P1253037260qgfMO - PowerPoint PPT Presentation

Title: P1253037260qgfMO


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Renewing Cell Population
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
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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)
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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

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• 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)

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

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

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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

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

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

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Modeling of Neurogenesis
Neurogenic Cascade d ?? apoptosis rate ??
rate of G2M in ANP2
differentiating into NB
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The m matrix where mik is the expected number of
progeny of type k at time t of a cell of type i
dij is the probability of its corresponding type
of cells committed to apoptosis, is
the chance that cell differentiated to NB
directly from the phase G2M in the process of ANP2
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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 ? per unit of time, thus
we obtain
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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
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Parameters in the inverse matrix are defined as
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Example

• A computational example in simulating pulse
  labeling experiment with parameters

T110 hr, T28 hr, T34 hr, T496 hr, T51hr,
dij0.15, a0.5
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Comparison suggests non-identical distribution of
the mitotic cycle duration for amplifying
neuroprogenitors (ANP) and unevenly distributed
cell death rates