Question

Patients in a clinical trial for a new cancer treatment are classified as permanently cured, in remission (the cancer is present, but not growing), sick (the cancer is growing), or deceased. After one year of treatment, 1/3 of sick patients go into remission, another 1/3 are permanently cured, and, unfortunately, 1/3 are deceased. Patients in remission also receive treatment after a year of which 1/4 remain in remission, 1/2 are permanently cured, and 1/4 decline to the sick condition

 (a) Formulate a Markov Chain model for the health of a patient in the trial described above.

 (b) Does your model from part (a) have a steady-state distribution? Explain why or why not?

 (c) Use your model from part (a) to determine the probability that a patient in the trial is eventually cured.

 (d) Use your model from part (a) to determine the average amount of time patients in the trial receive the new treatment.

Answer 1

(a)

Let the four states of the Markov chain be C (permanently cured) , R (remission), S (Sick) and D (deceased)
Once the patients reaches the state C or D, he/she will remain in the state. So, State C and D are absorbing states. The transition probability from State C to State C is 1. The transition probability from State D to State D is 1.

The transition probability from State S to State R is 1/3. The transition probability from State S to State C is 1/3. The transition probability from State S to State D is 1/3.

The transition probability from State R to State R is 1/4. The transition probability from State R to State C is 1/2. The transition probability from State R to State S is 1/4.

So, the transition matrix of the Markov chain model is,

(b)

As, the markov model contains absorbing states, the model would not have steady state distribution.

(c)

The matrix P is in canonical form. So, matrix Q and R from the matrix P are

where i is 2x2 unit matrix

The elements of matrix B (bij) gives the probability that an absorbing chain will be absorbed in the absorbing state sj if it starts in the transient state si.

So, the probability that a patient in the sick state is cured is 0.625 and the probability that a patient in the remission state is cured is 0.875

(d)

Let ti be the expected number of steps before the chain is absorbed, given that the chain starts in state si, and let t be the column vector whose ith entry is ti. Then t = Nc where c is the column vectors with all entries as 1.

So, the average amount of time for a sick patient to receive the new treatment is 1.625 years and the average amount of time for a patient in remission state to receive the new treatment is 1.875 years