Math Insight

Tumor growth project, part 1

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A small cancerous mass has begun to grow in a person's body. For this project, we will first analyze the cancer growth in the initial stages. Once it has grown into a sizable tumor, we will analyze the influence of drugs on tumor growth and examine the likelihood of the emergence of drug-resistant tumor cells.

The overarching questions of this project are:

  1. What factors influence the likelihood that cancer will disappear?
  2. What features of a drug will decrease the likelihood that drug-resistant cells proliferate?

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.

  1. 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 $\mu n$).

      Need help?
    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.
      Need help?

  2. Step 2: analyze the model
    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 $\mu$.
      Need help?
    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?

  3. Step 3: interpret the model analysis biologically
    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.

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Math 2241, Spring 2016