Video: A stochastic process introduction
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A stochastic process introduction
Let's think about possible models for the growth of a tumor over time.
We'll let n(t) be the number of tumor cells at time t. We'll let lambda be the growth rate of the cell population.
One approach is to use a differential equation model. A simple linear model is
dn/dt = lambda n
with initial condition n(0) = n_0
We can solve this differential equation for the number of cells as a function of time.
n(t) = n_0 * e^(lambda t)
Simple enough. Now we have function that will give us the number of cells at any time.
What if we solved the equation again with the same initial condition n(0) = n_0?
We'd get the same solution.
This model is what we call a deterministic model. There is no source of randomness, so we get the same answer every time.
We can plot the number of tumor cells n(t) versus time. It is a nice smoothly growing cell population.
How do the results of this model compare to real cells?
Here's a summary of some data of the growth of a population of cells. In this case, the cells are Paramecium cells.
This plot has three curves from three replicates of the same experiment. In constrast to our differential equation solution, the data jumps up and down and the population growth was varied among the experiments.
The variation among the experiments might be explained by different external factors. However, in these experiments, the external conditions were carefully controlled, so the variations are likely not due to different external conditions but are instead a sign that the growth process may include sources of randomness.
Another difference from our differential equation solution is that the data has only integer numbers of cells. Since the deterministic solution was continuous, we have to allow 3.7 cells as the population size is increasing.
To better model tumor cell growth when we have a small number of cells, we will develop a stochastic model, i.e., a model where the growth occurs via a random process.
In this stochastic model, we'll use capital letters for the random quantities, and let capital N(t) be the number of cells at time t. The number of cells will always be an integer and will jump up by one at the random times when a new cell is born.
To derive the stochastic model, first recall the differential equation model, where dn/dt = lambda n
Let's create a stochastic version this growth model.
First, let Delta t be a small interval of time. We'll denote by Delta N the change in the number of tumor cells during that small interval of time. Our goal is to have a rate of change analogous to the differential equation model, so we start by letting Delta N over Delta t be equal to lambda times N.
We solve for the change in number of cells Delta N by multiplying through by Delta t so that Delta N is lambda times N times interval length Delta t. This result seems reasonable, as the increase in the number of cells scales with the length of the interval. If you double the length of the interval, you double the change, at least if we keep the interval short.
But, one important criterion is that we need to the number of cells to be an integer; the change Delta N must be an integer. This expression is certainly not an integer, especially when Delta t is small.
To keep keep N an integer, we'll interpet this condition in a stochastic or random sense. Rather than N increasing exactly by this amount, it will increase by a random integer which, on average will be this amount.
To accomplish this, we let the probability that N(t) increases by 1 during the inteval Delta t be lambda times N times Delta t.
We are implicitly assuming that the interval Delta t is sufficiently small so that N(t) will never increase by more than 1 during the interval. It will either increase by one or stay at the same value.
To derive the stochastic model, let's be concrete and suppose that at time t, we have N(t)=5 cells. How many cells could we have after a time interval Delta t? (Remember that Delta t is assumed to be small.)
In this case, we have only two options. Either one cell was born so that we now have 6 cells. Or no cells were born so that we still have 5 cells.
What is the probability that a cell was born. According to our model, it will be lambda times N times the time interval Delta t. Since we are assuming that we have 5 cells, the probability will be lambda times 5 times Delta t. (Note how, if we assume Delta t is small enough, this number will be less than one, so it can be a probability.)
We write the result as a conditional probability on the value of N(t+Delta t) at the end of the interval conditioned on the previous value N(t). The probability that N(t+Delta t) is six, conditioned on the fact that N(t) is five, is lambda times five times Delta t.
What about the probability that a cell was not born. Since we are assuming the only possiblities are that one cell was born or zero cells were born, the probability that zero cells were born must be one minus the probability that one cell was born, or one minus lambda times 5 times Delta t. The probability that N(t+Delta t) is five, conditioned on the fact that N(t) is five, is one minus lambda times five times Delta t.
Of course, we don't really just care about the case where we are starting with 5 cells. We'll use the parameter lower case n (to distinguish it from the random quantity capital N(t)) instead of the number 5.
Our stochastic model becomes: The probability that capital N(t+Delta t) is lower case n plus one, conditioned on the fact that capital N(t) is lower case n, is lambda times n times Delta t.
The probability that N(t+Delta t) is n, conditioned on the fact that N(t) is n, is one minus lambda times n times Delta t.
The first equation is the probability that one cell is born; the second equation is the probability that no cells were born in the interval from time t to time t plus Delta t.
We have derived a stochastic birth model for tumor cell growth. We should add an initial condition, just like we do with a deterministic model. Let's denote the initial conditon by n-nought, so that the number of cells at time zero is n-nought.
Note that this model has only two parameters. The growth rate lambda and the initial condition n-nought. Let's give them values so we can look at simulations. We let lambda equal 0.2 and n-nought be 2.
It might look like Delta t is a parameter, but it's really just an abstract quantity that we have to use to define the model. Think of Delta t as being arbitrarily small. The equations have to hold for any sufficiently small interval Delta t.
Let's simulate the model and see what happens. Here are the results from one simulation. Starting with 2 tumor cells, we see that, at random times, the cell count jumped to three, four, and then five.
If we repeat the simulation, we won't get the same results, as the model is fundamentally stochastic. In this second simulation, we observe only one birth, while in the third simulation, we again observe three cell births, but the timing is dramatically different than in the first simulation.
A summary of 10 simulations gives more of an idea of the variation of this stochastic system.
We can compare these simulations to the solution of the deterministic system with the same average birth rate lambda equals 0.1 and the same initial condition n-nought equals 2. We see that the smooth curve of the deterministic dynamical system does capture the general behavior of the aggregate stochastic simulations, though its continuous solution doesn't behave like any of the individual stochastic simulations.
The stochastic model is more realistic than the deterministic model, as it captures the fluctuations in the number of cells and always has an integer number of cells. In that case, why would one ever use a deterministic model?
The best model to use would depend on your objectives for the model. Maybe the fluctuations aren't important for the problem you are interested in. In that case, maybe the fact that the deterministic model is simpler and easier to understand might outweigh the benefits of a stochastic model. In choosing an appropriate model, there's typically a tradeoff between the complexity of the model and one's ability to analyze and understand the model. Often a good question to ask is what's the simplest model you can use that captures what you need?