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TOWARDS THE ESTIMATION OF THE FRACTAL DIMENSION OF HEART RATE VARIABILITY DATA.Hernández Cáceres JL, Foyaca Sibat H, Hong R, Garcia L, Sautié M, Namugowa V.Department of Physiology and Department of Neurology,

x  a_{0}  a_{1}  a_{2}  n  Variance explained 
LRS  2.1193 ± 0.0034  0.9995 ± 0.014  0  10  99.90% 
ARDI  1.9154 ± 0.0047  0  0.000021 ± 0.000001  100  87% 
The relation between ARDI and FD is illustrated in fig (1).
FD_{LRS}, and FD_{ARDI} are obtained from (11) with the respective coefficients provided in Table I. respectively.
The degree of concordance between different FD estimates was high when purely fractal signals were evaluated, thus for FD_{ARDI} and FD_{H} the correlation coefficient was r = 0.96 (n=10).
Addition of a nonfractal signal to a purely fractal signal was a cause for FD increase with all the three indices. As shown in table I adding up to about 30% of a nonfractal component to the total variance increases the estimated FD value in about 57%. According to different authors, in HRV data from healthy subjects about 22% of the total variance is increased by nonfractal components^{3}. At the same time, it seems plausible to expect that most of the clinical conditions related to changes in the spectral peaks (LF, and HF) imply changes in the range from the "normal" value of 22% to lower values around 18 %^{4}. Following the results shown in table I, it means that changes in nonfractal spectral peaks may not account for more than a 1% change in FD values estimated by any of these indices. Thus, our results suggest that all the three proposed FD estimates might be pertinent for HRV fractal dimension evaluation.
% non fractal variance added  FD_{PLI}  FD_{LRS}  FD_{ARDI} 
0  100  100  100 
1  100  101  100 
10  102  102  101 
16  105  105  108 
29  105  106  107 
HRV data: In table III the estimates of FD according to each index are shown for each of the HRV recording analysed.
Table III
FD values for the 10 recordings from the "Fantasia" database. Recording’s codes correspond to the original file names as presented at the website.
Recording’s code  FD_{PLI}  FD_{LRS}  FD_{ARDI} 
01  1.64  1.85  1.61 
02  1.49  1.5  1.34 
03  1.46  1.45  1.36 
04  1.62  1.88  1.49 
05  1.78  1.9  1.66 
Y1  1.11  1  1.11 
Y2  1.47  1.15  1.29 
Y3  1.36  1  1.46 
Y4  1.29  1.22  1.29 
Y5  1.36  1.3  1.31 
In table IV, a summary is provided for the statistical processing of the data.
Group  FD_{H}  FD_{LRS}  FD_{ARDI} 
Y  1.8730 ± 0.04  1.9167 ± 0.0177  1.8837 ± 0.0360 
O  1.7656 ± 0.077  1.7403 ± 0.0688  1.8080 ± 0.06 
P_{I} (Gaussianity assumption)  0.0002  0.0003  0.0215 
P_{0}(Permutations)  0.004  0.004  0.0197 
According to the three indices considered FD is decreased in the group of elderly subjects. Statically significant differences were obtained using either conventional or nonparametric permutation analysis. These results agree with ample literature evidence mainly using other FD estimation methods^{11}.
DISCUSSION
Here three indices for time domain estimation of FD in HRV data were proposed. Higuchi had previously proposed one of these indices (FD_{H}). The second index (LRS) is a modification of the original DFA method. DFA indices often are used for statistical discrimination between groups of patients. In particular our group has used LRS for few years. Even when recognized that DFA indices are related to some properties of fractal signals, this is, to our knowledge, the first attempt to use one of these indices for FD estimation. The third index (ARDI) appeared from our early attempts to characterize HRV by a nonlinear identification approach^{23}.
As our result revealed, the two indices are functionally related to fractal dimension when purely fractal signals were analysed. The "robustness" of all three indices was similar when purely fractal signals were mixed with nonfractal components. However, we do agree that more specific numerical experiments are needed for a more complete characterization of these indices’ robustness.
All the three indices documented in a similar way a result with ample literature support: the decrease with age of FD from HRV recordings^{11}.
A further step in our research might be the possibility to use these estimates for assessing FD from short duration traces, as well as from traces sampled al lower frequencies. Our preliminary results (not shown) point to a preservation of the predictive power of these estimates even for 2min duration traces, whereas only one of the indices (FDH) loses its clinical predictability as the sampling frequency is reduced from 1000 to 100 Hz.
Recent experience with the advent of chaos theory shows that we must be extremely cautious when trying to apply results of mathematical theories into real objects. An avalanche of chaos "demystification" followed the early "epidemics" of chaos finding in almost any area of physiology^{17}, and the sad sensation that our understanding of different physiological mechanisms remains obscure had become a reason for pessimism.
In our opinion, the sole use of power spectrum measures for assessing the putative fractality of a time series whose mechanism is unknown may appear misleading. Even with mathematical objects this can take place. One example is the known intermittent PomeauManneville map^{24}. This map is a deterministic nonlinear low dimensional process. However, its power spectrum is undistinguishable from that of a typical fractal with "1/f" noise. The nonlinear identification approach (from which ARDI is derived) is capable of distinguishing the both. Thus we may expect that these indices will provide not only new ways for assessing HRV fractal properties, but also for understanding some of HRV underlying dynamics.
The fractal properties of HRV may open avenues for radically new promising views to the heart beat regulation. Concepts such as selforganized criticality increase our understanding of physiological regulatory mechanisms not encased in the classical framework of the feedback theory scenario^{25, 26}. However, for these views being productive, we need a simple conviction: that fractallike processes take place in the human body. Our work has been an attempt in that direction. On the other hand, there is no doubt regarding the presence of feedback mechanisms, very likely nonlinear, acting as result of the influences of the both branches of the Autonomic Nervous System. To quantitatively handle HRV in a framework that includes all of these processes is a major task.
FIGURE CAPTIONS
Figure 1. Relationship between FD and ARDI. Data corresponds to the Fractal Dimension evaluated from a simulated signal whose ARDI had been previously estimated. The solid line is calculated on the basis of the coefficients provided in table I (100 data points). Each dot is the average of at least three NE replicate.
Acknowledgements. We thank to Ms Allison Swanby for discussing the paper as well as for redaction corrections. Especially we thank an unknown referee that managed to be strict and encouraging at the same time.
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Comment by Douglas McLean. Department of Computing Science and Mathematics University of Stirling. Stirling, Scotland. UK drm@cs.stir.ac.uk
I am now satisfied that the authors have answered sufficiently my queries and recommend it for publication in The Electronic Journal of Biomedicine. The work deals with the estimation fractal dimension from heart rate variability data. The work is new and interesting in that these indices have not been applied in this context before. Their work allows HRV to be characterised as a part fractal, part deterministic signal.
The practical benefits to medicine afforded by this understanding should be pursued and, perhaps the authors would like to mention or speculate as to how the indices might be employed in a clinical context in their discussion.
Comment by Antonio Núñez Reiz. Unidad de Investigación. Hospital General Yagüe. Burgos. España anunez at uninet.edu
(Spanish below)
Studying the electrical properties of the heart has forced scientists to enter in a field rather unexplored and unknown for many of us, mixing physiology with mathematics and specially numerical analysis and control engineering. The inherent complexity of the elements composing the system which generates and conducts the electrical impulses in the human heart creates the need to use mathematical tools and investigation methods that has allowed us to capture that complexity and to analyze it in a rigorous way.
For a long time we have known that the heart rate is controlled by the signals which arrive at the heart from the vegetative nervous system, being the variability of the cardiac frequency partially explained by the influence of the sympathetic and parasympathetic nervous system. Pathologies of these systems (ShyDrager syndrome, multiple sclerosis, diabetic neuropathy, and others) may affect this variability, and can be studied by studying it. Heart rate variability reduction is an effect of the sympathetic predominance and it is associated to a greater mortality, for example in patients at Intensive Care Units.
This study explores the possibility that other factors exist that affect this variability, and it presents the elegant concept of the fractals applied to the description of the electrical cardiac system cardiac. Although something complicated for nonmathematical minds, the fractal dimension of the variability of the frequency cardiac concept can help to complete the list of the factors that influence this variability, and to explain under a deeper perspective the clinical findings.
As described in the article, fractals are figures or objects that are selfcontained in a recursive way, so if we observe the figure on greater or smaller scale we will be with seen the same (see Figure 1). The reader can ask "and what has this to do with the variability of the heart rate?". A little patience and everything will be clarified.
Figure 1
Saying that the variability of the cardiac frequency has a fractal component, we are in fact saying that if we measure RR intervals during a period of time in a subject, we will be see variability patterns that repeat themselves if we multiply the period of time by a certain factor. This factor is indeed the fractal dimension of the variability of the cardiac frequency.
There is an anatomical base, that can be observed in the system of connections that compose the Hiss bundle, that could explain the presence of a fractal dimension in the variability of the frequency cardiac. The authors plead for a possible component of automatic control of the system provided by the presence of a fractal component, that could provide advantageous characteristics in absence of a suitable control operation by autonomous system, that is the fundamental component in normal conditions.
Using numerical analysis the authors determine three different indexes to evaluate the fractal dimension of the heart rate variability and its importance to explain this variability, arriving to the conclusion, after applying them to young and older healthy subjects, that the importance of the fractal component of this variability increases with age. Without being able definitively to exclude the possibility that there is an associated numerical calculation bias to this finding, they provide convincing reasons to think that this mechanism may have relevance in the study of pathologies of the heart rate and others like epilepsy.
SPANISH
El estudio de las propiedades eléctricas del corazón nos obliga a adentrarnos en un campo para muchos de nosotros inexplorado, como es el que mezcla la fisiología con las matemáticas y en concreto con el análisis numérico y la ingeniería de control. La propia complejidad inherente a los elementos que componen el sistema de generación y conducción de los impulsos eléctricos en el corazón humano nos obliga a utilizar herramientas y métodos de investigación que nos permitan capturar esa complejidad y analizarla de manera rigurosa.
Desde hace tiempo se sabe que el ritmo cardiaco está controlado por las señales que llegan al corazón desde el sistema nervioso vegetativo, siendo la variabilidad de la frecuencia cardiaca explicable en parte por el influjo de simpático y el parasimpático. Existen patologías de dichos sistemas (síndrome de ShyDrager, esclerosis múltiple, neuropatía diabética, etc) que se manifiestan con frecuencia a este nivel y que pueden ser estudiadas explorando dicha variabilidad. La reducción en la variabilidad del ritmo cardiaco es un efecto del predominio simpático y se asocia a una mayor mortalidad, por ejemplo en pacientes ingresados en Unidades de Cuidados Intensivos.
El presente estudio explora la posibilidad de que existan otros factores que afecten dicha variabilidad, e nos presenta el elegante concepto de los fractales aplicados a la descripción del sistema eléctrico cardiaco. Aunque algo complicado para mentes no matemáticas, el concepto de dimensión fractal de la variabilidad de la frecuencia cardiaca puede ayudar a completar la lista de los factores que influyen sobre dicha variabilidad, y a explicar de una manera más completa los hallazgos de la clínica.
Como se describe en el artículo, los fractales son figuras u objetos que se autocontienen a si mismos, de tal forma que si observamos la figura a mayor o menor escala nos encontraremos con lo mismo (ver figura 1). El lector puede preguntarse "¿y que tiene esto que ver con la variabilidad del ritmo cardiaco?". Un poco de paciencia y todo quedará aclarado.
Al decir que la variabilidad de la frecuencia cardiaca tiene un componente fractal, estamos en realidad diciendo que si medimos durante un periodo de tiempo los intervalos RR del ritmo cardiaco en un sujeto, nos encontraremos patrones de variabilidad que se repiten si multiplicamos el periodo de tiempo por un factor determinado. Este factor es precisamente la dimensión fractal de la variabilidad de la frecuencia cardiaca.
Existe una base anatómica, que se puede observar en el sistema de conexiones que componen el haz de Hiss, que podría explicar la presencia de una dimensión fractal en la variabilidad de la frecuencia cardiaca. Los autores abogan por un posible componente de autocontrol del sistema proporcionado por la presencia de un componente fractal, que podría proporcionar características ventajosas en ausencia de un adecuado funcionamiento del control por parte del sistema autónomo, que es el componente fundamental en condiciones normales.
Mediante el empleo de análisis numérico los autores determinan tres índices diferentes que permiten evaluar la dimensión fractal de la variabilidad del ritmo cardiaco y su importancia para explicar dicha variabilidad, llegando a la conclusión, tras aplicarlos a sujetos sanos jóvenes y mayores, que la importancia del componente fractal de dicha variabilidad aumenta con la edad. Sin poder excluir definitivamente la posibilidad de que haya un artefacto de cálculo numérico asociado a este hallazgo, proporcionan razones convincentes para creer que este mecanismo puede llegar a tener relevancia en el estudio de patologías del ritmo cardiaco e incluso de otras como la epilepsia.