A Mathematical Model of Cancer Stem Cell Lineage ...€¦ · (i 1)th mutation class will produce, by acquiring a mutation during division, a cell in the j thtelomere, i mutation class.

Biological PreliminariesThe Mathematical Model

Discussion

A Mathematical Model of Cancer Stem CellLineage Population Dynamics with Mutation

Accumulation and Telomere LengthHierarchies

Georgi Kapitanov

Vanderbilt University

Feb 17, 2012

Georgi Kapitanov A Mathematical Model of Cancer Stem Cell Lineage Population Dynamics with Mutation Accumulation and Telomere Length Hierarchies


Discussion

Outline

1 Biological PreliminariesTelomeresStem Cells and DifferentiationCell Mutations and Cancer

2 The Mathematical ModelThe ModelModel analysis

3 Discussion



Discussion

TelomeresStem Cells and DifferentiationCell Mutations and Cancer

Telomeres and Cell Division

Definition: repeated sequence of DNA that protectsimportant DNA during the process of cell division.Cell Division leads to loss of telomeres.

Figure: The process of asymmetrical telomere shortening as a cell divides



Discussion


Stem Cells

Properties of stem cell: self-renewal, ability to differentiate.Progenitor cells: medium stage of differentiation.Mature (differentiated) cells: they have specific functions.



Discussion


Mutation accumulation

Vogelgram - represents the sequence of mutations in a cellthat eventually leads to a cancerous cell.

Figure: A Genetic Model for Colorectal Tumorigenesis. This is an example of a Vogelgram - multistepcancer progression model (http://www.hopkinscoloncancercenter.org)



Discussion

The ModelModel analysis

Questions to address

Considering cell mutation as a dynamic populationprocess, rather than a one-time random event, what canwe show about cancer cell population growth in relation tothe growth of the populations of non-cancer cells?What is the role of stem cells in the cell populationdynamics?Is the cancer stem cell count as small as scientists haveclaimed (some results claim that only one in ten thousandcancer cells is a cancer stem cell[32][4])?



Discussion


Equations

∂uj,i(a, t)∂t +

∂uj,i(a, t)∂a = −(µj,i(a) + βj,i(a))uj,i(a, t)

uj,i(0, t) = 2n∑

k=j

(pj,k ,i

∫ ∞0

βk ,i(a)uk ,i(a, t)da +

qj,k ,i−1

∫ ∞0

βk ,i−1(a)uk ,i−1(a, t)da)

uj,i(a,0) = φj,i(a)



Discussion


Explanation of the Terms

j = 1, ...,n represents the number of telomeres of a cell.

i = 0, ...,m − 1 is the number of mutations a cell hasaccumulated.

For t ≥ 0,uj,i (a, t) ∈ L1([0,∞)), represents the density of cellswith age a at time t , in the j th telomere class, with i mutations.

µj,i (a) ≥ 0, is the age-specific mortality rate of cells in the j th

telomere, i th mutation class.

βj,i (a) > 0, is the age-specific proliferation rate of cells in the j th


pj,k,i > 0, is the probability that one of the daughters of a cell inthe k th telomere, i th mutation class will be a cell in the j th


qj,k,i−1 > 0, is the probability that a cell in the k th telomere,(i − 1)th mutation class will produce, by acquiring a mutationduring division, a cell in the j th telomere, i th mutation class.



Discussion


Hypotheses

pj,j,i = 12 ,∀1 ≤ j ≤ n,0 ≤ i ≤ m − 1.

pj,k ,i = 0 for j > k ,∀2 ≤ j ≤ n,0 ≤ i ≤ m − 1.qj,k ,i = 0 for j > k ,∀2 ≤ j ≤ n,0 ≤ i ≤ m − 1.∑n

k=j+1 pj,k ,i +∑n

k=j qj,k ,i = 12 , ∀1 ≤ j ≤ n; 0 ≤ i ≤ m − 2.

µj,i(a) = µj,i ≥ 0, ∀1 ≤ j ≤ n; 0 ≤ i ≤ m − 1βj,i(a) = βj,i > 0,∀1 ≤ j ≤ n; 0 ≤ i ≤ m − 1



Discussion


Recasting the problem

New system of equations: ~U ′(t) = A~U(t)

Initial conditions: ~U(0) = ~Φ

Solution: ~U(t) = etA~ΦP0 0 0Q1 P1 00 Q2 P2

−µ1,0 2p1,2,0β2,0 0 0 0 00 −µ2,0 0 0 0

2q1,1,0β1,0 2q1,2,0β2,0 −µ1,1 2p1,2,1β2,1 0 00 2q2,2,0β2,0 0 −µ2,1 0 00 0 2q1,1,1β1,1 2q1,2,1β2,1 −µ1,2 2p1,2,2β2,20 0 0 2q2,2,1β2,1 0 −µ2,2



Discussion


Linear Case Results

If, for every 1 ≤ j ≤ n and for every 0 ≤ i ≤ m − 1, µj,i > 0,then limt→∞Uj,i(t) = 0.If, for every 1 ≤ j ≤ n and for every 0 ≤ i ≤ m − 1, µj,i = 0,then Uj,i(t) is a polynomial in t of degree n − j + i .Furthermore, the coefficient of tn−j+i of this polynomial is amultiple of Φn,0.



Discussion


Numerical Results for Linear Model - Figure 1

Figure: Linear model with n = 3 maximum number of telomeres and m = 3 mutation classes (2 mutationsnecessary to reach malignancy). Polynomial growth of cells with one mutation (i = 1 mutation). Stem cells (j = 3telomeres) grow linearly, progenitor cells (j = 2 telomeres) in t2, and differentiated cells (j = 1 telomere) in t3.



Discussion



Figure: Linear model with n = 3 maximum number of telomeres and m = 3 mutation classes (2 mutations

necessary to reach malignancy). Polynomial growth (t4) of differentiated cancer cells(j = 1 telomere, i = 2mutations).



Discussion




necessary to reach malignancy). Polynomial growth (t3) of progenitor cancer cells (j = 2 telomeres, i = 2mutations).



Discussion




necessary to reach malignancy). Polynomial growth (t2) of cancer stem cells (j = 3 telomeres, i = 2 mutations).



Discussion


Nonlinear Case

~U ′(t) = A~U(t)− F (~U(t))~U(t)F is a positive linear functional from L1(RN

+) to R+



Discussion


Assumptions for the Nonlinear Case

µj,i(a) = µj,i > 0, ∀1 ≤ j ≤ n; 0 ≤ i ≤ m − 1.βj,i(a) = βj,i > 0,∀1 ≤ j ≤ n; 0 ≤ i ≤ m − 1.pj,k ,i = 0 for j > k ,∀1 ≤ j ≤ n; 0 ≤ i ≤ m − 1.

Note: pj,j,i need not equal 12 ,∀1 ≤ j ≤ n,0 ≤ i ≤ m − 1.

λ0 = −µn,m−1 − βn,m−1 + 2pn,n,m−1βn,m−1.



Discussion


Result for Nonlinear Case

There is a unique solution to the equation above and theeigenspace of the dominant eigenvalue λ0 of A is onedimensional. Further, the first n(m − 1) entries of ~Ψ are 0,the last n are non-zero, and limt→∞ ~U(t) = λ0Π0~Φ

F (Π0~Φ)= λ0~Ψ

F (~Ψ),

where Π0 is the eigenprojection associated with λ0, ~U(t) isthe unique solution to the equation, and ~Ψ is aneigenvector of λ0.



Discussion


Numerical Result for Nonlinear Model

Figure: Nonlinear model with n = 8 maximum number of telomeres and m = 6 mutation classes (5 mutationsnecessary to reach malignancy). Cancer cells (i = 5 mutations) taking over the tissue environment according to theasymptotic steady state result.



Discussion

Summary and Discussion

Question 1: Considering cell mutation as a dynamicpopulation process, rather than a one-time random event,what can we show about cancer cell population growth inrelation to the growth of the populations of non-cancercells?

Answer: The theorem for the linear model proves that thenumber of cancer cells grows faster polynomially than anyother type of cell and it is the nature of mutation acquisitionthat explains the higher population growth of cancer cells.However, cancer cells do need to exhibit high proliferationrate in order for their population to grow to levels dangerousfor the organism in a realistic time frame.



Discussion

Question 2: What is the role of stem cells in the cellpopulation dynamics?

Answer: Stem cells are crucial for the development of allother cell classes and are also important for the rate atwhich those different cell populations grow.



Discussion

Question 3: Is the cancer stem cell count as small asscientists have claimed?

Answer: A relatively small subpopulation of cancer stemcells can generate the total population of cancer cells.



Discussion

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A Mathematical Model of Cancer Stem Cell Lineage ...€¦ · (i 1)th mutation class will produce, by acquiring a mutation during division, a cell in the j thtelomere, i mutation class.

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