Ankush Aggarwal

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 Youngs 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.

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.

Self-assembly of artificial microtubules

Understanding the complex self-assembly of biomacromolecules is a major outstanding question. Microtubules are one example of a biopolymer that possesses characteristics quite distinct from standard synthetic polymers that are derived from its hierarchical structure. In order to understand how to design and build artificial polymers that possess features similar to those of microtubules, we have initially studied the self-assembly of model monomers into a tubule geometry. Our model monomer has a wedge shape with lateral and vertical binding sites that are designed to form tubules. We used molecular dynamics simulations to study the assembly process for a range of binding site interaction strengths. In addition to determining the optimal regime for obtaining tubules, we have calculated a diagram of the structures that form over a wide range of interaction strengths. Unexpectedly, we find that the helical tubules form, even though the monomer geometry is designed for nonhelical tubules. We present the detailed dynamics of the tubule self-assembly process and show that the interaction strengths must be in a limited range to allow rearrangement within clusters. We extended previous theoretical methods to treat our system and to calculate the boundaries between different structures in the diagram.

Mitral Valves: A Computational Framework

The mitral valve (MV) is one of the four heart valves which locates in between the left atrium and left ventricle and regulates the unidirectional blood flow and normal functioning of the heart during cardiac cycles. Alternation of any component of the MV apparatus will typically lead to abnormal MV function. Currently, 40,000 patients in the United States receive MV repair or replacement annually according to the American Heart Association. Clinically, this can be achieved iteratively by surgical repair that reinstates normal annular geometry (size and shape) and restores mobile leaflet tissue, resulting in reduced annular and chordae force distribution. High-fidelity computer simulations provide a means to connect the cellular function with the organ-level MV tissue mechanical responses, and to help the design of optimal MV repair strategies. As in many physiological systems, one can approach heart valve biomechanics from using multiscale modeling (MSM) methodologies, since mechanical stimuli occur and have biological impact at the organ, tissue, and cellular levels. Yet, MSM approaches of heart valves are scarce, largely due to the major difficulties in adapting conventional methods to the areas where we simply do not have requisite data. There also remains both theoretical and computational challenges to applying traditional MSM techniques to heart valves. Moreover, existing physiologically realistic computational models of heart valve function make many assumptions, such as a simplified microstructural and anatomical representation of the MV apparatus, and thorough validations with in-vitro or in-vivo data are still limited. In the following, we present the details of the state of the art of mitral valve modeling techniques, with an emphasis on what is known and investigated at various length scales.

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.

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.

A Framework for Determination of Heart Valves’ Mechanical Properties Using Inverse-Modeling Approach

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. Relationship between the biomechanical properties of valve leaflets and their function has long been established, however, determining these properties in a non-invasive manner remains a challenge. Here we present a framework for such a tool for biomechanical properties determination. We use an inverse-modeling approach, where the only input is through imaging the leaflet tissue as it is loaded naturally during its functional cycle. Using a structural model for the leaflet material behavior allows us to reduce the number of parameters to be determined to only two, which in addition to dramatically reducing the computational time also allows one to visualize the cost function and the minimization process. We close with discussion about the contributions of the current framework and other constituents needed to make it a clinically viable tool.

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 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.