Fractals and Chaos in the Hemodynamics of Intracranial Aneurysms

Computing the emerging flow in blood vessel sections by means of computational fluid dynamics is an often applied practice in hemodynamics research. One particular area for such investigations is related to the cerebral aneurysms, since their formation, pathogenesis, and the risk of a potential rupture may be flow-related. We present a study on the behavior of small advected particles in cerebral vessel sections in the presence of aneurysmal malformations. These malformations cause strong flow disturbances driving the system toward chaotic behavior. Within these flows, the particle trajectories can form a fractal structure, the properties of which are measurable by quantitative techniques. The measurable quantities are well established chaotic properties, such as the Lyapunov exponent, escape rate, and information dimension. Based on these findings, we propose that chaotic flow within blood vessels in the vicinity of the aneurysm might be relevant for the pathogenesis and development of this malformation.

Doubly transient chaos: generic form of chaos in autonomous dissipative systems

Chaos is an inherently dynamical phenomenon traditionally studied for trajectories that are either permanently erratic or transiently influenced by permanently erratic ones lying on a set of measure zero. The latter gives rise to the final state sensitivity observed in connection with fractal basin boundaries in conservative scattering systems and driven dissipative systems. Here we focus on the most prevalent case of undriven dissipative systems, whose transient dynamics fall outside the scope of previous studies since no time-dependent solutions can exist for asymptotically long times. We show that such systems can exhibit positive finite-time Lyapunov exponents and fractal-like basin boundaries which nevertheless have codimension one. In sharp contrast to its driven and conservative counterparts, the settling rate to the (fixed-point) attractors grows exponentially in time, meaning that the fraction of trajectories away from the attractors decays superexponentially. While no invariant chaotic sets exist in such cases, the irregular behavior is governed by transient interactions with transient chaotic saddles, which act as effective, time-varying chaotic sets.

Driving a conceptual model climate by different processes: snapshot attractors and extreme events

In a low-order chaotic model of global atmospheric circulation the effects of driving, i.e., time-dependent (periodic, chaotic, and noisy) forcing, are investigated, with particular interest in extremal behavior. An approach based on snapshot attractors formed by a trajectory ensemble is applied to represent the time-dependent likelihood of extreme events in terms of a physical observable. A single trajectory-based framework, on the other hand, is used to determine the maximal value and the kurtosis of the distribution of the same observable. We find the most significant effect of the driving on the magnitude, relative frequency, and variability of extreme events when its characteristic time scale becomes comparable to that of the model climate. Extreme value statistics is pursued by the method of block maxima, and found to follow Weibull distributions. Deterministic drivings result in shape parameters larger in modulus than stochastic drivings, but otherwise strongly dependent on the particular type of driving. The maximal effects of deterministic drivings are found to be more pronounced, both in magnitude and variability of the extremes, than white noise, and the latter has a stronger effect than red noise.

Are the fractal skeletons the explanation for the narrowing of arteries due to cell trapping in a disturbed blood flow?

We show that common circulatory diseases, such as stenoses and aneurysms, generate chaotic advection of blood particles. This phenomenon has major consequences on the way the biochemical particles behave. Chaotic advection leads to a peculiar filamentary particle distribution, which in turn creates a favorable environment for particle reactions. Furthermore, we argue that the enhanced stretching dynamics induced by chaos can lead to the activation of platelets, particles involved in the thrombus formation. In particular, we vary the size of both stenoses and aneurysms, and model them under resting and exercising conditions. We show that the filamentary particle distribution, governed by the fractal skeleton, depends on the size of the vessel wall irregularity, and investigate how it varies under resting or exercising conditions.

Fractal snapshot components in chaos induced by strong noise

In systems exhibiting transient chaos in coexistence with periodic attractors, the inclusion of weak noise might give rise to noise-induced chaotic attractors. When the noise amplitude exceeds a critical value, an extended attractor appears along the fractal unstable manifold of the underlying nonattracting chaotic set. A further increase of noise leads to a fuzzy nonfractal pattern. By means of the concept of snapshot attractors and random maps, we point out that the fuzzy pattern can be decomposed into well-defined fractal components, the snapshot attractors belonging to a given realization of the noise and generated by following an ensemble of noisy trajectories. The pattern of the snapshot attractor and its characteristic numbers, such as the finite time Lyapunov exponents and numerically evaluated fractal dimensions, change continuously in time. We find that this temporal fluctuation is a robust property of the system which hardly changes with increasing ensemble size. The validity of the Kaplan-Yorke formula is also investigated. A superposition of about 100 snapshot attractors provides a good approximant to the fuzzy noise-induced attractor at the same noise strength.

Fractal structures in stenoses and aneurysms in blood vessels

Recent advances in the field of chaotic advection provide the impetus to revisit the dynamics of particles transported by blood flow in the presence of vessel wall irregularities. The irregularity, being either a narrowing or expansion of the vessel, mimicking stenoses or aneurysms, generates abnormal flow patterns that lead to a peculiar filamentary distribution of advected particles, which, in the blood, would include platelets. Using a simple model, we show how the filamentary distribution depends on the size of the vessel wall irregularity, and how it varies under resting or exercise conditions. The particles transported by blood flow that spend a long time around a disturbance either stick to the vessel wall or reside on fractal filaments. We show that the faster flow associated with exercise creates widespread filaments where particles can get trapped for a longer time, thus allowing for the possible activation of such particles. We argue, based on previous results in the field of active processes in flows, that the non-trivial long-time distribution of transported particles has the potential to have major effects on biochemical processes occurring in blood flow, including the activation and deposition of platelets. One aspect of the generality of our approach is that it also applies to other relevant biological processes, an example being the coexistence of plankton species investigated previously.

Onset of chaotic advection in open flows

In this paper we investigate the transition to chaos in the motion of particles advected by open flows with obstacles. By means of a topological argument, we show that the separation points on the surface of the obstacle imply the existence of a saddle point downstream from the obstacle, with an associated heteroclinic orbit. We argue that as soon as the flow becomes time periodic, these orbits give rise to heteroclinic tangles, causing passively advected particles to experience transient chaos. The transition to chaos thus coincides with the onset of time dependence in open flows with stagnant points, in contrast with flows with no stagnant points. We also show that the nonhyperbolic nature of the dynamics near the walls causes anomalous scalings in the vicinity of the transition. These results are confirmed by numerical simulations of the two-dimensional flow around a cylinder.

Effective dimensions and chemical reactions in fluid flows

We show that chemical activity in hydrodynamical flows can be understood as the outcome of three basic effects: the stirring protocol of the flow, the local properties of the reaction, and the global folding dynamics which also depends on the geometry of the container. The essence of each of these components can be described by simple functional relations. In an ordinary differential equation approach, they determine a new chemical rate equation for the concentration, which turns out to be coupled to the dynamics of an effective fractal dimension. The theory predicts an exponential convergence to the asymptotic chemical state. This holds even in flows characterized by a linear stirring protocol where transient fractal patterns are shown to exist despite the lack of any chaotic set of the advection dynamics. In the exponential case the theory applies to flows of chaotic time dependence (chaotic flows) as well.

Coexistence of inertial competitors in chaotic flows

We investigate the dynamics of inertial particles immersed in open chaotic flows. We consider the generic problem of competition between different species, e.g., phytoplankton populations in oceans. The strong influence from inertial effects is shown to result in the persistence of different species even in cases when the passively advected species cannot coexist. Multispecies coexistence in the ocean can be explained by the fact that the unstable manifold is different for each advected competitor of different size.

Conservative spatial chaos of buckled elastic linkages

Buckling of an elastic linkage under general loading is investigated. We show that buckling is related to an initial value problem, which is always a conservative, area-preserving mapping, even if the original static problem is nonconservative. In some special cases, we construct the global bifurcation diagrams, and argue that their complicated structure is a consequence of spatial chaos. We characterize spatial chaos by the associated initial value problem's topological entropy, which turns out to be related to the number of buckled configurations.

Chemical transients in closed chaotic flows: the role of effective dimensions

We investigate chemical activity in hydrodynamical flows in closed containers. In contrast to open flows, in closed flows the chemical field does not show a well-defined fractal property; nevertheless, there is a transient filamentary structure present. We show that the effect of the filamentary patterns on the chemical activity can be modeled by the use of time-dependent effective dimensions. We derive a new chemical rate equation, which turns out to be coupled to the dynamics of the effective dimension, and predicts an exponential convergence. Previous results concerning activity in open flows are special cases of this new rate equation.

Rock-scissors-paper game in a chaotic flow: The effect of dispersion on the cyclic competition of microorganisms

Laboratory experiments and numerical simulations have shown that the outcome of cyclic competition is significantly affected by the spatial distribution of the competitors. Short-range interaction and limited dispersion allows for coexistence of competing species that cannot coexist in a well-mixed environment. In order to elucidate the mechanisms that destroy species diversity we study the intermediate situation of imperfect mixing, typical in aquatic media, in a model of cyclic competition between toxin producing, sensitive and resistant phenotypes. It is found, that chaotic mixing, by changing the character of the spatial distribution, induces coherent oscillations in the populations. The magnitude of the oscillations increases with the strength of mixing, leading to the extinction of some species beyond a critical mixing rate. When mixing is non-uniform in space, coexistence can be sustained at much stronger mixing by the formation of partially isolated regions, that prevent global extinction. The heterogeneity of mixing may enable toxin producing and sensitive strains to coexist for very long time at strong mixing.

Growth induced curve dynamics for filamentary micro-organisms

The growth of filamentary micro-organisms is described in terms of the geometry of evolving planar curves in which the dynamics is determined by an underlying growth process. Steadily propagating tip shapes in two and three dimensions are found that are consistent with experimentally observed growth sequences.

Fractal scaling of microbial colonies affects growth

The growth dynamics of filamentary microbial colonies is investigated. Fractality of the fungal or actinomycetes colonies is shown both theoretically and in numerical experiments to play an important role. The growth observed in real colonies is described by the assumption of time-dependent fractality related to the different ages of various parts of the colony. The theoretical results are compared to a simulation based on branching random walks.

Reactive particles in random flows

We study the dynamics of chemically or biologically active particles advected by open flows of chaotic time dependence, which can be modeled by a random time dependence of the parameters on a stroboscopic map. We develop a general theory for reactions in such random flows, and derive the reaction equation for this case. We show that there is a singular enhancement of the reaction in random flows, and this enhancement is increased as compared to the nonrandom case. We verify our theory in a model flow generated by four point vortices moving chaotically.

Spatial models of prebiotic evolution: soup before pizza?

The problem of information integration and resistance to the invasion of parasitic mutants in prebiotic replicator systems is a notorious issue of research on the origin of life. Almost all theoretical studies published so far have demonstrated that some kind of spatial structure is indispensable for the persistence and/or the parasite resistance of any feasible replicator system. Based on a detailed critical survey of spatial models on prebiotic information integration, we suggest a possible scenario for replicator system evolution leading to the emergence of the first protocells capable of independent life. We show that even the spatial versions of the hypercycle model are vulnerable to selfish parasites in heterogeneous habitats. Contrary, the metabolic system remains persistent and coexistent with its parasites both on heterogeneous surfaces and in chaotically mixing flowing media. Persistent metabolic parasites can be converted to metabolic cooperators, or they can gradually obtain replicase activity. Our simulations show that, once replicase activity emerged, a gradual and simultaneous evolutionary improvement of replicase functionality (speed and fidelity) and template efficiency is possible only on a surface that constrains the mobility of macromolecule replicators. Based on the results of the models reviewed, we suggest that open chaotic flows ('soup') and surface dynamics ('pizza') both played key roles in the sequence of evolutionary events ultimately concluding in the appearance of the first living cell on Earth.

Competing populations in flows with chaotic mixing

We investigate the effects of spatial heterogeneity on the coexistence of competing species in the case when the heterogeneity is dynamically generated by environmental flows with chaotic mixing properties. We show that one effect of chaotic advection on the passively advected species (such as phytoplankton, or self-replicating macro-molecules) is the possibility of coexistence of more species than that limited by the number of niches they occupy. We derive a novel set of dynamical equations for competing populations.

Chaotic flow: the physics of species coexistence

Hydrodynamical phenomena play a keystone role in the population dynamics of passively advected species such as phytoplankton and replicating macromolecules. Recent developments in the field of chaotic advection in hydrodynamical flows encourage us to revisit the population dynamics of species competing for the same resource in an open aquatic system. If this aquatic environment is homogeneous and well-mixed then classical studies predict competitive exclusion of all but the most perfectly adapted species. In fact, this homogeneity is very rare, and the species of the community (at least on an ecological observation time scale) are in nonequilibrium coexistence. We argue that a peculiar small-scale, spatial heterogeneity generated by chaotic advection can lead to coexistence. In open flows this imperfect mixing lets the populations accumulate along fractal filaments, where competition is governed by an "advantage of rarity" principle. The possibility of this generic coexistence sheds light on the enrichment of phytoplankton and the information integration in early macromolecule evolution.

[Corneal arcus and life expectancy]

In 1964-66, the authors completed the comprehensive medical screening of 1412 persons. Apart from the indicators of health state they also recorded their social and cultural parameters. The diagnoses they registered included AC, which has been covered in literature in rather contradicting ways. They found no data concerning survival; as analysing such a correlation is only possible within the frame-work of a several-decade follow-up study. By the end of the follow-up stage (31:12:1994), after 30 years, 1375 persons had died. Their death certificates and--if there were any--necropsy records have been processed and thoroughly analysed. They examined the occurrence of AC, life duration and survival probability--all in correlation with age, gender, constitution, certain diseases (hypertonia, ostheoarthrosis) and diagnoses at death (ischaemic heart diseases, acute myocardiac infarction, cerebrovascular diseases). They point it out that the occurrence of AC is significantly higher among males, but it increases in strong correlation with age in both sexes. Those who had AC were found to be older at the time of death, but it doesn't mean that AC correlates with better life expectancy--it means that AC occurs at older age. The survival probability of men over 75 was better than that of women. On the whole, AC is unfavourable concerning life expectancy, but the later it occurs, the less it can be used as an indicator of life expectancy. It was found that greater average weight correlated with longer average life duration, while among females the more a person weighed, the less frequent AC became. It was true for each weight group that those with AC had worse life expectancy. Altogether those with no AC were found to suffer from hypertonia significantly more frequently. The life expectancy of those with both AC and hypertonia, however, was always worse than those with hypertonia only, regardless of age and the type of hypertonia. Generally women are in a more favourable position, but in the 'serious' and 'very serious' hypertonia groups there is practically no difference in the survival of the two sexes. The authors have also found that AC has a significant negative prognostical value concerning survival and correlation with ostheoarthrosis, ischaemic heart diseases and cerebrovascular diseases. The correlation of AC with acute myocardiac infarction could not be proved convincingly.

Chemical or biological activity in open chaotic flows

We investigate the evolution of particle ensembles in open chaotic hydrodynamical flows. Active processes of the type A+B-->2B and A+B-->2C are considered in the limit of weak diffusion. As an illustrative advection dynamics we consider a model of the von Kármán vortex street, a time-periodic two-dimensional flow of a viscous fluid around a cylinder. We show that a fractal unstable manifold acts as a catalyst for the process, and the products cover fattened-up copies of this manifold. This may account for the observed filamental intensification of activity in environmental flows. The reaction equations valid in the wake are derived either in the form of dissipative maps or differential equations depending on the regime under consideration. They contain terms that are not present in the traditional reaction equations of the same active process: the decay of the products is slower while the productivity is much faster than in homogeneous flows. Both effects appear as a consequence of underlying fractal structures. In the long time limit, the system locks itself in a dynamic equilibrium state synchronized to the flow for both types of reactions. For particles of finite size an emptying transition might also occur leading to no products left in the wake.

[Longitudinal socio-gerontologic geronto-epidemiologic study in Hajdúszoboszló (SZOLOVI, 1964-1991)]

The tendencies of the demographic changes in Hungary, the trend and pace of the expected development of the age distribution and social stratification anticipated even in the early sixties that the problem of the ageing population (of the agriculture) would raise to national level and a considerable part of its consequent tasks will be the share of the national health care. So began in 1964 in Hajdúszoboszló a social gerontological-geroepidemiological study of those who were 60 and over and had agricultural profession. This study was developed into a longitudinal one (in the years 1986, 1990, 1991 and 1993), and is being continued even today.

Value of mortality data and necropsy records in monitoring morbidity in a population

STUDY OBJECTIVE

The aim was to compare the value of four sources of data in assessing morbidity in a population: (1) data from a screening programme including follow up records, (2) death certifications by attending physicians, (3) death certifications by doctor-coroners, and (4) necropsy reports.

DESIGN

The study was a cohort analysis of health and mortality in a sample of agricultural workers first examined in 1964-66 when they were aged 60 years or older. Follow up examinations enabled morbidity assessment to be made and ICD diagnostic categories to be compared with data available on persons in the cohort who had died.

SETTING

Hajdúszoboszló, a small town in eastern Hungary.

PARTICIPANTS

1412 persons (96.1% of those aged greater than or equal to 60 years) were examined in 1964-6. Those still alive and available in 1989 were examined again. Necropsy records were available for 144 persons from the cohort in 1989 and were extensively reviewed in comparison with data available from other sources.

MEASUREMENTS AND MAIN RESULTS

Comparison of causes of death established at necropsy showed marked differences from those registered by attending physicians and doctor-coroners, deviations ranging from -91.6% to +74.8%; 19.4% of underlying causes of death occurred exclusively in the necropsy group. Major divergencies in diagnostic classification occurred in the three data sources, particularly for diseases of the circulatory system, where hypertensive renal disease, old myocardial infarction, acute cerebrovascular disease, and venous thrombosis were rarely documented by physicians/coroners. When necropsy data were used the number of diagnostic categories increased strikingly over the other sources of information. Necropsy records revealed quantitatively similar information on morbidity to follow up examination though there were qualitative differences, necropsy being less likely to document diagnoses of endocrine disorders, mental and neurological diseases, digestive disorders, and musculosketal disorders.

CONCLUSIONS

Necropsy records contain much valuable material not available from other sources, exceeding by ninefold the amount of information reported at present. A way should be found to make use of this large data pool.