Symmetry (Oct 2021)
Mathematical Analysis for the Effects of Medicine Supplies to a Solid Tumor
Abstract
Objective: 1. Interpretation of the variations of solute medicine amount in blood vessels and TAF concentration with respect to the flow rates of injected drugs into liver and heart. 2. Description of the alteration of tumor cell density versus the time and radius variations. Methodology: Step 1. Compartmental analysis is adopted for the concentration of chemotaxis caused by injected substances L and H based on the assumption: two different medicines I1 and I2 are injected into heart and liver to recover the functions of each organ, respectively, without any side effects. Step 2. A partial differential equation is derived for the growth of TAF considering the diffusion of TAF and the rate of decay of TAF according to the disturbance of medicine M in blood vessels. Step 3. A partial differential equation is derived for the motion of tumor cells in the lights of random motility and chemotaxis in response to TAF gradients. Step 4. Exact solutions are obtained for the concentration of chemotaxis caused by injected substances L and H under the assumption that the loss of mass is proportional to mass itself. Step 5. Exact solution is obtained for the partial differential equation describing the growth of TAF using the separation of variables. Step 6. A finite volume approach is executed to search approximated solutions due to the complexity of the partial differential equation describing the motion of tumor cells. Results: 1. The concentration of medicine (M) decreases as the ratio of flow rate from heart into vessel to flow rate from liver into heart (k1k2) increases. 2. TAF concentration increases with the growth of the value of ratio k1k2 and TAF shows the smallest concentration when the flow rate of each injected medicine is similar. 3. Tumor cells react highly sensitive as soon as medicine supplies and tumor cell’s density is decreased drastically at the moment of medicine injection. 4. Tumor cell density decreases exponentially at an early stage and the density decrease is developed in a fluctuating manner along the radius. Conclusions: 1. The presented mathematical approach has the potential for the profound analysis of the variations of solute medicine amount in blood vessels, TAF concentration, and the alteration of tumor cell density according to the functional recoveries of liver and heart. 2. The mathematical approach may be applicable in the investigation of tumor cell’s behavior on the basis of complex interaction among five represented organs: kidney, liver, heart, spleen, and lung. A mathematical approach is developed to describe the variation of a solid tumor cell density in response to drug supply. The investigation is progressed based on the assumption that two different medicines, I1 and I2, are injected into heart and liver with flow rates k1 and k2 to recover the functions of each organ, respectively. A medicine function system for the reactions of tumor angiogenic factors (TAF) to medicine injection is obtained using a compartmental analysis. The mathematical governing equations for tumor cells motion are derived taking into account random motility and chemotaxis in response to TAF gradients and a finite volume method with time-changing is adopted to obtain numerical solutions due to the complexity of the governing equations. The variation of the flow rates k1 and k2 exerts profound influences on the concentration of medicine, and similar flow rate of k1 and k2 produces the greatest amount of medicine in blood vessels and suppresses strong inhibition in TAF movement. Tumor cells react very sensitively to drug injection and the tumor cell density decreases to less than 20% at an early stage of administration. However, the density of tumor cell diminishes slowly after the early stage of sudden change and the duration for complete therapy of tumor cells requires a long time.
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