Cellular Automata: Some Applications in Detail

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Cellular Automata Some Applications in Detail

This is lecture 16 of Biologically Inspired
Computing Flu material from Beauchemin et al,
JTB, 2005 thanks to John Owen for bringing my
attention to it.
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Some additional things about CAs
A simple 2D CA to illustrate these
points States 0 and 1 Wraparound 2D array of
30 cells Rules if both neighbours are 1,
become 1 if both neighbours are
0, become 0 otherwise, stay the
same.
Synchronous update most CAs operate this way.
Each cells new state for time t1 is worked out
in parallel based on the situation at t.
Start 101001010001101000101010010001
T1 110000000001110000010100000001
T2 110000000001110000001000000001
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Some additional things about CAs
Asynchronous update Sometimes applied in
preference it is arguably a more valid way to
simulate some systems. Here, at each time step,
one cell is chosen at random and updated.
Start 101001010001101000101010010001
T1 101001000001101000101010010001
T2 101001000001101000101010010001
T3 111001000001101000101010010001
T4 111001000001101000101010000001
T5 etc ...
Clearly if there are n cells, then n timesteps in
an asynchronous CA corresponds to the 1 timestep
of a synchronous CA.
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Boundary conditions
But what about cells on the edge?
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The Flu

• Influenza, in humans, is caused by a virus that
  infects the upper respiratory tract, nose and
  throat, and sometimes the lungs
• Annual epidemics affect 515 of the worlds
  population, causing around 250,000500,000
  deaths.
• There are three different types, A, B and C, and
  further division among the types.
• Influenza A has two subtypes, which are the most
  important for humans A(H3N2) and A(H1N1) the H
  and N refer to different molecules on the surface
  of the virus, and the numbers indicate variants
  of these.

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Images from the wikipedia article
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The virus particle (virion) enters the healthy
cell, which is the infected. The virus then
tricks the cell into expressing the viral
proteins, so that many copies of the virus become
made in the cell. Soon the cell
becomes infectious, and eventually it just dies
from overcrowding of virus copies.
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Defense against it

• Our immune system normally manages to evolve a
  response that recognises the H molecule, which
  then gets destroyed by our white blood cells.
• But, the flu is very good at mutation and keeps
  coming back with new strains, corresponding to
  new variations on the H and N variants that we
  have not yet developed resistance to.
• When a strain emerges that has a major change in
  its H molecule, this tends to cause a global
  pandemic has happened 3 times in the last
  century.

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Modelling Influenza A

• It is useful to model Flu A (or any other
  disease) for many reasons
• Better understanding leads to better ways to
  treat and control it.
• When the model seems to work (reflect reality),
  we can do some research with it that may lead to
  useful new treatments.
• If the model doesnt work, then we know that our
  understanding of the disease is flawed, so this
  guides further in vivo/in vitro research that
  needs to be done towards improving our
  understanding (and our model).
• Modelling of most diseases is traditionally done
  using differential equations, but
• CAs are now beginning to show that they produce
  more useful and accurate models.
• Which is good, because it means that we dont
  have to leave saving-the-world to mathematicians.

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Beauchemon et als CA/Flu model

• Two types of cell in the model
• epithelial cells (the ones that flu infects)
• immune system cells (that try to attack and
  destroy infected cells)

• Simplifications in the model
• the infection spreads directly from one infected
  cell to another (virus particles themselves are
  not part of the CA model)
• 2 Dimensional CA (is this too much of a
  simplification?)
• Each position in the CA grid represents an
  epithelial cell. Immune system cells are mobile
  over the grid and vary in number.
• It is updated synchronously.
• The Moore neighbourhood is used, the grid is
  toroidal.

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States and Rules

• Epithelial cells can be
• Healthy Infected Expressing
  Infectious Dead

• Transition rules
• An epithelial cell becomes Dead when it is
  older than CELL_LIFESPAN
• A Healthy epithelial cell becomes Infected with
  probability
• INN
  INFECT_RATE/8
• where INN is the number of infectious cells
  in its neighbourhood
• An Infected cell becomes Expressing after
  EXPRESS_DELAY timesteps
• An Expressing cell becomes Infectious after it
  has been infected for
• INFECT_DELAY timesteps.
• Infected, Expressing and Infectious cells become
  dead after being infected
• for INFECT_LIFESPAN
• Expressing and Infected cells become dead after
  they have been recognised
• by an immune cell.
• A Dead cell is replaced by a new Healthy cell
  with probability
• Healthy / ( Dead x
  DIVISION_TIME)
• The new healthy cell may immediately be
  Infected according to 2nd rule above.

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The Immune Cells
At the start each epithelial cell has a
random age between 0 and CELL_LIFESPAN a
fraction INFECT_INIT are set to Infected, the
rest are Healthy

• Meanwhile, Immune cells have two states
  Virgin, and Mature
• Virgin an immune cell that has not encountered
  the virus before,
• so it does not recognise it, and therefore
  does not attack it.
• Mature an immune cell that could attack an
  infected cell.
• At the start, BASE_IMM_CELL immune cells are
  placed at random
• on the lattice, all Virgin

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The Immune Cell Rules

• At each timestep, each immune cell moves to a
  random neighbouring cell.
• An immune cell is removed when it reaches
  IMM_LIFESPAN
• A Virgin immune cell becomes Mature if the
  epithelial cell at its site is Expressing or
  Infectious.
• A Mature immune cell occupying the site of an
  Expressing or Infectious epithelial cell
  recognises it, and causes it to become Dead.
• When a recognition occurs, as above, a number
  RECRUITMENT of Mature immune cells are added at
  random site locations after RECRUIT_DELAY
  timesteps (this models the immune system
  response)
• If at any timestep there are fewer than
  BASED_IMM_CELL Virgin immune cells on the
  lattice, new ones are added at random locations
  to make up the numbers.

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What is known
Beauchemon et al wanted to compare their CA model
with a more complex model based on Differential
Equations. Parameters and observations gleaned
from real cases offered the following data to
evaluate the models

• The infection should peak on day 2 (after 48 hrs
  (timesteps))
• The fraction of dead epithelial cells should be
• 10 at 24h, 40 at 48hr, 10 at 120hr
• Virus level should decline to inoculation level
  on day 6.
• The number of immune cells should peak anywhere
  between
• days 2 and 7.

Using biologically plausible parameters, the
behaviour was as shown next
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Solid line dead Dashed line healthy Dotted
line infected Dot-dash immune cells per
healthy cell
Infection should peak on day 2
Solid line gets 2 out of 3
Virus long gone by day 6
Immune peak seems in right place
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Conclusions/discussion

• The paper goes on to see the effect of varying
  many of the parameters within biologically
  realistic ranges. But the main things to note
  are
• the model seems to have a good qualitative fit
  to the cell dynamics of the natural infection.
• the fit is not perfect, but there are plausible
  explanations that suggest improvements to the
  model. E.g. the infection dies away too quickly,
  however this is because the model leaves out the
  initial nonspecific immune response.
• the DE model is a better fit in some respects
  and worse in others but it is a much richer
  model (including different types of immune cells)
  in fact the CA model has 12 parameters, and the
  DE model has 60 parameters. This suggests the CA
  is the better approach.

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Modelling brain tumour growthKansal et al, 2000,
Journal of Theoretical Biology
Incidence of primary malignant brain tumours is
8/100,000 p.a. 3D CA, modelling brain tumour
growth Shows that Macroscopic tumour behaviour
can be predicted via microscopic parameters Uses
only 4 parameters Makes predictions that match
the biological reality
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MRI scan showing a tumour the white
area Represents blood leakage around the tumour
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Kansal et al use the Delaunay Tesselation as
their lattice on the right we see blackened
cells representing the tumour, in a simplified 2D
version
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States and Rules
Not easy to glean from the paper, but cells are
either healthy (empty lattice site) or
tumour. Tumour cells are either proliferative
(they divide into additional tumour cells) or
not. When a proliferative tumour cell wants to
divide, it fills a healthy space with a new
tumour cell if it can find one within delta_p of
its position. If it cant find one, it becomes
non-proliferative.
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1.5M lattice sites Initial tumour is 1000
proliferative cells at centre of lattice
Result seems realistic
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Very good fit to real data The lines are the CA
model predictions of tumour radius and volume
against time The plotted points are measurements
from real cases of untreated tumours
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Read a couple more applications for yourself.
See the www site for the Influenza CA paper
Tumour CA paper A Traffic Simulation CA paper
Historic urban growth in the San Francisco bay
area CA
Not examinable reading, but recommended