This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems. Although we discuss the main points in the application of the finite element method to electromagnetic design, including formulation and implementation, those who seek deeper understanding of the finite element method should consult some of the works listed in the bibliography section.

Introduction to the Finite Element Method

The flow of homogeneous fluids through porous media: analogies with other physical problems

The topics of verification and validation have increasingly been discussed in the field of computational biomechanics, and many recent articles have applied these concepts in an attempt to build credibility for models of complex biological systems. Verification and validation are evolving techniques that, if used improperly, can lead to false conclusions about a system under study. In basic science, these erroneous conclusions may lead to failure of a subsequent hypothesis, but they can have more profound effects if the model is designed to predict patient outcomes. While several authors have reviewed verification and validation as they pertain to traditional solid and fluid mechanics, it is the intent of this paper to present them in the context of computational biomechanics. Specifically, the task of model validation will be discussed, with a focus on current techniques. It is hoped that this review will encourage investigators to engage and adopt the verification and validation process in an effort to increase peer acceptance of computational biomechanics models.

/pdf/validation-of-computational-models-in-biomechanics-4fkakyjeth.pdf

Validation of computational models in biomechanics.

We survey recent progress in the application of nanoindentation to characterize the local mechanical properties of polymer gels and biological tissues. We review the theories, analytical models based thereon, and data processing techniques commonly used to determine elastic properties of these classes of materials by instrumented nanoindentation. Examples from the testing of synthetic and biological gels are used to illustrate the limitations of existing theories and approaches. Emphasis is placed on the need for contact mechanics models that more accurately represent the large-strain behaviour of soft matter.

Nanomechanics of polymer gels and biological tissues: A critical review of analytical approaches in the Hertzian regime and beyond

The distinct spatial architecture of the apical actin cables (or actin cap) facilitates rapid biophysical signaling between extracellular mechanical stimuli and intracellular responses, including nuclear shaping, cytoskeletal remodeling, and the mechanotransduction of external forces into biochemical signals. These functions are abrogated in lamin A/C-deficient mouse embryonic fibroblasts that recapitulate the defective nuclear organization of laminopathies, featuring disruption of the actin cap. However, how nuclear lamin A/C mediates the ability of the actin cap to regulate nuclear morphology remains unclear. Here, we show that lamin A/C expressing cells can form an actin cap to resist nuclear deformation in response to physiological mechanical stresses. This study reveals how the nuclear lamin A/C-mediated formation of the perinuclear apical actin cables protects the nuclear structural integrity from extracellular physical disturbances. Our findings highlight the role of the physical interactions between the cytoskeletal network and the nucleus in cellular mechanical homeostasis.

/pdf/nuclear-lamin-a-c-harnesses-the-perinuclear-apical-actin-yyk17bszxo.pdf

Nuclear lamin A/C harnesses the perinuclear apical actin cables to protect nuclear morphology

The variations in mechanical properties of cells obtained from experimental and theoretical studies can be overcome only through the development of a sound mathematical framework correlating the derived mechanical property with the cellular structure. Such a formulation accounting for the inhomogeneity of the cytoplasm due to stress fibers and actin cortex is developed in this work. The proposed model is developed using the Mori-Tanaka method of homogenization by treating the cell as a fiber-reinforced composite medium satisfying the continuum hypothesis. The validation of the constitutive model using finite element analysis on atomic force microscopy (AFM) and magnetic twisting cytometry (MTC) has been carried out and is found to yield good correlation with reported experimental results. It is observed from the study that as the volume fraction of the stress fiber increases, the stiffness of the cell increases and it alters the force displacement behavior for the AFM and MTC experiments. Through this model, we have also been able to find the stress fiber as a likely cause of the differences in the derived mechanical property from the AFM and MTC experiments. The correlation of the mechanical behavior of the cell with the cell composition, as obtained through this study, is an important observation in cell mechanics.

Constitutive material modeling of cell: a micromechanics approach.

The mechanical behavior of biological tissues has been studied using a number of mechanical models. Due to the relatively high fluid content and mobility, many biological tissues have been modeled as poroelastic materials. Diseases such as cancers are known to alter the poroelastic response of a tissue. Tissue poroelastic properties such as compressibility, interstitial permeability and fluid pressure also play a key role for the assessment of cancer treatments and for improved therapies. At the present time, however, a limited number of poroelastic models for soft tissues are retrievable in the literature, and the ones available are not directly applicable to tumors as they typically refer to uniform tissues. In this paper, we report the analytical poroelastic model for a non-uniform tissue under stress relaxation. Displacement, strain and fluid pressure fields in a cylindrical poroelastic sample containing a cylindrical inclusion during stress relaxation are computed. Finite element simulations are then used to validate the proposed theoretical model. Statistical analysis demonstrates that the proposed analytical model matches the finite element results with less than 0.5% error. The availability of the analytical model and solutions presented in this paper may be useful to estimate diagnostically relevant poroelastic parameters such as interstitial permeability and fluid pressure, and, in general, for a better interpretation of clinically-relevant ultrasound elastography results.

https://iopscience.iop.org/article/10.1088/1361-6560/aa9631/pdf

An analytical poroelastic model for ultrasound elastography imaging of tumors.

Review on the Constitutive Models of Tumor Tissue for Computational Analysis

Elastography is a well-established imaging modality. While a number of studies aimed at evaluating the performance of elastographic techniques are retrievable in the literature, very little information is available on the effects that the presence of an underlying permeability contrast in the tissue may have on the resulting elastograms. Permeability is a fundamental tissue parameter, which characterizes the ease with which fluid can move within a tissue. This parameter plays a central role both biomechanically in the description of the temporal behavior of fluid-filled tissues and clinically in the development of a number of diagnostic and therapeutic modalities. In this paper, we present a simulation study that investigates selected elastographic image quality factors in nonhomogeneous materials, modeled as poroelastic media with different geometries and permeability contrasts. The results of this study indicate that the presence of an underlying permeability contrast may create a new contrast mechanism in the spatial and temporal distributions of the axial strains and the effective Poisson's ratios experienced by the tissue and as imaged by the corresponding elastograms. The effect of permeability on the elastographic image quality factors analyzed in this study was found to be a nonsymmetric function of the underlying mechanical contrast between background and target, the geometry of the material and the boundary conditions.

G. U. Unnikrishnan

Papers

Constitutive material modeling of cell: a micromechanics approach.

An analytical poroelastic model for ultrasound elastography imaging of tumors.

Review on the Constitutive Models of Tumor Tissue for Computational Analysis

Effect of Permeability on the Performance of Elastographic Imaging Techniques

Effect of Interstitial Fluid Pressure on Ultrasound Axial Strain and Axial Shear Strain Elastography