Functional mechanical behavior of the murine pulmonary heart valve.

Genetically modified mouse models provide a versatile and efficient platform to extend our understanding of the underlying disease processes and evaluate potential treatments for congenital heart valve diseases. However, applications have been limited to the gene and molecular levels due to the small size of murine heart valves, which prohibits the use of standard mechanical evaluation and in vivo imaging methods. We have developed an integrated imaging/computational mechanics approach to evaluate, for the first time, the functional mechanical behavior of the murine pulmonary heart valve (mPV). We utilized extant mPV high resolution µCT images of 1-year-old healthy C57BL/6J mice, with mPVs loaded to 0, 10, 20 or 30 mmHg then chemically fixed to preserve their shape. Individual mPV leaflets and annular boundaries were segmented and key geometric quantities of interest defined and quantified. The resulting observed inter-valve variations were small and consistent at each TVP level. This allowed us to develop a high fidelity NURBS-based geometric model. From the resultant individual mPV geometries, we developed a mPV shape-evolving geometric model (SEGM) that accurately represented mPV shape changes as a continuous function of transvalvular pressure. The SEGM was then integrated into an isogeometric finite element based inverse model that estimated the individual leaflet and regional mPV mechanical behaviors. We demonstrated that the mPV leaflet mechanical behaviors were highly anisotropic and nonlinear, with substantial leaflet and regional variations. We also observed the presence of strong axial mechanical coupling, suggesting the important role of the underlying collagen fiber architecture in the mPV. When compared to larger mammalian species, the mPV exhibited substantially different mechanical behaviors. Thus, while qualitatively similar, the mPV exhibited important functional differences that will need to accounted for in murine heart valve studies. The results of this novel study will allow detailed murine tissue and organ level investigations of semi-lunar heart valve diseases.

Estimation of aortic valve interstitial cell-induced 3D remodeling of poly(ethylene glycol) hydrogel environments using an inverse finite element approach.

Aortic valve interstitial cells (AVICs) reside within the leaflet tissues of the aortic valve and maintain and remodel its extracellular matrix components. Part of this process is a result of AVIC contractility brought about by underlying stress fibers whose behaviors can change in various disease states. Currently, it is challenging to directly investigate AVIC contractile behaviors within dense leaflet tissues. As a result, optically clear poly (ethylene glycol) hydrogel matrices have been used to study AVIC contractility via 3D traction force microscopy (3DTFM). However, the local stiffness of the hydrogel is difficult to measure directly and is further confounded by the remodeling activity of the AVIC. Ambiguity in hydrogel mechanics can lead to large errors in computed cellular tractions. Herein, we developed an inverse computational approach to estimate AVIC-induced remodeling of the hydrogel material. The model was validated with test problems comprised of an experimentally measured AVIC geometry and prescribed modulus fields containing unmodified, stiffened, and degraded regions. The inverse model estimated the ground truth data sets with high accuracy. When applied to AVICs assessed via 3DTFM, the model estimated regions of significant stiffening and degradation in the vicinity of the AVIC. We observed that stiffening was largely localized at AVIC protrusions, likely a result of collagen deposition as confirmed by immunostaining. Degradation was more spatially uniform and present in regions further away from the AVIC, likely a result of enzymatic activity. Looking forward, this approach will allow for more accurate computation of AVIC contractile force levels. STATEMENT OF SIGNIFICANCE: The aortic valve (AV), positioned between the left ventricle and the aorta, prevents retrograde flow into the left ventricle. Within the AV tissues reside a resident population of aortic valve interstitial cells (AVICs) that replenish, restore, and remodel extracellular matrix components. Currently, it is technically challenging to directly investigate AVIC contractile behaviors within the dense leaflet tissues. As a result, optically clear hydrogels have been used to study AVIC contractility through means of 3D traction force microscopy. Herein, we developed a method to estimate AVIC-induced remodeling of PEG hydrogels. This method was able to accurately estimate regions of significant stiffening and degradation induced by the AVIC and allows a deeper understanding of AVIC remodeling activity, which can differ in normal and disease conditions.

Three-dimensional analysis of hydrogel-imbedded aortic valve interstitial cell shape and its relation to contractile behavior.

Cell-shape is a conglomerate of mechanical, chemical, and biological mechanisms that reflects the cell biophysical state. In a specific application, we consider aortic valve interstitial cells (AVICs), which maintain the structure and function of aortic heart valve leaflets. Actomyosin stress fibers help determine AVIC shape and facilitate processes such as adhesion, contraction, and mechanosensing. However, detailed 3D assessment of stress fiber architecture and function is currently impractical. Herein, we assessed AVIC shape and contractile behaviors using hydrogel-based 3D traction force microscopy to intuit the orientation and behavior of AVIC stress fibers. We utilized spherical harmonics (SPHARM) to quantify AVIC geometries through three days of incubation, which demonstrated a shift from a spherical shape to forming substantial protrusions. Furthermore, we assessed changes in post-three day AVIC shape and contractile function within two testing regimes: (1) normal contractile level to relaxation (cytochalasin D), and (2) normal contractile level to hyper-contraction (endothelin-1). In both scenarios, AVICs underwent isovolumic shape changes and produced complex displacement fields within the hydrogel. AVICs were more elongated when relaxed and more spherical in hyper-contraction. Locally, AVIC protrusions contracted along their long axis and expanded in their circumferential direction, indicating predominately axially aligned stress fibers. Furthermore, the magnitude of protrusion displacements was correlated with protrusion length and approached a consistent displacement plateau at a similar critical length across all AVICs. This implied that stress fiber behavior is conserved, despite great variations in AVIC shapes. We anticipate our findings will bolster future investigations into AVIC stress fiber architecture and function. STATEMENT OF SIGNIFICANCE: Within the aortic valve there exists a population of aortic valve interstitial cells, which orchestrate the turnover, secretion, and remodeling of its extracellular matrix, maintaining tissue integrity and ultimately sustaining the proper mechanical function. Alterations in these processes are thought to underlie diseases of the aortic valve, which affect hundreds of thousands domestically and world-wide. Yet, to date, there are no non-surgical treatments for aortic heart valve disease, in part due to our limited understanding of the underlying disease processes. In the present study, we built upon our previous study to include a full 3D analysis of aortic valve interstitial cell shapes at differing contractile levels. The resulting detailed shape and deformation analysis provided insight into the underlying stress-fiber structures and mechanical behaviors.

On the shape and structure of the murine pulmonary heart valve.

Murine animal models are an established standard in translational research and provides a potential platform for studying heart valve disease. To date, studies on heart valve disease using murine models have been hindered by a lack of appropriate methodologies due to their small scale. In the present study, we developed a multi-scale, imaging-based approach to extract the functional structure and geometry for the murine heart valve. We chose the pulmonary valve (PV) to study, due to its importance in congenital heart valve disease. Excised pulmonary outflow tracts from eleven 1-year old C57BL/6J mice were fixed at 10, 20, and 30 mmHg to simulate physiological loading. Micro-computed tomography was used to reconstruct the 3D organ-level PV geometry, which was then spatially correlated with serial en-face scanning electron microscopy imaging to quantify local collagen fiber distributions. From the acquired volume renderings, we obtained the geometric descriptors of the murine PV under increasing transvalvular pressures, which demonstrated remarkable consistency. Results to date suggest that the preferred collagen orientation was predominantly in the circumferential direction, as in larger mammalian valves. The present study represents a first step in establishing organ-level murine models for the study of heart valve disease.

A Coupled Mass Transport and Deformation Theory of Multi-constituent Tumor Growth.

We develop a general class of thermodynamically consistent, continuum models based on mixture theory with phase effects that describe the behavior of a mass of multiple interacting constituents. The constituents consist of solid species undergoing large elastic deformations and compressible viscous fluids. The fundamental building blocks framing the mixture theories consist of the mass balance law of diffusing species and microscopic (cellular scale) and macroscopic (tissue scale) force balances, as well as energy balance and the entropy production inequality derived from the first and second laws of thermodynamics. A general phase-field framework is developed by closing the system through postulating constitutive equations (i.e., specific forms of free energy and rate of dissipation potentials) to depict the growth of tumors in a microenvironment. A notable feature of this theory is that it contains a unified continuum mechanics framework for addressing the interactions of multiple species evolving in both space and time and involved in biological growth of soft tissues (e.g., tumor cells and nutrients). The formulation also accounts for the regulating roles of the mechanical deformation on the growth of tumors, through a physically and mathematically consistent coupled diffusion and deformation framework. A new algorithm for numerical approximation of the proposed model using mixed finite elements is presented. The results of numerical experiments indicate that the proposed theory captures critical features of avascular tumor growth in the various microenvironment of living tissue, in agreement with the experimental studies in the literature.

Calibrating a Predictive Model of Tumor Growth and Angiogenesis with Quantitative MRI.

The spatiotemporal variations in tumor vasculature inevitably alters cell proliferation and treatment efficacy. Thus, rigorous characterization of tumor dynamics must include a description of this phenomenon. We have developed a family of biophysical models of tumor growth and angiogenesis that are calibrated with diffusion-weighted magnetic resonance imaging (DW-MRI) and dynamic contrast-enhanced (DCE-) MRI data to provide individualized tumor growth forecasts. Tumor and blood volume fractions were evolved using two, coupled partial differential equations consisting of proliferation, diffusion, and death terms. To evaluate these models, rats (n = 8) with C6 gliomas were imaged seven times. The tumor volume fraction was estimated using DW-MRI, while DCE-MRI provided estimates of the blood volume fraction. The first three time points were used to calibrate model parameters, which were then used to predict growth at the remaining four time points and compared directly to the measurements. The best performing model predicted tumor growth with less than 10.3% error in tumor volume and with less than 9.4% error at the voxel-level at all prediction time points. The best performing model resulted in less than 9.3% error in blood volume at the voxel-level. This pre-clinical study demonstrates the potential for image-based, mechanistic modeling of tumor growth and angiogenesis.

Quantitative reconstruction of time-varying 3D cell forces with traction force optical coherence microscopy.

Cellular traction forces (CTFs) play an integral role in both physiological processes and disease, and are a topic of interest in mechanobiology. Traction force microscopy (TFM) is a family of methods used to quantify CTFs in a variety of settings. State-of-the-art 3D TFM methods typically rely on confocal fluorescence microscopy, which can impose limitations on acquisition speed, volumetric coverage, and temporal sampling or coverage. In this report, we present the first quantitative implementation of a new TFM technique: traction force optical coherence microscopy (TF-OCM). TF-OCM leverages the capabilities of optical coherence microscopy and computational adaptive optics (CAO) to enable the quantitative reconstruction of 3D CTFs in scattering media with minute-scale temporal sampling. We applied TF-OCM to quantify CTFs exerted by isolated NIH-3T3 fibroblasts embedded in Matrigel, with five-minute temporal sampling, using images spanning a 500 × 500 × 500 µm3 field-of-view. Due to the reliance of TF-OCM on computational imaging methods, we have provided extensive discussion of the equations, assumptions, and failure modes of these methods. By providing high-throughput, label-free, volumetric imaging in scattering media, TF-OCM is well-suited to the study of 3D CTF dynamics, and may prove advantageous for the study of large cell collectives, such as the spheroid models prevalent in mechanobiology.

An adjoint-based method for a linear mechanically-coupled tumor model: Application to estimate the spatial variation of murine glioma growth based on diffusion weighted magnetic resonance imaging.

We present an efficient numerical method to quantify the spatial variation of glioma growth based on subject-specific medical images using a mechanically-coupled tumor model. The method is illustrated in a murine model of glioma in which we consider the tumor as a growing elastic mass that continuously deforms the surrounding healthy-appearing brain tissue. As an inverse parameter identification problem, we quantify the volumetric growth of glioma and the growth component of deformation by fitting the model predicted cell density to the cell density estimated using the diffusion-weighted magnetic resonance imaging (DW-MRI) data. Numerically, we developed an adjoint-based approach to solve the optimization problem. Results on a set of experimentally measured, in vivo rat glioma data indicate good agreement between the fitted and measured tumor area and suggest a wide variation of in-plane glioma growth with the growth-induced Jacobian ranging from 1.0 to 6.0.

Mathematical models of tumor cell proliferation: A review of the literature.

INTRODUCTION: A defining hallmark of cancer is aberrant cell proliferation. Efforts to understand the generative properties of cancer cells span all biological scales: from genetic deviations and alterations of metabolic pathways to physical stresses due to overcrowding, as well as the effects of therapeutics and the immune system. While these factors have long been studied in the laboratory, mathematical and computational techniques are being increasingly applied to help understand and forecast tumor growth and treatment response. Advantages of mathematical modeling of proliferation include the ability to simulate and predict the spatiotemporal development of tumors across multiple experimental scales. Central to proliferation modeling is the incorporation of available biological data and validation with experimental data. Areas covered: We present an overview of past and current mathematical strategies directed at understanding tumor cell proliferation. We identify areas for mathematical development as motivated by available experimental and clinical evidence, with a particular emphasis on emerging, non-invasive imaging technologies. Expert commentary: The data required to legitimize mathematical models are often difficult or (currently) impossible to obtain. We suggest areas for further investigation to establish mathematical models that more effectively utilize available data to make informed predictions on tumor cell proliferation.

Incorporating drug delivery into an imaging-driven, mechanics-coupled reaction diffusion model for predicting the response of breast cancer to neoadjuvant chemotherapy: theory and preliminary clinical results.

Clinical methods for assessing tumor response to therapy are largely rudimentary, monitoring only temporal changes in tumor size. Our goal is to predict the response of breast tumors to therapy using a mathematical model that utilizes magnetic resonance imaging (MRI) data obtained non-invasively from individual patients. We extended a previously established, mechanically coupled, reaction-diffusion model for predicting tumor response initialized with patient-specific diffusion weighted MRI (DW-MRI) data by including the effects of chemotherapy drug delivery, which is estimated using dynamic contrast-enhanced (DCE-) MRI data. The extended, drug incorporated, model is initialized using patient-specific DW-MRI and DCE-MRI data. Data sets from five breast cancer patients were used-obtained before, after one cycle, and at mid-point of neoadjuvant chemotherapy. The DCE-MRI data was used to estimate spatiotemporal variations in tumor perfusion with the extended Kety-Tofts model. The physiological parameters derived from DCE-MRI were used to model changes in delivery of therapy drugs within the tumor for incorporation in the extended model. We simulated the original model and the extended model in both 2D and 3D and compare the results for this five-patient cohort. Preliminary results show reductions in the error of model predicted tumor cellularity and size compared to the experimentally-measured results for the third MRI scan when therapy was incorporated. Comparing the two models for agreement between the predicted total cellularity and the calculated total cellularity (from the DW-MRI data) reveals an increased concordance correlation coefficient from 0.81 to 0.98 for the 2D analysis and 0.85 to 0.99 for the 3D analysis (p < 0.01 for each) when the extended model was used in place of the original model. This study demonstrates the plausibility of using DCE-MRI data as a means to estimate drug delivery on a patient-specific basis in predictive models and represents a step toward the goal of achieving individualized prediction of tumor response to therapy.

Fibrous nonlinear elasticity enables positive mechanical feedback between cells and ECMs.

In native states, animal cells of many types are supported by a fibrous network that forms the main structural component of the ECM. Mechanical interactions between cells and the 3D ECM critically regulate cell function, including growth and migration. However, the physical mechanism that governs the cell interaction with fibrous 3D ECM is still not known. In this article, we present single-cell traction force measurements using breast tumor cells embedded within 3D collagen matrices. We recreate the breast tumor mechanical environment by controlling the microstructure and density of type I collagen matrices. Our results reveal a positive mechanical feedback loop: cells pulling on collagen locally align and stiffen the matrix, and stiffer matrices, in return, promote greater cell force generation and a stiffer cell body. Furthermore, cell force transmission distance increases with the degree of strain-induced fiber alignment and stiffening of the collagen matrices. These findings highlight the importance of the nonlinear elasticity of fibrous matrices in regulating cell-ECM interactions within a 3D context, and the cell force regulation principle that we uncover may contribute to the rapid mechanical tissue stiffening occurring in many diseases, including cancer and fibrosis.

An adaptive algorithm for tracking 3D bead displacements: application in biological experiments.

This paper presents a feature-vector-based relaxation method (FVRM) to track bead displacements within a three-dimensional (3D) volume. FVRM merges the feature vector method, a technique used in tracking bead displacements in biological gels, with the relaxation method, an algorithm employed successfully in tracking bead pairs in fluids. More specifically, FVRM evaluates the probability of a bead pairing event based on the quasi-rigidity condition between the feature vectors of a bead and its candidate positions within a searching domain. Computational efficiency is improved via the introduction of an adaptive searching domain size and mismatches are reduced via a two-directional matching strategy. The algorithm is validated using simulated 3D bead displacements caused by a force dipole within a linear elastic gel. Results demonstrate a consistently high recovery ratio (above 98%) and low mismatch ratio (below 0.1%) for tracking parameter (mean bead distance/maximum bead displacement) greater than 0.73.

Toward single cell traction microscopy within 3D collagen matrices.

Mechanical interaction between the cell and its extracellular matrix (ECM) regulates cellular behaviors, including proliferation, differentiation, adhesion, and migration. Cells require the three-dimensional (3D) architectural support of the ECM to perform physiologically realistic functions. However, current understanding of cell-ECM and cell-cell mechanical interactions is largely derived from 2D cell traction force microscopy, in which cells are cultured on a flat substrate. 3D cell traction microscopy is emerging for mapping traction fields of single animal cells embedded in either synthetic or natively derived fibrous gels. We discuss here the development of 3D cell traction microscopy, its current limitations, and perspectives on the future of this technology. Emphasis is placed on strategies for applying 3D cell traction microscopy to individual tumor cell migration within collagen gels.