Biological Preliminaries The Mathematical Model Discussion A Mathematical Model of Cancer Stem Cell Lineage Population Dynamics with Mutation Accumulation and Telomere Length Hierarchies Georgi Kapitanov Vanderbilt University Feb 17, 2012 Georgi Kapitanov A Mathematical Model of Cancer Stem Cell Lineage Population Dynamics
Biological Preliminaries The Mathematical Model Discussion Outline Biological Preliminaries 1 Telomeres Stem Cells and Differentiation Cell Mutations and Cancer The Mathematical Model 2 The Model Model analysis Discussion 3 Georgi Kapitanov A Mathematical Model of Cancer Stem Cell Lineage Population Dynamics
Biological Preliminaries Telomeres The Mathematical Model Stem Cells and Differentiation Discussion Cell Mutations and Cancer Telomeres and Cell Division Definition: repeated sequence of DNA that protects important 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 Georgi Kapitanov A Mathematical Model of Cancer Stem Cell Lineage Population Dynamics
Biological Preliminaries Telomeres The Mathematical Model Stem Cells and Differentiation Discussion Cell Mutations and Cancer Stem Cells Properties of stem cell: self-renewal, ability to differentiate. Progenitor cells: medium stage of differentiation. Mature (differentiated) cells: they have specific functions. Georgi Kapitanov A Mathematical Model of Cancer Stem Cell Lineage Population Dynamics
Biological Preliminaries Telomeres The Mathematical Model Stem Cells and Differentiation Discussion Cell Mutations and Cancer Mutation accumulation Vogelgram - represents the sequence of mutations in a cell that eventually leads to a cancerous cell. Figure: A Genetic Model for Colorectal Tumorigenesis. This is an example of a Vogelgram - multistep cancer progression model (http://www.hopkinscoloncancercenter.org) Georgi Kapitanov A Mathematical Model of Cancer Stem Cell Lineage Population Dynamics
Biological Preliminaries The Model The Mathematical Model Model analysis Discussion Questions to address Considering cell mutation as a dynamic population process, rather than a one-time random event, what can we show about cancer cell population growth in relation to the growth of the populations of non-cancer cells? What is the role of stem cells in the cell population dynamics? Is the cancer stem cell count as small as scientists have claimed (some results claim that only one in ten thousand cancer cells is a cancer stem cell[32][4])? Georgi Kapitanov A Mathematical Model of Cancer Stem Cell Lineage Population Dynamics
Biological Preliminaries The Model The Mathematical Model Model analysis Discussion Equations ∂ u j , i ( a , t ) + ∂ u j , i ( a , t ) = − ( µ j , i ( a ) + β j , i ( a )) u j , i ( a , t ) ∂ t ∂ a n � ∞ � u j , i ( 0 , t ) = 2 ( p j , k , i β k , i ( a ) u k , i ( a , t ) da + 0 k = j � ∞ q j , k , i − 1 β k , i − 1 ( a ) u k , i − 1 ( a , t ) da ) 0 u j , i ( a , 0 ) = φ j , i ( a ) Georgi Kapitanov A Mathematical Model of Cancer Stem Cell Lineage Population Dynamics
Biological Preliminaries The Model The Mathematical Model Model analysis 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 has accumulated. For t ≥ 0 , u j , i ( a , t ) ∈ L 1 ([ 0 , ∞ )) , represents the density of cells with age a at time t , in the j th telomere class, with i mutations. µ j , i ( a ) ≥ 0, is the age-specific mortality rate of cells in the j th telomere, i th mutation class. β j , i ( a ) > 0, is the age-specific proliferation rate of cells in the j th telomere, i th mutation class. p j , k , i > 0, is the probability that one of the daughters of a cell in the k th telomere, i th mutation class will be a cell in the j th telomere, i th mutation class. q j , 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 mutation during division, a cell in the j th telomere, i th mutation class. Georgi Kapitanov A Mathematical Model of Cancer Stem Cell Lineage Population Dynamics
Biological Preliminaries The Model The Mathematical Model Model analysis Discussion Hypotheses p j , j , i = 1 2 , ∀ 1 ≤ j ≤ n , 0 ≤ i ≤ m − 1. p j , k , i = 0 for j > k , ∀ 2 ≤ j ≤ n , 0 ≤ i ≤ m − 1. q j , k , i = 0 for j > k , ∀ 2 ≤ j ≤ n , 0 ≤ i ≤ m − 1. � n k = j + 1 p j , k , i + � n k = j q j , k , i = 1 2 , ∀ 1 ≤ j ≤ n ; 0 ≤ i ≤ m − 2. µ j , i ( a ) = µ j , i ≥ 0 , ∀ 1 ≤ j ≤ n ; 0 ≤ i ≤ m − 1 β j , i ( a ) = β j , i > 0 , ∀ 1 ≤ j ≤ n ; 0 ≤ i ≤ m − 1 Georgi Kapitanov A Mathematical Model of Cancer Stem Cell Lineage Population Dynamics
Biological Preliminaries The Model The Mathematical Model Model analysis Discussion Recasting the problem New system of equations: � U ′ ( t ) = A � U ( t ) Initial conditions: � U ( 0 ) = � Φ Solution: � U ( t ) = e tA � Φ P 0 0 0 Q 1 P 1 0 0 Q 2 P 2 − µ 1 , 0 2 p 1 , 2 , 0 β 2 , 0 0 0 0 0 0 − µ 2 , 0 0 0 0 2 q 1 , 1 , 0 β 1 , 0 2 q 1 , 2 , 0 β 2 , 0 − µ 1 , 1 2 p 1 , 2 , 1 β 2 , 1 0 0 0 2 q 2 , 2 , 0 β 2 , 0 0 − µ 2 , 1 0 0 0 0 2 q 1 , 1 , 1 β 1 , 1 2 q 1 , 2 , 1 β 2 , 1 − µ 1 , 2 2 p 1 , 2 , 2 β 2 , 2 0 0 0 2 q 2 , 2 , 1 β 2 , 1 0 − µ 2 , 2 Georgi Kapitanov A Mathematical Model of Cancer Stem Cell Lineage Population Dynamics
Biological Preliminaries The Model The Mathematical Model Model analysis Discussion Linear Case Results If, for every 1 ≤ j ≤ n and for every 0 ≤ i ≤ m − 1, µ j , i > 0, then lim t →∞ U j , i ( t ) = 0 . If, for every 1 ≤ j ≤ n and for every 0 ≤ i ≤ m − 1, µ j , i = 0, then U j , i ( t ) is a polynomial in t of degree n − j + i . Furthermore, the coefficient of t n − j + i of this polynomial is a multiple of Φ n , 0 . Georgi Kapitanov A Mathematical Model of Cancer Stem Cell Lineage Population Dynamics
Biological Preliminaries The Model The Mathematical Model Model analysis Discussion Numerical Results for Linear Model - Figure 1 Figure: Linear model with n = 3 maximum number of telomeres and m = 3 mutation classes (2 mutations necessary to reach malignancy). Polynomial growth of cells with one mutation ( i = 1 mutation). Stem cells ( j = 3 telomeres) grow linearly, progenitor cells ( j = 2 telomeres) in t 2 , and differentiated cells ( j = 1 telomere) in t 3 . Georgi Kapitanov A Mathematical Model of Cancer Stem Cell Lineage Population Dynamics
Biological Preliminaries The Model The Mathematical Model Model analysis Discussion Numerical Results for Linear Model - Figure 2 Figure: Linear model with n = 3 maximum number of telomeres and m = 3 mutation classes (2 mutations necessary to reach malignancy). Polynomial growth ( t 4 ) of differentiated cancer cells( j = 1 telomere, i = 2 mutations). Georgi Kapitanov A Mathematical Model of Cancer Stem Cell Lineage Population Dynamics
Biological Preliminaries The Model The Mathematical Model Model analysis Discussion Numerical Results for Linear Model - Figure 3 Figure: Linear model with n = 3 maximum number of telomeres and m = 3 mutation classes (2 mutations necessary to reach malignancy). Polynomial growth ( t 3 ) of progenitor cancer cells ( j = 2 telomeres, i = 2 mutations). Georgi Kapitanov A Mathematical Model of Cancer Stem Cell Lineage Population Dynamics
Biological Preliminaries The Model The Mathematical Model Model analysis Discussion Numerical Results for Linear Model - Figure 4 Figure: Linear model with n = 3 maximum number of telomeres and m = 3 mutation classes (2 mutations necessary to reach malignancy). Polynomial growth ( t 2 ) of cancer stem cells ( j = 3 telomeres, i = 2 mutations). Georgi Kapitanov A Mathematical Model of Cancer Stem Cell Lineage Population Dynamics
Biological Preliminaries The Model The Mathematical Model Model analysis Discussion Nonlinear Case � U ′ ( t ) = A � U ( t ) − F ( � U ( t )) � U ( t ) F is a positive linear functional from L 1 ( R N + ) to R + Georgi Kapitanov A Mathematical Model of Cancer Stem Cell Lineage Population Dynamics
Biological Preliminaries The Model The Mathematical Model Model analysis Discussion Assumptions for the Nonlinear Case µ j , i ( a ) = µ j , i > 0 , ∀ 1 ≤ j ≤ n ; 0 ≤ i ≤ m − 1. β j , i ( a ) = β j , i > 0 , ∀ 1 ≤ j ≤ n ; 0 ≤ i ≤ m − 1. p j , k , i = 0 for j > k , ∀ 1 ≤ j ≤ n ; 0 ≤ i ≤ m − 1. Note: p j , j , i need not equal 1 2 , ∀ 1 ≤ j ≤ n , 0 ≤ i ≤ m − 1. λ 0 = − µ n , m − 1 − β n , m − 1 + 2 p n , n , m − 1 β n , m − 1 . Georgi Kapitanov A Mathematical Model of Cancer Stem Cell Lineage Population Dynamics
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