Modelling the electrical properties of tissue as a porous medium

Magnetic Resonance Imaging-Derived Microvascular Perfusion Modeling to Assess Peripheral Artery Disease

Background Computational fluid dynamics has shown good agreement with contrast-enhanced magnetic resonance imaging measurements in cardiovascular disease applications. We have developed a biomechanical model of microvascular perfusion using contrast-enhanced magnetic resonance imaging signal intensities derived from skeletal calf muscles to study peripheral artery disease (PAD). Methods and Results The computational microvascular model was used to study skeletal calf muscle perfusion in 56 individuals (36 patients with PAD, 20 matched controls). The recruited participants underwent contrast-enhanced magnetic resonance imaging and ankle-brachial index testing at rest and after 6-minute treadmill walking. We have determined associations of microvascular model parameters including the transfer rate constant, a measure of vascular leakiness; the interstitial permeability to fluid flow which reflects the permeability of the microvasculature; porosity, a measure of the fraction of the extracellular space; the outflow filtration coefficient; and the microvascular pressure with known markers of patients with PAD. Transfer rate constant, interstitial permeability to fluid flow, and microvascular pressure were higher, whereas porosity and outflow filtration coefficient were lower in patients with PAD than those in matched controls (all P values ≤0.014). In pooled analyses of all participants, the model parameters (transfer rate constant, interstitial permeability to fluid flow, porosity, outflow filtration coefficient, microvascular pressure) were significantly associated with the resting and exercise ankle-brachial indexes, claudication onset time, and peak walking time (all P values ≤0.013). Among patients with PAD, interstitial permeability to fluid flow, and microvascular pressure were higher, while porosity and outflow filtration coefficient were lower in treadmill noncompleters compared with treadmill completers (all P values ≤0.001). Conclusions Computational microvascular model parameters differed significantly between patients with PAD and matched controls. Thus, computational microvascular modeling could be of interest in studying lower extremity ischemia.

Magnetic resonance imaging based modeling of microvascular perfusion in patients with peripheral artery disease

Peripheral artery disease (PAD) is associated with an increased risk of adverse cardiovascular events, impaired lower extremity blood flow and microvascular perfusion abnormalities in the calf muscles which can be determined with contrast-enhanced magnetic resonance imaging (CE-MRI). We developed a computational model of the microvascular perfusion in the calf muscles. We included 20 patients (10 PAD, 10 controls) and utilized the geometry, mean signal intensity and arterial input functions from CE-MRI calf muscle perfusion scans. The model included the microvascular pressure (pv), outflow filtration coefficient (OFC), transfer rate constant (kt), porosity (φ), and the interstitial permeability (Ktissue). Parameters were fitted and the simulations were compared across PAD patients and controls. Intra-observer reproducibility of the simulated mean signal intensities was excellent (intraclass correlation coefficients >0.995). kt and Ktissue were higher in PAD patients compared with controls (4.72 interquartile range (IQR) 3.33, 5.56 vs. 2.47 IQR 2.10, 2.85; p = 0.003; and 3.68 IQR 3.18, 4.41 vs. 1.81 IQR 1.81, 1.81; p < 0.001). Conversely, porosity (φ) was lower in PAD patients compared with controls (0.52 IQR 0.49, 0.54 vs. 0.61 IQR 0.58, 0.64; p = 0.016). Porosity (φ) was correlated with the ankle brachial index (r = 0.64, p = 0.011). The proposed computational microvascular model is robust and reproducible, and essential model parameters differ significantly between PAD patients and controls.

Phase transitions in pancreatic islet cellular networks and implications for type-1 diabetes

In many aspects the onset of a chronic disease resembles a phase transition in a complex dynamic system: Quantitative changes accumulate largely unnoticed until a critical threshold is reached, which causes abrupt qualitative changes of the system. In this study we examine a special case, the onset of type-1 diabetes (T1D), a disease that results from loss of the insulin-producing pancreatic islet β cells. Within each islet, the β cells are electrically coupled to each other via gap-junctional channels. This intercellular coupling enables the β cells to synchronize their insulin release, thereby generating the multiscale temporal rhythms in blood insulin that are critical to maintaining blood glucose homeostasis. Using percolation theory we show how normal islet function is intrinsically linked to network connectivity. In particular, the critical amount of β-cell death at which the islet cellular network loses site percolation is consistent with laboratory and clinical observations of the threshold loss of β cells that causes islet functional failure. In addition, numerical simulations confirm that the islet cellular network needs to be percolated for β cells to synchronize. Furthermore, the interplay between site percolation and bond strength predicts the existence of a transient phase of islet functional recovery after onset of T1D and introduction of treatment, potentially explaining the honeymoon phenomenon. Based on these results, we hypothesize that the onset of T1D may be the result of a phase transition of the islet β-cell network.

A genetic algorithm for optimizing multi-pole Debye models of tissue dielectric properties

Models of tissue dielectric properties (permittivity and conductivity) enable the interactions of tissues and electromagnetic fields to be simulated, which has many useful applications in microwave imaging, radio propagation, and non-ionizing radiation dosimetry. Parametric formulae are available, based on a multi-pole model of tissue dispersions, but although they give the dielectric properties over a wide frequency range, they do not convert easily to the time domain. An alternative is the multi-pole Debye model which works well in both time and frequency domains. Genetic algorithms are an evolutionary approach to optimization, and we found that this technique was effective at finding the best values of the multi-Debye parameters. Our genetic algorithm optimized these parameters to fit to either a Cole-Cole model or to measured data, and worked well over wide or narrow frequency ranges. Over 10 Hz-10 GHz the best fits for muscle, fat or bone were each found for ten dispersions or poles in the multi-Debye model. The genetic algorithm is a fast and effective method of developing tissue models that compares favourably with alternatives such as the rational polynomial fit.

A multiscale model for bioimpedance dispersion of liver tissue

Radio-frequency ablation (RFA) has been used in liver surgery to minimize blood loss during tissue division. However, the current RFA tissue division method lacks an effective way of determining the stoppage of blood flow. There is limitation on the current state-of-the-art laser Doppler flow sensor due to its small sensing area. A new technique was proposed to use bioimpedance for blood flow sensing. This paper discusses a new geometrical multiscale model of the liver bioimpedance incorporating blood flow impedance. This model establishes correlation between the physical tissue structure and bioimpedance measurement. The basic Debye structure within a multilevel framework is used in the model to account for bioimpedance dispersion. This dispersion is often explained by the Cole-Cole model that includes a constant phase element without physical explanation. Our model is able to account for reduced blood flow in its output with changes in permittivity in gamma dispersion that is mainly due to the polarization of water molecules. This study demonstrates the potential of a multiscale model in determining the stoppage of blood flow during surgery.

Mechanistic modeling of drug elimination by the liver using local volume averaging method

Local volume averaging method and local mass (drug) equilibrium were used for developing a mathematical model for transient drug transport and elimination in the liver. Taking into account the liver porosity and tortuosity, physio-chemical properties of the drug, the drug effective diffusivity, dispersion, convection, local plasma-hepatocyte equilibrium and hepatocellular drug metabolism, the governing partial differential equation was developed and numerically solved to describe a transient drug transfer and elimination across the liver following intravenous (IV) administration. The predicted values of hepatic clearance and bioavailability had very good agreement with the reported observations for different drugs. Unlike the well-stirred, parallel tube and dispersion models of hepatic clearance, the proposed mechanistic model is able to predict the drug concentration gradient across the liver with time and position in very dynamic conditions associated with drug absorption process in the intestine.

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.