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

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

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 c 1 to R diffusion 0c1c1 c2c2 R Necrotic Hypoxic High O 2

Model Equations (2) a = chemoattractant production in region from c 1 to c 2 diffusion decay 0c1c1 c2c2 R Necrotic Hypoxic High O 2

Model Equations (3) m = macrophages diffusionchemotaxis

Boundary Conditions At x=0 At x=R 0c1c1 c2c2 R Necrotic Hypoxic High O 2

Solution for oxygen Case 1 Case 2 Case 3 c1c1 c2c2 c(x) c2c2 c2c2 c1c1 c1c1

Solution for chemoattractant Case 1 Case 2 Case 3 a(x) x2x2 x2x2 x1x1 Nothing / Nada / Rien / Ling / Kusni

x2x2 Macrophage: The Movie concentration steady state

x1x1 x2x concentration Morning in Park City steady state

x1x1 x2x2 Afternoon on Great Salt Lake steady state

x1x1 x2x Night of the 4 th of July steady state

Model extension: tumor growth Tumor radius evolves over time Impact of macrophages by killing tumor R= tumor radius growth Death due to macrophages in region from c 1 to c 2 0c1c1 c2c2 R Necrotic Hypoxic High O 2

Solutions for R (tumor radius) R with treatment x2x2 x1x1 R without treatment x2x2 x1x1 R Time 

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 O 2

Open problems 2, continued. Macrophage induced vasculature - death Chemotaxisproduction

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

x 2 = y(y-1) 2 x = 0.3(y 2 -1) x = -0.2((y-0.3) ) ( R cos , R sin  ) Thank you Rui Zhao Peter Kim Natasha Li Applied math in action