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.

Some additional things about CAsA simple 2D CA to illustrate these points:

States 0 and 1: Wraparound 2D array of 30 cellsRules: 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 cell’s newstate for time t+1 is worked out in parallel based on the situation at t.

Start: 101001010001101000101010010001

T=1 : 110000000001110000010100000001

T=2 : 110000000001110000001000000001

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

T=1 : 101001000001101000101010010001T=2 : 101001000001101000101010010001

Clearly if there are n cells, then n timesteps in an asynchronous CAcorresponds to the 1 timestep of a synchronous CA.

T=3 : 111001000001101000101010010001T=4 : 111001000001101000101010000001T=5 : etc ...

Boundary conditions

Rules for a cell’s state transitions are usually defined in terms ofthe cell’s neighbourhood. E.g. this is the Moore neighbourhood:

But what about cells on the edge?

The common approach in 2D is to treat the CA surface as a Toroid

This just means wraparound inthe way indicated by theblue and green neighbourhoods illustrated

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 5—15% of the world’s population, causing around 250,000—500,000 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.

Images from the wikipedia article

The virus particle (virion) enters the healthy cell, which is the infected.The virus then tricks the cell into expressing the viral proteins, so thatmany copies of the virus become made in the cell. Soon the cell becomesinfectious, and eventually it just dies from overcrowding of virus copies.

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.

Modelling Influenza AIt 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 doesn’t 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 don’t have to leave saving-the-world to mathematicians.

Beauchemon et al’s 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.

States and RulesEpithelial 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.

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 randomon the lattice, all Virgin

The Immune Cells

• 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.

The Immune Cell Rules

What is knownBeauchemon 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 shownnext:

Solid line: deadDashed line: healthyDotted line: infectedDot-dash: immune cells per healthy cell

Infection should peak on day 2

Solid line gets 2 out of 3

Virus long gone by day 6Immune peakseems in right place

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.

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

MRI scan showing a tumour; the white areaRepresents blood leakagearound the tumour

Kansal et al use the Delaunay Tesselation as their lattice – on the rightwe see blackened cells representing the tumour, in a simplified 2D version

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 can’t find one, it becomes non-proliferative.

1.5M lattice sitesInitial tumour is 1000 proliferative cells at centre of lattice

Result seems realistic

Very good fit to real data;The lines are the CA model predictions of tumour radius andvolume against timeThe plotted points are measurements from real cases of untreated tumours

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