Bayesian Optimization-Based Inverse Finite Element Analysis for Atrioventricular Heart Valves.

Inverse finite element analysis (iFEA) of the atrioventricular heart valves (AHVs) can provide insights into the in-vivo valvular function, such as in-vivo tissue strains; however, there are several limitations in the current state-of-the-art that iFEA has not been widely employed to predict the in-vivo, patient-specific AHV leaflet mechanical responses. In this exploratory study, we propose the use of Bayesian optimization (BO) to study the AHV functional behaviors in-vivo. We analyzed the efficacy of Bayesian optimization to estimate the isotropic Lee-Sacks material coefficients in three benchmark problems: (i) an inflation test, (ii) a simplified leaflet contact model, and (iii) an idealized AHV model. Then, we applied the developed BO-iFEA framework to predict the leaflet properties for a patient-specific tricuspid valve under a congenital heart defect condition. We found that the BO could accurately construct the objective function surface compared to the one from a [Formula: see text] grid search analysis. Additionally, in all cases the proposed BO-iFEA framework yielded material parameter predictions with average element errors less than 0.02 mm/mm (normalized by the simulation-specific characteristic length). Nonetheless, the solutions were not unique due to the presence of a long-valley minima region in the objective function surfaces. Parameter sets along this valley can yield functionally equivalent outcomes (i.e., closing behavior) and are typically observed in the inverse analysis or parameter estimation for the nonlinear mechanical responses of the AHV. In this study, our key contributions include: (i) a first-of-its-kind demonstration of the BO method used for the AHV iFEA; and (ii) the evaluation of a candidate AHV in-silico modeling approach wherein the chordae could be substituted with equivalent displacement boundary conditions, rendering the better iFEA convergence and a smoother objective surface.

Strain energy density as a Gaussian process and its utilization in stochastic finite element analysis: application to planar soft tissues.

Data-based approaches are promising alternatives to the traditional analytical constitutive models for solid mechanics. Herein, we propose a Gaussian process (GP) based constitutive modeling framework, specifically focusing on planar, hyperelastic and incompressible soft tissues. The strain energy density of soft tissues is modeled as a GP, which can be regressed to experimental stress-strain data obtained from biaxial experiments. Moreover, the GP model can be weakly constrained to be convex. A key advantage of a GP-based model is that, in addition to the mean value, it provides a probability density (i.e. associated uncertainty) for the strain energy density. To simulate the effect of this uncertainty, a non-intrusive stochastic finite element analysis (SFEA) framework is proposed. The proposed framework is verified against an artificial dataset based on the Gasser-Ogden-Holzapfel model and applied to a real experimental dataset of a porcine aortic valve leaflet tissue. Results show that the proposed framework can be trained with limited experimental data and fits the data better than several existing models. The SFEA framework provides a straightforward way of using the experimental data and quantifying the resulting uncertainty in simulation-based predictions.

Strain estimation in aortic roots from 4D echocardiographic images using medial modeling and deformable registration.

Even though the central role of mechanics in the cardiovascular system is widely recognized, estimating mechanical deformation and strains in-vivo remains an ongoing practical challenge. Herein, we present a semi-automated framework to estimate strains from four-dimensional (4D) echocardiographic images and apply it to the aortic roots of patients with normal trileaflet aortic valves (TAV) and congenital bicuspid aortic valves (BAV). The method is based on fully nonlinear shell-based kinematics, which divides the strains into in-plane (shear and dilatational) and out-of-plane components. The results indicate that, even for size-matched non-aneurysmal aortic roots, BAV patients experience larger regional shear strains in their aortic roots. This elevated strains might be a contributing factor to the higher risk of aneurysm development in BAV patients. The proposed framework is openly available and applicable to any tubular structures.

A Bayesian constitutive model selection framework for biaxial mechanical testing of planar soft tissues: Application to porcine aortic valves.

A variety of constitutive models have been developed for soft tissue mechanics. However, there is no established criterion to select a suitable model for a specific application. Although the model that best fits the experimental data can be deemed the most suitable model, this practice often can be insufficient given the inter-sample variability of experimental observations. Herein, we present a Bayesian approach to calculate the relative probabilities of constitutive models based on biaxial mechanical testing of tissue samples. Forty-six samples of porcine aortic valve tissue were tested using a biaxial stretching setup. For each sample, seven ratios of stresses along and perpendicular to the fiber direction were applied. The probabilities of eight invariant-based constitutive models were calculated based on the experimental data using the proposed model selection framework. The calculated probabilities showed that, out of the considered models and based on the information available through the utilized experimental dataset, the May-Newman model was the most probable model for the porcine aortic valve data. When the samples were further grouped into different cusp types, the May-Newman model remained the most probable for the left- and right-coronary cusps, whereas for non-coronary cusps two models were found to be equally probable: the Lee-Sacks model and the May-Newman model. This difference between cusp types was found to be associated with the first principal component analysis (PCA) mode, where this mode's amplitudes of the non-coronary and right-coronary cusps were found to be significantly different. Our results show that a PCA-based statistical model can capture significant variations in the mechanical properties of soft tissues. The presented framework is applicable to other tissue types, and has the potential to provide a structured and rational way of making simulations population-based.

An intricate interplay between stent drug dose and release rate dictates arterial restenosis.

Since the introduction of percutaneous coronary intervention (PCI) for the treatment of obstructive coronary artery disease (CAD), patient outcomes have progressively improved. Drug eluting stents (DES) that employ anti-proliferative drugs to limit excess tissue growth following stent deployment have proved revolutionary. However, restenosis and a need for repeat revascularisation still occurs after DES use. Over the last few years, computational models have emerged that detail restenosis following the deployment of a bare metal stent (BMS), focusing primarily on contributions from mechanics and fluid dynamics. However, none of the existing models adequately account for spatiotemporal delivery of drug and the influence of this on the cellular processes that drive restenosis. In an attempt to fill this void, a novel continuum restenosis model coupled with spatiotemporal drug delivery is presented. Our results indicate that the severity and time-course of restenosis is critically dependent on the drug delivery strategy. Specifically, we uncover an intricate interplay between initial drug loading, drug release rate and restenosis, indicating that it is not sufficient to simply ramp-up the drug dose or prolong the time course of drug release to improve stent efficacy. Our model also shows that the level of stent over-expansion and stent design features, such as inter-strut spacing and strut thickness, influence restenosis development, in agreement with trends observed in experimental and clinical studies. Moreover, other critical aspects of the model which dictate restenosis, including the drug binding site density are investigated, where comparisons are made between approaches which assume this to be either constant or proportional to the number of smooth muscle cells (SMCs). Taken together, our results highlight the necessity of incorporating these aspects of drug delivery in the pursuit of optimal DES design.

Do we really understand how drug eluted from stents modulates arterial healing?

The advent of drug-eluting stents (DES) has revolutionised the treatment of coronary artery disease. These devices, coated with anti-proliferative drugs, are deployed into stenosed or occluded vessels, compressing the plaque to restore natural blood flow, whilst simultaneously combating the evolution of restenotic tissue. Since the development of the first stent, extensive research has investigated how further advancements in stent technology can improve patient outcome. Mathematical and computational modelling has featured heavily, with models focussing on structural mechanics, computational fluid dynamics, drug elution kinetics and subsequent binding within the arterial wall; often considered separately. Smooth Muscle Cell (SMC) proliferation and neointimal growth are key features of the healing process following stent deployment. However, models which depict the action of drug on these processes are lacking. In this article, we start by reviewing current models of cell growth, which predominantly emanate from cancer research, and available published data on SMC proliferation, before presenting a series of mathematical models of varying complexity to detail the action of drug on SMC growth in vitro. Our results highlight that, at least for Sodium Salicylate and Paclitaxel, the current state-of-the-art nonlinear saturable binding model is incapable of capturing the proliferative response of SMCs across a range of drug doses and exposure times. Our findings potentially have important implications on the interpretation of current computational models and their future use to optimise and control drug release from DES and drug-coated balloons.

A framework for incorporating 3D hyperelastic vascular wall models in 1D blood flow simulations.

We present a novel framework for investigating the role of vascular structure on arterial haemodynamics in large vessels, with a special focus on the human common carotid artery (CCA). The analysis is carried out by adopting a three-dimensional (3D) derived, fibre-reinforced, hyperelastic structural model, which is coupled with an axisymmetric, reduced order model describing blood flow. The vessel transmural pressure and lumen area are related via a Holzapfel-Ogden type of law, and the residual stresses along the thickness and length of the vessel are also accounted for. After a structural characterization of the adopted hyperelastic model, we investigate the link underlying the vascular wall response and blood-flow dynamics by comparing the proposed framework results against a popular tube law. The comparison shows that the behaviour of the model can be captured by the simpler linear surrogate only if a representative value of compliance is applied. Sobol's multi-variable sensitivity analysis is then carried out in order to identify the extent to which the structural parameters have an impact on the CCA haemodynamics. In this case, the local pulse wave velocity (PWV) is used as index for representing the arterial transmission capacity of blood pressure waveforms. The sensitivity analysis suggests that some geometrical factors, such as the stress-free inner radius and opening angle, play a major role on the system's haemodynamics. Subsequently, we quantified the differences in haemodynamic variables obtained from different virtual CCAs, tube laws and flow conditions. Although each artery presents a distinct vascular response, the differences obtained across different flow regimes are not significant. As expected, the linear tube law is unable to accurately capture all the haemodynamic features characterizing the current model. The findings from the sensitivity analysis are further confirmed by investigating the axial stretching effect on the CCA fluid dynamics. This factor does not seem to alter the pressure and flow waveforms. On the contrary, it is shown that, for an axially stretched vessel, the vascular wall exhibits an attenuation in absolute distension and an increase in circumferential stress, corroborating the findings of previous studies. This analysis shows that the new model offers a good balance between computational complexity and physics captured, making it an ideal framework for studies aiming to investigate the profound link between vascular mechanobiology and blood flow.

Effect of Residual and Transformation Choice on Computational Aspects of Biomechanical Parameter Estimation of Soft Tissues.

Several nonlinear and anisotropic constitutive models have been proposed to describe the biomechanical properties of soft tissues, and reliably estimating the unknown parameters in these models using experimental data is an important step towards developing predictive capabilities. However, the effect of parameter estimation technique on the resulting biomechanical parameters remains under-analyzed. Standard off-the-shelf techniques can produce unreliable results where the parameters are not uniquely identified and can vary with the initial guess. In this study, a thorough analysis of parameter estimation techniques on the resulting properties for four multi-parameter invariant-based constitutive models is presented. It was found that linear transformations have no effect on parameter estimation for the presented cases, and nonlinear transforms are necessary for any improvement. A distinct focus is put on the issue of non-convergence, and we propose simple modifications that not only improve the speed of convergence but also avoid convergence to a wrong solution. The proposed modifications are straightforward to implement and can avoid severe problems in the biomechanical analysis. The results also show that including the fiber angle as an unknown in the parameter estimation makes it extremely challenging, where almost all of the formulations and models fail to converge to the true solution. Therefore, until this issue is resolved, a non-mechanical-such as optical-technique for determining the fiber angle is required in conjunction with the planar biaxial test for a robust biomechanical analysis.

Mechanics of the Tricuspid Valve-From Clinical Diagnosis/Treatment, In-Vivo and In-Vitro Investigations, to Patient-Specific Biomechanical Modeling.

Proper tricuspid valve (TV) function is essential to unidirectional blood flow through the right side of the heart. Alterations to the tricuspid valvular components, such as the TV annulus, may lead to functional tricuspid regurgitation (FTR), where the valve is unable to prevent undesired backflow of blood from the right ventricle into the right atrium during systole. Various treatment options are currently available for FTR; however, research for the tricuspid heart valve, functional tricuspid regurgitation, and the relevant treatment methodologies are limited due to the pervasive expectation among cardiac surgeons and cardiologists that FTR will naturally regress after repair of left-sided heart valve lesions. Recent studies have focused on (i) understanding the function of the TV and the initiation or progression of FTR using both in-vivo and in-vitro methods, (ii) quantifying the biomechanical properties of the tricuspid valve apparatus as well as its surrounding heart tissue, and (iii) performing computational modeling of the TV to provide new insight into its biomechanical and physiological function. This review paper focuses on these advances and summarizes recent research relevant to the TV within the scope of FTR. Moreover, this review also provides future perspectives and extensions critical to enhancing the current understanding of the functioning and remodeling tricuspid valve in both the healthy and pathophysiological states.

Determination of prestress and elastic properties of virus capsids.

Virus capsids are protein shells that protect the virus genome, and determination of their mechanical properties has been a topic of interest because of their potential use in nanotechnology and therapeutics. It has been demonstrated that stresses exist in virus capsids, even in their equilibrium state, due to their construction. These stresses, termed "prestresses" in this study, closely affect the capsid's mechanical behavior. Three methods-shape-based metric, atomic force microscope indentation, and molecular dynamics-have been proposed to determine the capsid elastic properties without fully accounting for prestresses. In this paper, we theoretically analyze the three methods used for mechanical characterization of virus capsids and numerically investigate how prestresses affect the capsid's mechanical properties. We consolidate all the results and propose that by using these techniques collectively, it is possible to accurately determine both the mechanical properties and prestresses in capsids.

A multiscale active structural model of the arterial wall accounting for smooth muscle dynamics.

Arterial wall dynamics arise from the synergy of passive mechano-elastic properties of the vascular tissue and the active contractile behaviour of smooth muscle cells (SMCs) that form the media layer of vessels. We have developed a computational framework that incorporates both these components to account for vascular responses to mechanical and pharmacological stimuli. To validate the proposed framework and demonstrate its potential for testing hypotheses on the pathogenesis of vascular disease, we have employed a number of pharmacological probes that modulate the arterial wall contractile machinery by selectively inhibiting a range of intracellular signalling pathways. Experimental probes used on ring segments from the rabbit central ear artery are: phenylephrine, a selective α1-adrenergic receptor agonist that induces vasoconstriction; cyclopiazonic acid (CPA), a specific inhibitor of sarcoplasmic/endoplasmic reticulum Ca2+-ATPase; and ryanodine, a diterpenoid that modulates Ca2+ release from the sarcoplasmic reticulum. These interventions were able to delineate the role of membrane versus intracellular signalling, previously identified as main factors in smooth muscle contraction and the generation of vessel tone. Each SMC was modelled by a system of nonlinear differential equations that account for intracellular ionic signalling, and in particular Ca2+ dynamics. Cytosolic Ca2+ concentrations formed the catalytic input to a cross-bridge kinetics model. Contractile output from these cellular components forms the input to the finite-element model of the arterial rings under isometric conditions that reproduces the experimental conditions. The model does not account for the role of the endothelium, as the nitric oxide production was suppressed by the action of L-NAME, and also due to the absence of shear stress on the arterial ring, as the experimental set-up did not involve flow. Simulations generated by the integrated model closely matched experimental observations qualitatively, as well as quantitatively within a range of physiological parametric values. The model also illustrated how increased intercellular coupling led to smooth muscle coordination and the genesis of vascular tone.

An improved parameter estimation and comparison for soft tissue constitutive models containing an exponential function.

Motivated by the well-known result that stiffness of soft tissue is proportional to the stress, many of the constitutive laws for soft tissues contain an exponential function. In this work, we analyze properties of the exponential function and how it affects the estimation and comparison of elastic parameters for soft tissues. In particular, we find that as a consequence of the exponential function there are lines of high covariance in the elastic parameter space. As a result, one can have widely varying mechanical parameters defining the tissue stiffness but similar effective stress-strain responses. Drawing from elementary algebra, we propose simple changes in the norm and the parameter space, which significantly improve the convergence of parameter estimation and robustness in the presence of noise. More importantly, we demonstrate that these changes improve the conditioning of the problem and provide a more robust solution in the case of heterogeneous material by reducing the chances of getting trapped in a local minima. Based upon the new insight, we also propose a transformed parameter space which will allow for rational parameter comparison and avoid misleading conclusions regarding soft tissue mechanics.

Biomechanical Behavior of Bioprosthetic Heart Valve Heterograft Tissues: Characterization, Simulation, and Performance.

The use of replacement heart valves continues to grow due to the increased prevalence of valvular heart disease resulting from an ageing population. Since bioprosthetic heart valves (BHVs) continue to be the preferred replacement valve, there continues to be a strong need to develop better and more reliable BHVs through and improved the general understanding of BHV failure mechanisms. The major technological hurdle for the lifespan of the BHV implant continues to be the durability of the constituent leaflet biomaterials, which if improved can lead to substantial clinical impact. In order to develop improved solutions for BHV biomaterials, it is critical to have a better understanding of the inherent biomechanical behaviors of the leaflet biomaterials, including chemical treatment technologies, the impact of repetitive mechanical loading, and the inherent failure modes. This review seeks to provide a comprehensive overview of these issues, with a focus on developing insight on the mechanisms of BHV function and failure. Additionally, this review provides a detailed summary of the computational biomechanical simulations that have been used to inform and develop a higher level of understanding of BHV tissues and their failure modes. Collectively, this information should serve as a tool not only to infer reliable and dependable prosthesis function, but also to instigate and facilitate the design of future bioprosthetic valves and clinically impact cardiology.

In-vivo heterogeneous functional and residual strains in human aortic valve leaflets.

Residual and physiological functional strains in soft tissues are known to play an important role in modulating organ stress distributions. Yet, no known comprehensive information on residual strains exist, or non-invasive techniques to quantify in-vivo deformations for the aortic valve (AV) leaflets. Herein we present a completely non-invasive approach for determining heterogeneous strains - both functional and residual - in semilunar valves and apply it to normal human AV leaflets. Transesophageal 3D echocardiographic (3DE) images of the AV were acquired from open-heart transplant patients, with each AV leaflet excised after heart explant and then imaged in a flattened configuration ex-vivo. Using an established spline parameterization of both 3DE segmentations and digitized ex-vivo images (Aggarwal et al., 2014), surface strains were calculated for deformation between the ex-vivo and three in-vivo configurations: fully open, just-coapted, and fully-loaded. Results indicated that leaflet area increased by an average of 20% from the ex-vivo to in-vivo open states, with a highly heterogeneous strain field. The increase in area from open to just-coapted state was the highest at an average of 25%, while that from just-coapted to fully-loaded remained almost unaltered. Going from the ex-vivo to in-vivo mid-systole configurations, the leaflet area near the basal attachment shrank slightly, whereas the free edge expanded by ~10%. This was accompanied by a 10° -20° shear along the circumferential-radial direction. Moreover, the principal stretches aligned approximately with the circumferential and radial directions for all cases, with the highest stretch being along the radial direction. Collectively, these results indicated that even though the AV did not support any measurable pressure gradient in the just-coapted state, the leaflets were significantly pre-strained with respect to the excised state. Furthermore, the collagen fibers of the leaflet were almost fully recruited in the just-coapted state, making the leaflet very stiff with marginal deformation under full pressure. Lastly, the deformation was always higher in the radial direction and lower along the circumferential one, the latter direction made stiffer by the preferential alignment of collagen fibers. These results provide significant insight into the distribution of residual strains and the in-vivo strains encountered during valve opening and closing in AV leaflets, and will form an important component of the tool that can evaluate valve׳s functional properties in a non-invasive manner.

Nonuniform elastic properties of macromolecules and effect of prestrain on their continuum nature.

Many experimental and theoretical methods have been developed to calculate the coarse-grained continuum elastic properties of macromolecules. However, all of those methods assume uniform elastic properties. Following the continuum mechanics framework, we present a systematic way of calculating the nonuniform effective elastic properties from atomic thermal fluctuations obtained from molecular dynamics simulation at any coarse-grained scale using a potential of the mean-force approach. We present the results for a mutant of Sesbania mosaic virus capsid, where we calculate the elastic moduli at different scales and observe an apparent problem with the chosen reference configuration in some cases. We present a possible explanation using an elastic network model, where inducing random prestrain results in a similar behavior. This phenomenon provides a novel insight into the continuum nature of macromolecules and defines the limits on details that the elasticity theory can capture. Further investigation into prestrains could elucidate important aspects of conformational dynamics of macromolecules.

An inverse modeling approach for semilunar heart valve leaflet mechanics: exploitation of tissue structure.

Determining the biomechanical behavior of heart valve leaflet tissues in a noninvasive manner remains an important clinical goal. While advances in 3D imaging modalities have made in vivo valve geometric data available, optimal methods to exploit such information in order to obtain functional information remain to be established. Herein we present and evaluate a novel leaflet shape-based framework to estimate the biomechanical behavior of heart valves from surface deformations by exploiting tissue structure. We determined accuracy levels using an "ideal" in vitro dataset, in which the leaflet geometry, strains, mechanical behavior, and fibrous structure were known to a high level of precision. By utilizing a simplified structural model for the leaflet mechanical behavior, we were able to limit the number of parameters to be determined per leaflet to only two. This approach allowed us to dramatically reduce the computational time and easily visualize the cost function to guide the minimization process. We determined that the image resolution and the number of available imaging frames were important components in the accuracy of our framework. Furthermore, our results suggest that it is possible to detect differences in fiber structure using our framework, thus allowing an opportunity to diagnose asymptomatic valve diseases and begin treatment at their early stages. Lastly, we observed good agreement of the final resulting stress-strain response when an averaged fiber architecture was used. This suggests that population-averaged fiber structural data may be sufficient for the application of the present framework to in vivo studies, although clearly much work remains to extend the present approach to in vivo problems.

An immersogeometric variational framework for fluid-structure interaction: application to bioprosthetic heart valves.

In this paper, we develop a geometrically flexible technique for computational fluid-structure interaction (FSI). The motivating application is the simulation of tri-leaflet bioprosthetic heart valve function over the complete cardiac cycle. Due to the complex motion of the heart valve leaflets, the fluid domain undergoes large deformations, including changes of topology. The proposed method directly analyzes a spline-based surface representation of the structure by immersing it into a non-boundary-fitted discretization of the surrounding fluid domain. This places our method within an emerging class of computational techniques that aim to capture geometry on non-boundary-fitted analysis meshes. We introduce the term "immersogeometric analysis" to identify this paradigm. The framework starts with an augmented Lagrangian formulation for FSI that enforces kinematic constraints with a combination of Lagrange multipliers and penalty forces. For immersed volumetric objects, we formally eliminate the multiplier field by substituting a fluid-structure interface traction, arriving at Nitsche's method for enforcing Dirichlet boundary conditions on object surfaces. For immersed thin shell structures modeled geometrically as surfaces, the tractions from opposite sides cancel due to the continuity of the background fluid solution space, leaving a penalty method. Application to a bioprosthetic heart valve, where there is a large pressure jump across the leaflets, reveals shortcomings of the penalty approach. To counteract steep pressure gradients through the structure without the conditioning problems that accompany strong penalty forces, we resurrect the Lagrange multiplier field. Further, since the fluid discretization is not tailored to the structure geometry, there is a significant error in the approximation of pressure discontinuities across the shell. This error becomes especially troublesome in residual-based stabilized methods for incompressible flow, leading to problematic compressibility at practical levels of refinement. We modify existing stabilized methods to improve performance. To evaluate the accuracy of the proposed methods, we test them on benchmark problems and compare the results with those of established boundary-fitted techniques. Finally, we simulate the coupling of the bioprosthetic heart valve and the surrounding blood flow under physiological conditions, demonstrating the effectiveness of the proposed techniques in practical computations.

Architectural trends in the human normal and bicuspid aortic valve leaflet and its relevance to valve disease.

The bicuspid aortic valve (AV) is the most common cardiac congenital anomaly and has been found to be a significant risk factor for developing calcific AV disease. However, the mechanisms of disease development remain unclear. In this study we quantified the structure of human normal and bicuspid leaflets in the early disease stage. From these individual leaflet maps average fiber structure maps were generated using a novel spline based technique. Interestingly, we found statistically different and consistent regional structures between the normal and bicuspid valves. The regularity in the observed microstructure was a surprising finding, especially for the pathological BAV leaflets and is an essential cornerstone of any predictive mathematical models of valve disease. In contrast, we determined that isolated valve interstitial cells from BAV leaflets show the same in vitro calcification pathways as those from the normal AV leaflets. This result suggests the VICs are not intrinsically different when isolated, and that external features, such as abnormal microstructure and altered flow may be the primary contributors in the accelerated calcification experienced by BAV patients.

Patient-Specific Modeling of Heart Valves: From Image to Simulation.

Heart valves play a very important role in the functioning of the heart and many of the heart failures are related to the valvular dysfunctions, e.g. aortic stenosis and mitral regurgitation. As the medical field is moving towards a patient-specific diagnosis and treatment procedures, modeling of heart valves with patient-specific information is becoming a significant tool in medical field. Here we present the ingredients for valve simulation specifically the aortic valve, with a main focus on a novel spline-based mapping technique which solves many issues in generating patient-specific models - the microstructural mapping, the pre-strain calculations, prescribing dynamic boundary conditions, validation and inverse-modeling to obtain material parameters.

Viral capsid equilibrium dynamics reveals nonuniform elastic properties.

The long wavelength, low-frequency modes of motion are the relevant motions for understanding the continuum mechanical properties of biomolecules. By examining these low-frequency modes, in the context of a spherical harmonic basis set, we identify four elastic moduli that are required to describe the two-dimensional elastic behavior of capsids. This is in contrast to previous modeling and theoretical studies on elastic shells, which use only the two-dimensional Young's modulus (Y) and the bending modulus (κ) to describe the system. Presumably, the heterogeneity of the structure and the anisotropy of the biomolecular interactions lead to a deviation from the homogeneous, isotropic, linear elastic shell theory. We assign functional relevance of the various moduli governing different deformation modes, including a mode primarily sensed in atomic force microscopy nanoindentation experiments. We have performed our analysis on the T = 3 cowpea chlorotic mottle virus and our estimate for the nanoindentation modulus is in accord with experimental measurements.