Inference for an age-dependent,multitype branching-process model of mast cells

We consider an age-dependent, multitype model for the growth of mast cells in culture. After a colony of cells is established by an initiator type, the two possible types of cells are resting and proliferative. Using novel inferential procedures, we estimate the generation-time distribution and the offspring distribution of proliferative cells, and the waiting-time distribution of resting cells.List of Notations B _i cumulative distribution function for the time until branching of a cell of type i - b _i probability density function for the time until branching of a cell of type i - b _i b _i (1–D _i ) - D _i cumulative distribution function for the time until death of a cell of type i - d _i probability density function for the time until death of a cell of type i - probability density function of a gamma distribution - G _i cumulative distribution function for the lifetime of a cell of type i - G _1*2 Convolution of G ₁ and G ₂ - ¯G _i 1–G _i - g _i probability density function for the lifetime of a cell of type i - L _i likelihood of a history of type i - m average number of proliferative daughters produced by dividing cells - M _ij (t) the expected number of type-j cells in a colony at time t if that colony began at time 0 with one type-i cell - M _i+ (t) M _i0 (t) + M _i 1(t) + M _i 2(t) - p _rs probability that a dividing cell produces r proliferative and s resting daughters - t _i times defining colony histories. See IV.2.1 - T ₀ time to division of an initiator cell - T ₁, T ₂ times from birth to division of the two daughters of an initiator cell - T ₍₁₎, T ₍₂₎ order statistics of T ₁ and T ₂ - minimum value of a gamma distribution - scale parameter of a gamma distribution or of an exponential distribution - probability per unit time of death for proliferative and resting cells - _rs expected value of p _rs when there is heterogeneity - shape parameter of a gamma distribution