Biofilms, which are colonies of bacteria that can adhere to living or inert surf

Question

Biofilms, which are colonies of bacteria that can adhere to living or inert surfaces, can cause a wide range of human infections. Infections caused by bacteria living inside the biofilm are often chronic, as antibiotics applied to the surface of a biofilm have difficulty penetrating through the film thickness. Consider a biofilm that is associated with a cutaneous infection. An antibiotic (species A) is applied to the top layer of a biofilm (species B), such that a fixed concentration of the drug, CA,0 = 4 x 10-3 kmol/m3, is present on the top surface of the film. The diffusion coefficient of the drug within the biofilm is DAB = 2 x 10-12 m2/s. The antibiotic is consumed by biochemical reactions inside the film and the rate of consumption depends on the local concentration of drug, in the form RA = - kCA, where k = 0.1 s-1. To eliminate the bacteria, the antibiotic has to be consumed at a rate of at least 0.2 x 10-3 kmol/(m3 s), because at lower consumption rates, the bacteria will be able to grow faster than it is destroyed. Given this context, determine the maximum biofilm thickness that can be successfully treated by the antibiotic.

Explanation / Answer

Diffusion coefficient or DAB = 2 x 10-12 m2/s

Concentartion of drug = 4 x 10-3 kmol/m3

Rate of consumption of drug = RA = - kCA, where k = 0.1 s-1

Required rate to eliminate bacteria = 0.2 x 10-3 kmol/(m3 s)

Rate of consumption of drug = -0.1 * 4 x 10-3

= 4x10-4 kmol/(m3 s) = 0.4* 10-3 kmol/(m3 s)

This is two times higher than the required rate to eliminate the bacteria.

So, it will require just 0.5 seconds reaching the biofilm.

And we know that

T90 = 1.03L2/DAB

Where T is the time required by solute to attain 90% of the bulk fluid concentration at the base of flat slab on biofilm.

L = thickness of biofilm

D = diffusion constant

So, 0.5 = L2/2 x 10-12 m2/s

L2 = 0.5*2 x 10-12 m2

= 1*10-12m2

Therefore L = sq. root of 1*10-12m2

= 1* 10-6 m

Therefore the thickness of the biofilm is 1 micrometre

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