Conflicting Biomedical Assumptions for Mathematical Modeling: The Case of Cancer Metastasis
Heat maps that visualize the transition probability matrices in our Markov model of metastasis after n Markov chain transitions (n = 1, 2, 10); and a plot of probabilities associated with reaching each state after n chain transitions.
(A) Stochastic matrices are organized as in Figure 7G, H and I: each stochastic matrix is square, non-negative and organized such that the ith row of the matrix provides probabilities of transitions from state i to all other states in the model and the sum of each row is equal to 1. The three matrices show the transition probabilities from states i to state j after one (P1), two (P2), and ten (P10), transition steps of the Markov process. The labels, “Start” and “End” correspond to the generating and absorbing states of the Markov chain, respectively. The rest of the labels indicate metastasis stages that were suggested by the authors (in blue) and by individual experts (in red). (B) The probability of finding the Markov process in a given state at transition n (n = 1, 2, …, 50) is given by . Here Λ is the distribution over all states at the beginning of the chain, and Pn = [pij(n)] is the transition probability matrix after the nth transition (obtained by raising matrix P to nth power). This figure demonstrates the decreasing probability that expert intuition about metastasis lands in any particular state at any particular stage. Note that even after 50 transitions there is a substantial probability that the chain would not reach the “End” state.