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# Tumor growth dynamics with nutrient limitation and cell proliferation time delay

• It is known that avascular spherical solid tumors grow monotonically, often tends to a limiting final size. This is repeatedly confirmed by various mathematical models consisting of mostly ordinary differential equations. However, cell growth is limited by nutrient and its proliferation incurs a time delay. In this paper, we formulate a nutrient limited compartmental model of avascular spherical solid tumor growth with cell proliferation time delay and study its limiting dynamics. The nutrient is assumed to enter the tumor proportional to its surface area. This model is a modification of a recent model which is built on a two-compartment model of cancer cell growth with transitions between proliferating and quiescent cells. Due to the limitation of resources, it is imperative that the population values or densities of a population model be nonnegative and bounded without any technical conditions. We confirm that our model meets this basic requirement. From an explicit expression of the tumor final size we show that the ratio of proliferating cells to the total tumor cells tends to zero as the death rate of quiescent cells tends to zero. We also study the stability of the tumor at steady states even though there is no Jacobian at the trivial steady state. The characteristic equation at the positive steady state is complicated so we made an initial effort to study some special cases in details. We find that delay may not destabilize the positive steady state in a very extreme situation. However, in a more general case, we show that sufficiently long cell proliferation delay can produce oscillatory solutions.

Mathematics Subject Classification: Primary:34K20, 92C50;Secondary:92D25.

 Citation: • • Figure 1.  Bifurcation diagrams of model (2.4) with $f(r)= \frac{kr}{ar+1}$ and $g(r)= \frac{c}{r+m}$ using the cell proliferation time delay $\tau$ as the bifurcation parameter. The parameter values are $k=2, a=1, m=2, \mu=0.1, c=1, \theta=2/3.$ The positive steady state appears to be globally attractive for short time delay but lost its stability for larger values of $\tau$. As cell proliferation time delay increases, tumor size oscillates more noticeably and at a lower lever. Notice that the percentage of the average amount of proliferating cells decreases as $\tau$ increases

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