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      A Finite Element Model for Mixed Porohyperelasticity with Transport, Swelling, and Growth

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          Abstract

          The purpose of this manuscript is to establish a unified theory of porohyperelasticity with transport and growth and to demonstrate the capability of this theory using a finite element model developed in MATLAB. We combine the theories of volumetric growth and mixed porohyperelasticity with transport and swelling (MPHETS) to derive a new method that models growth of biological soft tissues. The conservation equations and constitutive equations are developed for both solid-only growth and solid/fluid growth. An axisymmetric finite element framework is introduced for the new theory of growing MPHETS (GMPHETS). To illustrate the capabilities of this model, several example finite element test problems are considered using model geometry and material parameters based on experimental data from a porcine coronary artery. Multiple growth laws are considered, including time-driven, concentration-driven, and stress-driven growth. Time-driven growth is compared against an exact analytical solution to validate the model. For concentration-dependent growth, changing the diffusivity (representing a change in drug) fundamentally changes growth behavior. We further demonstrate that for stress-dependent, solid-only growth of an artery, growth of an MPHETS model results in a more uniform hoop stress than growth in a hyperelastic model for the same amount of growth time using the same growth law. This may have implications in the context of developing residual stresses in soft tissues under intraluminal pressure. To our knowledge, this manuscript provides the first full description of an MPHETS model with growth. The developed computational framework can be used in concert with novel in-vitro and in-vivo experimental approaches to identify the governing growth laws for various soft tissues.

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          Most cited references29

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          Perspectives on biological growth and remodeling.

          The continuum mechanical treatment of biological growth and remodeling has attracted considerable attention over the past fifteen years. Many aspects of these problems are now well-understood, yet there remain areas in need of significant development from the standpoint of experiments, theory, and computation. In this perspective paper we review the state of the field and highlight open questions, challenges, and avenues for further development.
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            Stress-modulated growth, residual stress, and vascular heterogeneity.

            A simple phenomenological model is used to study interrelations between material properties, growth-induced residual stresses, and opening angles in arteries. The artery is assumed to be a thick-walled tube composed of an orthotropic pseudoelastic material. In addition, the normal mature vessel is assumed to have uniform circumferential wall stress, which is achieved here via a mechanical growth law. Residual stresses are computed for three configurations: the unloaded intact artery, the artery after a single transmural cut, and the inner and outer rings of the artery created by combined radial and circumferential cuts. The results show that the magnitudes of the opening angles depend strongly on the heterogeneity of the material properties of the vessel wall and that multiple radial and circumferential cuts may be needed to relieve all residual stress. In addition, comparing computed opening angles with published experimental data for the bovine carotid artery suggests that the material properties change continuously across the vessel wall and that stress, not strain, correlates well with growth in arteries.
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              Physiological transport forces govern drug distribution for stent-based delivery.

              The first compounds considered for stent-based delivery, such as heparin, were chosen on the basis of promising tissue culture and animal experiments, and yet they have failed to stop restenosis clinically. More recent compounds, such as paclitaxel, are of a different sort, being hydrophobic in nature, and their effects after local release seem far more profound. This dichotomy raises the question of whether drugs that have an effect when released from a stent do so because of differences in biology or differences in physicochemical properties and targeting. We applied continuum pharmacokinetics to examine the effects of transport forces and device geometry on the distribution of stent-delivered hydrophilic and hydrophobic drugs. We found that stent-based delivery invariably leads to large concentration gradients, with drug concentrations ranging from nil to several times the mean tissue concentration over a few micrometers. Concentration variations were a function of the Peclet number (Pe), the ratio of convective to diffusive forces. Although hydrophobic drugs exhibited greater variability than hydrophilic drugs, they achieved higher mean concentrations and remained closer to the intima. Inhomogeneous strut placement influenced hydrophilic drugs more negatively than hydrophobic drugs, dramatically affecting local concentrations without changing mean concentrations. Because local concentrations and gradients are inextricably linked to biological effect, our results provide a potential explanation for the variable success of stent-based delivery. We conclude that mere proximity of delivery devices to tissues does not ensure adequate targeting, because physiological transport forces cause local concentrations to deviate significantly from mean concentrations.
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                Author and article information

                Contributors
                Role: Editor
                Journal
                PLoS One
                PLoS ONE
                plos
                plosone
                PLoS ONE
                Public Library of Science (San Francisco, CA USA )
                1932-6203
                2016
                14 April 2016
                : 11
                : 4
                : e0152806
                Affiliations
                [1 ]Graduate Interdisciplinary Program in Applied Mathematics, The University of Arizona, Tucson, AZ, United States of America
                [2 ]Department of Mechanical Engineering, Stanford University, Stanford, CA, United States of America
                [3 ]Department of Aerospace and Mechanical Engineering, The University of Arizona, Tucson, AZ, United States of America
                [4 ]Graduate Interdisciplinary Program of Biomedical Engineering, The University of Arizona, Tucson, AZ, United States of America
                [5 ]BIO5 Institute for Biocollaborative Research, The University of Arizona, Tucson, AZ 85721, United States of America
                [6 ]Department of Biomedical Engineering, The University of Arizona, Tucson, AZ 85721, United States of America
                [7 ]Department of Bioengineering, The University of Pittsburgh, Pittsburgh, PA 15219, United States of America
                University of Washington, UNITED STATES
                Author notes

                Competing Interests: The authors have declared that no competing interests exist.

                Conceived and designed the experiments: MHA ABT EK BRS JPVG. Performed the experiments: MHA. Analyzed the data: MHA JPVG. Contributed reagents/materials/analysis tools: MHA ABT EK BRS JPVG. Wrote the paper: MHA ABT EK BRS JPVG. Designed the software used in analysis: MHA. Formulated the theory: MHA ABT EK BRS JPVG.

                Article
                PONE-D-15-36382
                10.1371/journal.pone.0152806
                4831841
                27078495
                6cd3aaa0-0f61-49d4-bb67-f256e0730d93
                © 2016 Armstrong et al

                This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

                History
                : 18 August 2015
                : 18 March 2016
                Page count
                Figures: 11, Tables: 4, Pages: 35
                Funding
                Funded by: funder-id http://dx.doi.org/10.13039/100000053, National Eye Institute;
                Award ID: 1R01EY020890
                Award Recipient :
                Funded by: funder-id http://dx.doi.org/10.13039/100000050, National Heart, Lung, and Blood Institute;
                Award ID: NHLBI-1R21HL111990
                Award Recipient :
                Funded by: funder-id http://dx.doi.org/10.13039/100008227, Achievement Rewards for College Scientists Foundation;
                Award Recipient :
                Funded by: funder-id http://dx.doi.org/10.13039/100007852, Whitaker International Program;
                Award Recipient :
                Funded by: funder-id http://dx.doi.org/10.13039/100000001, National Science Foundation;
                Award ID: DGE0841234
                Award Recipient :
                This work was supported by National Eye Institute 1R01EY020890 ( http://www.nih.gov/, JPVG); National Heart, Lung, and Blood Institute 1R21HL111990 ( http://www.nih.gov/, JPVG); Whitaker International Summer Grant ( http://www.whitaker.org/, MHA); National Science Foundation 0841234 ( http://www.nsf.gov/); and ARCS Foundation ( https://www.arcsfoundation.org/, MHA). Support for MHA was partly provided by the National Science Foundation under award No 0841234.
                Categories
                Research Article
                Physical Sciences
                Materials Science
                Material Properties
                Porosity
                Physical Sciences
                Physics
                Classical Mechanics
                Deformation
                Physical Sciences
                Physics
                Classical Mechanics
                Damage Mechanics
                Deformation
                Physical Sciences
                Materials Science
                Materials by Attribute
                Porous Materials
                Physical Sciences
                Materials Science
                Material Properties
                Density
                Physical Sciences
                Materials Science
                Materials Physics
                Density
                Physical Sciences
                Physics
                Materials Physics
                Density
                Physical Sciences
                Chemistry
                Chemical Elements
                Physical Sciences
                Mathematics
                Applied Mathematics
                Finite Element Analysis
                Biology and Life Sciences
                Anatomy
                Biological Tissue
                Soft Tissues
                Medicine and Health Sciences
                Anatomy
                Biological Tissue
                Soft Tissues
                Physical Sciences
                Physics
                Classical Mechanics
                Mechanical Stress
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                All relevant data are within the paper and its Supporting Information files.

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