Question

Chemotherapy In an experimental study at Dartmouth College, mice with tumors were treated with the chemotherapeutic drug Cisplatin. Before treatment, the tumors consisted entirely of clonogenic cells that divide rapidly, causing the tumors to double in size every 2.9 days. Immediately after treatment, 99% of the cells in the tumor became quiescent cells which do not divide and lose 50% of their volume every 5.7 days. For a particular mouse, assume the tumor size is 0.5 cm³ at the time of treatment.
d. Plot a graph of V(t) for 0 ≤ t ≤ 15. What happens to the size of the tumor, assuming there are no follow-up treatments with Cisplatin?

Accepted Answer

Identify the two phases of tumor volume change: before treatment, the tumor doubles every 2.9 days due to rapidly dividing clonogenic cells; after treatment, 99% of cells become quiescent and lose 50% of their volume every 5.7 days. At time $$t=0$$, the tumor volume is $$V(0) = 0.5 cm$$³. Immediately after treatment, 1% of the tumor volume continues to grow as before, and 99% becomes quiescent and shrinks over time. Express the tumor volume $$V(t) as $$the sum of two components for $$t \geq 0$$: the growing part and the shrinking part. The growing part is $$0.01 	imes 0.5 	imes 2$$^{\frac{t}$${2.9}}$$, since it doubles every 2.9 days. The shrinking part is $$0.99 	imes 0.5 	imes \left(\frac{1}{2}\$$right)^{\frac{t}$${5.7}}$$, since it halves every 5.7 days. Write the full expression for $$V(t) as$$:

$$V(t) = 0.01 	imes 0.5 	imes 2$$^{\frac{t}$${2.9}} + 0.99 	imes 0.5 	imes \left(\frac{1}{2}\$$right)^{\frac{t}$${5.7}} To $$plot $$V(t)$$ for $$0 \leq t \leq 15$$, calculate values of $$V(t) at $$various points in this interval using the formula above, then sketch or use graphing software. Observe how the tumor volume changes over time without further treatment.

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Key Concepts

Exponential Growth and Decay

Clonogenic vs. Quiescent Cells

Piecewise Function Modeling