Dermal Surface Dose and Skin Permeation

From Applied Surface Dose to Skin Entry

A dermal dose is placed on a surface, but the amount applied is not itself the rate at which drug enters the body. Entry depends on the concentration available at the skin surface, the concentration gradient across the skin, the skin’s permeability, and the area exposed. A PBPK model therefore needs a mass-per-time input function, rather than only the applied dose.

The key distinction is between applied surface loading, usually expressed as mass per area, and formulation concentration, expressed as mass per volume. The loading constrains how much drug is available overall. Concentration, together with the skin’s permeability, determines the permeation flux under the assumptions below. A loading value alone is not enough to calculate that flux.

Deriving Flux From the Skin Concentration Gradient

For a simple barrier model, represent the skin as a homogeneous layer of thickness \(h\), with a constant effective diffusivity \(D_{\mathrm{skin}}\). Under steady-state diffusion, Fick’s law states that flux is proportional to the concentration gradient through the skin. Let \(C_0\) and \(C_h\) be the drug concentrations in the skin at its outer and inner boundaries. If concentration changes linearly across the layer, the gradient is \((C_h-C_0)/h\), so the flux directed inward is:

This expression uses concentrations within the skin. To relate flux to a formulation concentration \(C_{\mathrm{d}}\), suppose the drug partitions at the outer boundary so that \(C_0=K C_{\mathrm{d}}\), where \(K\) is a dimensionless skin-to-formulation partition coefficient. Under a sink condition at the inner boundary, \(C_h\approx 0\). Substitution gives:

Here \(P_{\mathrm{skin}}\) is the permeability coefficient, with units of length per time. Since \(D_{\mathrm{skin}}\) has units of length squared per time and \(h\) has units of length, \(P_{\mathrm{skin}}\) has units of length per time. Multiplying it by \(C_{\mathrm{d}}\), in mass per volume, gives flux in mass per area per time.

The sink assumption means that drug concentration at the receiving side is low enough that it does not materially oppose inward transport. If the inner boundary is not a sink, the concentration gradient must reflect that nonzero boundary concentration; using \(P_{\mathrm{skin}}C_{\mathrm{d}}\) would then overstate flux. More generally, flux is determined by the actual concentration difference across the skin, not by the donor concentration alone.

Converting Flux to PBPK Input

If the concentration at the skin surface changes with time, write it as \(C_{\mathrm{d}}(t)\). With constant permeability and a sink at the receiving side, the flux and total rate entering skin are:

The cumulative mass that crosses the skin over an interval from time 0 to time \(T\) is the integral of the rate:

For a systemic compartment that receives this mass immediately, the model may use \(R_{\mathrm{skin}}(t)\) as its input. If a constant fraction \(F_{\mathrm{escape}}\) of skin-crossing mass is assumed to escape local loss and reach systemic circulation, then \(R_{\mathrm{sys}}(t)=F_{\mathrm{escape}}R_{\mathrm{skin}}(t)\). Use this scaling only when local loss is represented as that fixed fraction; if a PBPK model represents intervening skin or tissue compartments explicitly, the input should enter the appropriate compartment instead.

These equations define an input rate conditional on the donor concentration. They do not predict how the formulation concentration changes as drug leaves it. In particular, treating \(C_{\mathrm{d}}\) as constant describes a maintained-concentration boundary condition; a fixed applied mass does not, by itself, establish that the condition will persist.

Time-Varying Surface Concentration

A time-varying donor concentration produces a time-varying entry rate, even if area and permeability remain constant. A convenient calculation is to divide time into intervals over which \(C_{\mathrm{d}}(t)\) is constant or can be approximated by a simple function. Integrate the rate within each interval, then sum the interval amounts. The resulting cumulative amount should be checked against the available applied mass.

Relating Applied Loading to Donor Concentration

A formulation description may give a surface loading rather than its concentration. Conversion requires additional information about the formulation. For example, if a uniform liquid layer has volume \(V_{\mathrm{form}}\) over area \(A\), its volume per area is \(v_{\mathrm{form}}=V_{\mathrm{form}}/A\). If the applied mass is uniformly distributed and dissolved in that volume, then \(C_{\mathrm{d}}=q_{\mathrm{app}}/v_{\mathrm{form}}\). This conversion relies on the stated distribution and dissolution assumptions; it is not a general identity between loading and concentration.

Choosing an Input Model and Checking It

A permeability-based input is useful when surface concentration, exposed area, and a defensible permeability estimate are available. It connects formulation conditions to a mechanistic flux and makes the roles of area and concentration explicit. A simpler alternative is to specify an empirical absorption-rate function, such as a constant input over a stated interval or a first-order input from a surface depot. That may be practical when the aim is to reproduce observed input and mechanistic skin data are unavailable, but its rate parameters do not separately identify permeability, concentration, or area.

Useful checks include verifying units at every step, confirming that the modeled rate is nonnegative for inward input, and comparing integrated skin entry with the applied mass. If cumulative modeled entry exceeds the available applied mass, the assumed concentration history, area, units, or input implementation is inconsistent with mass balance. Compare the predicted rate with any available permeation or concentration-time data, while recognizing that agreement in a plasma profile alone may not uniquely establish the skin input.

The steady-state approximation also deserves scrutiny. If the skin concentration profile has not had time to approach steady state, the instantaneous flux may differ from \(P_{\mathrm{skin}}C_{\mathrm{d}}\). Changes in skin condition, formulation, or exposed area can also invalidate a constant permeability or fixed-area assumption. A poor fit should therefore prompt checks of the boundary concentrations and assumptions, not just adjustment of an unrelated systemic clearance parameter.

Common Pitfall: Treating Applied Dose as Flux

A common incorrect calculation is to take an applied loading such as \(0.20\ \mathrm{mg\,cm^{-2}}\) and call it a flux of \(0.20\ \mathrm{mg\,cm^{-2}\,h^{-1}}\). This silently adds a time unit and assumes, without evidence, that the entire applied mass crosses each unit area in exactly one hour. Loading is mass per area; flux is mass per area per time. To calculate flux, one needs a concentration gradient and a transport property, or an independently specified rate model. To calculate a rate from flux, one then multiplies by exposed area.