How difficult is it for a needle-free injection to reach the intramuscular level?

I recently answered this question on Quora, but I thought that this topic warranted further discussion on this blog for  seasoned professionals.  Drug transport is quite complicated, and is many times misunderstood.

I’ve always thought that this happens because most of the standard mass transport modeling assumptions don’t apply to drug transport!  We can’t assume high levels of convective flow (thus medium is not well mixed leading to gradient formation), we can’t assume point sources and sinks (these are in fact distributed over significant areas), and we can’t assume that one effect dominates over all others (consumption by cells, partitioning to ECM, partitioning to blood, etc. all have fairly significant roles).   The end result is that the mass transport is quite complicated and many times defies common sense!

For a book chapter that I have written on the topic, please look up:

Hydrogels in Cell-Based Therapies; Edited by Che J. Connon and Ian W. Hamley; Chapter 1: Soluble Molecule Transport Within Synthetic Hydrogels in Comparison to the Native Extracellular Matrix

For the original answer to the Quora question, please click here.  The original text of the Quora article can be found there, and for the sake of brevity, I will summarize it here.  Basically, the original poster asked about how difficult it was for needle-free drug delivery from the skin into the intramuscular level.  This in an interesting question for a number of reasons, but especially since they are focusing on needle-free delivery, the entire focus is on drug transport via diffusion.

Our general discussion will use the following general solution:

J = -D*dC/dx – E’

Where J is the mass flux into the tissue from the delivery mechanism, D is the drug diffusion constant, C is drug concentration, x is position into the tissue, and E’ is a rate function (i.e. derivative of C with respect to x) describing the rate of elimination of the drug product.  This model is Fick’s Law of Diffusion with the extra function of drug elimination subtracted at the end.

One must note right away that the more detail we build into the model, the more complex it will become, but it will also be more descriptive.  Alternatively, the simpler the model is, the easier it will be to use, but the less descriptive it will become.  In this and following posts, we will discuss situations of increasing complexity and the necessary changes to the model to accommodate these complexities.

For today, let’s assume that the drug has very simple properties.  So simple in fact that all methods of drug elimination can be compiled into one rate constant, K.  Thus, E’ = K.  Such a situation would apply when elimination from the tissue is a linear function of x, position into the tissue.  The equation would now be as follows:

J = -D*dC/dx – K

J*dx = -D*dC – K*dx ; J*dx + K*dx = -D*dC

∫(J+K)*dx = ∫-D*dC ; (J+K)*x + Initial Concentration = -D * C

C = – (J+K)/D * x + Initial Concentration

We now have a function that defines concentration into the tissue dependent on mass flux, J; elimination constant, K; drug diffusion constant, D; and position, x.  Initial Concentration (the integration constant) is defined by the boundary conditions of the problem.  One thing to clarify that may be confusing to some readers is the “+” sign between J and K.  Note that while this may on the surface read as elimination and mass flux are positive, the “-” sign from the front of the equation can be distributed to make the signs work out correctly.

Lastly, another simplification that is commonly performed is the division of K/D.  Since K is a constant and D is a constant, we can simply perform this operation to simplify the expression.  We will state that K/D = k.

C = – J/D*x – k*x + Initial Concentration

Some interesting situations are graphed out below.  Please don’t worry about the values at this time, but instead look at the general trends.

J > D and K

J = D and K

J < D and K

You should notice 1 big trend: as J gets smaller with respect to D and K, the slope of the graph gets shallower and shallower.  A great qualitative way to think of this is a traffic jam.  If J is large (i.e. lots of cars coming into the city), there is plenty of drug molecule to go around, but it is limited by the ability to diffuse and/or be eliminated (i.e. the cars are limited by jamming up at intersections).  Thus, the slope is very steep.  Conversely, if J is small (i.e. very few cars coming into the city), there is no limitation on the ability to diffuse and/or be eliminated (i.e. no traffic jams at all).  Thus, the slope is very shallow.

Now it’s your turn.  Try working out other scenarios and post them in the comments below!  Let’s thoroughly explore this scenario before moving on to more complicated analysis.

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