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April  2020, 19(4): 1875-1890. doi: 10.3934/cpaa.2020082

## A mathematical model of chemotherapy with variable infusion

 Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA

*Corresponding author

Dedicated to Prof. Dr. Tomás Caraballo's 60th birthday

Received  June 2019 Revised  September 2019 Published  January 2020

Fund Project: This work was partially supported by Simons Foundation, USA (Collaboration Grants for Mathematicians No. 429717).

A nonautonomous mathematical model of chemotherapy cancer treatment with time-dependent infusion concentration of the chemotherapy agent is developed and studied. In particular, a mutual inhibition type model is adopted to describe the interactions between the chemotherapy agent and cells, in which the chemotherapy agent is modeled as the prey being consumed by both cancer and normal cells, thereby reducing the population of both. Properties of solutions and detailed dynamics of the nonautonomous system are investigated, and conditions under which the treatment is successful or unsuccessful are established. It can be shown both theoretically and numerically that with the same amount of chemotherapy agent infused during the same period of time, a treatment with variable infusion may over perform a treatment with constant infusion.

Citation: Ismail Abdulrashid, Xiaoying Han. A mathematical model of chemotherapy with variable infusion. Communications on Pure & Applied Analysis, 2020, 19 (4) : 1875-1890. doi: 10.3934/cpaa.2020082
##### References:

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##### References:
Chemotherapy with time-dependent infusion $\mu(t) = 4 + 2 \sin 0.04 t$, resulting a successful treatment where all cancer cells are removed and normal cells remain
Chemotherapy with time-dependent infusion $\hat{\mu} = 4$, resulting a failed treatment where all normal cells are removed and cancer cells remain
Description of parameters in the chemotherapy model
 Parameter Description $b_{1}$ (1/time) Per capita growth rate of cancer cells $b_{2}$ (1/time) Per capita growth rate of normal cells $\kappa_{1}$ (mass/vol) Environmental carrying capacity of cancer cells $\kappa_{2}$ (mass/vol) Environmental carrying capacity of normal cells $d_1$ (vol/time$\cdot$mass) Intraspecific competition coefficient of cancer on normal cells $d_2$ (vol/time$\cdot$mass) Intraspecific competition coefficient of normal on cancer cells $r_1$ (1) Consumption effectiveness of cancer cells on the agent $r_2$ (1) Consumption effectiveness of normal cells on the agent
 Parameter Description $b_{1}$ (1/time) Per capita growth rate of cancer cells $b_{2}$ (1/time) Per capita growth rate of normal cells $\kappa_{1}$ (mass/vol) Environmental carrying capacity of cancer cells $\kappa_{2}$ (mass/vol) Environmental carrying capacity of normal cells $d_1$ (vol/time$\cdot$mass) Intraspecific competition coefficient of cancer on normal cells $d_2$ (vol/time$\cdot$mass) Intraspecific competition coefficient of normal on cancer cells $r_1$ (1) Consumption effectiveness of cancer cells on the agent $r_2$ (1) Consumption effectiveness of normal cells on the agent
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