Novel analytical solutions for convolution in compartmental pharmacokinetic models and application to non-bioequivalent formulations

Deconvolution and convolution are powerful tools that allow decomposition and reconstruction, respectively, of plasma versus time profiles from input and impulse functions. While deconvolution have commonly used compartmental approaches (e.g., Wagner-Nelson or Loo-Riegelman), convolution most typically used the convolution integral which can be solved with numerical methods. In 2005, an analytical solution for one-compartment pharmacokinetic was proposed and has been widely used ever since. However, to the best of our knowledge, analytical solutions for drugs distributed in more than one compartment have not been reported yet. In this paper, analytical solutions for compartmental convolution from both original and exact Loo-Riegelman approaches were developed and evaluated for different scenarios. While convolution from original approach was slightly more precise than that from the exact Loo-Riegelman, both methods were extremely accurate for reconstruction of plasma profiles after respective deconvolutions. Nonetheless, convolution from exact Loo-Riegelman was easier to interpret and to be manipulated mathematically. In fact, convolution solutions for three and more compartments can be easily written with this approach. Finally, our convolution analytical solution was applied to predict the failure in bioequivalence for levonorgestrel, demonstrating that equations in this paper may be useful tools for pharmaceutical scientists.