Fluid Dynamics of Ventricular Filling in the Embryonic Heart

The vertebrate embryonic heart first forms as a valveless tube that pumps blood using waves of contraction. As the heart develops, the atrium and ventricle bulge out from the heart tube, and valves begin to form through the expansion of the endocardial cushions. As a result of changes in geometry, conduction velocities, and material properties of the heart wall, the fluid dynamics and resulting spatial patterns of shear stress and transmural pressure change dramatically. Recent work suggests that these transitions are significant because fluid forces acting on the cardiac walls, as well as the activity of myocardial cells that drive the flow, are necessary for correct chamber and valve morphogenesis. In this article, computational fluid dynamics was used to explore how spatial distributions of the normal forces acting on the heart wall change as the endocardial cushions grow and as the cardiac wall increases in stiffness. The immersed boundary method was used to simulate the fluid-moving boundary problem of the cardiac wall driving the motion of the blood in a simplified model of a two-dimensional heart. The normal forces acting on the heart walls increased during the period of one atrial contraction because inertial forces are negligible and the ventricular walls must be stretched during filling. Furthermore, the force required to fill the ventricle increased as the stiffness of the ventricular wall was increased. Increased endocardial cushion height also drastically increased the force necessary to contract the ventricle. Finally, flow in the moving boundary model was compared to flow through immobile rigid chambers, and the forces acting normal to the walls were substantially different.     

Aggregation and disaggregation of red blood cells

The aggregation of red blood cells may be analyzed as an interaction of an adhesive surface energy and the elastic stored energy which results from deformation of the cell. The adhesive surface energy is the work required to separate a unit adhered area and is the resultant of adhesive forces due to the bridging molecules that bind the cells together and the electrostatic repulsion due to surface charge. The elastic strain energy in the case of the red blood is associated with the membrane elasticity only since the interior of the cell is liquid. The membrane elasticity is due both to bending stiffness and shear. The area expansion is small and may be neglected. These assumptions allow realistic computation of red cell shapes in rouleaux. The disaggregation of rouleaux requires an external force which must overcome the adhesive energy and also supply additional elastic energy of deformation. Depending on the geometry, the initial effect of elastic energy may tend to aid disaggregation. In a shear flow, the stresses on a suspended rouleau alternately tend to compress and to disaggregate the cells if they are free to rotate. This introduces a time dependence so that viscous effects due to the viscosity of the cell membrane, the cell cytoplasm and the external fluid may play a role in determining whether disaggregation proceeds to completion or not.     

Muscle contraction: viscous-like frictional forces and the impulsive model

Apart from a few experimental studies muscle viscosity has not received much recent analytical attention as a determinant of the contractile process. This is surprising, since any muscle cell is 80% water, and may undergo large shape changes during its working cycle. Intuitively one might expect the viscosity of the solvent to be an important determinant of the physiological activity of muscle tissue. This was apparent to pioneers of the study of muscle contraction such as Hill and his contemporaries, whose putative theoretical formulations contained terms related to muscle viscosity. More recently, though, a hydrodynamic calculation by Huxley, using a solvent viscosity close to that of water, has been held to demonstrate that viscous forces are negligible in muscle contraction. We have re-examined the role of viscosity in contraction, postulating impulsive acto-myosin forces that are opposed by a viscous resistance between the filaments. The viscous force required, 10⁴ times the hydrodynamic estimate, is close to recent experimental measurements, themselves 10²–10³ times the hydrodynamic estimate. This also agrees with contemporary measurements of cytoplasmic viscosity in other biological cells using magnetic bead micro-rheometry. These are several orders of magnitude greater than the viscosity of water. In the course of the analysis we have derived the force-velocity equation for an isolated half-sarcomere containing a single actin filament for the first time, and from first principles. We conclude that muscle viscosity is indeed important for the contractile process, and that it has been too readily discounted.     

Cell-level finite element studies of viscous cells in planar aggregates

A new cell-level finite element formulation is presented and used to investigate how epithelia and other planar collections of viscous cells might deform during events such as embryo morphogenesis and wound healing. Forces arising from cytoskeletal components, cytoplasm viscosity, and cell-cell adhesions are included. Individual cells are modeled using multiple finite elements, and cell rearrangements can occur. Simulations of cell-sheet stretching indicate that the initial stages of sheet stretching are characterized by changes in cell shape, while subsequent stages are governed by cell rearrangement. Inferences can be made from the simulations about the forces that act in real cell sheets when suitable experimental data are available.     

Jet collisions and vortex reversal in the human left ventricle

Unnatural dynamics of the notorious vortex in the left ventricle is often associated with cardiac disease. Understanding how different cardiac diseases alter the flow physics in the left ventricle may therefore provide a powerful tool for disease detection. In this work, the fluid dynamics in the left ventricle subject to different severities of aortic regurgitation is experimentally investigated by performing time-resolved particle image velocimetry in a left heart duplicator. Diastolic vortex reversal was observed in the left ventricle accompanied by an increase in viscous energy dissipation. Vortex dynamics and energy dissipation may provide useful insights on sub-optimal flow patterns in the left ventricle.     

Micro-environmentally restricted cell growth dynamics – modeling considerations

Various modeling approaches have been applied to describe the rearrangement of immobilized cell clusters within the extracellular matrix. The cell rearrangement has been related with the micro-environmental restrictions to cell growth. Herein, an attempt is made to discuss and connect various modeling approaches on various time scales which have been proposed in the literature in order to shed further light to this complex phenomenon which induces micro-environmental restrictions to cell growth. The rearrangement is driven by internal stress generated within the cluster. The internal stress represents a consequence of the matrix rheological response to cell expansion. The rearrangement includes the interplay between the processes of: (1) single and collective cell migrations, (2) cell deformation and orientation, (3) decrease of cell-to-cell separation distances and (4) cell growth. It has been considered on two time scales: a short time scale (i.e. the rearrangement time) and a long time scale (i.e. the growing time). The results indicate that short and long times cell rearrangement induces energy dissipation. The dissipation provokes biological responses of cells which cause the resistance effects to cell growth. Deeper insight in the anomalous nature of the energy dissipation would be useful for understanding the biological mechanisms which causes the resistance effects to cell growth.     

Viscous flow through slowly expanding or contracting porous walls with low seepage Reynolds number: a model for transport of biological fluids through vessels

In this article, the problem of laminar, isothermal, incompressible and viscous flow in a rectangular domain bounded by two moving porous walls, which enable the fluid to enter or exit during successive expansions or contractions, is investigated. The governing non-linear equations and their associated boundary conditions are transformed into a highly non-linear ordinary differential equation. The series solution of the problem is obtained by utilising the homotopy perturbation method. Graphical results are presented to investigate the influence of the non-dimensional wall dilation rate and seepage Reynolds number (Re) on the velocity, normal pressure distribution and wall shear stress. Since the transport of biological fluids through contracting or expanding vessels is characterised by low seepage Res, the current study focuses on the viscous flow driven by small wall contractions and expansions of two weakly permeable walls.     

Left ventricular wall stress compendium

Left ventricular (LV) wall stress has intrigued scientists and cardiologists since the time of Lame and Laplace in 1800s. The left ventricle is an intriguing organ structure, whose intrinsic design enables it to fill and contract. The development of wall stress is intriguing to cardiologists and biomedical engineers. The role of left ventricle wall stress in cardiac perfusion and pumping as well as in cardiac pathophysiology is a relatively unexplored phenomenon. But even for us to assess this role, we first need accurate determination of in vivo wall stress. However, at this point, 150 years after Lame estimated left ventricle wall stress using the elasticity theory, we are still in the exploratory stage of (i) developing left ventricle models that properly represent left ventricle anatomy and physiology and (ii) obtaining data on left ventricle dynamics. In this paper, we are responding to the need for a comprehensive survey of left ventricle wall stress models, their mechanics, stress computation and results. We have provided herein a compendium of major type of wall stress models: thin-wall models based on the Laplace law, thick-wall shell models, elasticity theory model, thick-wall large deformation models and finite element models. We have compared the mean stress values of these models as well as the variation of stress across the wall. All of the thin-wall and thick-wall shell models are based on idealised ellipsoidal and spherical geometries. However, the elasticity model's shape can vary through the cycle, to simulate the more ellipsoidal shape of the left ventricle in the systolic phase. The finite element models have more representative geometries, but are generally based on animal data, which limits their medical relevance. This paper can enable readers to obtain a comprehensive perspective of left ventricle wall stress models, of how to employ them to determine wall stresses, and be cognizant of the assumptions involved in the use of specific models.     

Left ventricular wall stress compendium

Left ventricular (LV) wall stress has intrigued scientists and cardiologists since the time of Lame and Laplace in 1800s. The left ventricle is an intriguing organ structure, whose intrinsic design enables it to fill and contract. The development of wall stress is intriguing to cardiologists and biomedical engineers. The role of left ventricle wall stress in cardiac perfusion and pumping as well as in cardiac pathophysiology is a relatively unexplored phenomenon. But even for us to assess this role, we first need accurate determination of in vivo wall stress. However, at this point, 150 years after Lame estimated left ventricle wall stress using the elasticity theory, we are still in the exploratory stage of (i) developing left ventricle models that properly represent left ventricle anatomy and physiology and (ii) obtaining data on left ventricle dynamics. In this paper, we are responding to the need for a comprehensive survey of left ventricle wall stress models, their mechanics, stress computation and results. We have provided herein a compendium of major type of wall stress models: thin-wall models based on the Laplace law, thick-wall shell models, elasticity theory model, thick-wall large deformation models and finite element models. We have compared the mean stress values of these models as well as the variation of stress across the wall. All of the thin-wall and thick-wall shell models are based on idealised ellipsoidal and spherical geometries. However, the elasticity model's shape can vary through the cycle, to simulate the more ellipsoidal shape of the left ventricle in the systolic phase. The finite element models have more representative geometries, but are generally based on animal data, which limits their medical relevance. This paper can enable readers to obtain a comprehensive perspective of left ventricle wall stress models, of how to employ them to determine wall stresses, and be cognizant of the assumptions involved in the use of specific models.     

Adaptation of passive rat left ventricle in diastolic dysfunction.

This article deals with providing a theoretical explanation for quantitative changes in the geometry, the opening angle and the deformation parameters of the rat ventricular wall during adaptation of the passive left ventricle in diastolic dysfunction. A large deformation theory is applied to analyse transmural stress and strain distribution in the left ventricular wall considering it to be made of homogeneous, incompressible, transversely isotropic, non-linear elastic material. The basic assumptions made for computing stress distributions are that the average circumferential stress and strain for the adaptive ventricle is equal to the average circumferential stress and strain in the normotensive ventricle, respectively.All the relevant parameters, such as opening angle, twist per unit length, axial extension, internal and external radii and others, in the stress-free, unloaded and loaded states of normotensive, hypertensive and adaptive left ventricle are determined. The circumferential stress and strain distribution through the ventricular wall are also computed. Our analysis predicts that during adaptation, wall thickness and wall mass of the ventricle increase. These results are consistent with experimental findings and are the indications of initiation of congestive heart failure.     

Pressure drop and flow rate measurements in a human aortic bifurcation cast for steady and pulsatile flow

Pressure drop and flow rate measurements in a rigid cast of a human aortic bifurcation under both steady and physiological pulsatile flow conditions are reported. Integral momentum and mechanical energy balances are used to calculate impedance, spatially averaged wall shear stress and viscous dissipation rate from the data. In the daughter branches, steady flow impedance is within 30% of the Poiseuille flow prediction, while pulsatile flow impedance is within a factor of 2 of fully developed, oscillatory, straight tube flow theory (Womersley theory). Estimates of wall shear stress are in accord with measurements obtained from velocity profiles. Mean pressure drop and viscous dissipation rate are elevated in pulsatile flow relative to steady flow at the mean flow rate, and the exponents of their Reynolds number dependence are in accord with available theory.     

Viscous flow through slowly expanding or contracting porous walls with low seepage Reynolds number: a model for transport of biological fluids through vessels

In this article, the problem of laminar, isothermal, incompressible and viscous flow in a rectangular domain bounded by two moving porous walls, which enable the fluid to enter or exit during successive expansions or contractions, is investigated. The governing non-linear equations and their associated boundary conditions are transformed into a highly non-linear ordinary differential equation. The series solution of the problem is obtained by utilising the homotopy perturbation method. Graphical results are presented to investigate the influence of the non-dimensional wall dilation rate and seepage Reynolds number (Re) on the velocity, normal pressure distribution and wall shear stress. Since the transport of biological fluids through contracting or expanding vessels is characterised by low seepage Res, the current study focuses on the viscous flow driven by small wall contractions and expansions of two weakly permeable walls.     

Plasma release-cell layering theory for blood flow

Significant internal structural changes occur in flowing blood when shear strain exceeds the critical value of 1 (unit strain), forcing alignment of the erythrocytes and releasing trapped plasma, which in turn leads to the formation of multiple layers of plasma on which oriented and compacted cells slide. These effects are identified in the inflections in the shear rate dependence of viscoelasticity of normal blood and in the viscous and elastic stress-to-strain relationships. Theoretical factors for plasma release and cell compaction allow calculation of the viscous and elastic properties of the cell layers from measured whole blood viscoelasticity and plasma viscosity. The new plasma release-cell layering theory encompasses, reinterprets and unifies many diverse previous observations relating to how blood flows, and provides a new understanding of the roles of red cell deformability and aggregation tendency.     

A nonlinear model of passive muscle viscosity

The material properties of passive skeletal muscle are critical to proper function and are frequently a target for therapeutic and interventional strategies. Investigations into the passive viscoelasticity of muscle have primarily focused on characterizing the elastic behavior, largely neglecting the viscous component. However, viscosity is a sizeable contributor to muscle stress and extensibility during passive stretch and thus there is a need for characterization of the viscous as well as the elastic components of muscle viscoelasticity. Single mouse muscle fibers were subjected to incremental stress relaxation tests to characterize the dependence of passive muscle stress on time, strain and strain rate. A model was then developed to describe fiber viscoelasticity incorporating the observed nonlinearities. The results of this model were compared with two commonly used linear viscoelastic models in their ability to represent fiber stress relaxation and strain rate sensitivity. The viscous component of mouse muscle fiber stress was not linear as is typically assumed, but rather a more complex function of time, strain and strain rate. The model developed here, which incorporates these nonlinearities, was better able to represent the stress relaxation behavior of fibers under the conditions tested than commonly used models with linear viscosity. It presents a new tool to investigate the changes in muscle viscous stresses with age, injury and disuse.     

Adhesion Strength of Settled Spores of the Green Alga Enteromorpha

Strengths of attachment of spores of the green fouling alga Enteromorpha to glass have been measured using a modified water jet apparatus. Surface pressures of ～250 kPa were required to quantitatively remove attached spores after 4 h contact with a surface. The development of adhesive and cohesive strength is highly time-dependent; after 8 h in contact with a surface spores did not detach, even at pressures in excess of 250 kPa. Spores settled in groups are more resistant to detachment than single spores, which suggests that the adaptive value of gregarious settlement behaviour may lie in the greater resistance of groups to detachment forces in a naturally turbulent environment. The interfacial forces exerted as water impinges on the surface and the derivation of adhesion strength values in terms of wall shear stress are discussed and compared with those obtained by other methods. A surface pressure of 250 kPa approximates to 325 Pa wall shear stress. Calculation using the power-law formula predicts that detachment forces of this magnitude are unlikely to be realized at operating speeds for most vessels and that most Enteromorpha spores would not detach from untreated hulls.     

Fluid‐driven interfacial instabilities and turbulence in bacterial biofilms

Biofilms are thin layers of bacteria embedded within a slime matrix that live on surfaces. They are ubiquitous in nature and responsible for many medical and dental infections, industrial fouling and are also evident in ancient fossils. A biofilm structure is shaped by growth, detachment and response to mechanical forces acting on them. The main contribution to biofilm versatility in response to physical forces is the matrix that provides a platform for the bacteria to grow. The interaction between biofilm structure and hydrodynamics remains a fundamental question concerning biofilm dynamics. Here, we document the appearance of ripples and wrinkles in biofilms grown from three species of bacteria when subjected to high‐velocity fluid flows. Linear stability analysis suggested that the ripples were Kelvin–Helmholtz Instabilities. The analysis also predicted a strong dependence of the instability formation on biofilm viscosity explaining the different surface corrugations observed. Turbulence through Kelvin–Helmholtz instabilities occurring at the interface demonstrated that the biofilm flows like a viscous liquid under high flow velocities applied within milliseconds. Biofilm fluid‐like behavior may have important implications for our understanding of how fluid flow influences biofilm biology since turbulence will likely disrupt metabolite and signal gradients as well as community stratification.     

A finite element model of the human left ventricular systole

Local wall stress is the pivotal determinant of the heart muscle's systolic function. Under in vivo conditions, however, such stresses cannot be measured systematically and quantitatively. In contrast, imaging techniques based on magnetic resonance (MR) allow the determination of the deformation pattern of the left ventricle (LV) in vivo with high accuracy. The question arises to what extent deformation measurements are significant and might provide a possibility for future diagnostic purposes. The contractile forces cause deformation of LV myocardial tissue in terms of wall thickening, longitudinal shortening, twisting rotation and radial constriction. The myocardium is thereby understood to act as a densely interlaced mesh. Yet, whole cycle image sequences display a distribution of wall strains as function of space and time heralding a significant amount of inhomogeneity even under healthy conditions. We made similar observations previously by direct measurement of local contractile activity. The major reasons for these inhomogeneities derive from regional deviations of the ventricular walls from an ideal spheroidal shape along with marked disparities in focal fibre orientation. In response to a lack of diagnostic tools able to measure wall stress in clinical routine, this communication is aimed at an analysis and functional interpretation of the deformation pattern of an exemplary human heart at end-systole. To this end, the finite element (FE) method was used to simulate the three-dimensional deformations of the left ventricular myocardium due to contractile fibre forces at end-systole. The anisotropy associated with the fibre structure of the myocardial tissue was included in the form of a fibre orientation vector field which was reconstructed from the measured fibre trajectories in a post mortem human heart. Contraction was modelled by an additive second Piola-Kirchhoff active stress tensor. As a first conclusion, it became evident that longitudinal fibre forces, cross-fibre forces and shear along with systolic fibre rearrangement have to be taken into account for a useful modelling of systolic deformation. Second, a realistic geometry and fibre architecture lead to typical and substantially inhomogeneous deformation patterns as they are recorded in real hearts. We therefore, expect that the measurement of systolic deformation might provide useful diagnostic information.     

Determinants of Fluidlike Behavior and Effective Viscosity in Cross-Linked Actin Networks

The actin cortex has a well-documented ability to rapidly remodel and flow while maintaining long-range connectivity, but how this is achieved remains poorly understood. Here, we use computer simulations to explore how stress relaxation in cross-linked actin networks subjected to extensional stress depends on the interplay between network architecture and turnover. We characterize a regime in which a network response is nonaffine and stress relaxation is governed by the continuous dissipation of elastic energy via cyclic formation, elongation, and turnover of tension-bearing elements. Within this regime, for a wide range of network parameters, we observe a constant deformation (creep) rate that is linearly proportional to the rate of filament turnover, leading to a constant effective viscosity that is inversely proportional to turnover rate. Significantly, we observe a biphasic dependence of the creep rate on applied stress: below a critical stress threshold, the creep rate increases linearly with applied stress; above that threshold, the creep rate becomes independent of applied stress. We show that this biphasic stress dependence can be understood in terms of the nonlinear force-extension behavior of individual force-transmitting network elements. These results have important implications for understanding the origins and control of viscous flows both in the cortex of living cells and in other polymer networks.     

Determinants of Fluidlike Behavior and Effective Viscosity in Cross-Linked Actin Networks

The actin cortex has a well-documented ability to rapidly remodel and flow while maintaining long-range connectivity, but how this is achieved remains poorly understood. Here, we use computer simulations to explore how stress relaxation in cross-linked actin networks subjected to extensional stress depends on the interplay between network architecture and turnover. We characterize a regime in which a network response is nonaffine and stress relaxation is governed by the continuous dissipation of elastic energy via cyclic formation, elongation, and turnover of tension-bearing elements. Within this regime, for a wide range of network parameters, we observe a constant deformation (creep) rate that is linearly proportional to the rate of filament turnover, leading to a constant effective viscosity that is inversely proportional to turnover rate. Significantly, we observe a biphasic dependence of the creep rate on applied stress: below a critical stress threshold, the creep rate increases linearly with applied stress; above that threshold, the creep rate becomes independent of applied stress. We show that this biphasic stress dependence can be understood in terms of the nonlinear force-extension behavior of individual force-transmitting network elements. These results have important implications for understanding the origins and control of viscous flows both in the cortex of living cells and in other polymer networks.