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.
a. Find an exponential decay function V₁(t) that equals the total volume of the quiescent cells in the tumor t days after treatment.

Accepted Answer

Identify the initial volume of the quiescent cells immediately after treatment. Since 99% of the tumor cells become quiescent, multiply the total tumor volume at treatment (0.5 cm³) by 0.99 to find the initial volume of quiescent cells: $$V_0 = 0.5$$ 	imes 0.99$$ cm³. Recognize that the quiescent cells decrease in volume exponentially over time, losing 50% of their volume every 5.7 days. This means the volume halves every 5.7 days, which is a characteristic of exponential decay. Write the general form of the exponential decay function for volume V_1$(t) as$$:

$$V_1(t$$) = $$V_0$$ 	imes $$2^{-\frac{t}$${T}}$$

where T$ is the half-life (5.7 days) and t$ is the time in days after treatment. Substitute the known values into the formula:

V_1$(t) = (0.5 	imes 0.99) 	imes 2$$^{-\frac{t}$${5.7}}. $$This function $$V_1(t$$)$$ now models the total volume of quiescent cells in the tumor at any time t$ days after treatment.

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Key Concepts

Exponential Decay Function

Decay Constant and Half-Life Relationship

Initial Conditions and Volume Partitioning