  Studying the development of malignant tumours, it is important to know and
predict the proportions of different cell types in tissue samples. Knowing the
expected temporal evolution of the proportion of normal tissue cells, compared
to stem-like and non-stem like cancer cells, gives an indication about the
progression of the disease and indicates the expected response to interventions
with drugs. Such processes have been modeled using Markov processes. An
essential step for the simulation of such models is then the determination of
state transition probabilities. We here consider the experimentally more
realistic scenario in which the measurement of cell population sizes is noisy,
leading to a particular hidden Markov model. In this context, randomness in
measurement is related to noisy measurements, which are used for the estimation
of the transition probability matrix. Randomness in sampling, on the other
hand, is here related to the error in estimating the state probability from
small cell populations. Using aggregated data of fluorescence-activated cell
sorting (FACS) measurement, we develop a minimum mean square error estimator
(MMSE) and maximum likelihood (ML) estimator and formulate two problems to find
the minimum number of required samples and measurements to guarantee the
accuracy of predicted population sizes using a transition probability matrix
estimated from noisy data. We analyze the properties of two estimators for
different noise distributions and prove an optimal solution for Gaussian
distributions with the MMSE. Our numerical results show, that for noisy
measurements the convergence mechanism of transition probabilities and steady
states differ widely from the real values if one uses the standard
deterministic approach in which measurements are assumed to be noise free.
