Stem cells (SC): low frequency, not accessible to direct observation provide inexhaustible supply of cells Progenitor cells (PC): immediate downstream of SC, identifiable with cell surface markers, provide a quick proliferative response Terminally differentiated cells: mature cells, represent a final cell type, do not divide All cell types are susceptible to death Renewing Cell Population

Example of renewing cell systems Development of Oligodendrocytes Kinetics of Leukemic cells To: Stem cell T1: Glial-restricted precursor (GRP) T2: oligodendrocyte/type-2 astrocyte (O-2A)/oligodendrocyte precursor cell (OPC) (O-2A/OPC) To: Leukemic stem cell (LSC) T1: Leukemic progenitor (LP) T2: Leukemic blast (LB)

Age-dependent Branching Process of Progenitor Cell Evolution without Immigration Evolution of an individual PC from birth to leaving Every PC with probability p has a random life-time with probability 1 - p it differentiates into another cell type At the end of its life, every PC gives rise to v offsprings v characterized by pgf generally, it takes a random time for differentiation to occur

Stochastic processes Z(t), Z(t, x) Z(t) ~ total number of PCs Z(t, x) ~ number of PCs of note that if then, pgfs of Z(t), Z(t, x) Applying the law of total probability (LTP)

Using notations: with initial conditions: From (1) and (2): with initial conditions :

(3) is a renewal type equation with solution: where, and is the k-fold convolution renewal function

Renewal-type Non-homogeneous Immigration Let Y(t) be the number of PCs at time t with the same evolution of Z(t) in the presence of immigration Y(t, x) number of PCs of pgfs of Y(t) and Y(t, x) with initial conditions time periods between the successive events of immigration: i.i.d, r.v.’s with c.d.f. at any given t, number of immigrants is random with distribution

pgf of number of immigrants at time t mean number of immigrants at time t Applying LTP (10) (11) with initial conditions:

(12) (13) with initial conditions: Whenever is bounded, (12) has a unique solution which is bounded on any finite interval The solution is: whereis the renewal function

Neurogenic Cascade d  apoptosis rate  rate of G 2 M in ANP2 differentiating into NB Modeling of Neurogenesis

The m matrix where m ik is the expected number of progeny of type k at time t of a cell of type i d ij is the probability of its corresponding type of cells committed to apoptosis, is the chance that cell differentiated to NB directly from the phase G 2 M in the process of ANP2

Model Construction We obtain the fundamental solution of the model where M is a matrix, with number of cells at time t in compartment j, given the population was seeded by a single cell in compartment i under physiological conditions, the system is fed by a stationary influx of freshly activated ANPs, which may be represented by a Poisson process with constant intensity ω per unit of time, thus we obtain

To calculate the stationary distribution of M we need to derive the inverse matrix of I-m The inverse of an upper triangular matrix is also an upper triangular matrix

Parameters in the inverse matrix are defined as

Example A computational example in simulating pulse labeling experiment with parameters T1=10 hr, T2=8 hr, T3=4 hr, T4=96 hr, T5=1hr, dij=0.15, α=0.5

Comparison suggests non-identical distribution of the mitotic cycle duration for amplifying neuroprogenitors (ANP) and unevenly distributed cell death rates