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Drug diffusion through the skin is usually modelled using the diffusion equation $∂t∂c(x,t) =D∂x_{2}∂_{2}c(x,t) $ where $c(x,t)$ is the concentration of drug at depth $x$ and time $t$. The skin is modelled as a homogenous membrane of thickness $h$, with the following boundary conditions: - At the upper skin surface $(x=0)$, the concentration is constant, equal to the concentration $c_{0}$ applied to the skin (this is because such a small fraction of drug enters the skin) - At the lower skin surface $(x=h)$, the concentration is zero (because any drug that makes it though the skin membrane is cleared away by the blood stream almost immediately - Initially $(t=0)$, there is no drug in the skin (a) Show from the diffusion equation that the Laplace transform of $c(x,t)$ in the skin must take the form $C(x,s)=a(s)sinh(D xs )+b(s)cosh(D xs )$ (b) Show that the upper and lower surface boundary conditions lead to the following expression for $C(x,s)$ : $C(x,s)=c_{0}stanh(D hs )cosh(D xs )tanh(D hs )−sinh(D xs ) $ (c) From this expression, the Laplace transform of the total amount of drug $q(t)$ penetrated through an area $A$ of skin can be shown to be $e_{s}$ $Q(s)=−AD∂x∂ sC(x,s) ∣∣ _{x=h}=s_{2}K sinhst_{d} st_{d} $ for constants $K$ and $t_{d}$. Using complex integration, invert the Laplace transform to derive a series expansion for $q(t)$

Hence the above explained answer

(a) To find the Laplace transform of c(x,t), we will take the Laplace transform of both sides of the diffusion equation with respect to t, using the notation $\mathcal{L} \{f(t)\} = F(s)$ for the Laplace transform of f(t), to get:

$$\mathcal{L}\left\{\frac{\partial c(x,t)}{\partial t}\right\} = \mathcal{L}\left\{D \frac{\partial^2 c(x,t)}{\partial x^2}\right\}$$

Using the linearity of the Laplace transform and the fact that it is a continuous linear operator, we can interchange the order of differentiation and applying the Laplace transform to yield:

$$s C(x,s) - c(x,0) = D \frac{\partial^2 C(x,s)}{\partial x^2}$$

Since we have the initial condition that there is no drug in the skin at t=0, we have $c(x,0)=0$, and thus:

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