Mathematical Modelling of The Liver Microcirculation

The models of the microcirculation of blood and interstitial fluid in the human liver lobule are developed based on the classical hexagon model of Kiernan. Both blood and interstitial flows in the lobule are treated as flows in porous medium connected via the fenestrated membrane of sinusoids. Several important physiological components are developed and included in the models. The lobule with tissue elasticity shows that the pressure-flux relationship is non-linear and the poroelastic model has more compliance than the solid elastic model. Models of the interstitial flow in both a single lobule and the whole liver are also developed. The results show that our models can predict the amount of interstitial fluid drainage including the ascites. From the parameter studies, we find that the permeabilities of the sinusoids and the interstitial space, and the portal pressure are the most important factors on ascites production. We further investigate the oxygen transportation and uptake by liver cells using the advection–diffusion equations and Michaelis-Menten kinetics. The studies show that the main mechanism of oxygen transportation within the sinusoids is advection; however, the transportations within the interstitial space and across the fenestrated endothelial cells are mainly from diffusion process. The effect of the arrangement of the vessels and the geometry of the lobule on blood perfusion and oxygen distribution is also studied. The results show that the classical hexagonal lobule with the vascular septa provides the optimal perfusion compared to other geometries of the lobule. In summary, this thesis contributes to the development of mathematical models of several important features in the liver microcirculation such as the tissue elasticity, the interstitial flow, the oxygen distribution, and the arrangement of the vessels in the lobule

Mathematical model of blood and interstitial flow and lymph production in the liver.

We present a mathematical model of blood and interstitial flow in the liver. The liver is treated as a lattice of hexagonal \u2018classic\u2019 lobules, which are assumed to be long enough that end effects may be neglected and a two-dimensional problem considered. Since sinusoids and lymphatic vessels are numerous and small compared to the lobule, we use a homogenized approach, describing the sinusoidal and interstitial spaces as porous media. We model plasma filtration from sinusoids to the interstitium, lymph uptake by lymphatic ducts, and lymph outflow from the liver surface. Our results show that the effect of the liver surface only penetrates a depth of a few lobules\u2019 thickness into the tissue. Thus, we separately consider a single lobule lying sufficiently far from all external boundaries that we may regard it as being in an infinite lattice, and also a model of the region near the liver surface. The model predicts that slightly more lymph is produced by interstitial fluid flowing through the liver surface than that taken up by the lymphatic vessels in the liver and that the on-peritonealized region of the surface of the liver results in the total lymph production (uptake by lymphatics plus fluid crossing surface) being about 5 % more than if the entire surface were covered by the Glisson\u2013peritoneal membrane. Estimates of lymph outflow through the surface of the liver are in good agreement with experimental data. We also study the effect of non-physiological values of the controlling parameters, particularly focusing on the conditions of portal hypertension and ascites. To our knowledge, this is the first attempt to model lymph production in the liver. The model provides clinically relevant information about lymph outflow pathways and predicts the systemic response to pathological variations

Modelling the Influence of Foot-and-Mouth Disease Vaccine Antigen Stability and Dose on the Bovine Immune Response

Foot and mouth disease virus causes a livestock disease of significant global socio-economic importance. Advances in its control and eradication depend critically on improvements in vaccine efficacy, which can be best achieved by better understanding the complex within-host immunodynamic response to inoculation. We present a detailed and empirically parametrised dynamical mathematical model of the hypothesised immune response in cattle, and explore its behaviour with reference to a variety of experimental observations relating to foot and mouth immunology. The model system is able to qualitatively account for the observed responses during in-vivo experiments, and we use it to gain insight into the incompletely understood effect of single and repeat inoculations of differing dosage using vaccine formulations of different structural stability

<i>In-vivo</i> experimental results for cattle inoculated with a regular dose of vaccine at 0 and 29 days, giving the resultant IgM (left) and IgG (right) levels recorded: (top: blue) normal vaccine producing a regular immune response; (bottom: green) vac

<p>Plots give the median value (central bar), 25th–75th percentile (box) and extreme values (whiskers) unless considered outliers, in which case they are plotted separately (cross) for four (bottom: T-cell independent) or five (top: T-cell dependent) replicates (individual cattle). Data from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0030435#pone.0030435-Grant1" target="_blank">[9]</a>. Note the significant differences in magnitude between the T-cell dependent and T-cell independent cases. Results presented on a log-scale.</p

Parameters.

<p>Definitions of system parameters and values used in simulations (with justification and/or reference source). Here generally refers to production rates, to conversion rates, to decay rates and to temporal delays. For simulations we consider .</p>†<p>wk: week; c: particle (vaccine, complex or cell) concentration.</p

Experimental and simulation results for IgM () and IgG () for the full system (blue) and for the system with a reduced T-cell dependent response only (green).

<p>Here the mean and range of each of the datasets from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0030435#pone-0030435-g006" target="_blank">Figure 6</a> are plotted, together with the simulation results from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0030435#pone-0030435-g007" target="_blank">Figure 7</a>, with the model outputs suitably scaled (the peak of the experimental mean for each of the two immunoglobulins matched by the peak of the full T-dependent system).</p

Model results for the system with stable (blue) and unstable (red) vaccine, the latter having a decay rate roughly twice that of the former (see Table 2).

<p>The benefits of stability are not fully realised until a booster dose is applied (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0030435#pone-0030435-g005" target="_blank">Figure 5</a>).</p

LHS applied to the immunological model with parameter ranging from to 4 times the nominal values shows that qualitative behaviour is maintained.

<p>Here the median (solid line) is plotted together with the range of possible results, in 5 percentile steps (shaded) from 410 replicates (axes upper bound set at maximum of 95^th percentile range) on a log scale.</p

Model results for the full system (blue) and for the system with a reduced T-cell dependent response only (green), where and have been reduced to 1% of their original values.

<p>All figures present variables on a log scale as percentages of their peak value.</p

Schematic of the short-term and long-term dynamics of the immune response, as stimulated by vaccine antigen.

<p>For variable definitions see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0030435#pone-0030435-t001" target="_blank">Table 1</a>; for parameter definitions and values see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0030435#pone-0030435-t002" target="_blank">Table 2</a>.</p