(596ap) A Three Compartment Model for Pharmacodynamics and Anomalous Diffusion of Engineered Nanoparticles | AIChE

(596ap) A Three Compartment Model for Pharmacodynamics and Anomalous Diffusion of Engineered Nanoparticles


Ubaque, L. - Presenter, Universidad de los Andes
Vargas, R. D., Universidad de los Andes
Vargas, W. L., Universidad de los Andes

Application of nanocarriers as a solution for targeted drug delivery is a topic of increasing interest [1-4]. Transport of these nano-particles through biological tissues has been a problematic area since describing its dynamics has proven to be inefficient using classical approaches due to the anomalous diffusion (Podlubny, 1999) that often times is observed in such complex media.

Anomalous diffusion is characterized by a nonlinear evolution of the solute Mean Square Displacement (MSD) in time as is supposed by classical transport laws (e.g. Fick law). As a solution to these kind of anomalies in MSD evolution, fractional calculus has emerged as a new branch of applied mathematics capable of involving memory in transport processes, an important number of works has been recently published incorporating fractional temporal derivatives [6] over the classical Flick’s Law, therefore, arriving to the so-called Time Fractional Diffusion Equation (TFDE).

In this work a three compartment Physiologically Based Pharmacokinetic Model (PBPK) is presented with the incorporation of the TFDE that takes into account memory effects in transport through biological tissues. The compartments are connected in series with only mass transfer by diffusion as the only mechanism available for transport. The first of the compartments is a liquid cavity assumed to be perfectly well mixed and initialized with a high particle concentration of nanoparticles. The second is a biological tissue where one-dimensional anomalous diffusion takes place and with an initially null particle concentration. Finally the third compartment is another perfectly mixed liquid cavity with no particle concentration at the beginning of the process.

Parameters of the model (including the fractional derivative order) are adjusted to experimental data from a Franz diffusion cell via lest squares optimization over the evolution of the third compartment particle concentration. Future directions for Improving the PK of nanoparticles in complex tissues are discussed.

Keywords: Time delay, pharmacodynamics, anomalous diffusion, fractional derivatives.


  1. Kohji Ohno, Tatsuki Akashi, Yoshinobu Tsujii, Masaya Yamamoto, and Yasuhiko Tabata. Blood Clearance and Biodistribution of Polymer Brush-Afforded Silica Particles Prepared by Surface-Initiated Living Radical Polymerization. Biomacromolecules 2012, 13, 927−936
  2. Rochelle R. Arvizo, Oscar R. Miranda, Daniel F. Moyano, Chad A. Walden, Karuna Giri, Resham Bhattacharya, J. David Robertson, Vincent M. Rotello, Joel M. Reid, Priyabrata Mukherjee. Modulating Pharmacokinetics, Tumor Uptake and Biodistribution by Engineered Nanoparticles PLoS ONE | www.plosone.org  September 2011 | Volume 6 | Issue 9 | e24374.
  3. Mingguang Li, Zoi Panagi, Konstantinos Avgoustakis, Joshua Reineke. Physiologically based pharmacokinetic modeling of PLGA nanoparticles with varied mPEG content. International Journal of Nanomedicine 2012:7 1345–1356
  4. Shyh-Dar Li and Leaf Huang. Pharmacokinetics and Biodistribution of Nanoparticles. Molecular pharmaceutics  2008, VOL. 5, NO. 4, 496–504
  5. Podlubny, Fractional Differential Equations, Academic Press, San Diego (1999).
  6. Ivo Petraš, Richard L. Magin. Simulation of drug uptake in a two compartmental fractional model for a biological system. Commun Nonlinear Sci Numer Simulat 16 (2011) 4588–4595