Optimization of Tissue-Engineered Vascular Graft Design Using Computational Modeling

Abstract

Tissue-engineered vascular grafts hold great promise in many clinical applications, especially in pediatrics wherein growth potential is critical. A continuing challenge, however, is identification of optimal scaffold parameters for promoting favorable neovessel development. In particular, given the countless design parameters available, including those related to polymeric microstructure, material behavior, and degradation kinetics, the number of possible scaffold designs is almost limitless. Advances in computationally modeling the growth and remodeling of native blood vessels suggest that similar simulations could help reduce the search space for candidate scaffold designs in tissue engineering. In this study, we meld a computational model of in vivo neovessel formation with a surrogate management framework to identify optimal scaffold designs for use in the extracardiac Fontan circulation while comparing the utility of different objective functions. We show that evolving luminal radius and graft compliance can be matched to that of the native vein by the end of the simulation period with judicious combinations of scaffold parameters, although the inability to match these metrics at all times reveals constraints engendered by current materials. We emphasize further that there is yet a need to examine additional metrics, and combinations thereof, when seeking to optimize functionality and reduce the potential for adverse outcomes.

Impact Statement

Tissue-engineered vascular grafts have considerable promise for treating myriad conditions, and multiple designs are now in FDA-approved trials. Nevertheless, the search continues for the optimal design of the underlying polymeric scaffold. We present a novel melding of a computational model of vascular adaptation and a formal method of optimization that can aid in identifying optimal design parameters, with potential to save development time and costs while improving clinical outcomes.

Introduction

Tremendous advances have emerged over the past two decades in the design of tissue-engineered vascular grafts (TEVGs) using different strategies.^1,2 One promising approach uses an implanted biodegradable polymeric scaffold that is seeded with autologous bone marrow-derived cells. This TEVG has proved to be safe and effective in extracardiac Fontan procedures used to treat children with congenital heart disease, with the graft connecting the thoracic inferior vena cava (IVC) to the pulmonary arteries.³ In brief, cells seeded within the polymeric scaffold promote the infiltration of host vascular cells, which deposit neotissue within the porous scaffold.⁴ From a tissue engineering perspective, the primary goal is to achieve a progressive deposition of neotissue that balances the degradation of polymer so as to maintain structural integrity. From a clinical perspective, the ultimate goal is to achieve a neovessel that yields appropriate hemodynamics while growing and adapting as the child matures. Realization of these goals would transform clinical care.

Deposition of neotissue in vivo results primarily from two basic processes, immunobiological and mechanobiological. Although the polymeric scaffolds are biocompatible, they elicit a foreign body response. Inflammatory cells, especially monocytes/macrophages, infiltrate the porous scaffold and via paracrine signaling promote host cell infiltration and deposition of extracellular matrix to isolate the graft from the host.⁵ This inflammatory response subsides as the polymer degrades, which also transfers more of the hemodynamic load to the accumulating neotissue. The intramural cells sense this changing mechanical environment and respond via further matrix deposition and degradation. Achieving sufficient stiffness and strength throughout neovessel development thus requires a coordinated balance between immuno-driven and mechano-mediated processes.^6,7 Too much deposition results in stenosis or scarring and thus suboptimal neovessel function; too much degradation results in structural vulnerability, potentially leading to aneurysmal dilatation or rupture.

The initial polymer properties and degradation profile thus play key roles in neovessel development. Pore size and fiber diameter are particularly important parameters for they modulate the inflammatory response and dictate the initial mechanical properties of the scaffold. Fiber alignment and intrinsic polymer stiffness similarly play important roles in both processes. Despite the almost limitless possible combination of scaffold geometries, microstructures, and methods of fabrication, trial- and-error has been the method of choice over decades to arrive at current designs,² which fortunately are both safe and effective. Nevertheless, numerous complications remain, including local luminal narrowing in patients and animal models. There is a need, therefore, to explore additional scaffold designs with the hope of eliminating, or at least minimizing, such complications. Recent advances in computational biomechanical modeling as well as formal methods of optimization promise to contribute to the search for improved, ultimately optimal, scaffold designs. In this study, we combine for the first time a novel vascular growth (changes in mass) and remodeling (changes in microstructure) model and formal methods of optimization to identify improved polymeric scaffold designs for TEVGs in a low-pressure model of the Fontan procedure.

Methods

All computational models must be informed by data. In this study, we focus on optimizing polymeric scaffolds to be implanted as interposition grafts within the IVC of SCID/bg mice, for which extensive data exist and there are clinically acceptable outcomes with no instances of occlusive thrombus, severe narrowing, or rupture through 2 years of implantation.^7–9 Specifically, the present computational growth and remodeling (G&R) model was developed based upon data from n = 15 in vitro degradation studies plus n = 33 in vivo studies using female CB-17 SCID/bg mice (5 nonimplanted controls plus 28 implanted with grafts), with data collected at 0, 2, 6, 12, 24, and 104 weeks of implantation.^8,9 Moreover, similar data were collected for n = 14 female C57BL/6 mice (5 controls plus 9 implanted), which enabled refinement of the inflammation modeling.⁷ The reader is referred to the original articles for further details on the experiments. While an IVC-interposition graft does not model the hemodynamics of the Fontan circulation, it provides significant information on neovessel development in a low-pressure environment.

G&R framework

To simulate time-dependent changes in the graft and to predict effects of design changes on key outcomes, we used a constrained mixture G&R model that describes the evolving geometry, composition, and biomechanical behavior of the implanted TEVGs.^6,7,9 The underlying general theoretical framework has been refined over the past 15 years, and similarly describes well mechano-mediated and immuno-driven adaptations of native arteries.¹⁰ While overall scaffold geometry (e.g., caliber and wall thickness) plays important roles in TEVGs,¹¹ particularly at different sites within the vasculature, other graft features, including polymer microstructure,¹² mechanical behavior,¹³ and degradation rate,¹⁴ largely determine the inflammatory response and neotissue turnover by infiltrating cells. Thus, constitutive relationships were developed to focus on some of these key material parameters.

Details of the theoretical framework and associated methods of solution can be found elsewhere,^6,7,10,15 hence, we briefly review key equations and model parameters for the optimization study performed herein. Importantly, the contribution of the degrading polymeric scaffold to the overall mechanical properties of the TEVG is given by a constituent stored energy density, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \rho { W^p } (s) = { \rho ^p } (0) { Q^p } (s) {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}} {W } { ^ p } } ({ { { \bf { C } } ^p } (s)}) , \end{align*} \end{document}

where microstructural analyses of such a cellular solid¹⁶ suggest that the shear modulus \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \mu ^p} = 0.03E_B^p{ ( 1 - { \varepsilon ^p} ) ^2}$$ \end{document} , with \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$E_B^p$$ \end{document} a Young's-like modulus for the bulk sealant-dominated polymer having porosity \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \varepsilon ^p}$$ \end{document} . Also, let the associated degradation be described by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { Q^p } (s) = { \frac { 1 + \exp ({ - { k^p } { \zeta ^p } }) } {1 + \exp ({ { k^p } ({ s - { \zeta ^p } }) } ) \; \; \; \; \; \; }} \end{align*} \end{document}

where k^p is a rate-type parameter and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \zeta ^p}$$ \end{document} governs the shape of the degradation profile.

Conversely, let the evolving stored energy of any constituent \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\alpha$$ \end{document} of the neotissue (e.g., different families of fibrillar collagen) be \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \rho { W^ \alpha } (s) = \mathop \int \limits_0^s {m^ \alpha } (\tau) { q^ \alpha } ({ s , \tau }) {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}} {W} { ^ \alpha } } ( { { \bf{ C } } ^ \alpha } (s)) d \tau , \end{align*} \end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${m^ \alpha } \left( \tau \right) > 0$$ \end{document} is the cell-mediated true mass density production rate at time \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\tau \in \left[ {0 , s} \right]$$ \end{document} , \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${q^ \alpha } \in \left[ {0 , 1} \right]$$ \end{document} is the fraction of neotissue produced at time \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\tau$$ \end{document} that survives at time s (Fig. 1), and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}} {W} { ^ \alpha } } ( { { \bf {C}} ^ \alpha } \left( s \right))$$ \end{document} is the constituent-specific stored energy. The interested reader is referred to prior papers for details on appropriate functional forms for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${q^ \alpha }$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}} {W} { ^ \alpha } }$$ \end{document} ^6,7 while we focus herein on the neotissue production \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${m^ \alpha } \left( \tau \right)$$ \end{document} that is governed by either immuno-driven (early) or mechano-mediated (late) processes, with the latter assumed to yield native venous properties¹⁷ and the former assumed to yield stiffer matrix, as identified previously from experimental data^7,9 and is common in inflammation-driven fibrosis. As in previous G&R formulations,^6,7,10 deviations from the homeostatic circumferential wall stress \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\Delta \sigma$$ \end{document} and luminal wall shear stress \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\Delta { \tau _w}$$ \end{document} drive the mechano-mediated matrix production, with a form ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\alpha = { \rm{m}}$$ \end{document} )

FIG. 1.

Idealized evolution of constituent mass density through G&R processes that result from gradual polymer degradation and neotissue deposition. In particular, inflammatory-driven matrix production (solid line) tends to dominate early graft behavior, but as the inflammation wanes with a loss of polymer mass (dotted lines), mechano-mediated matrix production (dashed lines) can contribute to long-term graft behavior and its eventual steady state in the absence of significant chronic inflammation. G&R, growth and remodeling.

\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {m^{ \rm{m}}} (\tau) = m_h^{ \rm{m}} ({1 - {e^{ - \tau }}}) \left({K_{ \rm{ \sigma }}^{ \rm{m}} \Delta \sigma (\tau) - K_{{\tau _w}}^{ \rm{m}} \Delta { \tau _w} (\tau) }\right) , \end{align*} \end{document}

The ability of inflammatory cells to infiltrate a scaffold is governed by its microstructure, particularly pore size r^p, which can be related to porosity \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \varepsilon ^p}$$ \end{document} and fiber diameter \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \omega ^p}$$ \end{document} via: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${r ^p} = - \sqrt \pi / 4 \cdot \left( {1 + \pi / \left( {2 \ln \left( {{ \varepsilon ^p}} \right) } \right) } \right) { \omega ^p}$$ \end{document} .¹⁸ Although scaffold porosity affects the inflammatory response, we parameterize immuno-driven kinetics in terms of r^p and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \omega ^p}$$ \end{document} , which influence cellular behavior directly, hence, we assumed immuno-driven ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\alpha = i$$ \end{document} ) neotissue production, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {m^i} \left( \tau \right) = m_h^i \left( {K_{ \rm{ \Gamma }}^i{{ \rm{ \Gamma }}^i} \left( \tau \right) + K_{ \rm{ \Upsilon }}^i{{ \rm{ \Upsilon }} ^i} \left( { \rm{ \tau }} \right) } \right) , \end{align*} \end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${{ \rm{ \Gamma }}^i} \left( \tau \right)$$ \end{document} describes an early transient response and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${{ \rm{ \Upsilon }} ^i} \left( { \rm{ \tau }} \right)$$ \end{document} a long-term persistent response, with gain-type parameters \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$K_{ \rm{ \Gamma }}^i \ge 0$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$K_{ \rm{ \Upsilon }}^i \ge 0$$ \end{document} depending on scaffold pore size and fiber diameter, which were normalized using values from the literature. \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$K_{ \rm{ \Gamma }}^i = K_h^i \left( {{r ^p}{ \omega ^p} / {r_n}{ \omega _n}} \right) + K_w^i$$ \end{document} with \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \rm{ \Gamma }} \left( \tau \right) = { \delta ^ \beta }{ \tau ^{ \beta - 1}} \exp \left( { - \delta \tau } \right)$$ \end{document} , which is normalized so that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${{ \rm{ \Gamma }}^i} \left( \tau \right) = { \rm{ \Gamma }} \left( \tau \right) / { \rm{max}} \left( {{ \rm{ \Gamma }} \left( \tau \right) } \right)$$ \end{document} ; \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$K_h^i$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$K_w^i$$ \end{document} determine microstructurally dependent and independent inflammatory production, respectively, while the gamma distribution parameter \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\delta$$ \end{document} affects the duration of the transient inflammatory response and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta$$ \end{document} is the skewness of the distribution. Conversely, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$K_{ \rm{ \Upsilon }}^i = K_h^i \left( {{r_n} / {r ^p} - 1} \right) + K_h^i \left( {{ \omega ^p} / { \omega _n} - 1} \right)$$ \end{document} with its onset determined by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${{ \rm{ \Upsilon }} ^i} \left( \tau \right) = 1 - \exp \left( { - \delta \tau } \right)$$ \end{document} . Parameter values are in Table 1.

Table 1.

Description, Values, and Units of Key Parameters that Control the Inflammatory-Driven Growth and Remodeling Processes (First Four Rows) as Well as Two Key Scaffold Parameters and Their Values Used for Normalization

G&R parameter	Description	Value	Units
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$K_h^i$$ \end{document}	Gain on microstructurally dependent inflammatory matrix production	3.31	—
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$K_w^i$$ \end{document}	Gain on microstructurally independent inflammatory matrix production	1.00	—
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\delta$$ \end{document}	Inflammatory production rate parameter	0.05	days⁻¹
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta$$ \end{document}	Inflammatory production shape parameter	3.08	—
r_n	Normalizing pore size	10	μm
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \omega _n}$$ \end{document}	Normalizing fiber diameter	4	μm

G&R, growth and remodeling.

Note that these functional forms for the inflammatory gains \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$K_{ \rm{ \Gamma }}^i$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$K_{ \rm{ \Upsilon }}^i$$ \end{document} account for several observations in the literature. For \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$K_{ \rm{ \Gamma }}^i$$ \end{document} , matrix production increases with increases in pore size¹⁴ and fiber diameter,¹⁹ which motivates the hypothesized proportionality in the transient matrix production equations. Yet, production of a fibrous cap around an implant has been observed for sufficiently small pores and large fibers,²⁰ thus leading to an inverse proportionality for pore size and direct correlation for fiber diameter with the magnitude of the chronic production term. Choice of normalizing values r_n and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \omega _n}$$ \end{document} also stemmed from observations in the literature. For example, limited cellular infiltration and matrix production have been observed for pore sizes less than \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$10 \; \mu \rm m$$ \end{document} ,^12,14 thus, we set \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${r_n} = 10 \;\mu \rm m$$ \end{document} .^6,7 It has also been observed that there is limited matrix production around fibers less than \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$4 \; \mu \rm m$$ \end{document} in size²⁰ and that cells undergo morphological changes for fibers larger than \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$5 \; \mu \rm m$$ \end{document} ,²¹ which together motivated \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \omega _n} = 4 \; \mu \rm m$$ \end{document} .

Objective function formulation

Consider a general objective function of the form \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$J = \sqrt { \sum {w_Y}{J_Y} \left( \textbf{\textit{A}} \right) } ,$$ \end{document} where w_Y are weights and J_Y are specific functions that depend on output metrics Y, each of which could depend on multiple design parameters A . Next, consider a common normalized mean squared error of the form \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J_Y} = ( { \left( {Y \left( \textbf{\textit{A}} \right) - {Y_h} ) / {Y_n}} \right) ^2}$$ \end{document} , where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$Y \left( \textbf{\textit{A}} \right)$$ \end{document} is the calculated output for parameter set A , Y_h is a (homeostatic) target value, and Y_n is a normalization value, chosen as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${Y_n} = {Y_h}$$ \end{document} subsequently. Given the absence of thrombotic occlusion and catastrophic rupture in our SCID/bg mouse model, we focused on optimizing the evolving geometry and mechanical properties of the TEVG toward a normal state. Thus, let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {J_R} \left( \textbf{\textit{A}} \right) = 1 / {n_s} \mathop \sum \limits_0^{{n_s}} { ( \left( {R \left( {\textbf{\textit{A}} , {n_ \tau }} \right) - {R_h} ) / {R_h}} \right) ) ^2} , \end{align*} \end{document}

where TEVG compliance \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$C = \left( {{R_2} - {R_1}} \right) / {R_1} \left( {{P_2} - {P_1}} \right)$$ \end{document} , with R₂ the luminal radius at pressure P₂ and R₁ that at P₁, is an important clinical and previously identified computational metric²² that can be used to minimize the mechanical mismatch between graft and adjacent vessels and thus promote normal hemodynamics. Since evolving thickness affects radial changes during loading, compliance quantifies the structural stiffness of the construct. We let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${C_h} = 0.042$$ \end{document} mmHg⁻¹ at the homeostatic target, with \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${P_1} = 2$$ \end{document} mmHg and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${P_2} = 3$$ \end{document} mmHg. Finally, let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {J_S} \left( \textbf{\textit{A}} \right) = 1 / {n_s} \mathop \sum \limits_0^{{n_s}} { ( \left( {S \left( {\textbf{\textit{A}} , {n_ \tau }} \right) - {S_h} ) / {S_h}} \right) ) ^2} , \end{align*} \end{document}

where circumferential material stiffness S defines TEVG behavior independent of geometry, with \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${S_h} = 80$$ \end{document} kPa for a native vein. Since neotissue resulting from inflammatory stimuli appears to be stiffer than that produced in response to mechanobiological stimuli,⁷ seeking to match material stiffness can serve as a surrogate for matching the composition of the graft to that of the native wall.

Given these three descriptors, we considered optimizations based on various combinations thereof, as, for example, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {J^{ \left( {RCS} \right) }} = \sqrt { \left( {{J_R} \left( \textbf{\textit{A}} \right) + {J_C} \left( \textbf{\textit{A}} \right) + {J_S} \left( \textbf{\textit{A}} \right) } \right) / 3} , \end{align*} \end{document}

with \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\textbf{\textit{A}} = \left[ {E_B^p , {r ^p} , { \omega ^p} , {k^p} , { \zeta ^p}} \right]$$ \end{document} defining the five independent design parameters of interest, and similarly for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RS} \right) }}$$ \end{document} , \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RC} \right) }}$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {CS} \right) }}$$ \end{document} . Deviations for each metric were computed daily from 0 to 2 years,⁹ nearly the full life span of a mouse, and included in the different objective functions used for minimization. All optimizations proceeded by parsing a parameter space encompassing each entry in A with bounds on each parameter reflecting scaffolds that could be fabricated using techniques reported in the literature (Table 2). Time courses for each metric were compared with those observed experimentally^7–9 as well as with homeostatic target values.

Table 2.

Values of Each Parameter that Defines the Polymeric Scaffold as Identified by a Particular Optimization Procedure (Third to Sixth Rows), Given Prescribed Bounds for the Search (First Row)

Objective function	Modulus \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$E_B^p$$ \end{document} (MPa)	Pore size r^p (μm)	Fiber diameter \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \omega ^p}$$ \end{document} (μm)	Degradation rate k^p (1/days)	Degradation shape \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \zeta ^p}$$ \end{document} (days)
Bounds	[1, 10,000]	[0.75, 80]	[1, 21]	[0.02, 0.62]	[0, 1000]
Experimental	100	11.21	4.01	0.55	32.60
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RCS} \right) }}$$ \end{document}	1.46	6.81	5.77	0.30	25.78
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RS} \right) }}$$ \end{document}	3.80	8.47	16.00	0.27	813.10
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RC} \right) }}$$ \end{document}	8.20	5.79	3.04	0.06	60.25
\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {CS} \right) }}$$ \end{document}	1.10	6.53	5.31	0.40	31.75

The previously used experimental graft design (second row) is also included for comparison. Note in particular that each of the optimizations desired a much lower value of the intrinsic elastic modulus \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$E_B^p$$ \end{document} and smaller pore size.

Surrogate management framework

Although many different methods are available for optimization,²³ we use a nonintrusive, derivative-free method called the Surrogate Management Framework, which has guaranteed convergence to a local minimum via pattern search theory and has been useful for parameter identification in diverse vascular remodeling simulations^7,24 as well as applications in cardiovascular surgery design.^25,26 In this study, however, we focus for the first time on optimizing the scaffold design to improve neovessel development. A surrogate management algorithm using a mesh adaptive direct search²⁷ is initiated with a set of randomly initialized points using Latin Hypercube Sampling (LHS). True objective function evaluations are performed at these initial points in the parameter space via repeated calls to the G&R simulation code, which outputs each metric Y for a given input parameter set A . All subsequent function evaluations are restricted to lie on a mesh in the parameter space, as dictated by pattern search convergence theory.

The algorithm proceeds through a series of SEARCH and POLL steps. In the SEARCH step, current values of J are interpolated over the discretized parameter space using kriging to generate a high-dimensional response surface. Potential minima are identified on the interpolated surface, and true objective function evaluations are performed for the corresponding parameter values. The kriging surface is updated after every true function evaluation for successive SEARCH steps. When a SEARCH step fails to produce an improved minimum, a POLL step evaluates candidate points around the current minimum in a positive spanning set using mesh adaptive direct search and seeks to find a better objective function value. If a POLL step succeeds in identifying an improved point, the algorithm proceeds back to the SEARCH step. Otherwise, if the POLL step fails, a mesh local optimizer has been found and the algorithm performs a mesh refinement or terminates after reaching a predefined mesh threshold. Optimal values are verified in each case considered with five independent runs of the optimization algorithm for different random initializations of the LHS, as the convergence is only guaranteed to a local minimum.

Results

We first simulated the evolving inner radius, compliance, and material stiffness based on scaffold parameters used in the experimentally implanted grafts (Table 2), which performed well according to classic metrics of outcome (e.g., suture retention strength, burst pressure, patency) as there were no catastrophic failures through 2 years in the SCID/bg mouse model.⁹ Model simulations based on experimental data (dashed line, Fig. 2) yet highlighted marked differences from the geometry and mechanical properties of the native IVC. Graft radius increased beyond that of the native vein as immuno-driven neotissue degraded before mechano-mediated neotissue could stabilize the geometry (dashed line, Fig. 3). Compliance increased from minimal on implantation to above native values, but then decreased with the onset of mechano-mediated neotissue turnover (dashed line, Fig. 2); after a recovery, remodeling left the compliance higher than that of native, which could adversely affect the hemodynamics and, in turn, associated continued graft remodeling. Circumferential material stiffness was initially several orders of magnitude larger than that of the native vein, due primarily to the intrinsic stiffness of the polymeric scaffold, but it dropped below that of the native vein as the polymer degraded and neotissue was deposited in a wall that remained thicker than native, thus leading to a lower stretch from the unloaded to loaded state and an associated lower (stretch-dependent) stiffness. Recall that this design emerged from years of prior trial-and-error studies and was found to be safe and effective.^8,9

FIG. 2.

Simulated evolution of graft inner radius (top), compliance (middle), and circumferential material stiffness (bottom) based on model predictions informed directly by the experimental scaffold (Exp, dashed line) as well as an optimization of objective function \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RCS} \right) }}$$ \end{document} (solid line), which equally weighs the desire for radius, compliance, and material stiffness to mimic native values (native, gray line). Note the logarithmic scale for stiffness.

FIG. 3.

Simulated inflammatory-driven (top) and mechano-mediated (bottom) mass density evolutions for a polymeric scaffold that was implanted experimentally (Exp, dashed line) versus that for a scaffold design that was optimized according to objective function \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RCS} \right) }}$$ \end{document} (solid line).

In an attempt to identify a scaffold design that could improve on outcomes for the experimental graft, consider predictions associated with minimization of objective function \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RCS} \right) }}$$ \end{document} (solid line, Fig. 2), which accounts for differences in radius, compliance, and stiffness throughout the entire 2-year G&R period. In the optimal case, we found an initial dilatation due to a loss of mechanical integrity of the polymer, as in the experimental case, although this dilatation was faster due to an initially lower polymer modulus, a higher rate of degradation, and an earlier onset of degradation (Table 2). A robust inflammatory response can halt such dilatations, however, and production of inflammation-stimulated neotissue (solid line, Fig. 3) stabilized luminal radius near the target value even though its persistence limited the production of more native mechano-mediated neotissue. This chronic immuno-driven matrix production was caused by a higher fiber diameter and a smaller pore size relative to the experimental scaffold (Table 2). Compliance for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RCS} \right) }}$$ \end{document} followed an initial time course similar to that for the experimental scaffold, with an early increase as the scaffold degraded, followed by a decrease with neotissue production. In this case, compliance remained closer to, although lower than, that of the native vessel once graft geometry reached a steady state. The material stiffness for the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RCS} \right) }}$$ \end{document} scaffold remained closer to the target value than did the experimental case throughout much of the development, but remained less than 50% of the target value by the end of remodeling.

Due to the inability of the model to match the target stiffness when minimizing \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RCS} \right) }}$$ \end{document} , the objective function was modified to remove the dependence on graft compliance since an inverse correlation between compliance and material stiffness suggests a tradeoff that involves wall thickening. Minimizing \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RS} \right) }} = \sqrt { \left( {{J_R} \left( \textbf{\textit{A}} \right) + {J_S} \left( \textbf{\textit{A}} \right) } \right) / 2}$$ \end{document} resulted in a near steady-state stiffness that approached the target value (solid line, Fig. 4). Despite improved stiffness matching, there was an early period of substantial narrowing and the steady state radius remained farther from the target than for the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RCS} \right) }}$$ \end{document} case. Compliance was also vastly reduced. These differences resulted from a nearly 3-fold larger fiber diameter in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RS} \right) }}$$ \end{document} compared to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RCS} \right) }}$$ \end{document} , while pore size increased 1.25-fold (Table 2), leading to both increased transient and chronic inflammatory matrix accumulations, which in turn suppressed mechano-mediated matrix production (Fig. 5). The increased value of the degradation shape parameter greatly reduced the extent of polymer degradation; hence, the sealant remained structurally significant to the end of the simulation.

FIG. 4.

Simulated changes in inner radius (top), compliance (middle), and circumferential material stiffness (bottom) for a polymeric scaffold optimized according to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RS} \right) }}$$ \end{document} (i.e., optimized for radius and stiffness; solid line), \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RC} \right) }}$$ \end{document} (i.e., for radius and compliance; dashed line), and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {CS} \right) }}$$ \end{document} (i.e., for compliance and material stiffness; dotted line), all in comparison to the homeostatic native values for an IVC (gray line). Note the logarithmic scale for stiffness. IVC, inferior vena cava.

FIG. 5.

Simulated evolution of inflammatory-driven (top) and mechano-mediated (bottom) mass density for scaffolds optimized according to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RS} \right) }}$$ \end{document} (solid line), \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RC} \right) }}$$ \end{document} (dashed line), or \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {CS} \right) }}$$ \end{document} (dotted line). Note that mechano-mediated mass density curves overlap for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {CS} \right) }}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RS} \right) }}$$ \end{document} .

To identify the potential for improved compliance over the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RCS} \right) }}$$ \end{document} case, we next examined \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RC} \right) }} = \sqrt { \left( {{J_R} \left( \textbf{\textit{A}} \right) + {J_C} \left( \textbf{\textit{A}} \right) } \right) / 2}$$ \end{document} . Resulting evolving compliances matched the target value well (dashed line, Fig. 4), but predicted values of radius were slightly worse than for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RCS} \right) }}$$ \end{document} , although close to the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RS} \right) }}$$ \end{document} case (Fig. 6). Stiffness matching was worse in the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RC} \right) }}$$ \end{document} case than for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RCS} \right) }}$$ \end{document} , as expected by the tradeoff between these two metrics. The changes for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RC} \right) }}$$ \end{document} were driven by reductions in both pore size and fiber diameter (Table 2), which resulted in a decreased inflammatory neotissue production and a corresponding increased mechano-mediated matrix production, both relative to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RCS} \right) }}$$ \end{document} . Furthermore, sealant degradation was delayed for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RC} \right) }}$$ \end{document} with lower degradation rates and an increased degradation shape parameter, although degradation was complete within 12 weeks of the G&R simulation.

FIG. 6.

Absolute percent deviation from native homeostatic values for three key G&R metrics–inner radius, overall compliance, and circumferential material stiffness–after a simulated period of 2 years for each scaffold design optimized according to the specified objective functions, all compared to results for the original experimental (implanted) TEVG. TEVG, tissue-engineered vascular grafts.

Finally, minimizing \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {CS} \right) }} = \sqrt { \left( {{J_C} \left( \textbf{\textit{A}} \right) + {J_S} \left( \textbf{\textit{A}} \right) } \right) / 2}$$ \end{document} yielded a surprising result (dotted line, Fig. 4): the associated optimal parameter values (Table 2) and simulated time courses (Fig. 4) were similar to those for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RCS} \right) }}$$ \end{document} , including the evolving luminal radius. Indeed, long-term radius matching was better for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {CS} \right) }}$$ \end{document} than for either \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RC} \right) }}$$ \end{document} or \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RS} \right) }}$$ \end{document} , despite their explicit dependence on radius; in contrast, compliance and stiffness were largely reduced for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RC} \right) }}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RS} \right) }}$$ \end{document} , respectively (Fig. 4). This finding suggests that improvements in compliance and material stiffness were possible for \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RC} \right) }}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RS} \right) }}$$ \end{document} via tradeoffs in radius, and all of these metrics are coupled. Figure 6 provides an overall summary of the long-term efficacy afforded by the different objective functions.

Discussion

With successful clinical trials of TEVGs underway,^28,29 the promise of vascular tissue engineering has never been greater. Nevertheless, although proved to be safe and effective, current clinical TEVGs are not yet optimal; complications include local narrowing (stenosis) and dilatation (aneurysm). While the clinical trials arose from trial-and-error empirical approaches, computational models have also become increasingly sophisticated and recognized as tools for guiding the design of implants for tissue engineering applications.^22,30 Thus, we sought to exploit the advances in computational modeling to establish a new approach to scaffold design. We previously showed that computational models can describe⁶ and predict⁹ in vivo neovessel development that starts from a biodegradable polymeric scaffold implanted as an IVC-interposition model, a reasonable surrogate for an extracardiac Fontan conduit. We further showed that these models can be used to identify preferred values of scaffold parameters via informal parameter sensitivity studies.¹⁵ Yet, seeking optimal designs via brute force methods (i.e., grid-based search methods) is computationally inefficient and likely intractable for models having the complexity needed to capture evolving TEVG behavior. As in experimental studies, brute force methods require one to hold all parameters constant while varying a single parameter and then to repeat this process for the different parameters of interest. For example, for the five key parameters examined herein, comparison of results for five different values, assuming the need for at least three experimental times (e.g., 2, 6, 24 weeks in mouse studies) and five replicates at each time for each combination of parameters, would necessitate 375 long-term mouse studies using 25 different graft designs. In contrast, in this study, we coupled numerical simulations of TEVG development with a formal surrogate management framework-based optimization to refine the experimental search space, that is, to identify promising combinations of parameters and thereby to reduce the requisite number of time-consuming and expensive mouse studies, noting that experimental validation of potentially improved designs is ultimately necessary. We suggest that such a reduction in iterative testing could accelerate the design process.

Complicating the search for the best TEVG is the coexistence of two overlapping processes that govern in vivo neovessel development: immuno-driven and mechano-mediated deposition and degradation of neotissue. Mechanobiological cues can guide the development of TEVGs, with increasing evidence that the loads present when cells infiltrate and differentiate can impact the outcome.³¹ Yet, in vivo TEVG development is initially driven primarily by a foreign body response to the implanted polymeric scaffold.^4,7 Iterative experiments examining effects of changing scaffold parameters on inflammatory cell function and polarization¹² have laid the groundwork for computational studies that seek to describe the inflammatory response to different scaffold microstructures. Noting the fundamental importance of initial experimental studies for informing a computational model, we focused on TEVG implants in immuno-compromised SCID/bg mice that yielded high patency rates without the need for anticoagulant therapy or seeding with bone marrow-derived mononuclear cells.^8,9 Because our simulations are informed by data from mouse IVC models, we also focused on matching uniform proximal and distal IVC properties and geometry over the ∼2-year lifespan of the mouse. Finally, we focused on responses in maturity, not adaptations needed in response to somatic growth since our data were for adult SCID/bg mice. These simplifications allowed us to consider potential differences in evolving behavior due solely to changes in graft microstructure and material properties.

Examination of the predicted optimal parameter sets for different objective functions (Table 2) revealed that each optimization produced a microstructure more prone to inflammation than did the experimental case. Since these simulations were based on data from immuno-compromised SCID/bg mice, such an enhancement of the inflammatory response may have been necessary to compensate for the impaired monocyte/macrophage function inherent to these mice. Indeed, previous studies in SCID/bg mice using grafts seeded with bone marrow-derived mononuclear cells demonstrated that release of chemokines such as MCP-1 during early stages of graft development helps increase monocyte/macrophage recruitment and improves remodeling.⁴ Mild inflammatory responses, as for the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RCS} \right) }}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RC} \right) }}$$ \end{document} objective functions, can prevent excessive dilatation via a compensatory early mechano-mediated neotissue production, as seen experimentally (Figs. 3 and 5). Yet, care must be taken to avoid excessive inflammation, as in the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RS} \right) }}$$ \end{document} case, where early exuberant neotissue production led to graft narrowing, and chronic inflammation-stimulated neotissue production prevented remodeling toward a native luminal radius or compliance (Fig. 4). Excessive inflammation leading to stenosis and occlusion has also been observed in vivo.⁵

For optimizations without excessive narrowing ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RCS} \right) }}$$ \end{document} , \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RC} \right) }}$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {CS} \right) }}$$ \end{document} ), rapid degradation of the polymeric sealant appeared advantageous since geometry was stabilized by inflammation-stimulated neotissue. In addition, predicted optimal scaffold moduli were at least an order of magnitude less than that of the experimental sealant and many orders of magnitude less than that of other common tissue engineering polymers, such as poly(L-lactide). These findings suggest that less stiff polymers, such as poly(glycerol sebacate), may be better candidates for venous scaffolds.³² Of course, loads experienced by an IVC-interposition graft are low relative to arterial grafts, with luminal pressures generally less than 5 mmHg, hence reinforcing the idea that scaffold mechanical and degradation properties must be tailored for the application of interest.

As demonstrated by possible stenosis when optimizing with \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RS} \right) }}$$ \end{document} , choice of objective function is also critical in ensuring clinically acceptable outcomes, particularly when multiple effects compete. For example, graft stiffness depends not only on the amount of neotissue produced but also whether it is produced in response to normal mechanical or excessive inflammatory stimuli. If polymer persists, as in the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${J^{ \left( {RS} \right) }}$$ \end{document} case, graft stiffness also continues to depend, in part, on the mechanical properties of the specific polymer. Importantly, the inability of the current computational model to match well both the material stiffness and overall compliance of the evolving graft by modifying the initial scaffold design suggests that either additional mechanisms of remodeling, such as collagen cross-linking and compaction, may need to be considered directly or that remodeling processes observed experimentally do not support the generation of truly native structures. Exact matching of native structure may be unnecessary for a functional Fontan graft, however, as evidenced by the clinical standard, Gore-Tex™ grafts. Trying to match native tissue properties should nonetheless remain as our ultimate goal. As there were large deviations from the native properties during early neovessel development, despite close matching at the end of the time course, the need for novel biomaterials and methods of controlling the foreign body response is clear, as this will prevent extreme early states with the potential for adverse clinical outcomes.

Although our approach-combining methods of biomechanical simulation with formal optimization-promises to reduce the experimental search space, many other aspects of neovessel development must be explored. The target population for Fontan procedures is young children for whom neither a single basal rate of matrix production nor a target homeostatic mechanical state exists. Moreover, gene expression and cellular activity can be very different in development versus maturity, as, for example, most vascular elastin is deposited and cross-linked before adolescence. There is a need to understand better the influence of normal vascular development. We also assumed that the distending blood pressure and associated blood flow remained constant throughout neovessel development. These, too, can change as the child grows. Among more general issues, it would be useful to examine theoretically whether the predicted quasistatic equilibrium solutions are mechanobiologically stable.³³ That is, would the resulting neovessels be expected to mechano-adapt to subsequent perturbations in mechanical loading? Such stability would be fundamental to ensuring appropriate growth potential. These and additional considerations remind us of the need for even more experimental data to inform the development of refined constitutive relationships, particularly in terms of scaffold microstructural parameters and for human cohorts of children.

Footnotes

Acknowledgments

This work was supported by grants from the NIH (HL098228, HL128602, HL139796), DoD (PR170976), and NSF (DGE1122492). We appreciated discussions with Ramak Khosravi.

Disclosure Statement

C.K.B. receives support from Gunze, Ltd., which did not influence this study. None of the other authors has disclosures.

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