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Mathematical Approach to
Predict the Drug Effects on
Cancer Stem cell Models
M. Amin Ghavami
Fall 95
Model structure
 Many cancers are driven by small groups of Cancer Stem
Cells (CSCs)
 In normal tissues:
 CSCs are self-renewing
 Capable of tissue regeneration
 Rise to non-CSCs
 In our ODEs system we also account for the drug effect on
the cancer in order to simulate the behavior of the cells,
considering different drug concentrations.
Model structure
 Traditional view
 That the malignant tumor consists of a unique type of cell population
characterized by the ability to divide without limit.
 New heterogeneity concept
 CSCs that can differentiate into heterogeneous cancer
subpopulations
 Progenitor cells (PCs)
 Terminally differentiated cells (TCs)
Model structure
Model structure
 Proliferation rate is: ω
 Differentiation rate is: η
 Ability of progenitor cells to acquire CSC phenotype: ϒ
 die rate: d
 NCSC, NPCi and NTC are the numbers of cancer stem cells
 Drug effects for each individual population i: Θi
Model structure
Model structure
Equilibrium analysis
Z’= AZ
 matrix A contains the coefficients of the biological system
 vector Z refers to the variables
Equilibrium analysis
Zi = Wi eλit
 Wi is the i − th eigenvector of A
 λi is its corresponding eigenvalue
 Our description of the dynamics of tumor progression builds
on Zhu’s model
Equilibrium analysis
 This result confirms the key role of CSCs in tumor growth: all
cell populations reach a steady state if and only if the
proliferation of CSCs is kept under control
PsyωCSC = d1
 in real cases, instead, the control of the proliferation of CSCs depends
also on differentiation, feedback, and drug treatments
Equilibrium analysis
 The difficulty of this task is mainly caused by the presence of
the feedback effect which can be however overcome by
following a step-by-step procedure
 we consider three variants of the model:
 Var1: corresponds to the system of ODEs with nine subpopulations
but without the representation of the feedback
 var2: is the system of ODEs accounting only for the CSC and PC1
subpopulations, but with the representation of the feedback
 var3: is the whole system corresponding
Equilibrium analysis
 Var 1
PsyωCSC = d1 + η1 +Θ1
Equilibrium analysis
 Var2
 It has to be balanced by the CSC surrogate production given
by the feedback action that is proportional to the
concentration of PC1
Equilibrium analysis
 Var3
 solve a cubic polynomial
 The analytic solution of this polynomial is difficult to manage,
hence we adopt a graphical approach to determine its roots
Results
 The model parameters are
tuned by using
experimental data; the
resulting model setting is
used to investigate several
drug effects and their
combination
Mathematical approach to predict the drug effects on cancer stem cell models

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Mathematical approach to predict the drug effects on cancer stem cell models

  • 1. An article presentation Mathematical Approach to Predict the Drug Effects on Cancer Stem cell Models M. Amin Ghavami Fall 95
  • 2.
  • 3. Model structure  Many cancers are driven by small groups of Cancer Stem Cells (CSCs)  In normal tissues:  CSCs are self-renewing  Capable of tissue regeneration  Rise to non-CSCs  In our ODEs system we also account for the drug effect on the cancer in order to simulate the behavior of the cells, considering different drug concentrations.
  • 4. Model structure  Traditional view  That the malignant tumor consists of a unique type of cell population characterized by the ability to divide without limit.  New heterogeneity concept  CSCs that can differentiate into heterogeneous cancer subpopulations  Progenitor cells (PCs)  Terminally differentiated cells (TCs)
  • 6. Model structure  Proliferation rate is: ω  Differentiation rate is: η  Ability of progenitor cells to acquire CSC phenotype: ϒ  die rate: d  NCSC, NPCi and NTC are the numbers of cancer stem cells  Drug effects for each individual population i: Θi
  • 8.
  • 10. Equilibrium analysis Z’= AZ  matrix A contains the coefficients of the biological system  vector Z refers to the variables
  • 11. Equilibrium analysis Zi = Wi eλit  Wi is the i − th eigenvector of A  λi is its corresponding eigenvalue  Our description of the dynamics of tumor progression builds on Zhu’s model
  • 12. Equilibrium analysis  This result confirms the key role of CSCs in tumor growth: all cell populations reach a steady state if and only if the proliferation of CSCs is kept under control PsyωCSC = d1  in real cases, instead, the control of the proliferation of CSCs depends also on differentiation, feedback, and drug treatments
  • 13. Equilibrium analysis  The difficulty of this task is mainly caused by the presence of the feedback effect which can be however overcome by following a step-by-step procedure  we consider three variants of the model:  Var1: corresponds to the system of ODEs with nine subpopulations but without the representation of the feedback  var2: is the system of ODEs accounting only for the CSC and PC1 subpopulations, but with the representation of the feedback  var3: is the whole system corresponding
  • 14. Equilibrium analysis  Var 1 PsyωCSC = d1 + η1 +Θ1
  • 15. Equilibrium analysis  Var2  It has to be balanced by the CSC surrogate production given by the feedback action that is proportional to the concentration of PC1
  • 16. Equilibrium analysis  Var3  solve a cubic polynomial  The analytic solution of this polynomial is difficult to manage, hence we adopt a graphical approach to determine its roots
  • 17. Results  The model parameters are tuned by using experimental data; the resulting model setting is used to investigate several drug effects and their combination