Math Insight

During the initial stages of growth, each cancer cell will divide at a random moment in time, with a probability per day of $\lambda = 0.01$ (equivalently, a probability per hour of $0.00041667$ or a probability per week of $0.07$). When a cell divides, it gives birth to a new cell, thereby increasing the number of cells by one. During this stage, each cancer cell will also die at a random moment in time, with a probability per day of $\mu = 0.005$, decreasing the number of cells by one when the death occurs.

Step 1: map from biology to math

Since the cells are dividing and dying at random moments of time (rather than at discrete points in time), we will develop a continuous time model of the cell division and death. We will develop two versions of the model: a stochastic model and a deterministic model. For both models, we will let $c(t)$ be the number of cells at time $t$, measured in days.

1. For the stochastic model, we will use a continuous time birth-death process. In this context, we'll often refer to $\lambda$ as the birth rate and $\mu$ as the death-rate. For convenience, we can use the language of rates, rather than probability per day, saying each cell divides at the rate of $\lambda=0.01$ per day and dies at the rate $\mu=0.005$ per day. The rate language makes even more sense when we talk about the whole population. If there are $c(t)$ cells, then the overall birth rate is $\lambda c(t)$ and the overall death rate is $\mu c(t)$.

   To create a model, we will assume that a birth or death of a cell is independent of the behavior of all other cells, as well as independent of its history. (A newly born cell is just as likely to die or give birth as an older cell, and the timing of its birth or death is not influenced by the times of other cells' births or deaths.) The number $c(t)$ will jump up or down by one when a random event of a birth or death occurs. Since the events are happening in continuous time, there is a probability of zero that two events occur exactly at the same time and cause $c(t)$ to jump up or down by two.

   Write down a stochastic model for the growth of $c(t)$ in continuous time. This model will allow us to march forward in time, changing $c(t)$ at every moment. We formulate the model by assuming that at time $t$, the cell count is $c(t)=n$. Then, we will write down a probability distribution for the number of cells a short time later, conditioned on the fact that $c(t)=n$. Let the length of this short interval be $\Delta t$. Hence, we specify the model by prescribing the conditional distribution $$p(c(t+\Delta t) = m \,|\, c(t) = n).$$ In words, we assign a probability that the number of cells changes from $n$ to $m$ in a short interval $\Delta t$.

   Since we are thinking of $\Delta t$ as being really small, and no two events can occur at the same time, we can ignore the possibility that two events occurred in the time interval $\Delta t$. In your model, allow $m=c(t+\Delta t)$ to be at most one different from $n=c(t)$. Your model should be based on the overall birth rate $c(t)\lambda$ (which you'll write as $\lambda n$) and the overall death rate $c(t)\mu$ (which you'll write as $\mu n$).

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   Hint

   Since $\lambda$ is the birth rate, or the probability per day of a birth, if one multiplies by a short time interval $\Delta t$, then $\lambda \Delta t$ is the probability of a birth happening that that interval. If there are $n$ cells, each with a birth rate of $\lambda$, what is the probability that one of those cells had a birth in a short interval $\Delta t$? If at time $t$, there are $c(t)=n$ cells, how cells are there at time $t+\Delta t$ when there is a birth?

   Since $\Delta t$ is small, it is most likely that there were no births or deaths in the interval. What is the probability that there were no births or deaths?

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2. We can approximate the random process by a deterministic one. Write down a deterministic model for the growth of $c(t)$ in continuous time with growth rate $\lambda$ and death rate $\mu$. In other words, write down a differential equation for $\diff{c}{t}$. Unlike in the random process, $c(t)$ will change continuously, taking on fractional values.
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   Hint

   As in the probabilistic model, the overall birth rate $\lambda c(t)$ and the overall death rate is $\mu c(t)$ are the rates that will influence the change in $c(t)$.

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1. It's easier to analyze and simulate the stochastic model by first combining the birth and death rates together into a total rate of events, and then later determining whether the event was a birth or a death. For a single cell, what is the rate at which birth or death occur? If there is a population of $c(t)=n$ cancer cells, what is the total rate at which events occur? What will be the average amount of time between events?

   The amount of time between events (also called the waiting time) will be a random variable. What type of random variable will this time interval be?

2. Given that an event occurred, what is the probability that the event is a birth? What is a probability the event is a death? Express these probabilities in terms of the parameters $\lambda$ and $\mu$; also, give numbers for the values of the parameter that we are considering. (These probabilities must add up to one, since we know an event did occur.)
3. Solve the deterministic approximation to the cancer growth to obtain a formula for $c(t)$ as a function of the initial condition $c(0)=c_0$ and the rates $\lambda$ and $\mu$.
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   Hint

   The differential equation for the deterministic model should be a simple linear differential equation whose solution is exponential growth or decay.

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4. Simulate the stochastic model with initial condition $c(0)=1$ cancer cell for 365 days. Since the results will be random, simulate the random process 20 times with the same initial condition. Plot all 20 curves on the same plot of $c$ versus $t$. Note how many of those 20 simulation resulted in extinction, i.e., where all the cancer cells died. On the plot, include the solution to the deterministic equation with the same initial condition $c(0)=1$.

   If you like, you can use the provided cancer growth R script. The script is missing 3 important details, which happen to be exactly what you calculated on in the previous 3 parts of this question. See the comments at the beginning of the file for how to complete the script with your answers so that the script will run.

5. Repeat the simulation for two more initial conditions: $c(0)=10$ cells and $c(0)=100$ cells. In each case, note how many of the 20 simulation resulted in extinction and include the deterministic solution on the same plot.
6. (optional) If we start with just $c(0)=1$ cancer cell, what is the probability that the population of cancer cells will go extinct? If we start with $c(0)=10$ cells? $c(0)=100$ cells?

1. For the 20 simulations with the initial condition of one cell, how much did the first year's result vary among the simulations? What you would interpret is the prognosis for someone who has one of these cancer cells?

   How well did the deterministic model capture the dynamics of the probabilistic model?

2. As the initial condition was increased to $c(0)=10$ and $c(0)=100$, what trend did you observe in the variability of the first year's result? What you would interpret is the prognosis for someone who already has 10-100 of these cancer cells?

   How did the accuracy of the deterministic model change as the initial condition was increased? Why did the accuracy change? Under what conditions can one get reasonable results by using a deterministic approximation?

3. We can use the deterministic model to gauge how fast the cancer is growing. Let's quantify the speed in two ways. First, calculate by what percent the number of cells increases in each month (say 30 days). Second, calculate how long it takes for the number of cells to double.

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.

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 cells 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”.)

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.)

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?

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.