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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
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
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!