Evolution of Resistance and Progression to Disease during Clonal Expansion of Cancer

Inspired by previous work of Iwasa et al. (2006) and Haeno et al. (2007), we consider an exponentially growing population of cancerous cells that will evolve resistance to treatment after one mutation or display a disease phenotype after two or more mutations. We prove results about the distribution of the first time when k mutations have accumulated in some cell, and about the growth of the number of type-k cells. We show that our results can be used to derive the previous results about a tumor grown to a fixed size.

[1]  S. Frank.  Dynamics of Cancer: Incidence, Inheritance, and Evolution , 2007 .

[2]  C. Nordling.  A New Theory on the Cancer-inducing Mechanism , 1953, British Journal of Cancer.

[3]  A. Knudson.  Mutation and cancer: statistical study of retinoblastoma. , 1971, Proceedings of the National Academy of Sciences of the United States of America.

[4]  Jason Schweinsberg,et al.  Waiting for $m$ mutations , 2008 .

[5]  Martin A Nowak,et al.  Evolution of Resistance During Clonal Expansion , 2006, Genetics.

[6]  P. Armitage,et al.  The age distribution of cancer and a multi-stage theory of carcinogenesis , 1954, British Journal of Cancer.

[7]  R. Durrett.  Probability: Theory and Examples , 1993 .

[8]  Nordling Co.  A New Theory on the Cancer-inducing Mechanism , 1953 .

[9]  M. Nowak,et al.  Stochastic Tunnels in Evolutionary Dynamics , 2004, Genetics.

[10]  A. Knudson,et al.  Two genetic hits (more or less) to cancer , 2001, Nature Reviews Cancer.

[11]  Tom Lenaerts,et al.  Dynamics of chronic myeloid leukemia under nilotinib therapy , 2010 .

[12]  J. C. FISHER,et al.  Multiple-Mutation Theory of Carcinogenesis , 1958, Nature.

[13]  Martin A Nowak,et al.  Population genetics of tumor suppressor genes. , 2005, Journal of theoretical biology.

[14]  Deena R. Schmidt,et al.  A waiting time problem arising from the study of multi-stage carcinogenesis , 2007, 0707.2057.

[15]  Jason Schweinsberg,et al.  The waiting time for m mutations , 2008, 0802.3485.

[16]  P. Armitage,et al.  A Two-stage Theory of Carcinogenesis in Relation to the Age Distribution of Human Cancer , 1957, British Journal of Cancer.

[17]  P. Armitage.  A Note on the Time-Homogeneous Birth Process , 1953 .

[18]  Yoh Iwasa,et al.  The Evolution of Two Mutations During Clonal Expansion , 2007, Genetics.