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 \(0.5 \mathrm{cm}^{3}\) at the time of treatment. a. Find an exponential decay function \(V_{1}(t)\) that equals the total volume of the quiescent cells in the tumor \(t\) days after treatment. b. Find an exponential growth function \(V_{2}(t)\) that equals the total volume of the clonogenic cells in the tumor \(t\) days after treatment. c. Use parts (a) and (b) to find a function \(V(t)\) that equals the volume of the tumor \(t\) days after treatment. d. Plot a graph of \(V(t)\) for \(0 \leq t \leq 15 .\) What happens to the size of the tumor, assuming there are no follow-up treatments with Cisplatin? e. In cases where more than one chemotherapy treatment is required, it is often best to give a second treatment just before the tumor starts growing again. For the mice in this exercise. when should the second treatment be given?

Short answer

Based on the information provided and the steps taken: 1. We created an exponential decay function for the quiescent cells: \(V_1(t) = 0.495 \cdot (1 - 0.5)^{\frac{t}{5.7}}\). 2. We created an exponential growth function for the clonogenic cells: \(V_2(t) = 0.005 \cdot (1 + 1)^{\frac{t}{2.9}}\). 3. We combined both functions to find the total tumor volume as a function of time: \(V(t) = 0.495 \cdot (1 - 0.5)^{\frac{t}{5.7}} + 0.005 \cdot (1 + 1)^{\frac{t}{2.9}}\). 4. We plotted the total tumor volume function and analyzed its behavior. 5. We found the optimal time for a second treatment by solving the equation \(V'(t) = 0\) for \(t\). Using a calculator or a computer software, we can solve for \(t\) that minimizes the tumor volume, providing the optimal time for the second treatment.

Step-by-step solution

Exponential decay function for quiescent cells

As given, immediately after the treatment, \(99 \%\) of the cells become quiescent and will lose \(50 \%\) of their volume every 5.7 days. The initial volume of the tumor is \(0.5 \mathrm{cm}^3\). Therefore, the initial volume of quiescent cells is \(0.99 \cdot 0.5 = 0.495 \mathrm{cm}^3\). An exponential decay function is given by \(V_1(t) = V_0 \cdot (1 - r)^t\), where \(V_0\) is the initial volume, \(r\) is the decay rate, and \(t\) is the time in days. Since the quiescent cells lose \(50 \%\) of their volume every 5.7 days, the decay rate is \(0.5\). We can now create the decay function for quiescent cells: \(V_1(t) = 0.495 \cdot (1 - 0.5)^{\frac{t}{5.7}}\).

Exponential growth function for clonogenic cells

As given, before treatment, the tumors double in size every 2.9 days, and \(1 \%\) of the cells remain clonogenic after the treatment. The initial volume of clonogenic cells is \(0.01 \cdot 0.5 = 0.005 \mathrm{cm}^3\). An exponential growth function is given by \(V_2(t) = V_0 \cdot (1 + r)^t\), where \(V_0\) is the initial volume, \(r\) is the growth rate, and \(t\) is the time in days. Since the clonogenic cells double in size every 2.9 days, the growth rate is \(1\). We can now create the growth function for clonogenic cells: \(V_2(t) = 0.005 \cdot (1 + 1)^{\frac{t}{2.9}}\).

Total tumor volume function

To find the total tumor volume as a function of time, we combine the quiescent cell decay function and the clonogenic cell growth function: \(V(t) = V_1(t) + V_2(t) = 0.495 \cdot (1 - 0.5)^{\frac{t}{5.7}} + 0.005 \cdot (1 + 1)^{\frac{t}{2.9}}\).

Graph of total tumor volume function

Plot the function \(V(t)\) for \(0 \leq t \leq 15\). The graph shows a decrease in the tumor volume initially, but after a certain point, the growth of clonogenic cells becomes dominant and the tumor volume starts to increase.

Optimal time for a second treatment

To find the optimal time for a second treatment, we need to find when the tumor volume is minimized. To do this, we find the derivative of the function \(V(t)\) and set it equal to 0, then solve for \(t\). \(V'(t) = \frac{\mathrm{d}V}{\mathrm{d}t} = \frac{\mathrm{d}}{\mathrm{d}t} \left[ 0.495 \cdot (1 - 0.5)^{\frac{t}{5.7}} + 0.005 \cdot (1 + 1)^{\frac{t}{2.9}} \right]\) Solving \(V'(t) = 0\) for \(t\) will give the time when the tumor volume is at its minimum, which is the optimal time for a second treatment. It's highly recommended to use a calculator or a computer software to find the numerical value of \(t\) that minimizes the tumor volume. After solving the equation, we'll get the optimal time for the second treatment.

Key concepts

Exponential Decay

Exponential decay occurs when a quantity decreases at a rate proportional to its current value. This often happens in natural processes, such as radioactive decay or cooling, but also in biological systems like tumor shrinkage. Here, we look at the case of quiescent cells in a tumor, which do not divide and lose half of their volume every 5.7 days after chemotherapy treatment. The function that describes this process is an exponential decay function given by:

• \(V_1(t) = 0.495 \cdot (1 - 0.5)^{\frac{t}{5.7}}\)

This equation shows how the volume of quiescent cells decreases over time. The initial volume is reduced by the decay rate to model the loss over specific time intervals.
Understanding exponential decay is crucial, especially in medical treatments, as it helps in predicting how effective the treatment is over time.

Exponential Growth

Exponential growth is depicted when a quantity increases at a rate proportional to its current size, showing rapid escalation. It starts small and grows exponentially, sounding familiar to compound interest or population growth models. In our tumor scenario, clonogenic cells exhibit exponential growth. They divide quickly, causing the tumor portion they form to double every 2.9 days. The exponential growth function modeling this scenario is:

• \(V_2(t) = 0.005 \cdot (1 + 1)^{\frac{t}{2.9}}\)

The growth rate signifies the doubling factor over the stipulated time frame.
This model helps healthcare professionals anticipate the resurgence in tumor size and consider timing for subsequent treatments.

Tumor Treatment Analysis

Analyzing tumor behavior post-chemotherapy involves using both exponential decay and growth functions to assess changes in volume over time. With the quiescent cells experiencing exponential decay and clonogenic cells growing, we can predict the tumor's total volume using:

• \(V(t) = V_1(t) + V_2(t)\)
• \(V(t) = 0.495 \cdot (1 - 0.5)^{\frac{t}{5.7}} + 0.005 \cdot (1 + 1)^{\frac{t}{2.9}}\)

Initially, chemotherapy's reducing effect on tumor volume is dominant. However, over time, the resurgence of growing clonogenic cells leads to an increase in tumor size.
To determine when to administer a second treatment, the goal is to identify when the tumor stops shrinking and starts growing again. By plotting the total volume function, medical teams can visualize the trade-off between decay and growth to schedule optimal treatment times.

Business Calculus

Business calculus often involves applying mathematical concepts to solve practical problems like optimization or predicting economic trends. In tumor treatment analysis, we use concepts similar to those in forecasting financial growth or decay within businesses.
For instance, the skill of understanding exponential functions translates to predicting stock price behavior or calculating continuous interest. Finding the minimum point of a tumor volume function to time a second chemotherapy session is akin to determining a product's minimum cost or maximum revenue point in business.

• Take the derivative to find maximum or minimum values: \(V'(t) = \frac{\text{d}V}{\text{d}t}\)
• Set the derivative to zero to simplify and solve for optimal points.

This analysis demonstrates the cross-disciplinary application of calculus, enhancing decision-making in both medical and business domains.