# Biomod/2014/UANLl/DiffusionModel

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# Diffusion Model

In order to make the reactants transport process into the liposome more efficient, it is important to develop a mathematical expression that takes into account the multiple variables that may affect the process.

Although a liposome is not a living cell, the osmotic transport through its lipidic membrane can be mathematically modeled the same way as a cell because of its resemblance to the external membrane structure.
There are two transport mechanisms for molecules to enter a cell: energy-independent or energy-dependent. Energy independent mechanisms are spontaneous; in this particular case, reactants enter the liposome by concentration gradient driven molecular diffusion.

The reactants diffusion into the liposome can be treated as a pseudosteady process and it can be expressed by Fick’s law of diffusion:

Equation 1 can be rewritten as:

where H represents the fraction of the diaphragm’s area that is available for diffusion. Then mass balances on both sides of the membrane can be expressed as[4]:

where A is the membrane’s area. By dividing equations 3 and 4 in V and subtract them, the result can be combined with the flux equation to obtain:

in which:

where β is a geometrical constant characteristic of the particular membrane. The differential equation can be solved assuming the following initial condition:

Integrating the differential equation gives the following result:

Rearranging the equation:

From the equation 6 and 8 it can be noticed that the diffusion coefficient (D) is inversely proportionate to the geometrical constant β, meaning that the membrane area (A) and longitude (L) must be as little as possible to increase the value of diffusivity. Also, as more reactants enter the liposome, the concentration increases so the difference in concentration between this and the concentration outside the membrane is lower as time pases. This reduces significantly the diffusivity of the reactants into the liposome as time goes on.

It can be implied by the mathematical expressions previously described that by developing smaller liposomes in large quantities would result in a faster and more extended diffusion process, allowing the reactants to enter the lipidic membrane in more efficient way.

## References:

[N1] http://www.news-medical.net/health/What-is-a-Liposome.aspx

[N2] “Bioprocess Engineering Basic Concepts ” Michael L. Shuler, Fikret Kargi. Pearson Education Limited 2nd Edition USA 1992 ISBN 1292025999

[N3] “Transport Phenomena” R. Byron Bird, Warren E. Steward, Edwin N. Lightfoot. John Wiley & Sons 2nd Edition USA 2007 ISBN 0470115394

[N4] “Diffusion-Mass Transfer in Fluid Systems” E.L. Culler. Cambridge University Press. England 1984 1st Edition ISBN 978-0-521-87121-1