Project Macrophage: Macrophages on the Move

Heather More, Rachel Psutka, Vishaal Rajani

Overview

Tumour grows quickly

Hypoxia

Release of chemoattractants

Macrophage migration

Release of angiogenic factors

Blood vessel growth

Tumours

• Abnormal growth of tissue

• Macrophages migrate via chemotaxis

• Engineered macrophages activated to release chemicals – kill tumour cells

Macrophages in Tumours

• Movement via chemotaxis – innate behavioural response to the addition of a chemical

• Tumour cells release chemoattractants under hypoxia

Chemotaxis Video

• Chemotaxis

Macrophage

• Originate from monocytes

• Phagocytotic, specific and non-specific defense

• Engulf cellular debris and pathogens

• Assist in wound healing, migrate to hypoxic regions

Macrophage Video

• Macrophage

Blood Vessel Growth

• Macrophages release angiogenic factors

• Promotes blood vessel growth

Angiogenesis Video

• Angiogenesis

Cells

For our 1-dimensional purposes...

Necrotic core

Hypoxic cells

Live cells

Building a model for chemotaxis

• Chemoattractants released by hypoxic cells attract macrophages

• Taking the diffusion equation of macrophage density at position x and time t;

x

tx

xt

tx ),(ma

),m(

where a is the diffusion coefficient.

• We add an “attracting” term which describes the attraction of macrophages to positions of highest chemoattractant concentration, becoming:

x

txtx

xx

tx

xt

tx ),(c),m(

),(ma

),(m

where X denotes the chemotaxis coefficient, c(x,t) denotes the chemoattractant concentration at position x, time t.

Diffusion Term Attraction Term

Exploring Chemotaxis

Hypoxic cells emit chemoattractants

What kind of function can we use to model this?

length of tumour

chemoattractant concentration

Exploring Chemotaxis

Using a Gaussian profile for the concentration of chemoattractants;

2

),( xetxc

we solved the attraction-diffusion equation of macrophages numerically, using MAPLE.

x

txtx

xx

tx

x

tx ),(c),m(

),(ma

t

),(m

t=0

What would happen if we introduced bioengineered macrophages that had a killing effect on tumour cells?

ra,

β

β

Ф

a,

Building the modelChemoattractant concentration spread over space is proportional to hypoxic cells over space...

x

txh

x

txc

),(),(

For our purposes .1

xxx

h

xt

m mam

Hypoxic Cells

killing effect of macrophages, where β is the proportion of

bioengineered macrophages

the proportion of living cells that become hypoxic

Living Cells

killing effect of macrophages, where β is the proportion of

bioengineered macrophages

living cells that become hypoxic

logistic growth of tumour cells

What determines r?

Becoming Hypoxic

When cells are lacking oxygen, they are hypoxic and do not divide.

We want a growth rate that applies only to cells that are on the outside layers of the cell.

BUT step functions are NOT continuous.

Let

001.0

)5.0tanh15.0

001.0

5.0tanh15.0)(

xxxr

The Resulting System (Boo Yah!)

xxx

h

xt

m mam

001.0

)5.0tanh15.0

001.0

5.0tanh15.0)(

xxxr

)(1 xr

Initial and Boundary Conditions

Initial Conditions

1 )0m( kx,

Boundary Conditions

Of the two possibilities,

we used these ones.

Observations

• decreases faster than

• and/or causes faster decrease of and

• causes slower decrease of and

a large amount of engineered macrophages with enhanced sensitivity to chemoattractant and low diffusion rates are required in order to have the eventual death of all cells

),(h tx ),(L tx

),(h tx ),(L tx

a

),(h tx ),(L tx

Lmhh

t

xxx

h

xt

m mam

constant cchemotacti

smacrophage normal : engineered of ratio

constantdiffusion a L-1LLmL

Lr

t

Observations

β = 0.5 β = 4

Observations

Assumptions

• No angiogenesis

• 1D tumour

• Killing effect was assumed to be the same for living and hypoxic cells

• Proliferation of hypoxic cells is negligible

Conclusions

• β has a threshold... a sufficient killing effect is needed to kill the tumour entirely

• Increasing macrophage sensitivity to chemoattractants would increase the killing effect of bioengineered macrophages

• Hypoxic cells die faster than living cells with the introduction of bioengineered macrophages

Future Projects/Questions

• Hardcore analysis of β

• Conservative PDE Solver

• Angiogenesis

• Necrotic core...spatial modelling of tumor

• 2D and 3D models

• Chemotherapy, Radiotherapy

Acknowledgements

• Byrne, H. The role of mathematics in solid tumour growth. Mathematics Today. April 1999, 48-53.

• Gerda de Vries, Caroline Bampfylde, Jim Keener, Tomas de-Camino-Beck

• Colin More (Heather’s brother, for Maple help) • All other instructors and volunteers for

background knowledge• Fellow students for motivation and emotional

support

The End!