Math Insight

Tumor growth project, part 2

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Total points: 30

For the second part of the tumor growth project, we imagine a cancerous tumor was detected after it reached a sizable mass, containing $c=10,000,000$ cells. At that point, to fight the cancer, a drug is administered to kill the cancer cells, increasing their death rate by a factor of three, so that the probability per day of a cancer cell dying becomes $\mu =0.015$.

However, the fear is that the application of the drug may lead to the emergence of cancer cells that are resistant to the drug. Given the presence of a drug, there is a tiny chance that, upon cell division, one of the daughter cells might contain a mutation that produces drug resistance. The probability that a division produces one drug-resistant cell is $m = 10^{ -8 }$. Since the drug-resistant cells don't respond to the drug, their probability of death per day is $0.005$, as was the case for all cancer cells before the drug was administered.

  1. Step 1: map from biology to math
    1. Given that the initial condition is a huge number of cancer cells, must we use a stochastic model for the dynamics of the number of cancer cells $c(t)$? Or, would be it be sufficient to use use a deterministic model? Explain your reasoning.
    2. Write down a deterministic model, i.e., a differential equation, for the dynamics of the number of cancer cells $c(t)$. Let $t$ be time in days since the drug was administered, so the initial condition is $c(0)=10,000,000$.

      Ignore any mutations at this point. Since we are simply investigating the emergence of drug-resistant cells, we'll assume that the number of drug-resistant cells, if there are any, is so small as to have virtually no effect on the dynamics of the number of cells. In the model, you can imagine that $c(t)$ represents the number of cancer that are susceptible to the drug. (We would have to change the model to investigate the dynamics in the case where the number of drug-resistant cells became large.)

    3. Given that there are $c(t)$ (susceptible) cells at time $t$, write down an expression for the rate of cell divisions per day. (If you actually plugged in a large value of $c(t)$, such as the initial condition, into your expression, the division rate will be larger than one. Even so, we can still think of it as a probability rate, or the probability per day of a cell division. It might be easier to imagine we were talking about the probability per millisecond of a cell division; in that case, number would be smaller than one.)

      Given that each of these cell divisions could produce a drug-resistant cell with probability $m$, write an an expression for the probability per day that a drug-resistant cell emerges. This expression should be a function of the number of cell $c(t)$. Denote this mutation rate as $\nu_m(c(t))$. (The symbol $\nu$ is the Greek letter “nu”.)

  2. Step 2: analyze the model
    1. Solve the model for $c(t)$.
    2. Now that you have determined the mutation rate $\nu_m(c(t))$ as a function of the number of cells $c(t)$ and solved the model for $c(t)$, you can calculate the probability that a drug-resistant cell emerges during at interval of time. You just need to “add up” the mutation probability in the interval to get the total probability. Since we are working in continuous time “adding up” means integrating.

      Calculate the probability that a drug-resistant mutation occurs during the first year, i.e., in the interval $0 \le t \le 365$: $$\int_0^{365} \nu_m(c(t)) dt.$$

      How many cancer cells are there after 1 year?

    3. Calculate the probability that a drug-resistant mutation occurs during the second year, i.e., in the interval $365 \le t \le 730$: $$\int_{365}^{730} \nu_m(c(t)) dt.$$

      How many cancer cells are there after 2 years?

    4. Calculate the probability that a drug-resistant mutation any time from year 3 through year 7, i.e., in the interval $730 \le t \le 2555$: $$\int_{730}^{2555} \nu_m(c(t)) dt.$$

      How many cancer cells are there after 7 years?

    5. Calculate the probability that a drug-resistant mutation any time in the first seven years, i.e., in the interval $0 \le t \le 2555$: $$\int_{0}^{2555} \nu_m(c(t)) dt.$$

    6. (optional) These probabilities of mutation slightly overstate the likelihood that drug-resistant cells become a problem. Just because a single drug-resistant cell appears, it doesn't mean it will actually proliferate. Since drug-resistant cells have die with a probability per day of 0.005, there is a probability that, starting with a single cell, the drug-resistant population of cells might become extinct. The parameters for a drug-resistant cell are identical to the cancer cells before introduction of the drug. If you calculated the probability that a single cancer becomes extinct in part 1 of this project, you can multiply the 7 year probability of mutation by one minus this extinction probability to calculate the probability that a drug-resistant cell emerges and then proliferates (i.e., its cell lineage doesn't go extinct.)

  3. Step 3: interpret the model analysis biologically
    1. After the drug treatment, how fast does the tumor shrink? Let's quantify the speed in two ways. First, calculate by what percent the number of cells decreases in each month (say 30 days). Second, calculate how long it takes for the number of cells to be cut in half.
    2. In which year is the chance of a mutation leading to drug-resistance the highest? Why is this probability so much higher in one year than in the other years?
    3. Given the number of cells left after 7 years, is the deterministic model valid for much longer than 7 years? Why or why not? How many cells does the deterministic model predict after 10 years?

      If one wanted to model the tumor shrinking beyond 7 years, what type of model should one use?

  4. Step 4: compare different drugs

    In addition to the drug analyzed so far (we'll call it drug A), two other drugs are available.

    • Drug B increases the death rate by only a factor of two to $\mu=0.01$ but at the same time cuts the division rate in half to $\lambda = 0.005$.
    • Drug C slightly increases the death rate to $\mu=0.007$ and decreases the division rate by a factor of five to $\lambda = 0.002$.

    You goal is to compare the risk of a mutation leading to drug-resistance for these three drugs.

    1. For each drug, write down and solve a deterministic model describing the evolution of the number of cancer cells $c(t)$ with initial condition $c(0)=10,000,000$. How do these models compare to the deterministic model with drug A? Compare the effectiveness of each of the three drugs for shrinking the tumor; which drug(s) cause the tumor to shrink the fastest?

    2. Assume that the probability of a cell division leading to a drug-resistant daughter cell is the same for all drugs; it is $m=10^{-8}$. Calculate the likelihood of the emergence of a drug-resistant cell in the first seven years for both drug B and drug C.

    3. How do these probabilities compare to that from drug A? Which drug(s) have the highest and lowest chance of leading to a drug-resistant cancer? Explain what causes the difference.
    4. Imagine that two drugs shrink the tumor at the same rate and have the same probability of a cell-division leading to drug resistance. If one drug act primarily by decreasing the birth rate and the other acts primarily by increasing the death rate, which drug will lead to a higher likelihood of the emergence of drug-resistant cells? Explain your reasoning.