An incremental deformation model of arterial dissection.

We develop a mathematical model for a small axisymmetric tear in a residually stressed and axially pre-stretched cylindrical tube. The residual stress is modelled by an opening angle when the load-free tube is sliced along a generator. This has application to the study of an aortic dissection, in which a tear develops in the wall of the artery. The artery is idealised as a single-layer thick-walled axisymmetric hyperelastic tube with collagen fibres using a Holzapfel-Gasser-Ogden strain-energy function, and the tear is treated as an incremental deformation of this tube. The lumen of the cylinder and the interior of the dissection are subject to the same constant (blood) pressure. The equilibrium equations for the incremental deformation are derived from the strain energy function. We develop numerical methods to study the opening of the tear for a range of material parameters and boundary conditions. We find that decreasing the fibre angle, decreasing the axial pre-stretch and increasing the opening angle all tend to widen the dissection, as does an incremental increase in lumen and dissection pressure.

Study of cardiovascular function using a coupled left ventricle and systemic circulation model.

To gain insight into cardio-arterial interactions, a coupled left ventricle-systemic artery (LV-SA) model is developed that incorporates a three-dimensional finite-strain left ventricle (LV), and a physiologically-based one-dimensional model for the systemic arteries (SA). The coupling of the LV model and the SA model is achieved by matching the pressure and the flow rate at the aortic root, i.e. the SA model feeds back the pressure as a boundary condition to the LV model, and the aortic flow rate from the LV model is used as the input for the SA model. The governing equations of the coupled system are solved using a combined immersed-boundary finite-element (IB/FE) method and a Lax-Wendroff scheme. A baseline case using physiological measurements of healthy subjects, and four exemplar cases based on different physiological and pathological scenarios are studied using the LV-SA model. The results of the baseline case agree well with published experimental data. The four exemplar cases predict varied pathological responses of the cardiovascular system, which are also supported by clinical observations. The new model can be used to gain insight into cardio-arterial interactions across a range of clinical applications.

Wavelength Selection in Gyrotactic Bioconvection.

We investigate pattern formation by swimming micro-organisms (bioconvection), when their orientation is determined by balance between gravitational and viscous torques (gyrotaxis), due to being bottom heavy. The governing equations, which consist of the Navier-Stokes equations for an incompressible fluid coupled with a micro-organism conservation equation, are solved numerically in a large cross section chamber with periodic boundary conditions in the horizontal directions. The influence of key parameters on wavelength selection in bioconvection patterns is investigated numerically. For realistic ranges of parameter values, the computed wavelengths are in good agreement with the experimental observations provided that the diffusion due to randomness in cell swimming behaviour is small, refuting a recently published claim that the mathematical model becomes inaccurate at long times. We also provide the first computational evidence of "bottom-standing" plumes in a three-dimensional simulation.

A computational study of pressure wave reflections in the pulmonary arteries.

Experiments using wave intensity analysis suggest that the pulmonary circulation in sheep and dogs is characterized by negative or open-end type wave reflections, that reduce the systolic pressure. Since the pulmonary physiology is similar in most mammals, including humans, we test and verify this hypothesis by using a subject specific one-dimensional model of the human pulmonary circulation and a conventional wave intensity analysis. Using the simulated pressure and velocity, we also analyse the performance of the P-U loop and sum of squares techniques for estimating the local pulse wave velocity in the pulmonary arteries, and then analyse the effects of these methods on linear wave separation in the main pulmonary artery. P-U loops are found to provide much better estimates than the sum of squares technique at proximal locations, but both techniques accumulate progressive error at distal locations away from heart, particularly near junctions. The pulse wave velocity estimated using the sum of squares method also gives rise to an artificial early systolic backward compression wave. Finally, we study the influence of three types of pulmonary hypertension viz. pulmonary arterial hypertension, chronic thromboembolic pulmonary hypertension and pulmonary hypertension associated with hypoxic lung disease. Simulating these conditions by changing the relevant parameters in the model and then applying the wave intensity analysis, we observe that for each group the early systolic backward decompression wave reflected from proximal junctions is maintained, whilst the initial forward compression and the late systolic backward compression waves amplify with increasing pathology and contribute significantly to increases in systolic pressure.

Numerical simulation of blood flow and pressure drop in the pulmonary arterial and venous circulation.

A novel multiscale mathematical and computational model of the pulmonary circulation is presented and used to analyse both arterial and venous pressure and flow. This work is a major advance over previous studies by Olufsen et al. (Ann Biomed Eng 28:1281-1299, 2012) which only considered the arterial circulation. For the first three generations of vessels within the pulmonary circulation, geometry is specified from patient-specific measurements obtained using magnetic resonance imaging (MRI). Blood flow and pressure in the larger arteries and veins are predicted using a nonlinear, cross-sectional-area-averaged system of equations for a Newtonian fluid in an elastic tube. Inflow into the main pulmonary artery is obtained from MRI measurements, while pressure entering the left atrium from the main pulmonary vein is kept constant at the normal mean value of 2 mmHg. Each terminal vessel in the network of 'large' arteries is connected to its corresponding terminal vein via a network of vessels representing the vascular bed of smaller arteries and veins. We develop and implement an algorithm to calculate the admittance of each vascular bed, using bifurcating structured trees and recursion. The structured-tree models take into account the geometry and material properties of the 'smaller' arteries and veins of radii ≥ 50 µm. We study the effects on flow and pressure associated with three classes of pulmonary hypertension expressed via stiffening of larger and smaller vessels, and vascular rarefaction. The results of simulating these pathological conditions are in agreement with clinical observations, showing that the model has potential for assisting with diagnosis and treatment for circulatory diseases within the lung.

Anisotropic behaviour of human gallbladder walls.

Inverse estimation of biomechanical parameters of soft tissues from non-invasive measurements has clinical significance in patient-specific modelling and disease diagnosis. In this paper, we propose a fully nonlinear approach to estimate the mechanical properties of the human gallbladder wall muscles from in vivo ultrasound images. The iteration method consists of a forward approach, in which the constitutive equation is based on a modified Hozapfel-Gasser-Ogden law initially developed for arteries. Five constitutive parameters describing the two orthogonal families of fibres and the matrix material are determined by comparing the computed displacements with medical images. The optimisation process is carried out using the MATLAB toolbox, a Python code, and the ABAQUS solver. The proposed method is validated with published artery data and subsequently applied to ten human gallbladder samples. Results show that the human gallbladder wall is anisotropic during the passive refilling phase, and that the peak stress is 1.6 times greater than that calculated using linear mechanics. This discrepancy arises because the wall thickness reduces by 1.6 times during the deformation, which is not predicted by conventional linear elasticity. If the change of wall thickness is accounted for, then the linear model can used to predict the gallbladder stress and its correlation with pain. This work provides further understanding of the nonlinear characteristics of human gallbladder.

A quasi-nonlinear analysis of the anisotropic behaviour of human gallbladder wall.

Estimation of biomechanical parameters of soft tissues from noninvasive measurements has clinical significance in patient-specific modeling and disease diagnosis. In this work, we present a quasi-nonlinear method that is used to estimate the elastic moduli of the human gallbladder wall. A forward approach based on a transversely isotropic membrane material model is used, and an inverse iteration is carried out to determine the elastic moduli in the circumferential and longitudinal directions between two successive ultrasound images of gallbladder. The results demonstrate that the human gallbladder behaves in an anisotropic manner, and constitutive models need to incorporate this. The estimated moduli are also nonlinear and patient dependent. Importantly, the peak stress predicted here differs from the earlier estimate from linear membrane theory. As the peak stress inside the gallbladder wall has been found to strongly correlate with acalculous gallbladder pain, reliable mechanical modeling for gallbladder tissue is crucial if this information is to be used in clinical diagnosis.

Rarefaction and blood pressure in systemic and pulmonary arteries.

The effects of vascular rarefaction (the loss of small arteries) on the circulation of blood are studied using a multiscale mathematical model that can predict blood flow and pressure in the systemic and pulmonary arteries. We augmented a model originally developed for the systemic arteries (Olufsen et al. 1998, 1999, 2000, 2004) to (a) predict flow and pressure in the pulmonary arteries, and (b) predict pressure propagation along the small arteries in the vascular beds. The systemic and pulmonary arteries are modelled as separate, bifurcating trees of compliant and tapering vessels. Each tree is divided into two parts representing the `large' and `small' arteries. Blood flow and pressure in the large arteries are predicted using a nonlinear cross-sectional area-averaged model for a Newtonian fluid in an elastic tube with inflow obtained from magnetic resonance measurements. Each terminal vessel within the network of the large arteries is coupled to a vascular bed of small `resistance' arteries, which are modelled as asymmetric structured trees with specified area and asymmetry ratios between the parent and daughter arteries. For the systemic circulation, each structured tree represents a specific vascular bed corresponding to major organs and limbs. For the pulmonary circulation, there are four vascular beds supplied by the interlobar arteries. This manuscript presents the first theoretical calculations of the propagation of the pressure and flow waves along systemic and pulmonary large and small arteries. Results for all networks were in agreement with published observations. Two studies were done with this model. First, we showed how rarefaction can be modelled by pruning the tree of arteries in the microvascular system. This was done by modulating parameters used for designing the structured trees. Results showed that rarefaction leads to increased mean and decreased pulse pressure in the large arteries. Second, we investigated the impact of decreasing vessel compliance in both large and small arteries. Results showed, that the effects of decreased compliance in the large arteries far outweigh the effects observed when decreasing the compliance of the small arteries. We further showed that a decrease of compliance in the large arteries results in pressure increases consistent with observations of isolated systolic hypertension, as occurs in ageing.

Cross-bridge apparent rate constants of human gallbladder smooth muscle.

This paper studies human gallbladder (GB) smooth muscle contractions. A two-state cross-bridge model was used to estimate the apparent attachment and detachment rate constants, as well as increased Ca2+ concentration from the peak active stress during the isometric contraction. The active stress was computed from a mechanical model based entirely on non-invasive routine ultrasound scans. In the two-state cross-bridge model, the two apparent rate constants, representing the total attached/detached cross-bridges, respectively, were estimated using active stress prediction for 51 subjects undergoing cholecystokinin-provocation test, together with estimates from the four-state cross-bridge model for a swine carotid, bovine tracheal and guinea pig GB smooth muscles. The study suggests that the apparent rate constants should be patient-specific, i.e. patients with a lower stress level are characterized by smaller apparent rate constants. In other words, the diseased GB may need to develop fast cycling cross-bridges to compensate in the emptying process. This is a first step towards more quantitative and non-invasive measures of GB pain, and may provide useful insight in understanding GB motility and developing effective drug treatments.

A mechanical model for CCK-induced acalculous gallbladder pain.

This study investigates the potential correlation between acalculous biliary pain and mechanical stress during the bile-emptying phase. This study is built on the previously developed mathematical model used to estimate stress in the gallbladder wall during emptying [Li, W. G., X. Y. Luo, et al. Comput. Math. Methods Med. 9(1):27-45, 2008]. Although the total stress was correctly predicted using the previous model, the contribution from patient-specific active stress induced by the cholecystokinin (CCK) test was overlooked. In this article, we evaluate both the active and passive components of pressure in a gallbladder, which undergoes isotonic refilling, isometric contraction and emptying during the infusion of CCK. The pressure is estimated from in vivo ultrasonographical scan measurements of gallbladder emptying during CCK tests, assuming that the gallbladder is a thin ellipsoidal membrane. The passive stress is caused by the volume and shape changes during refilling at the gallbladder basal pressure, whereas the active stress arises from the pressure rise during the isometric gallbladder contraction after the CCK infusion. The effect on the stress estimates of the gallbladder to the liver is evaluated to be small by comparing numerical simulations of a gallbladder model with and without a rigid 'flat top' boundary. The model was applied to 51 subjects, and the peak total stress was found to have a strong correlation with the pain stimulated by CCK, as measured by the patient pain score questionnaires. Consistent with our previous study for a smaller sample, it is found that the success rate in predicting of CCK-induced pain is over 75%.

Evolving mechanical properties of a model of abdominal aortic aneurysm.

The novel three-dimensional (3D) mathematical model for the development of abdominal aortic aneurysm (AAA) of Watton et al. Biomech Model Mechanobiol 3(2): 98-113, (2004) describes how changes in the micro-structure of the arterial wall lead to the development of AAA, during which collagen remodels to compensate for loss of elastin. In this paper, we examine the influence of several of the model's material and remodelling parameters on growth rates of the AAA and compare with clinical data. Furthermore, we calculate the dynamic properties of the AAA at different stages in its development and examine the evolution of clinically measurable mechanical properties. The model predicts that the maximum diameter of the aneurysm increases exponentially and that the ratio of systolic to diastolic diameter decreases from 1.13 to 1.02 as the aneurysm develops; these predictions are consistent with physiological observations of Vardulaki et al. Br J Surg 85:1674-1680 (1998) and Lanne et al. Eur J Vasc Surg 6:178-184 (1992), respectively. We conclude that mathematical models of aneurysm growth have the potential to be useful, noninvasive diagnostic tools and thus merit further development.

Non-Newtonian bile flow in elastic cystic duct: one- and three-dimensional modeling.

Bile flow is thought to play an essential role in the pathophysiological genesis of cholelithiasis (gallstone formation) and in gallbladder pain. In this paper, we extend our previous study of the human biliary system (Li et al., 2007, J. Biomech. Eng., 129:164-173) to include two important factors: the non-Newtonian properties of bile, and elastic deformation of the cystic duct. A one-dimensional (1D) model is analyzed and compared with three-dimensional (3D) fluid-structure interaction simulations. It is found that non-Newtonian bile raises resistance to the flow of bile, which can be augmented significantly by the elastic deformation (collapse) of the cystic duct. We also show that the 1D model predicts the pressure drop of the cystic duct flow well for all cases considered (Newtonian or non-Newtonian flow, rigid or elastic ducts), when compared with the full 3D simulations.

One-dimensional models of the human biliary system.

This paper studies two one-dimensional models to estimate the pressure drop in the normal human biliary system for Reynolds number up to 20. Excessive pressure drop during bile emptying and refilling may result in incomplete bile emptying, leading to stasis and subsequent formation of gallbladder stones. The models were developed following the group's previous work on the cystic duct using numerical simulations. Using these models, the effects of the biliary system geometry, elastic property of the cystic duct, and bile viscosity on the pressure drop can be studied more efficiently than with full numerical approaches. It was found that the maximum pressure drop occurs during bile emptying immediately after a meal, and is greatly influenced by the viscosity of the bile and the geometric configuration of the cystic duct, i.e., patients with more viscous bile or with a cystic duct containing more baffles or a longer length, have the greatest pressure drop. It is found that the most significant parameter is the diameter of the cystic duct; a 1% decrease in the diameter increases the pressure drop by up to 4.3%. The effects of the baffle height ratio and number of baffles on the pressure drop are reflected in the fact that these effectively change the equivalent diameter and length of the cystic duct. The effect of the Young's modulus on the pressure drop is important only if it is lower than 400 Pa; above this value, a rigid-walled model gives a good estimate of the pressure drop in the system for the parameters studied.

On the mechanical behavior of the human biliary system.

This paper reviews the progress made in understanding the mechanical behaviour of the biliary system. Gallstones and diseases of the biliary tract affect more than 10% of the adult population. The complications of gallstones, i.e. acute pancreatitis and obstructive jandice, can be lethal, and patients with acalculous gallbladder pain often pose diagnostic difficulties and undergo repeated ultrasound scans and oral cholecystograms. Moreover, surgery to remove the gallbladder in these patients, in an attempt to relieve the symptoms, gives variable results. Extensive research has been carried out to understand the physiological and pathological functions of the biliary system, but the mechanism of the pathogenesis of gallstones and pain production still remain poorly understood. It is believed that the mechanical factors play an essential role in the mechanisms of the gallstone formation and biliary diseases. However, despite the extensive literature in clinical studies, only limited work has been carried out to study the biliary system from the mechanical point of view. In this paper, we discuss the state of art knowledge of the fluid dynamics of bile flow in the biliary tract, the solid mechanics of the gallbladder and bile ducts, recent mathematical and numerical modelling of the system, and finally the future challenges in the area.

Patellofemoral joint kinematics in individuals with and without patellofemoral pain syndrome.

BACKGROUND: Patellofemoral pain syndrome is a prevalent condition in young people. While it is widely believed that abnormal patellar tracking plays a role in the development of patellofemoral pain syndrome, this link has not been established. The purpose of this cross-sectional case-control study was to test the hypothesis that patterns of patellar spin, tilt, and lateral translation make it possible to distinguish individuals with patellofemoral pain syndrome and clinical evidence of patellar malalignment from those with patellofemoral pain syndrome and no clinical evidence of malalignment and from individuals with no knee problems. METHODS: Three-dimensional patellofemoral joint kinematics in one knee of each of sixty volunteers (twenty in each group described above) were assessed with use of a new, validated magnetic resonance imaging-based method. Static low-resolution scans of the loaded knee were acquired at five different angles of knee flexion (ranging between -4 degrees and 60 degrees). High-resolution geometric models of the patella, femur, and tibia and associated coordinate axes were registered to the bone positions on the low-resolution scans to determine the patellar motion as a function of knee flexion angle. Hierarchical modeling was used to identify group differences in patterns of patellar spin, tilt, and lateral translation. RESULTS: No differences in the overall pattern of patellar motion were observed among groups (p>0.08 for all global maximum likelihood ratio tests). Features of patellar spin and tilt patterns varied greatly between subjects across all three groups, and no significant group differences were detected. At 19 degrees of knee flexion, the patellae in the group with patellofemoral pain and clinical evidence of malalignment were positioned an average of 2.25 mm more laterally than the patellae in the control group, and this difference was marginally significant (p=0.049). Other features of the pattern of lateral translation did not differ, and large overlaps in values were observed across all groups. CONCLUSIONS: It cannot be determined from our cross-sectional study whether the more lateral position of the patella in the group with clinical evidence of malalignment preceded or followed the onset of symptoms. It is clear from the data that an individual with patellofemoral pain syndrome cannot be distinguished from a control subject by examining patterns of spin, tilt, or lateral translation of the patella, even when clinical evidence of mechanical abnormality was observed.

Magnetic resonance imaging for in vivo assessment of three-dimensional patellar tracking.

We have developed a non-invasive measurement technique which can ultimately be used to quantify three-dimensional patellar kinematics of human subjects for a range of static positions of loaded flexion and assessed its accuracy. Knee models obtained by segmenting and reconstructing one high-resolution scan of the knee were registered to bone outlines obtained by segmenting fast, low-resolution scans of the knee in static loaded flexion. We compared patellar tracking measurements made using the new method to measurements made using Roentgen stereophotogrammetric analysis in three cadaver knee specimens loaded through a range of flexion in a test rig. The error in patellar spin and tilt measurements was less than 1.02 degrees and the error in lateral patellar shift was 0.88 mm. Sagittal plane scans provided more accurate final measurements of patellar spin and tilt, whereas axial plane scans provided more accurate measurements of lateral translation and patellar flexion. Halving the number of slices did not increase measurement error significantly, which suggests that scan times can be reduced without reducing accuracy significantly. The method is particularly useful for multiple measurements on the same subject because the high-resolution bone-models need only be created once; thus, the potential variability in coordinate axes assignment and model segmentation during subsequent measurements is removed.

Calculating spatial statistics for velocity jump processes with experimentally observed reorientation parameters.

Mathematical modelling of the directed movement of animals, microorganisms and cells is of great relevance in the fields of biology and medicine. Simple diffusive models of movement assume a random walk in the position, while more realistic models include the direction of movement by assuming a random walk in the velocity. These velocity jump processes, although more realistic, are much harder to analyse and an equation that describes the underlying spatial distribution only exists in one dimension. In this communication we set up a realistic reorientation model in two dimensions, where the mean turning angle is dependent on the previous direction of movement and bias is implicitly introduced in the probability distribution for the direction of movement. This model, and the associated reorientation parameters, is based on data from experiments on swimming microorganisms. Assuming a transport equation to describe the motion of a population of random walkers using a velocity jump process, together with this realistic reorientation model, we use a moment closure method to derive and solve a system of equations for the spatial statistics. These asymptotic equations are a very good match to simulated random walks for realistic parameter values.

Sampling rate effects on measurements of correlated and biased random walks.

When observing the two-dimensional movement of animals or microorganisms, it is usually necessary to impose a fixed sampling rate, so that observations are made at certain fixed intervals of time and the trajectory is split into a set of discrete steps. A sampling rate that is too small will result in information about the original path and correlation being lost. If random walk models are to be used to predict movement patterns or to estimate parameters to be used in continuum models, then it is essential to be able to quantify and understand the effect of the sampling rate imposed by the observer on real trajectories. We use a velocity jump process with a realistic reorientation model to simulate correlated and biased random walks and investigate the effect of sampling rate on the observed angular deviation, apparent speed and mean turning angle. We discuss a method of estimating the values of the reorientation parameters used in the original random walk from the rediscretized data that assumes a linear relation between sampling time step and the parameter values.

Accurate assessment of patellar tracking using fiducial and intensity-based fluoroscopic techniques.

Accuracies of a point-based and an intensity-based fluoroscopic methods of assessing patella tracking were determined by comparing the pattern of patellar motion with respect to orientation (flexion, internal rotation, and lateral tilt) and translation (lateral, proximal, and anterior) with the pattern of patellar motion measured using Roentgen stereophotogrammetric analysis in three cadaver knee specimens. Each pose in the patellar motion could be obtained from single as well as multiple calibrated fluoroscopic images. The errors using the intensity-based method were slightly higher than those of the point-based method, but they appear to be sufficiently low to detect clinically significant differences in patellar kinematics.

A mathematical model for the growth of the abdominal aortic aneurysm.

We present the first mathematical model to account for the evolution of the abdominal aortic aneurysm. The artery is modelled as a two-layered, cylindrical membrane using nonlinear elasticity and a physiologically realistic constitutive model. It is subject to a constant systolic pressure and a physiological axial prestretch. The development of the aneurysm is assumed to be a consequence of the remodelling of its material constituents. Microstructural 'recruitment' and fibre density variables for the collagen are introduced into the strain energy density functions. This enables the remodelling of collagen to be addressed as the aneurysm enlarges. An axisymmetric aneurysm, with axisymmetric degradation of elastin and linear differential equations for the remodelling of the fibre variables, is simulated numerically. Using physiologically determined parameters to model the abdominal aorta and realistic remodelling rates for its constituents, the predicted dilations of the aneurysm are consistent with those observed in vivo. An asymmetric aneurysm with spinal contact is also modelled, and the stress distributions are consistent with previous studies.