Transcript Tumor Growth and Radio Therapy

Tumor Growth and Radio
Therapy
Bettina Greese Biomathematics, University of Greifswald
Nuha Jabakhanji Bioinformatics , University of Alberta
Cancer: Background
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A cancer tumor is a mass of cells with uncontrolled cell
proliferation as a result of defective cell cycle control
mechanisms.
How it results: point mutations, DNA rearrangements,
gene amplification, translocation, mutations in tumor
suppressor genes.
Cancer cells are genetically unstable and are able to
become “worse” by accumulating mutations.
Why it’s important:
“An estimated 149,000 new cases of cancer and 69,500
deaths from cancer will occur in Canada in 2005.” National
Cancer Institute of Canada
Mathematical Model: Simple Model
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Two populations: Healthy and cancerous cells.
Logistic growth with intrinsic growth rate.
Competition for resources and space.
Initial conditions: 100 healthy cells, 1 cancerous cell.
 h(t )  a1c(t ) 
d
h(t )
h(t )  r1 1 
dt
K1
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 a2 h(t )  c(t ) 
d
c(t )
c(t )  r2 1 
dt
K2
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Mathematical Model: Simple Model
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Two populations: Healthy and cancerous cells.
Logistic growth with intrinsic growth rate.
Competition for resources and space.
Natural mutations (healthy to cancerous cell).
 h(t )  a1c(t ) 
d
h(t )  h(t )
h(t )  r1 1 
dt
K1
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 a2 h(t )  c(t ) 
d
c(t )  h(t )
c(t )  r2 1 
dt
K2
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Mathematical Model: Simple Model
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Two populations: Healthy and cancerous cells.
Logistic growth with intrinsic growth rate.
Competition for resources and space.
Mutations: natural and radiation induced.
Radiation induced death for cancerous and healthy cells.
 h(t )  a1c(t ) 
d
h(t )  h(t )  h(t )  h(t )
h(t )  r1 1 
dt
K1
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 a2 h(t )  c(t ) 
d
c(t )  h(t )  c(t )  h(t )
c(t )  r2 1 
dt
K2
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Simple Model: Analysis
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Calculated the steady states and their stability.
Plotted the phase plane for different parameter sets.
Extinction
Coexistence
Simple Model: Analysis
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Plots of healthy (red) and cancerous (green) cells
versus time.
Extinction
Coexistence
Simple Model: Analysis
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Thin lines: with effects of radiation and/or mutation.
Left: The effect of natural mutation on the populations.
Right: Cell deaths caused by constant radiation.
Simple Model: Analysis
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Left: Cell deaths caused by constant high radiation.
Right: Cell deaths caused by pulsed radiation.
  0.9
 (t )  2.5  sin10 (0.6t )
Mathematical Model: Final Model
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Three populations: Healthy, cancerous and aggressive
cancerous cells.
Logistic growth, competition, mutations, radiation induced
death are as before.
Initial conditions: 100 healthy, 1 cancerous and 0 aggressive.
 h(t )  a12 c(t )  a13 d (t ) 
h(t )  12    12 h(t )
h(t )  r1 1 
K1
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 a h(t )  c(t )  a23 d (t ) 
c(t )  12  12 h(t )   23     23 c(t )
c(t )  r2 1  21
K2
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 a31h(t )  a32 c(t )  d (t ) 
d (t )   23   23 c(t )  d (t )
d (t )  r3 1 
K3
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Final Model: Analysis
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Plot of healthy (red), cancerous (green) and aggressive
cancerous (blue) cells versus time.
Thin lines: effect of natural mutation.
Final Model: Analysis
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Plot of healthy (red), cancerous (green) and aggressive
cancerous (blue) cells versus time.
Thin lines: effect of radiation induced mutations and
death.
Final Model: Analysis
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Thick lines: natural mutations included.
Thin lines: effect of radiation induced mutations and
death in addition to natural mutations.
Conclusions and Limitations
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Biologically meaningful parameters result in extinction of
healthy cells.
Natural mutation accelerates extinction of healthy cells.
Radiation delays extinction.
High doses of radiation are needed to maintain a level of
healthy cells above cancer cells.
Pulsed radiation allows higher doses of radiation, thus a
higher level of healthy cells is maintained for a
significantly longer time.
Pulsed radiation includes breaks in radiation that result in
the recovery of the cells.
Conclusions and Limitations
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In the final model, the cancerous cells are driven to
extinction by the aggressive cancerous cells when natural
mutation is included.
Also, radiation does not change the qualitative behavior
but results in lower levels of cell populations.
We used same mutation rates and carrying capacities for
healthy and cancerous cells.
Pulsed radiation was not included in the final model.
Initial cell populations were low.
Further insight can be gained by varying parameters
within biological reason.
THANK YOU