1. Immunotherapy for Treating Cancer
Project A, proposed by Helen Byrne
Rui Zhao, Peter Kim, and Natasha Li
Advisor: Philip Maini

2. Macrophage Infiltration
Necrotic core x
High oxygen region
chemoattractant
macrophages
Hypoxic (low oxygen) region z
Hypoxic tumor cells release chemoattractant, causing macrophages to localize to hypoxic regions.

3. Possible Treatment
Engineered macrophages which release enzyme that promote tumor killing.
Macrophages are localized to hypoxic region.

4. Model Equations (1) c = Oxygen (or limiting nutrient)
Uptake of oxygen in region from c1 to R
diffusion c1 c2 R
Necrotic Hypoxic High O2

5. Model Equations (2) a = chemoattractant
production in region from c1 to c2
diffusion
decay c1 c2 R
Necrotic Hypoxic High O2

6. Model Equations (3)
m = macrophages
diffusion
chemotaxis

7. Boundary Conditions
At x=0
At x=R c1 c2 R
Necrotic Hypoxic High O2

8. Solution for oxygen c(x) c(x) c(x) c2 Case 1 c1 c2 Case 2 c1 c2 Case 3

9. Solution for chemoattractant
Nothing / Nada / Rien / Ling / Kusni
a(x)
Case 1
a(x)
Case 2 x2
Case 3
a(x) x1 x2

10. Macrophage: The Movie
1.4
steady state
concentration
1.0 x2 1

11. Morning in Park City 12
concentration
steady state x1 x2 3

12. Afternoon on Great Salt Lake
1.3
steady state
0.9 x1 x2 3

13. Night of the 4th of July
1.3
steady state
0.9 x1 x2 5

14. Model extension: tumor growth
Tumor radius evolves over time
Impact of macrophages by killing tumor
R= tumor radius
Death due to macrophages in region from c1 to c2
growth c1 c2 R
Necrotic Hypoxic High O2

15. Solutions for R (tumor radius)
R without treatment
R with treatment x2 x1
Time  R x2 x1

16. Open Problems 1: Macrophage populations and killing interactions
Include normal and cancer cells to assess efficiency of killing
Separate macrophage populations (normal vs. engineered)
What is the best treatment protocol?

17. Open Problems 2: Vasculature
V=vasculature
The “well mixed” model- Logistic growth
Simple model- Macrophage induced vasculature
Vessels bringing in O2

18. Open problems 2, continued.
Macrophage induced vasculature - death
Chemotaxis
production

19. Open problems 2, cont. What is the playoff between:
Cancer cell killing by engineered macrophages
Enhanced nutrient delivery caused by macrophages via new vasculature

20. Thank you Applied math in action x2 = y(y-1)2 x = 0.3(y2-1)
Rui Zhao
Peter Kim
Natasha Li
( R cos q, R sin q )