Mechanisms and constraints shaping the evolution of body plan segmentation.

Segmentation of the major body axis into repeating units is arguably one of the major inventions in the evolution of animal body plan pattering. It is found in current day vertebrates, annelids and arthropods. Most segmented animals seem to use a clock-and-wavefront type mechanism in which oscillations emanating from a posterior growth zone become transformed into an anterior posterior sequence of segments. In contrast, few animals such as Drosophila use a complex gene regulatory hierarchy to simultaneously subdivide their entire body axis into segments. Here I discuss how in silico models simulating the evolution of developmental patterning can be used to investigate the forces and constraints that helped shape these two developmental modes. I perform an analysis of a series of previous simulation studies, exploiting the similarities and differences in their outcomes in relation to model characteristics to elucidate the circumstances and constraints likely to have been important for the evolution of sequential and simultaneous segmentation modes. The analysis suggests that constraints arising from the involved growth process and spatial patterning signal--posterior elongation producing a propagating wavefront versus a tissue wide morphogen gradient--and the evolutionary history--ancestral versus derived segmentation mode--strongly shaped both segmentation mechanisms. Furthermore, this implies that these patterning types are to be expected rather than random evolutionary outcomes and supports the likelihood of multiple parallel evolutionary origins.

Organization of ventricular fibrillation in the human heart: experiments and models.

Sudden cardiac death is a major health problem in the industrialized world. The lethal event is typically ventricular fibrillation (VF), during which the co-ordinated regular contraction of the heart is overthrown by a state of mechanical and electrical anarchy. Understanding the excitation patterns that sustain VF is important in order to identify potential therapeutic targets. In this paper, we studied the organization of human VF by combining clinical recordings of electrical excitation patterns on the epicardial surface during in vivo human VF with simulations of VF in an anatomically and electrophysiologically detailed computational model of the human ventricles. We find both in the computational studies and in the clinical recordings that epicardial surface excitation patterns during VF contain around six rotors. Based on results from the simulated three-dimensional excitation patterns during VF, which show that the total number of electrical sources is 1.4 +/- 0.12 times greater than the number of epicardial rotors, we estimate that the total number of sources present during clinically recorded VF is 9.0 +/- 2.6. This number is approximately fivefold fewer compared with that observed during VF in dog and pig hearts, which are of comparable size to human hearts. We explain this difference by considering differences in action potential duration dynamics across these species. The simpler spatial organization of human VF has important implications for treatment and prevention of this dangerous arrhythmia. Moreover, our findings underline the need for integrated research, in which human-based clinical and computational studies complement animal research.

A computational study of mother rotor VF in the human ventricles.

Sudden cardiac death is one of the major causes of death in the industrialized world. It is most often caused by a cardiac arrhythmia called ventricular fibrillation (VF). Despite its large social and economical impact, the mechanisms for VF in the human heart yet remain to be identified. Two of the most frequently discussed mechanisms observed in experiments with animal hearts are the multiple wavelet and mother rotor hypotheses. Most recordings of VF in animal hearts are consistent with the multiple wavelet mechanism. However, in animal hearts, mother rotor fibrillation has also been observed. For both multiple wavelet and mother rotor VF, cardiac heterogeneity plays an important role. Clinical data of action potential restitution measured from the surface of human hearts have been recently published. These in vivo data show a substantial degree of spatial heterogeneity. Using these clinical restitution data, we studied the dynamics of VF in the human heart using a heterogeneous computational model of human ventricles. We hypothesized that this observed heterogeneity can serve as a substrate for mother rotor fibrillation. We found that, based on these data, mother rotor VF can occur in the human heart and that ablation of the mother rotor terminates VF. Furthermore, we found that both mother rotor and multiple wavelet VF can occur in the same heart depending on the initial conditions at the onset of VF. We studied the organization of these two types of VF in terms of filament numbers, excitation periods, and frequency domains. We conclude that mother rotor fibrillation is a possible mechanism in the human heart.

Effect of heterogeneous APD restitution on VF organization in a model of the human ventricles.

The onset of ventricular fibrillation (VF) has been associated with steep action potential duration restitution in both clinical and computational studies. Recently, detailed clinical restitution properties in cardiac patients were reported showing a substantial degree of heterogeneity in restitution slopes at the epicardium of the ventricles. The aim of the present study was to investigate the effect of heterogeneous restitution properties in a three-dimensional model of the ventricles using these clinically measured restitution data. We used a realistic model of the human ventricles, including detailed descriptions of cell electrophysiology, ventricular anatomy, and fiber direction anisotropy. We extended this model by mapping the clinically observed epicardial restitution data to our anatomic representation using a diffusion-based algorithm. Restitution properties were then fitted by regionally varying parameters of the electrophysiological model. We studied the effects of restitution heterogeneity on the organization of VF by analyzing filaments and the distributions of excitation periods. We found that the number of filaments and the excitation periods were both dependent on the extent of heterogeneity. An increased level of heterogeneity leads to a greater number of filaments and a broader distribution of excitation periods, thereby increasing the complexity and dynamics of VF. Restitution heterogeneity may play an important role in providing a substrate for cardiac arrhythmias.

Cell model for efficient simulation of wave propagation in human ventricular tissue under normal and pathological conditions.

In this paper, we formulate a model for human ventricular cells that is efficient enough for whole organ arrhythmia simulations yet detailed enough to capture the effects of cell level processes such as current blocks and channelopathies. The model is obtained from our detailed human ventricular cell model by using mathematical techniques to reduce the number of variables from 19 to nine. We carefully compare our full and reduced model at the single cell, cable and 2D tissue level and show that the reduced model has a very similar behaviour. Importantly, the new model correctly produces the effects of current blocks and channelopathies on AP and spiral wave behaviour, processes at the core of current day arrhythmia research. The new model is well over four times more efficient than the full model. We conclude that the new model can be used for efficient simulations of the effects of current changes on arrhythmias in the human heart.

Alternans and spiral breakup in a human ventricular tissue model.

Ventricular fibrillation (VF) is one of the main causes of death in the Western world. According to one hypothesis, the chaotic excitation dynamics during VF are the result of dynamical instabilities in action potential duration (APD) the occurrence of which requires that the slope of the APD restitution curve exceeds 1. Other factors such as electrotonic coupling and cardiac memory also determine whether these instabilities can develop. In this paper we study the conditions for alternans and spiral breakup in human cardiac tissue. Therefore, we develop a new version of our human ventricular cell model, which is based on recent experimental measurements of human APD restitution and includes a more extensive description of intracellular calcium dynamics. We apply this model to study the conditions for electrical instability in single cells, for reentrant waves in a ring of cells, and for reentry in two-dimensional sheets of ventricular tissue. We show that an important determinant for the onset of instability is the recovery dynamics of the fast sodium current. Slower sodium current recovery leads to longer periods of spiral wave rotation and more gradual conduction velocity restitution, both of which suppress restitution-mediated instability. As a result, maximum restitution slopes considerably exceeding 1 (up to 1.5) may be necessary for electrical instability to occur. Although slopes necessary for the onset of instabilities found in our study exceed 1, they are within the range of experimentally measured slopes. Therefore, we conclude that steep APD restitution-mediated instability is a potential mechanism for VF in the human heart.

Eikonal formulation of the minimal principle for scroll wave filaments.

Recently, Wellner et al. [Proc. Natl. Acad. Sci. U.S.A. 99, 8015 (2002)]] proposed a principle for predicting a stable scroll wave filament shape as a geodesic in a 3D space with a metric determined by the inverse diffusivity tensor of the medium. Using the Hamilton-Jacobi theory we show that this geodesic is the shortest path for a wave propagating through the medium. This allows the use of shortest path algorithms to predict filament shapes, which we confirm numerically for a medium with orthotropic anisotropy. Our method can be used in cardiac tissue experiments since it does not require knowledge of the tissue anisotropy.

A model for human ventricular tissue.

The experimental and clinical possibilities for studying cardiac arrhythmias in human ventricular myocardium are very limited. Therefore, the use of alternative methods such as computer simulations is of great importance. In this article we introduce a mathematical model of the action potential of human ventricular cells that, while including a high level of electrophysiological detail, is computationally cost-effective enough to be applied in large-scale spatial simulations for the study of reentrant arrhythmias. The model is based on recent experimental data on most of the major ionic currents: the fast sodium, L-type calcium, transient outward, rapid and slow delayed rectifier, and inward rectifier currents. The model includes a basic calcium dynamics, allowing for the realistic modeling of calcium transients, calcium current inactivation, and the contraction staircase. We are able to reproduce human epicardial, endocardial, and M cell action potentials and show that differences can be explained by differences in the transient outward and slow delayed rectifier currents. Our model reproduces the experimentally observed data on action potential duration restitution, which is an important characteristic for reentrant arrhythmias. The conduction velocity restitution of our model is broader than in other models and agrees better with available data. Finally, we model the dynamics of spiral wave rotation in a two-dimensional sheet of human ventricular tissue and show that the spiral wave follows a complex meandering pattern and has a period of 265 ms. We conclude that the proposed model reproduces a variety of electrophysiological behaviors and provides a basis for studies of reentrant arrhythmias in human ventricular tissue.

Reentry in heterogeneous cardiac tissue described by the Luo-Rudy ventricular action potential model.

Heterogeneity of cardiac tissue is an important factor determining the initiation and dynamics of cardiac arrhythmias. In this paper, we studied the effects of gradients of electrophysiological heterogeneity on reentrant excitation patterns using computer simulations. We investigated the dynamics of spiral waves in a two-dimensional sheet of cardiac tissue described by the Luo-Rudy phase 1 (LR1) ventricular action potential model. A gradient of action potential duration (APD) was imposed by gradually varying the local current density of K(+) current or inward rectifying K(+) current along one axis of the tissue sheet. We show that a gradient of APD resulted in spiral wave drift. This drift consisted of two components. The longitudinal (along the gradient) component was always directed toward regions of longer spiral wave period. The transverse (perpendicular to the gradient) component had a direction dependent on the direction of rotation of the spiral wave. We estimated the velocity of the drift as a function of the magnitude of the gradient and discuss its implications.

Influence of nonexcitable cells on spiral breakup in two-dimensional and three-dimensional excitable media.

We study spiral wave dynamics in the presence of nonexcitable cells in two-dimensional (2D) and three-dimensional (3D) excitable media, described by the Aliev-Panfilov model. We find that increasing the percentage of randomly distributed nonexcitable cells can prevent the breaking up of a spiral wave into a complex spatiotemporal pattern. We show that this effect is more pronounced in 2D than 3D excitable media. We explain the observed 2D vs 3D differences by a different dependence of the period and diastolic interval of the spiral wave on the percentage of nonexcitable cells in 2D and 3D excitable media.