Mathematical models and numerical simulation of drug release in the vascular system

Paolo Zunino
Politecnico di Milano

Carlo D'Angelo, Luca Formaggia, Sara Minisini, Christian Vergara)

The efficacy of the vascular system to transport oxygen, nutrients and
wastes can be also exploited to deliver drugs. However, for the specific case
of drugs, a uniform distribution in the body can be sometimes negative, and
a selective/localized delivery system could be preferable. For this reasons, we
develop and analyze different models for drug release. On one side, we study
models for systemic drug release, where the vascular system is exploited to
distribute the drug into tissues and organs. On the other side, we address
suitable technologies and the related models for local drug release, aiming
to deliver the drug to a specific location.
The first topic is based on the experience accumulated on the mathematical
models and simulation techniques of the cardiovascular system (see [1]
and related works). In particular, the governing equations for fluid dynamics
and mass transport in three dimensions are suitably reduced to obtain
simplified models that allow to handle the geometrical complexity of the
vascular tree. Then, to account for blood perfusion and the related mass
transfer, we address the coupling between small arteries, described by means
of the aforementioned reduced models, and the surrounding tissues (see[2,3]).
For the second topic, we address the specific case of drug eluting cardiovascular
stents (DES). In this case, mathematical models and numerical
simulation techniques play a relevant role in understanding what are the
most appropriate choices for the optimal design of DES, in terms of materials
and geometrical shape, (see [4,5] and references therein). The main
difficulties arise from the need to deal with phenomena that take place on
multiple scales in space and time. Concerning the space scales, we remind
that DES for cardiovascular applications are miniaturized metal structures
that are coated with a micro-_lm containing the drug that will be locally released
into the arterial walls for healing purposes. The thickness of this film
generally lays within the range of microns. As regards the time scales, the
release of drug generally persists until a few weeks after the stent implantation.
However, the local phenomena that influence the drug release take
place within much shorter time scales, typically minutes or even seconds.
Finally, for both both cases, we will address the interaction of the delivered
drug with the surrounding tissues, including diffusion advection and
reaction phenomena. In conclusion, the collection of these models can be seen as a starting point to study how to handle the complexity of the cardiovascular system
in order to effectively deliver a given drug.

[1] A. Quarteroni, L. Formaggia, Mathematical modelling and numerical
simulation of the cardiovascular system, Handbook of numerical analysis.
Vol. XII, pp. 3{127,North-Holland, Amsterdam, 2004.

[2] Carlo D'Angelo, Multiscale modelling of metabolism and transport phenomena
in living tissues, Ph.D. thesis 3803 (2007), EPFL.

[3] D'Angelo C., Quarteroni A., On the coupling of 1D and 3D di_usionreaction
equations. Application to tissue perfusion problems. Accepted
for publication in Mathematical Models and Methods in Applied Sciences
(M3AS), 2008.

[4] F. Migliavacca, F. Gervaso, M. Prosi, P. Zunino, S. Minisini, L. Formaggia,
and G. Dubini. Expansion and drug elution model of a coronary stent.
Comput. Methods. Biomech. Biomed. Engin., 10:63{73, 2007.

[5] C. Vergara and P. Zunino. Multiscale modeling and simulation of drug
release from cardiovascular stents. Technical Report 15, MOX, Department
of Mathematics, Politecnico di Milano, accepted on SIAM Multiscale Model.
Simul., 2008.

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