Modeling the Dynamics of Glioblastoma Multiforme and Cancer Stem Cells

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


Table of Contents

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




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




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




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




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




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




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.




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




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



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





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





Figure 2: Global stability of cancer persistence state




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




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




Periodic Treatment - Source Model

I Recurrence without Cancer Stem Cells


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




Comparing the Models

I Recurrence in both models


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




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




Future Work

I Find a condition for N that ensures a globally asymptoticallystable cure state

I Investigate local and global stability for internal equilibriumpoints




Acknowledgements

Dr. Zach Abernathy, Winthrop UniversityDr. Kristen Abernathy, Winthrop University

Winthrop University SURE ProgramSC INBRE Program

National Institutes of Health




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)



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Modeling the Dynamics of Glioblastoma Multiforme and Cancer Stem Cells

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