## Simplifying Deconvolution

Deconvolution is used to evaluate the absorption kinetics of a drug. Unfortunately the term can be confusing and explanations are generally even more confusing. While deconvolution is not a simple topic, I believe it can be understood so that more scientists can apply the principles to their work.

Before I define deconvolution, let me define the term “convolution”. Convolution describes a combination of two mathematical functions that create a third mathematical function. Let me over-simplify this for explanation purposes. The following mathematical equation is readily understood:

$x*y=z$

We commonly understand x, y, and z to be individual values or numbers. But let’s imagine that each one is a function such as the following:

$x = 2*n+1$

$y = 34*r+15$

When x and y are functions instead of numbers, we call x*y the convolution of x and y, resulting in z. So a convolution in simple terms is the combination of two functions that give rise to a third function. If you know x and y, you can perform a convolution to find z. If you know z and y, then you can perform a deconvolution to find x.

Deconvolution is the opposite of convolution. You know the resulting function and one of the starting functions and you want to deconvolve to get the other starting function. Deconvolution is used in many areas of science, but particularly in photo editing with blur and sharpen methods. Gaussian blur (as demonstrated in the picture of the butterfly) uses deconvolution methods.

So how does this apply to pharmacokinetics? Let’s take the same example as before, and rename things:

$x*y=z$

where z is the plasma concentration-time curve C(t) resulting from extravascular administration, y is the plasma concentration-time curve resulting from intravascular administration, and x is the input, or absorption function. Imagine a single particle of drug administered by oral ingestion. Once that particle is absorbed into the blood and passes through the liver (first pass effect), it acts the same way that a particle would if it had been injected into the blood stream directly. This means that a plasma concentration-time curve is simply a convolution of an absorption function and an IV disposition function. You can maybe think of the system of equations as the following:

$[text{Absorption Function}]*[text{IV Disposition Function}] = [text{Extravascular Disposition Function}]$

To use deconvolution methods, you will need concentration-time data from IV and oral administration. Point area methods of deconvolution are used to calculate the amount of drug absorbed in a given interval, the cumulative amount of drug absorbed, and the fraction remaining to be absorbed using deconvolution. Refer to pages 96 – 99 in Pharmacokinetics for the Pharmaceutical Scientist (Wagner, John G) for specific mathematical methods for deconvolution.

Convolution and deconvolution methods are frequently used to establish in vitro-in vivo correlations (IVIVCs). Watch this webinar to learn about two powerful approaches to developing IVIVCs.

### About the author

By: Nathan Teuscher
Dr. Teuscher has been involved in clinical pharmacology and pharmacometrics work since 2002. He holds a PhD in Pharmaceutical Sciences from the University of Michigan and has held leadership roles at biotechnology companies, contract research organizations, and mid-sized pharmaceutical companies. Prior to joining Certara, Dr. Teuscher was an active consultant for companies and authored the Learn PKPD blog for many years. At Certara, Dr. Teuscher developed the software training department, led the software development of Phoenix, and now works as a pharmacometrics consultant. He specializes in developing fit-for-purpose models to support drug development efforts at all stages of clinical development. He has worked in multiple therapeutic areas including immunology, oncology, metabolic disorders, neurology, pulmonary, and more. Dr. Teuscher is passionate about helping scientists leverage data to aid in establishing the safety and efficacy of therapeutics.