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

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.

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

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

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

Model Equations (3) m = macrophages diffusion chemotaxis

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

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

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

Macrophage: The Movie 1.4 steady state concentration 1.0 x2 1

Morning in Park City 12 concentration steady state x1 x2 3

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

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

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

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

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?

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

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

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

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 )