Cancer Research Applies a Markov Chain Monte Carlo Approach

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This is an example of a two-state Markov Chain : Image by Joxemai4

Recent cancer research developed a Markov chain model using a Monte Carlo simulation that has promising implications for medical research and eventually treatment of cancer. This leads to a few questions, including – but not limited to:

  • What is a Markov chain model and a what is a Monte Carlo simulation?
  • How does mathematics developed for probability theory and gambling model cancer metastasis?
  • What are the possible implications for cancer treatment?

What is a Markov Chain Model?

In math, a Markov Chain is a “collection of random variables” for which the probability of future values does not depend on previous values.

A simple example is the combination of a coin toss plus the roll of two dice. Each coin toss and each dice roll is independent of their previous results. If the coin and dice are “fair”, then the probability of “heads” equals “tails”. If the dice are lopsided, perhaps the probability of a ‘1’ result would be 1/4 rather than 1/6; but a Markov Chain model often does include different probabilities for different outcomes.

A Markov Chain model must have independent variables. Suppose that we roll dice but attach a wad of chewing gum to the side opposite the last result. That would change the probability of achieving the same result twice; this situation cannot use a Markov Chain model.

Estimate Pi with a Monte Carlo Simulation : image by CaitlinJo

You can estimate pi with a Monte Carlo Simulation : Image by CaitlinJo

What is a Monte Carlo Simulation?

A Monte Carlo simulation generates random numbers to model a process. This is a valuable approach when the situation is too complex to solve with some other mathematical analysis.

For example, it is possible to estimate the value of pi=π=3.14159… by randomly dropping or throwing objects onto a surface with two separate regions. The percentage of “successes”, that land in one region, versus “failures” can relate to the value for π. In this picture, randomly placed dots are either inside the circle or or outside the circle but inside the square.

Suppose the radius of the circle, which equals one-half the length of the side of the square, is ‘1’. Then the area of a circle is “C=π*r2=π”; the area of the square is “S=(2*r)2=4″. The ratio “C/S=π/4” is the ratio of the dots inside the circle to all the dots, if this Monte Carlo simulation is truly random and has a large number of dots.

The Cancer Research Found Spreaders and Sponges

Dr. Newton, et al, reviewed autopsy results from a large number of cancer patients. They set up a Markov Chain model with fifty distinct tumour sites, both primary and metastatic.

The previous one-directional model of cancer metastasis predicted that a primary cancer tumour might send a “seed” to a different part of the body, where it could develop into a metastatic tumour. The assumption was that a tumour would neither “plant a seed” into its current site, nor send a seed back from the metastatic tumour to the primary site.

This research team applied the autopsy data to the Markov Chain model for lung cancer as the primary tumour. They found that the path from lung to adrenal gland to a third site was relatively likely. However, if the lung cancer metastatized to a bone, then a third tumour growth was less likely. This allowed the research team to use the term “spreader” for a secondary cancer likely to send seeds again; versus a “sponge” from which the cancer is less likely to spread.

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