One of the main objectives of the project was to synthesize a practical device, the Snifferometer for tuberculosis detection. As a first step towards achieving this goal, we built a temporal model of the system using PDE's which was simulated in matlab. A 2D reaction-diffusion system was then implemented in COMSOL multiphysics using the knowledge obtained from single cell pathway model,ombining the behaviours of the which helped us get a better understanding of how such a device could be implemented and the response times involved in such a process.

Diffusion Model

Setup of diffusion model

Figure 1: Full structure of diffusion device.

The diffusion setup contains two parts: a petridish of odor liquid/solid in the bottom and agar of olfactory yeast on the top. The odor molecular diffuses through gas from petridish to agar.

Approach

The diffusion model is built by PDEs.

where C represents the concentration, x is the distance from the surface of petridish, and D is the diffusion coefficient.

In order to solve PDEs, numerical methods are used as approximation.

On the left side of the equation, the Euler forward method is taken:

On the right side of the equation, the central differential method is used:

Left side equals to right side

Therefore, the PDEs can be numerically replaced by the equation:

where .

Boundary conditions

In order to solve this equation, boundary conditions are required. Two boundary conditions are set for these two surfaces of petridish-gas and gas-agar.

• At x=0, the Dirichlet boundary condition was placed:

with Psat being the saturation pressure of the specific compound being IsoAmylAcetate as an example in our model. For IsoAmylAcetate this value is 533.3 Pa at room temperature.

• One can take Neumann boundary condition at x=l, where l is the distant between petridish and agar and assumed to be 0.01 meter.

In this part of the system the molecules bounce against the agar surface back into the gasphase. A van Neumann boundary is the way to give a mathematical workable structure to that. For the problem definition to be complete and be able to start programming in MATLAB we only need one more value, being the diffusion coefficient. For this an empirical relation from the [1] was used.

And you probably guessed but my value for h is ok in the model and in my code this translates to the line:

dcdt(i,1)= a*x(i-1,1)+a*x(i+1,1) -2*a*x(i,1);

and YES a=h here and YES that IS confusing considering the other above statements. What I basically did is grab a bunch of Ordinary differential equations and couples them So that it becomes a PDE ( partial differential equation ). H*time step should also not be bigger than 0.5 and that does not happen, so the solution is stable.

Analytical solutions for this problem appeared a little more troublesome than expected. There are solutions ( in Sum row form ), but I was anticipating a nice function. I will get a book at the library to see if there are any better ones, but for now this works out fine. I can also use a Cranck-Nicolson approach to the problem to have another numerical solution for verification. And last but not least.

Simulation result

The results:

Figure 2: Simulation of diffusion model.(in 3D Cartesian coordinate, X axis is time, Y axis is the distance from the petridish, Z axis is the concentration)

Figure 2 shows the simulation result with distance dimension 0 - 0.01 meter and the time dimension 0 - 100 seconds. From Figure2 it can be seen after 100 seconds it approximately reaches steady state at the x=0.01 boundary.

Snifferometer

One of the other goals of the diffusion modeling was to model the device which we intended to build for the project. We made use of the finite element analysis simulator Comsol Multiphysics[3]

Figure 1: Snifferometer - Device with the modified yeast cells for sensing

Different models were analyzed for their suitability before coming to the final design in Figure. The device at the bottom has a membrance preventing the yeast coming in direct contact with external substances, above which is a layer of agar in which yeast is placed, the nutrients for it's growth is fed in through a channel along the sides of the tube using which the yeast cells can also be replaced as and when needed. A micro-optrode is then used to sense the photons emitted by the fluoroscent proteins.

Saliva Model

File:Test1.swf

References

Chen, N. H. (1962). New Generalized Equation for Gas Diffusion Coefficient. J. Chem. Eng. Data, 37–41.