Background Information The Model Preliminary Results Outcomes and Analysis
Modeling the Dynamics of GlioblastomaMultiforme and Cancer Stem Cells
Stephen Steward1
1Winthrop University
Winthrop University, SC INBRE ProgramUNCG Regional Mathematics and Statistics Conference
Stephen Steward Winthrop University
Modeling the Dynamics of Glioblastoma Multiforme and Cancer Stem Cells
Background Information The Model Preliminary Results Outcomes and Analysis
Table of Contents
Background Information
The Model
Preliminary Results
Outcomes and Analysis
Stephen Steward Winthrop University
Modeling the Dynamics of Glioblastoma Multiforme and Cancer Stem Cells
Background Information The Model Preliminary Results Outcomes and Analysis
Glioblastoma Multiforme
I Rare, highly malignant type of brain tumor
I Resistant to most conventional treatment methodsI High mortality rate
I 50% of patients die within one yearI 90% of patients die within three years
Stephen Steward Winthrop University
Modeling the Dynamics of Glioblastoma Multiforme and Cancer Stem Cells
Background Information The Model Preliminary Results Outcomes and Analysis
Cancer Stem Cell Hypothesis
I Malignant tumors are created and maintained by specializedtumor cells with properties similar to healthy adult stem cells,called Cancer Stem Cells (CSCs)
I CSCs may divide symmetrically or asymmetrically
I CSCs are capable of regenerating their own population at theexpense of creating new tumor cells, making them difficult toeradicate completely
Stephen Steward Winthrop University
Modeling the Dynamics of Glioblastoma Multiforme and Cancer Stem Cells
Background Information The Model Preliminary Results Outcomes and Analysis
Immunotherapy
What is immunotherapy?
I Method of treatment that stimulates the patient’s immunesystem to fight off CSC and tumor cell populations
I Infusions of specialized immune cells called Cytotoxic TLymphocytes (CTLs) directly into the cancer site
I Requires certain antigens to be present on CSCs and tumorcells to be effective
Why immunotherapy?
I Less invasive and less harmful than chemotherapy, radiation,or surgery
I Boosts the body’s natural immune response, rather thanintroducing an external agent
Stephen Steward Winthrop University
Modeling the Dynamics of Glioblastoma Multiforme and Cancer Stem Cells
Background Information The Model Preliminary Results Outcomes and Analysis
Immunotherapy
What is immunotherapy?
I Method of treatment that stimulates the patient’s immunesystem to fight off CSC and tumor cell populations
I Infusions of specialized immune cells called Cytotoxic TLymphocytes (CTLs) directly into the cancer site
I Requires certain antigens to be present on CSCs and tumorcells to be effective
Why immunotherapy?
I Less invasive and less harmful than chemotherapy, radiation,or surgery
I Boosts the body’s natural immune response, rather thanintroducing an external agent
Stephen Steward Winthrop University
Modeling the Dynamics of Glioblastoma Multiforme and Cancer Stem Cells
Background Information The Model Preliminary Results Outcomes and Analysis
The Model
Cancer Stem Cells
dSdt
= r1S(1− S
K1
)− aS
MIMI+eS
·(aS,β +
eS,β (1−aS,β )
Fβ+eS,β
)· C ·ShS+S
Tumor Cells
dTdt
= r2S(
SK1
)(1− T
K2
)− aT
MIMI+eT
·(aT ,β +
eT,β (1−aT,β )
Fβ+eT,β
)· C ·ThT+T
− µT · T
CTLs
dCdt
=
(aC,MII
MII ·(T+S)
MII ·(T+S)+eC,MII
)·(aC ,β +
eC,β (1−aC,β )
Fβ+eC,β
)− µc · C + N
Stephen Steward Winthrop University
Modeling the Dynamics of Glioblastoma Multiforme and Cancer Stem Cells
Background Information The Model Preliminary Results Outcomes and Analysis
The Model
TGF-βdFβdt
=gβ + aβ,T ·(T + S
)− µβ · Fβ
IFN-γdFγdt
=aγ,C · C − µγ · Fγ
MHC Class IdMIdt
=gMI+
aMI ,γ·Fγ
Fγ+emI ,γ− µMI
·MI
MHC Class II
dMIIdt
=aMII ,γ
·FγFγ+eMII ,γ
·(
eMII ,β·(1−aMII ,β
)
Fβ+eMII ,β+ aMII ,β
)− µMII
·MII
Stephen Steward Winthrop University
Modeling the Dynamics of Glioblastoma Multiforme and Cancer Stem Cells
Background Information The Model Preliminary Results Outcomes and Analysis
Preliminary Results
I Existence/UniquenessI Invariance/Dissipativity
I Invariance guarantees that for nonnegative initial populations,all populations will remain nonnegative.
I Dissipativity guarantees all populations remain bounded.
Stephen Steward Winthrop University
Modeling the Dynamics of Glioblastoma Multiforme and Cancer Stem Cells
Background Information The Model Preliminary Results Outcomes and Analysis
Stability Analysis
I Investigate the behavior of the model around equilibriumsolutions, both with and without treatment.
I Local StabilityI Jacobian MatrixI EigenvaluesI Draw conclusions for cure or persistence
I Global StabilityI Comparisons and simplifications
Stephen Steward Winthrop University
Modeling the Dynamics of Glioblastoma Multiforme and Cancer Stem Cells
Background Information The Model Preliminary Results Outcomes and Analysis
Local Stability - No Treatment
I Unstable cure stateI Locally stable persistent state when the following inequality is
satisfied:
µCµγµMII>
aγ,CaMII ,γaC ,MII(K1 +
r2K1K1r2K1+K2µT
)
eMII ,γeC ,MII
· p,
where
p =
aMII ,β+
(1 − aMII ,β)eMII ,β
eMII ,β+
gβ+aβ,T ·(K1+r2K1K1
r2K1+K2µT)
µβ
aC,β +
(1 − aC,β )eC,β
eC,β +gβ+aβ,T (K1+
r2K1K1r2K1+K2µT
)
µβ
0 1 2 3 4 5 6 7 8 9 10 11Years
20
40
60
80
100Cells
CSCs (x108)
Tumor Cells (x109)
CTLs
Figure 1: Stable persistence stateStephen Steward Winthrop University
Modeling the Dynamics of Glioblastoma Multiforme and Cancer Stem Cells
Background Information The Model Preliminary Results Outcomes and Analysis
Global Stability - No Treatment
I Isolate C ,Fγ and MII
I Reduce to the following linear system:
dC
dt≤(aC ,MII
· (K1 +r2K1K1
r2K1+K2µT)
eC ,MII
)·MII − µc · C
dFγ
dt≤ aγ,C · C − µγ · Fγ
dMII
dt≤(aMII ,γ
eMII ,γ
)· Fγ − µMII
·MII
I Origin is globally stable when the following inequality issatisfied:
µCµγµMII>
aγ,CaMII ,γaC ,MII(K1 +
r2K1K1r2K1+K2µT
)
eMII ,γeC ,MII
Stephen Steward Winthrop University
Modeling the Dynamics of Glioblastoma Multiforme and Cancer Stem Cells
Background Information The Model Preliminary Results Outcomes and Analysis
Global Stability - No Treatment
I Isolate S and TI Reduce to the following system:
dS
dt= r1S
(1−
S
K1
)dT
dt= r2S
(S
K1
)(1−
T
K2
)− µT · T
I We use a nullcline plot to argue global stability
Stephen Steward Winthrop University
Modeling the Dynamics of Glioblastoma Multiforme and Cancer Stem Cells
Background Information The Model Preliminary Results Outcomes and Analysis
Global Stability - No Treatment
Figure 2: Global stability of cancer persistence state
Stephen Steward Winthrop University
Modeling the Dynamics of Glioblastoma Multiforme and Cancer Stem Cells
Background Information The Model Preliminary Results Outcomes and Analysis
Local Stability - Constant Treatment
I Persistent state is locally stable for N ≤ 172, 553
I Cure state is locally stable for N > 172, 553
1 2 3 4 5 6 7 8Years
20
40
60
80
100
Cells
CSCs (x107)
Tumor Cells (x108)
CTLs (x106)
(a) Persistence when N = 120, 000
0 1 2 3 4 5 6 7 8Years
20
40
60
80
100Cells
CSCs (x105)
Tumor Cells (x106)
CTLs (x106)
(b) Cure when N = 300, 000
Figure 3: Stability is dependent on N
Stephen Steward Winthrop University
Modeling the Dynamics of Glioblastoma Multiforme and Cancer Stem Cells
Background Information The Model Preliminary Results Outcomes and Analysis
Periodic Treatment - Source Model
I Recurrence without Cancer Stem Cells
I CTL immunotherapy schedule: (3×(3×108 aCTL q5d)+ 45d rest)×5
0 1Years
20
40
60
80
100Cells
Tumor Cells (x106)
CTLs (x107)
(a) Source Model
0 5 10 15 20 25Years
20
40
60
80
100Cells
Tumor Cells (x106)
CTLs (x107)
(b) Source Model - Extended
Stephen Steward Winthrop University
Modeling the Dynamics of Glioblastoma Multiforme and Cancer Stem Cells
Background Information The Model Preliminary Results Outcomes and Analysis
Periodic Treatment - Source Model
I Recurrence without Cancer Stem Cells
I CTL immunotherapy schedule: (3×(3×108 aCTL q5d)+ 45d rest)×5
0 1Years
20
40
60
80
100Cells
Tumor Cells (x106)
CTLs (x107)
(a) Source Model
0 5 10 15 20 25Years
20
40
60
80
100Cells
Tumor Cells (x106)
CTLs (x107)
(b) Source Model - Extended
Stephen Steward Winthrop University
Modeling the Dynamics of Glioblastoma Multiforme and Cancer Stem Cells
Background Information The Model Preliminary Results Outcomes and Analysis
Comparing the Models
I Recurrence in both models
I CTL immunotherapy schedule: (3×(3×108 aCTL q5d)+ 45d rest)×5
0 5 10 15 20 25Years
20
40
60
80
100Cells
Tumor Cells (x106)
CTLs (x107)
(a) Source Model - Extended
0 1 2 3 4Years
20
40
60
80
100Cells
CSCs (x106)
Tumor Cells (x106)
CTLs (x107)
(b) Our Model
Stephen Steward Winthrop University
Modeling the Dynamics of Glioblastoma Multiforme and Cancer Stem Cells
Background Information The Model Preliminary Results Outcomes and Analysis
Potential Treatment Regimen
I First year:CTL immunotherapy schedule: (3×(3×108 aCTL q5d)+ 45d rest)×5
I Recurring treatment every two years:CTL immunotherapy schedule: (3×(3×108 aCTL q5d)+ 45d rest)×3
0 1 2 3 4 5 6 7 8 9 10Years
20
40
60
80
100Cells
CSCs (x105)
Tumor Cells (x106)
CTLs (x107)
(a) Sufficient treatment
0 1 2 3 4 5 6 7 8 9 10Years
20
40
60
80
100Cells
CSCs (x107)
Tumor Cells (2x108)
CTLs (x107)
(b) Cancer Recurrence
Stephen Steward Winthrop University
Modeling the Dynamics of Glioblastoma Multiforme and Cancer Stem Cells
Background Information The Model Preliminary Results Outcomes and Analysis
Future Work
I Find a condition for N that ensures a globally asymptoticallystable cure state
I Investigate local and global stability for internal equilibriumpoints
Stephen Steward Winthrop University
Modeling the Dynamics of Glioblastoma Multiforme and Cancer Stem Cells
Background Information The Model Preliminary Results Outcomes and Analysis
Acknowledgements
Dr. Zach Abernathy, Winthrop UniversityDr. Kristen Abernathy, Winthrop University
Winthrop University SURE ProgramSC INBRE Program
National Institutes of Health
Stephen Steward Winthrop University
Modeling the Dynamics of Glioblastoma Multiforme and Cancer Stem Cells
Background Information The Model Preliminary Results Outcomes and Analysis
References
Natalie Kronik et al. “Improving alloreactive CTLimmunotherapy for malignant gliomas using a simulationmodel of their interactive dynamics” Cancer Immunology,Immunotherapy 57.3 (2008), pp. 425-439
Yuri Kogan et al. “Analysis of the immunotherapy model forglioblastoma multiforme brain tumour” Institute of AppliedMathematics and Mechanics UW 178 (2008)
Stephen Steward Winthrop University
Modeling the Dynamics of Glioblastoma Multiforme and Cancer Stem Cells