Key aspects for effective mathematical modelling of fractional-diffusion in cardiac electrophysiology: a quantitative study.

Microscopic structural features of cardiac tissue play a fundamental role in determining complex spatio-temporal excitation dynamics at the macroscopic level. Recent efforts have been devoted to the development of mathematical models accounting for non-local spatio-temporal coupling able to capture these complex dynamics without the need of resolving tissue heterogeneities down to the micro-scale. In this work, we analyse in detail several important aspects affecting the overall predictive power of these modelling tools and provide some guidelines for an effective use of space-fractional models of cardiac electrophysiology in practical applications. Through an extensive computational study in simplified computational domains, we highlight the robustness of models belonging to different categories, i.e., physiological and phenomenological descriptions, against the introduction of non-locality, and lay down the foundations for future research and model validation against experimental data. A modern genetic algorithm framework is used to investigate proper parameterisations of the considered models, and the crucial role played by the boundary assumptions in the considered settings is discussed. Several numerical results are provided to support our claims.

A microstructural model of cross-link interaction between collagen fibrils in the human cornea.

We propose a simplified micromechanical model of the fibrous reinforcement of the corneal tissue. We restrict our consideration to the structural function of the collagen fibrils located in the stroma and disregard the other all-important components of the cornea. The reinforcing structure is modelled with two sets of parallel fibrils, connected by transversal bonds within the single fibril family (inter-cross-link) and across the two families (intra-cross-link). The particular design chosen for this ideal structure relies on the fact that its ability to sustain loads is dependent on the degree of the cross-link and, therefore, on the density and stiffness of the bonds. We analyse the mechanical response of the system according to the type of interlacing and on the stiffness of fibres and bonds. Results show that the weakening of transversal bonds is associated with a marked increase of the deformability of the system. In particular, the deterioration of transversal bonds due to mechanical, chemical or enzymatic reasons can justify the loss of stiffness of the stromal tissue resulting in localized thinning and bulging typically observed in keratoconus corneas. This article is part of the theme issue 'Rivlin's legacy in continuum mechanics and applied mathematics'.

Viscoelastic computational modeling of the human head-neck system: Eigenfrequencies and time-dependent analysis.

A subject-specific 3-dimensional viscoelastic finite element model of the human head-neck system is presented and investigated based on computed tomography and magnetic resonance biomedical images. Ad hoc imaging processing tools are developed for the reconstruction of the simulation domain geometry and the internal distribution of bone and soft tissues. Material viscoelastic properties are characterized point-wise through an image-based interpolating function used then for assigning the constitutive prescriptions of a heterogenous viscoelastic continuum model. The numerical study is conducted both for modal and time-dependent analyses, compared with similar studies and validated against experimental evidences. Spatiotemporal analyses are performed upon different exponential swept-sine wave-localized stimulations. The modeling approach proposes a generalized, patient-specific investigation of sound wave transmission and attenuation within the human head-neck system comprising skull and brain tissues. Model extensions and applications are finally discussed.

Nonlinear diffusion and thermo-electric coupling in a two-variable model of cardiac action potential.

This work reports the results of the theoretical investigation of nonlinear dynamics and spiral wave breakup in a generalized two-variable model of cardiac action potential accounting for thermo-electric coupling and diffusion nonlinearities. As customary in excitable media, the common Q10 and Moore factors are used to describe thermo-electric feedback in a 10° range. Motivated by the porous nature of the cardiac tissue, in this study we also propose a nonlinear Fickian flux formulated by Taylor expanding the voltage dependent diffusion coefficient up to quadratic terms. A fine tuning of the diffusive parameters is performed a priori to match the conduction velocity of the equivalent cable model. The resulting combined effects are then studied by numerically simulating different stimulation protocols on a one-dimensional cable. Model features are compared in terms of action potential morphology, restitution curves, frequency spectra, and spatio-temporal phase differences. Two-dimensional long-run simulations are finally performed to characterize spiral breakup during sustained fibrillation at different thermal states. Temperature and nonlinear diffusion effects are found to impact the repolarization phase of the action potential wave with non-monotone patterns and to increase the propensity of arrhythmogenesis.

Multi-band decomposition analysis: application to cardiac alternans as a function of temperature.

OBJECTIVE: It has long been known that variations in temperature can facilitate the development of cardiac arrhythmias. Here, we aim to quantify the effects of temperature on cardiac alternans properties. APPROACH: in this work, we use optical mapping recordings of canine ventricular epicardial preparations to demonstrate that hypothermia can promote the formation of alternans, which is an important precursor to potentially lethal arrhythmias like fibrillation. We then present a novel quantification of alternans properties for a broad range of cycle lengths under different thermal states. Specifically, we apply the recently developed multi-band-decomposition analysis (MBDA) in the context of cardiac action potential dynamics. MAIN RESULTS: We show that the MBDA offers several advantages compared with traditional analysis of action potential durations. First, MBDA allows a depiction and quantification of the magnitude of alternans at all threshold values simultaneously and thus offers more information about how alternans relates to the action potential morphology while also removing the necessity of choosing a single threshold value. Second, the MBDA technique offers simple ways for assessing action potential amplitude alternans. Finally, MBDA provides a quantification of signal quality without any additional processing. SIGNIFICANCE: We find that the MBDA technique shows promise in leading to a deeper understanding of cardiac alternans properties.

Experimental evidence and mathematical modeling of thermal effects on human colonic smooth muscle contractility.

It has been shown, in animal models, that gastrointestinal tract (GIT) motility is influenced by temperature; nevertheless, the basic mechanism governing thermal GIT smooth muscle responses has not been fully investigated. Studies based on physiologically tuned mathematical models have predicted that thermal inhomogeneity may induce an electrochemical destabilization of peristaltic activity. In the present study, the effect of thermal cooling on human colonic muscle strip (HCMS) contractility was studied. HCMSs were obtained from disease-free margins of resected segments for cancer. After removal of the mucosa and serosa layers, strips were mounted in separate chambers. After 30 min, spontaneous contractions developed, which were measured using force displacement transducers. Temperature was changed every hour (37, 34, and 31°C). The effect of cooling was analyzed on mean contractile activity, oscillation amplitude, frequency, and contraction to ACh (10(-5) M). At 37°C, HCMSs developed a stable phasic contraction (~0.02 Hz) with a significant ACh-elicited mean contractile response (31% and 22% compared with baseline in the circular and longitudinal axis, respectively). At a lower bath temperature, higher mean contractile amplitude was observed, and it increased in the presence of ACh (78% and 43% higher than the basal tone in the circular and longitudinal axis, respectively, at 31°C). A simplified thermochemomechanical model was tuned on experimental data characterizing the stress state coupling the intracellular Ca(2+) concentration to tissue temperature. In conclusion, acute thermal cooling affects colonic muscular function. Further studies are needed to establish the exact mechanisms involved to better understand clinical consequences of hypothermia on intestinal contractile activity.

A stochastic mathematical model to study the autoimmune progression towards type 1 diabetes.

BACKGROUND: The integrity of the interactions and the 3D architecture among beta cell populations in pancreatic islets is critical for proper biosynthesis, storage and release of insulin. The aim of this study was to evaluate the effect on electrophysiological signalling of beta cells that is produced by progressive lymphocytic islet cell infiltration (insulitis), by modelling the disruption of pancreatic islet anatomy as a consequence of insulitis and altered glucose concentrations. METHODS: On the basis of histopathological images of murine islets from non-obese diabetic mice, we simulated the electrophysiological dynamics of a 3D cluster of mouse beta cells via a stochastic model. Progressive damage was modelled at different glucose concentrations, representing the different glycaemic states in the autoimmune progression towards type 1 diabetes. RESULTS: At 31% of dead beta cells (normoglycaemia) and 69% (hyperglycaemia), the system appeared to be biologically robust to maintain regular Ca(2+) ion oscillations guaranteeing an effective insulin release. Simulations at 84%, 94% and 98% grades (severe hyperglycemia) showed that intracellular calcium oscillations were absent. In such conditions, insulin pulsatility is not expected to occur. CONCLUSIONS: Our results suggest that the islet tissue is biophysically robust enough to compensate for high rates of beta cell loss. These predictions can be experimentally tested in vitro by quantifying space and time electrophysiological dynamics of animal islets kept at different glucose gradients. The model indicates the necessity of maintaining glycaemia within the physiological range as soon as possible after diabetes onset to avoid a dramatic interruption of Ca(2+) pulsatility and the consequent drop of insulin release.

Computational modeling and stress analysis of columellar biomechanics.

The open approach for rhinoplasty offers excellent exposure of the various components of the nose in situ. The biggest advantage of the external approach is the complete anatomic exposure, which allows the surgeon to inspect the osteo-cartilagineous framework, while the biggest disadvantage is represented by the transcolumellar scar. The goal of this study is to numerically quantify the stress induced on the scar of a human columella by a constant load, through a fine tuned finite elasticity continuum model. Specifically we want to determine the best shape of incision which would minimize this stress. The columellar portion of the nose, together with the various constituting tissues, has been modeled in a first approximation as a hyperelastic body and seven types of scars have been studied. The determination of the best incision must be a compromise among different factors: shape and size primarily, but also position with respect to the internal structures and external loads. From this point of view, the best class of scar appears to be, both at simulated and real levels, the V-shaped one, inducing a maximum logarithmic von Mises stress in tissue of 4.67 Pa, and an absolute minimum stress distribution on the scar of 4.17 Pa. Numerical simulations appear to be in agreement with the evidence-based results coming from surgical practice, thus confirming the necessity to minimize local stresses on the tissue. A parameters' sensitivity analysis further highlighted our optimal choice. The proposed mathematical model can be applied both to theoretically designed and numerically verified new non-conventional scar geometries.

Electroelastic unpinning of rotating vortices in biological excitable media.

Spiral waves in excitable biological media are associated with pathological situations. In the heart an action potential vortex pinned by an obstacle has to be removed through defibrillation protocols fine-tuned theoretically by using electrophysiological nonlinear mathematical models. Cardiac tissue, however, is an electroelastic medium whose electrical properties are strongly affected by large deformations. In this paper we specifically investigate the electroelastic pinning-unpinning mechanism in order to include cardiac contraction in the preexisting theoretically modeled defibrillation scenarios. Based on a two-dimensional minimal electromechanical model, we show numerically the existence of an unpinning band characterized by the size of the obstacle, the pacing site, and the frequency. Similar numerical simulations, performed in the absence of elastic coupling, show small differences in comparison with the electroelastic studies, suggesting for this specific scenario of pinning-unpinning dynamics a nonprominent role of elasticity.

Three-band decomposition analysis of wall shear stress in pulsatile flows.

Space-time patterns of wall shear stress (WSS) resulting from the numerical simulation of pulsating hemodynamic flows in semicoronal domains are analyzed, in the case of both regular semicoronal domains and semicoronal domains with bumpy insertions, mimicking aneurysm-like geometries. A new family of cardiovascular risk indicators, which we name three-band diagrams (TBDs), are introduced, as a sensible generalization of the two standard indicators, i.e., the time-averaged WSS and the oscillatory shear index. TBDs provide a handy access to additional information contained in the dynamic structure of the WSS signal as a function of the physiological risk threshold, thereby allowing a quick visual assessment of the risk sensitivity to individual fluctuations of the physiological risk thresholds. Due to its generality, TBD analysis is expected to prove useful for a wide host of applications in science, engineering, and medicine, where risk assessment plays a central role.

On the electrical intestine turbulence induced by temperature changes.

Paralytic ileus is a temporary syndrome with impairment of peristalsis and no passage of food through the intestine. Although improvements in supportive measures have been achieved, no therapy useful to specifically reduce or eliminate the motility disorder underlying postoperative ileus has been developed yet. In this paper, we draw a plausible, physiologically fine-tuned scenario, which explains a possible cause of paralytic ileus. To this aim we extend the existing 1D intestinal electrophysiological Aliev-Richards-Wikswo ionic model based on a double-layered structure in two and three dimensions. Thermal coupling is introduced here to study the influence of temperature gradients on intestine tissue which is an important external factor during surgery. Numerical simulations present electrical spiral waves similar to those experimentally observed already in the heart, brain and many other excitable tissues. This fact seems to suggest that such peculiar patterns, here electrically and thermally induced, may play an important role in clinically experienced disorders of the intestine, then requiring future experimental analyses in the search for possible implications for medical and physiological practice and bioengineering.