This is a presentation on "Mathematical Approach to Predict the Drug Effects on Cancer Stem Cell Models" article doi:10.1016/j.entcs.2011.09.033

1. An article presentation
Mathematical Approach to
Predict the Drug Effects on
Cancer Stem cell Models
M. Amin Ghavami
Fall 95

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)

5. Model structure

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

7. Model structure

9. Model structure

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