partial differential equations in terms of fundamental solutions, as well as on stochastic processes and stochastic differential equations.

View Article: PubMed Central - PubMed

Affiliation: Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands, F.J.Vermolen@tudelft.nl.

ABSTRACT
A phenomenological model is formulated to model the early stages of tumor formation. The model is based on a cell-based formalism, where each cell is represented as a circle or sphere in two-and three dimensional simulations, respectively. The model takes into account constituent cells, such as epithelial cells, tumor cells, and T-cells that chase the tumor cells and engulf them. Fundamental biological processes such as random walk, haptotaxis/chemotaxis, contact mechanics, cell proliferation and death, as well as secretion of chemokines are taken into account. The developed formalism is based on the representation of partial differential equations in terms of fundamental solutions, as well as on stochastic processes and stochastic differential equations. We also take into account the likelihood of seeding of tumors. The model shows the initiation of tumors and allows to study a quantification of the impact of various subprocesses and possibly even of various treatments.

No MeSH data available.


Related in: MedlinePlus

Front view of tumor development at four consecutive times at a later stage. The green cells denote the epithelial cells, the red ones are the tumor cells and finally the blue cells are the T-cells (not present here). Further, the large black dots represent the points at which T-cells possibly originate. Snapshots are at times 494, 496, 498 and 500 s.
© Copyright Policy - OpenAccess
Related In: Results  -  Collection


getmorefigures.php?uid=PMC4495267&req=5

Fig8: Front view of tumor development at four consecutive times at a later stage. The green cells denote the epithelial cells, the red ones are the tumor cells and finally the blue cells are the T-cells (not present here). Further, the large black dots represent the points at which T-cells possibly originate. Snapshots are at times 494, 496, 498 and 500 s.

Mentions: We consider a spherical domain with radius of 40 micrometer that is initially filled with endothelial cells. The epithelial cells are allowed to migrate, proliferate and allowed to die. Further, they may mutate to tumor cells. At six points of the domain edge, we assume that T-cells can invade the domain. The epithelial cells are depicted as green spheres, the tumor cell are red and finally the T-cells are blue spheres. Further, the small blood vessel points where T-cells originate from are depicted by large black dots. See Fig. 5 for the initial state. At later stages, it can be seen that the T-cells appear in the domain and start chasing the tumor cells. To this extent, we plot four consecutive snapshots of the front and rear view in Figs. 6 and 7. In Figs. 6 and 7, it can be seen that the tumor develops in a continuous way. Tumor cells are seen to be inside the cell colony, where the T-cells invade the region from the boundary. The T-cells have to make their way by chemotaxis while enduring the contact forces that they experience from the native epithelial cells. In Fig. 8, we see the cell pattern at a late stage where the tumor cells are dominant in the tissue. The number of red tumor-cells has increased tremendously. This stage could account for a decrease of performance of the body under consideration. One could insert a measure for disfunction of an organ to quantify the damage that the tumor is causing to the patient. Further, in Fig. 9, we plot the number of T-cells, tumor cells and epithelial cells vs. time, from which the proportion of the tumor cells can be determined, see also in Fig. 9. It can be seen that the calculation was stopped near the point where the fraction of the tumor cells is equal to 0.5 (that is 50 %). The number of tumor-cells increases almost exponentially and gradually take over the tissue by making the epithelial cell counts decrease as a result of the pressure that is exerted onto the latter. The number of T-cells does not increase sufficiently in this run to fight the tumor cells. Next to plotting the number of cell counts for the various phenotypes over time, we present the cumulative number of tumor-cells that transmigrate into the small blood vessel in Fig. 10. It can be seen that this number increases exponentially. This number of cells is related to the likelihood of tumor expansion into other anatomic parts of the body. The exponential nature of this quantity over time illustrates the danger of tumor expansion, which is partly kept in check by the immune system and various obstacles that the cells encounter.Figure 5


Towards a Mathematical Formalism for Semi-stochastic Cell-Level Computational Modeling of Tumor Initiation.

Vermolen FJ, Meijden RP, Es Mv, Gefen A, Weihs D - Ann Biomed Eng (2015)

Front view of tumor development at four consecutive times at a later stage. The green cells denote the epithelial cells, the red ones are the tumor cells and finally the blue cells are the T-cells (not present here). Further, the large black dots represent the points at which T-cells possibly originate. Snapshots are at times 494, 496, 498 and 500 s.
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4495267&req=5

Fig8: Front view of tumor development at four consecutive times at a later stage. The green cells denote the epithelial cells, the red ones are the tumor cells and finally the blue cells are the T-cells (not present here). Further, the large black dots represent the points at which T-cells possibly originate. Snapshots are at times 494, 496, 498 and 500 s.
Mentions: We consider a spherical domain with radius of 40 micrometer that is initially filled with endothelial cells. The epithelial cells are allowed to migrate, proliferate and allowed to die. Further, they may mutate to tumor cells. At six points of the domain edge, we assume that T-cells can invade the domain. The epithelial cells are depicted as green spheres, the tumor cell are red and finally the T-cells are blue spheres. Further, the small blood vessel points where T-cells originate from are depicted by large black dots. See Fig. 5 for the initial state. At later stages, it can be seen that the T-cells appear in the domain and start chasing the tumor cells. To this extent, we plot four consecutive snapshots of the front and rear view in Figs. 6 and 7. In Figs. 6 and 7, it can be seen that the tumor develops in a continuous way. Tumor cells are seen to be inside the cell colony, where the T-cells invade the region from the boundary. The T-cells have to make their way by chemotaxis while enduring the contact forces that they experience from the native epithelial cells. In Fig. 8, we see the cell pattern at a late stage where the tumor cells are dominant in the tissue. The number of red tumor-cells has increased tremendously. This stage could account for a decrease of performance of the body under consideration. One could insert a measure for disfunction of an organ to quantify the damage that the tumor is causing to the patient. Further, in Fig. 9, we plot the number of T-cells, tumor cells and epithelial cells vs. time, from which the proportion of the tumor cells can be determined, see also in Fig. 9. It can be seen that the calculation was stopped near the point where the fraction of the tumor cells is equal to 0.5 (that is 50 %). The number of tumor-cells increases almost exponentially and gradually take over the tissue by making the epithelial cell counts decrease as a result of the pressure that is exerted onto the latter. The number of T-cells does not increase sufficiently in this run to fight the tumor cells. Next to plotting the number of cell counts for the various phenotypes over time, we present the cumulative number of tumor-cells that transmigrate into the small blood vessel in Fig. 10. It can be seen that this number increases exponentially. This number of cells is related to the likelihood of tumor expansion into other anatomic parts of the body. The exponential nature of this quantity over time illustrates the danger of tumor expansion, which is partly kept in check by the immune system and various obstacles that the cells encounter.Figure 5

Bottom Line: The model is based on a cell-based formalism, where each cell is represented as a circle or sphere in two-and three dimensional simulations, respectively.Fundamental biological processes such as random walk, haptotaxis/chemotaxis, contact mechanics, cell proliferation and death, as well as secretion of chemokines are taken into account.The developed formalism is based on the representation of partial differential equations in terms of fundamental solutions, as well as on stochastic processes and stochastic differential equations.

View Article: PubMed Central - PubMed

Affiliation: Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands, F.J.Vermolen@tudelft.nl.

ABSTRACT
A phenomenological model is formulated to model the early stages of tumor formation. The model is based on a cell-based formalism, where each cell is represented as a circle or sphere in two-and three dimensional simulations, respectively. The model takes into account constituent cells, such as epithelial cells, tumor cells, and T-cells that chase the tumor cells and engulf them. Fundamental biological processes such as random walk, haptotaxis/chemotaxis, contact mechanics, cell proliferation and death, as well as secretion of chemokines are taken into account. The developed formalism is based on the representation of partial differential equations in terms of fundamental solutions, as well as on stochastic processes and stochastic differential equations. We also take into account the likelihood of seeding of tumors. The model shows the initiation of tumors and allows to study a quantification of the impact of various subprocesses and possibly even of various treatments.

No MeSH data available.


Related in: MedlinePlus