Jennifer H. Siggers

Mathematical Modeling of the Circulation in the Liver Lobule

In this paper, we develop a mathematical model of blood circulation in the liver lobule. We aim to find the pressure and flux distributions within a liver lobule. We also investigate the effects of changes in pressure that occur following a resection of part of the liver, which often leads to high pressure in the portal vein. The liver can be divided into functional units called lobules. Each lobule has a hexagonal cross-section, and we assume that its longitudinal extent is large compared with its width. We consider an infinite lattice of identical lobules and study the two-dimensional flow in the hexagonal cross-sections. We model the sinusoidal space as a porous medium, with blood entering from the portal tracts (located at each of the vertices of the cross-section of the lobule) and exiting via the centrilobular vein (located in the center of the cross-section). We first develop and solve an idealized mathematical model, treating the porous medium as rigid and isotropic and blood as a Newtonian fluid. The pressure drop across the lobule and the flux of blood through the lobule are proportional to one another. In spite of its simplicity, the model gives insight into the real pressure and velocity distribution in the lobule. We then consider three modifications of the model that are designed to make it more realistic. In the first modification, we account for the fact that the sinusoids tend to be preferentially aligned in the direction of the centrilobular vein by considering an anisotropic porous medium. In the second, we account more accurately for the true behavior of the blood by using a shear-thinning model. We show that both these modifications have a small quantitative effect on the behavior but no qualitative effect. The motivation for the final modification is to understand what happens either after a partial resection of the liver or after an implantation of a liver of small size. In these cases, the pressure is observed to rise significantly, which could cause deformation of the tissue. We show that including the effects of tissue compliance in the model means that the total blood flow increases more than linearly as the pressure rises.

Evaluation of droplet dispersion during non-invasive ventilation, oxygen therapy, nebuliser treatment and chest physiotherapy in clinical practice: implications for management of pandemic influenza and other airborne infections.

BACKGROUND Influenza viruses are thought to be spread by droplets, but the role of aerosol dissemination is unclear and has not been assessed by previous studies. Oxygen therapy, nebulised medication and ventilatory support are treatments used in clinical practice to treat influenzal infection are thought to generate droplets or aerosols. OBJECTIVES Evaluation of the characteristics of droplet/aerosol dispersion around delivery systems during non-invasive ventilation (NIV), oxygen therapy, nebuliser treatment and chest physiotherapy by measuring droplet size, geographical distribution of droplets, decay in droplets over time after the interventions were discontinued. METHODS Three groups were studied: (1) normal controls, (2) subjects with coryzal symptoms and (3) adult patients with chronic lung disease who were admitted to hospital with an infective exacerbation. Each group received oxygen therapy, NIV using a vented mask system and a modified circuit with non-vented mask and exhalation filter, and nebulised saline. The patient group had a period of standardised chest physiotherapy treatment. Droplet counts in mean diameter size ranges from 0.3 to > 10 µm were measured with an counter placed adjacent to the face and at a 1-m distance from the subject/patient, at the height of the nose/mouth of an average health-care worker. RESULTS NIV using a vented mask produced droplets in the large size range (> 10 µm) in patients (p = 0.042) and coryzal subjects (p = 0.044) compared with baseline values, but not in normal controls (p = 0.379), but this increase in large droplets was not seen using the NIV circuit modification. Chest physiotherapy produced droplets predominantly of > 10 µm (p = 0.003), which, as with NIV droplet count in the patients, had fallen significantly by 1 m. Oxygen therapy did not increase droplet count in any size range. Nebulised saline delivered droplets in the small- and medium-size aerosol/droplet range, but did not increase large-size droplet count. CONCLUSIONS NIV and chest physiotherapy are droplet (not aerosol)-generating procedures, producing droplets of > 10 µm in size. Due to their large mass, most fall out on to local surfaces within 1 m. The only device producing an aerosol was the nebuliser and the output profile is consistent with nebuliser characteristics rather than dissemination of large droplets from patients. These findings suggest that health-care workers providing NIV and chest physiotherapy, working within 1 m of an infected patient should have a higher level of respiratory protection, but that infection control measures designed to limit aerosol spread may have less relevance for these procedures. These results may have infection control implications for other airborne infections, such as severe acute respiratory syndrome and tuberculosis, as well as for pandemic influenza infection.

Unsteady flows in pipes with finite curvature

Motivated by the study of blood flow in a curved artery, we consider fluid flow through a curved pipe of uniform curvature, δ, driven by a prescribed oscillatory axial pressure gradient. The curved pipe has finite (as opposed to asymptotically small) curvature, and we determine the effects of both the centrifugal and Coriolis forces on the flow. In addition to δ, the flow is parameterized by the Dean number, D , the Womersley number, α, and a secondary streaming Reynolds number, R s . Asymptotic solutions are developed for the case when δ≪1, α≪1 and the magnitude of the axial pressure gradient is small, using regular perturbation techniques. For intermediate values of the governing parameters, a pseudospectral code is used to obtain numerical solutions. For flows driven by a sinusoidal pressure gradient ( D =0), we identify three distinct classes of stable solutions: 2π-periodic symmetric, 2π-periodic asymmetric, and asymmetric solutions that are either quasi-periodic, or periodic with period 2π k for k ∈ . The transition between solutions is dependent on the value of δ; thus pipes with finite curvature may exhibit qualitatively different transitions between the solution classes as the governing parameters are varied from those of curved pipes with asymptotically small curvature. When α≫1, matched asymptotic expansions are used to simplify the system, and the resulting equations are solved analytically for R s ≪1, δ≪1 and numerically for larger parameter values. We then determine the effect of a non-zero steady component of the pressure gradient ( D ≠0), and show that, for certain parameter values, when D is above a critical value the periodic asymmetric solutions regain spatial symmetry. Finally, we show that the effects of finite curvature can lead to substantial quantitative differences in the wall shear stress distribution and discuss briefly the physiological implications of the results for blood flow in arteries.

Techniques for automated local activation time annotation and conduction velocity estimation in cardiac mapping

Measurements of cardiac conduction velocity provide valuable functional and structural insight into the initiation and perpetuation of cardiac arrhythmias, in both a clinical and laboratory context. The interpretation of activation wavefronts and their propagation can identify mechanistic properties of a broad range of electrophysiological pathologies. However, the sparsity, distribution and uncertainty of recorded data make accurate conduction velocity calculation difficult. A wide range of mathematical approaches have been proposed for addressing this challenge, often targeted towards specific data modalities, species or recording environments. Many of these algorithms require identification of activation times from electrogram recordings which themselves may have complex morphology or low signal-to-noise ratio. This paper surveys algorithms designed for identifying local activation times and computing conduction direction and speed. Their suitability for use in different recording contexts and applications is assessed.

Intracellular Flow in Optic Nerve Axons: A Mechanism for Cell Death in Glaucoma

PURPOSE In glaucoma, elevated intraocular pressure causes a progressive loss of retinal ganglion cells and results in optic neuropathy. The authors propose a potential mechanism for cell death, whereby elevated intraocular pressure causes fluid to permeate axonal membranes, creating a passive intracellular fluid flow within the axons. It is hypothesized that this intracellular flow locally depletes the adenosine triphosphate (ATP) concentration, disrupting axonal transport and leading to cell death. METHODS A mathematical model was developed that takes into account the biomechanical principles underpinning the proposed hypothesis, and was solved to determine the implications of the mechanism. RESULTS The model suggests that the raised intraocular pressures present in glaucoma are adequate to produce significant intracellular fluid flow. In the periphery of the optic nerve head, this flow may be sufficient to disrupt the diffusion of ATP and hence interrupt active axonal transport. CONCLUSIONS The mathematical model demonstrates that it is physically plausible that a passive intracellular fluid flow could significantly contribute to the pathophysiology of the retinal ganglion cell axon in glaucoma.

Flow dynamics in a stented ureter.

Vesicorenal reflux is a major side effect associated with ureteric stent placement. In a stented upper urinary tract when the bladder pressure rises, such as during bladder spasms (due to irritation caused by the stent) or voiding of the bladder, it drives urine reflux up the ureter, which, in turn, may be a contributory factor for infections in the renal pelvis. We develop a mathematical model to examine urine flow in a stented ureter, assuming that it remains axisymmetric and treating the wall as a non-linear elastic membrane. The stent is modelled as a rigid, permeable, hollow, circular cylinder lying coaxially inside the ureter. The renal pelvis is treated as an elastic bag, whose volume increases in response to an increased internal pressure. Fluid enters the renal pelvis from the kidney with a prescribed flux. The stent, ureter and renal pelvis are filled with urine, and the bladder pressure is prescribed. We use the model to calculate the total volume of reflux generated during rises in bladder pressure and investigate how it is affected by the stent and ureter properties.

Ureteric stents: Investigating flow and encrustation

Blockages of the ureter, e.g. due to calculi (kidney stones), can result in an increase in renal pelvic pressure. This may be relieved by inserting a stent (essentially a permeable hollow tube). However, a number of complications are associated with stent use. Stents can result in reflux (backflow of urine along the ureter), which will promote recurrent urinary infection and possible renal parenchymal damage. Furthermore, long-term stent use is associated with infection and precipitation of salts from the urine, which can lead to a build-up of crystalline deposits on the stent surface, making stent removal difficult and painful. This paper examines factors governing urine flow in a stented ureter, the implications for reflux, and the processes by which the stent surface encrusts, in particular focusing on the influence of bacterial infection. An interdisciplinary approach is adopted, involving a combination of theoretical investigations and novel experiments.

A 3D porous media liver lobule model: the importance of vascular septa and anisotropic permeability for homogeneous perfusion

The hepatic blood circulation is complex, particularly at the microcirculatory level. Previously, 2D liver lobule models using porous media and a 3D model using real sinusoidal geometries have been developed. We extended these models to investigate the role of vascular septa (VS) and anisotropic permeability. The lobule was modelled as a hexagonal prism (with or without VS) and the tissue was treated as a porous medium (isotropic or anisotropic permeability). Models were solved using computational fluid dynamics. VS inclusion resulted in more spatially homogeneous perfusion. Anisotropic permeability resulted in a larger axial velocity component than isotropic permeability. A parameter study revealed that results are most sensitive to the lobule size and radial pressure drop. Our model provides insight into hepatic microhaemodynamics, and suggests that inclusion of VS in the model leads to perfusion patterns that are likely to reflect physiological reality. The model has potential for applications to unphysiological and pathological conditions.

Steady streaming within a periodically rotating sphere

We consider the flow in a spherical chamber undergoing periodic torsional oscillations about an axis through its centre, and analyse it both theoretically and experimentally. We calculate the flow in the limit of small-amplitude oscillations in the form of a series expansion in powers of the amplitude, finding that at second order, a steady streaming flow develops consisting of two toroidal cells. This streaming behaviour is also observed in our experiments. We find good quantitative agreement between theory and experiments, and we discuss the dependence of the steady streaming behaviour as both the oscillation frequency and amplitude are varied.

Reducing the data: Analysis of the role of vascular geometry on blood flow patterns in curved vessels

Three-dimensional simulations of blood flow usually produce such large quantities of data that they are unlikely to be of clinical use unless methods are available to simplify our understanding of the flow dynamics. We present a new method to investigate the mechanisms by which vascular curvature and torsion affect blood flow, and we apply it to the steady-state flow in single bends, helices, double bends, and a rabbit thoracic aorta based on image data. By calculating forces and accelerations in an orthogonal coordinate system following the centreline of each vessel, we obtain the inertial forces (centrifugal, Coriolis, and torsional) explicitly, which directly depend on vascular curvature and torsion. We then analyse the individual roles of the inertial, pressure gradient, and viscous forces on the patterns of primary and secondary velocities, vortical structures, and wall stresses in each cross section. We also consider cross-sectional averages of the in-plane components of these forces, which can be t...