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A Stochastic Model of Cancer Growth with lmmune Response |
A. Boondirek and Y. Lenbury
Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand
J. Wong-ekkabut
Department of Physics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand
W. Triampo
Capability Building Unit in Nanoscience and Nanotechnology,
Faculty of Science, Mahidol University, Bangkok, 10400, Thailand
I. M. Tang
Institute of Science and Technology for Research and Development,
Mahidol University, Nakhonpathom 73170, Thailand
P. Picha
National Cancer Institute of Thailand, Bangkok 10400, Thailand
(Received 1 April 2006, in final form 12 July 2006)
Abstract
A cellular automaton model for the growth of an avascular tumor on a two-dimensional square
lattice is presented. The pattern formation and the growth of the cell population are investigated
by using a Monte Carlo simulation. A microscopic description of the immune system response,
including cell proliferation, cell death, and cell degradation, is used to simulate the growth. In
particular, the escape rate for cancer from immune surveillance is included for consistency with
experimental observations. The simulation results give rise to a growth curve with an explanation
on a microscopic scale that is shown to agree well with experimental animal tumor growth and
relevant biological implications. Our model clearly shows that an increase in the lysis rate leads to
a decrease in the proliferation rate of cancer cells.
The spatial distribution of proliferated cell and
the fractal dimension of the boundary are also measured.
PACS numbers: 87.15.Aa, 87.17.Aa
Keywords: Cancer growth, Immune response, Cellular automaton, Monte Carlo, Gompertz curve
Selected results (Full paper..(pdf))
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Typical averaged growth curve, where the error bars are of the same magnitude as the size of the points. The simulation result is for an average over 1000 individual simulations with the parameters rprolif. = 0.25, rbinding = 0.04, rescape = 0.6, rlysis = 0.3, rdecay = 0.35, and K = 550
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Comparison between the simulated tumor growth (circles) and the experimental growth curves in vivo for the spontaneous mouse carcinoma C3H ( Steel [41] black solid line) with a coefficient of nonlinear regression r2 = 0.99985. The Gompertz parameters are Vo = 0.0376 (cm)3, A = 0.177 (day)-1, B = 0.0311(day)-1 , and Vmax= 11.12 (cm)3 . The parameter settings are rprolif. = 0.25, rbinding = 0.04, rescape = 0.6, rlysis = 0.3, rdecay = 0.35, and K = 550. with No = 6.02 and Nmax =551.18.
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Comparison between the simulated tumor growth and the experimental growth curves in vivo for mouse Ehrlich [41] carcinoma with the coefficient of nonlinear regression r2 = 0.9997. The Gompertz parameters are Vo = 0.0226 (cm)3, A = 0.456 (day)-1, B = 0.102 (day)-1 , and Vmax= 1.94 (cm)3 . The parameter settings are rprolif. = 0.85, rbinding = 0.1, rescape = 0.5, rlysis = 0.35, rdecay = 0.35, and K = 550 with No =8.381 and Nmax =627.379.
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Plots of the time evolution of the total numbers of tumor cells. The inset shows magnified time step of 5 to 10 and 20 to 30. The simulation results are averaged over 1000 individual realizations by varying the value of rbinding from 0.00 to 0.15 in steps of 0.05 while fixing the other values at rprolif. = 0.85, rescape= 0.5, rlysis = 0.35, rdecay = 0.35, and K = 550.
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Plots of the time evolutions of the numbers of proliferating cells. The simulation results are averaged over 1000 individual realizations with the same parameters
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Plots of the time evolution of the total number of tumor cells. The simulation results are averaged over 1000 individual realizations by varying the value of rescape from 0.0 to 0.4 in steps of 0.2 while fixing the other values at rprolif. =0.85, rbinding= 0.1, rlysis = 0.35, rdecay = 0.35, and K = 550.
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