Modeling multi-mutation and drug resistance: analysis of some case studies

Table 1 Parameters used in simulations in different therapeutic cases introduced above

Resistance Detected	Growth Parameters	Specifications of Drug I	τ ₂	τ _M → R	τ ₁	Specifications of Drug II
Drug-Induced	K_T = K_N = K_R = 10⁶ (cells) r_T = r_r = r_M = 0.25 r_n = 0.5 day⁻¹ κ = 0124 day⁻¹ T^* = 3*10⁵ cells	Constant Drug C = 0.2 (mg.m-²) a _T = 0.15 (day⁻¹) t(start) = 50 days	10⁻³/day Mutation starts at t = 0 day	10⁻⁴/day Conversion starts at t = 50 days	0 (No intrinsic resistance)	0 (No combination therapy)
Drug-Induced	Same as above	Constant Drug C = 0.2 (mg.m-²) a_T = 0.15 (day⁻¹) t(start) = 50 days	10⁻³/day Mutation starts at t = 0 day	10⁻⁴/day Conversion starts at t = 150 days	0 (No intrinsic resistance)	0 (No combination therapy)
Drug-Induced	Same as above	Decaying Drug C = 0.2exp(-0.001 t) (mg.m-²) a_T = 0.15 (day⁻¹) t(start) = 50 days	10⁻³/day Mutation starts at t = 0 day	10⁻⁴/day Conversion starts at t = 50 days	0 (No intrinsic resistance)	0 (No combination therapy)
Drug-Induced and Intrinsic	Same as above	Decaying Drug C = 0.2exp(-0.001 t) (mg.m-²) a_T = 0.15 (day⁻¹) t(start) = 50 days	10⁻³/day Mutation starts at t = 0 day	10⁻⁴/day Conversion starts at t = 50 days	10⁻⁴/day	C₂ = 0.6 (mg.m-²) a_TR = 0.15(day-1) t(start) = 50 days

The description of parameters are: τ ₂: Mutation rate of wild tumor cells to tumor cells with the gene that can potentially go through second mutation to drug-resistant cells as they interact with the drug. τ _M → R: Mutation/ conversion rate of mutated tumor cells with targeted gene to drug-resistant tumor cells. τ ₁: Mutation rate of wild tumor cells to drug-resistant tumor cells. Specifications of Drug I (Effective on wild and mutated tumor cells). Specifications of Drug II (Effective on drug resistant tumor cells). Also, the common parameters related to the tumor growth are: K_N, K_T, K _R: the carrying capacity of normal cells and two types of tumor cells. The per capita growth rate for the drug-responsive tumor cells, two groups of drug-resistant tumor cells, and normal cells are expressed by r_T, r_R, r_M, r_N. The T* is the critical size of the collection of tumor cells. κ is the parameter expresses interaction between tumor and normal cells

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