It is important to appreciate the role of mathematics in the analysis of physiological systems. From the ability to predict the future course of the system behaviour, the related goal of control follows naturally [1]. Probably the greatest appeal of chaos for physiology is the simple observation that so much physiological activity is highly variable, appearing random or noisy. A chaotic system can appear this way as well, but there is an underlying deterministic structure.

Erythrocyte deformability improves blood flow in microvessels and in large arteries at high shear rate. Physiologically, the erythrocyte deformability depends on the surface-volume ratio, internal viscosity and dynamics properties of the erythrocyte membrane.

Ektacytometry is a well established method in which cells, usually erythrocytes, are exposed to increasing shear stress and laser diffraction pattern through the suspension is recorded. The diffraction pattern, which is circular when the mammalian erythrocytes membrane is at rest, becomes elliptical when the cell undergoes shear stress. When laser is applied during creep and recovery process, light intensity dynamically changes along the major axes of the elliptical diffraction pattern. These experimental determinations are carried out with a home made device called Erythrodeformeter [2], which was developed and constructed for rheological measurements of red blood cells subjected to definite shear stress. This fluid shear stress is similar to the one in the capillaries. The corresponding time series (diffracted intensity measured in the major axis of the elliptical pattern under creep or recovery process) can be used in order to obtain same insight of the corresponding associated dynamics under healthy or illness conditions.

In the characterization of erythrocyte viscoelastic properties (time series) corresponding to healthy donors and hematological disorders, nonlinear dynamics tools and correlated random walk approach have been applied [3-7]. Diffractometric data belonging to healthy donors behave as white noise, while data series from different disease were found to be chaotic. Also, evidence of ordinary Brownian motion was found in the case of healthy donors. On the other hand, for samples corresponding to patients with hereditary spherocytosis, dyslipidemic and beta thalassemic a fractional Brownian motion was found [3-7].

In order to compress information contained in the diffractometric data, in such a way that emphasizes the most significant characteristics, we must not merely use observer's judgement but objective methods of analysis.

The clinical interpretation of erythrocytes deformation through the photometrically recorded series obtained measuring the diffraction pattern, attempts to link pathological features with the visual microscopy inspection of the cells samples.

The diabetes mellitus is studied in the present work. Diabetes mellitus induces several changes in the erythrocyte membrane and its cytoplasm, leading to alteration in the deformability. A decreasing trend of deformability in diabetes patients has been reported [8, 9]. Many studies have shown that diabetes mellitus is associated with increased whole blood viscosity and decreased erythrocytes deformability. It has been suggested that these abnormalities in blood rheology may play a causative role in the pathogenesis of diabetic vascular complications [10, 11].

Tools for nonlinear biosignal analysis are quite different from those used in the linear approach. The basic strategy is to determine appropriate characteristics of the recorded signal, which have changed associated with the creep and recovery process of the erythrocytes as the signal unfolds in time. During this process the question about order or chaos arises. To exclude amplitude information, the photometrically time series were normalized to have a variance of unity. All the series undergo Fourier analysis and appropriate smoothing was performed according to a frequency and amplitude dependent algorithms [12]. We also evaluate an essential aspect of the phase space trajectory, which is the Correlation Coefficient proposed by May and co-workers [13], which is a measure of the sensitivity of the process to initial conditions.

The results of the present study suggest that on random-walk approach the samples from healthy controls exhibit significant differences (ordinary Brownian motion) from those from diabetic patients (fractional Brownian motion) and these could allow us to claim that we have linked mathematical nonlinear tools with clinical aspects of diabetic erythrocytes’ rheological properties.

This is an active work, trying to use time series analysis for practical applications, this includes: identification of those erythrocytes of patients with severe anomalies, early diagnosis and also classification of the disease from the point of view of nonlinear dynamics. The mechanisms through which diabetic disease could induce vascular damage are both metabolic and mechanical. Hemorheological alterations in diabetes are result of changes affecting both red cell intrinsic structure and their interactions with the plasmatic components. Several hemorheological variables could influence and produce an impaired erythrocyte deformability determining an increased flow resistance in the microcirculation. The lipid-protein interaction can change the activity of membrane protein [17]. It was also reported that there were some changes in lipid component, stiffness and fluidity of diabetic erythrocytes membrane [18].

Once we can predict the future, it is natural to try to control that future, in order to guide the system to a preferred state or keep it away from undesired states. These basic goals of understanding, prediction, and control are closely related to the practical clinical goals of diagnosis and treatment, which underlie much of the rationale for research into physiological systems. Understanding a system’s behaviour and how it is altered under pathological conditions is of course a form of diagnosis, and control is effectively just another word for treatment [19]. Random behaviour is as one might expect unpredictable. Thus the question of randomness in a data series is a relative one, and more a question of mixtures of determinism and randomness. Since noise is present in all physical measurements, determining if randomness is inherent in the system dynamics or in the measurement process is not always straightforward.

The simplest and typical first test is the Fourier transform, in Fig. (3) two typical results are shown for healthy controls and for diabetic patients: 1) The first one is about the nature of the power spectrum, where the frequency constituents of the signals are the same for both populations, which could represent a 1/f dynamics. 2) The second result is about the intensity, it is bigger on healthy controls than on diabetic patients especially on low frequencies. It could be because healthy control erythrocytes are completely adapted at flow, while on diabetic erythrocytes the lipid protein interaction can change the activity of the membrane protein [18], which was also reported by Muzulu and co-workers [20]. But from the point of view of Fourier analysis it doesn't distinguish between chaos involving a small number of degrees of freedom and white noise, so this limits its application and leads us to turn to another method, notably that of studying phase space trajectories, which offers significant advantages.

Fig. (3). FFT – Fourier Power Spectrum (Intensity vs Frequency) for one Diabetic patient and one Healthy Control samples.

Certainly, the use of a nonlinear quantifier is not intended to replace conventional analyses, but to provide further insights into the underlying erythrocytes deformation mechanisms. In order to reconstruct the process dynamics we define the recovery and creep phase space using the technique of delay coordinates suggested by Takens and co workers. The phase space dimension was chosen applying Abarbanel method of false nearest neighbours. Thus, we obtained for four different and increasing embedding dimensions the %FNN and the results were: for m = 3 then %FNN = 19.74; for m = 4 then %FNN = 14.19; m = 5 then %FNN = 9.08; m = 6 then %FNN = 6.17, so we generated two phase space on delay coordinates for creep and recovery and one for the theoretical recovery space obtained from the creep one, all of them with m = 7.

Applying May algorithm to these phase space for the first nine steps s of the process we found the Correlation Coefficient C(s). Before computing any correlation coefficient for diffractometric data one should carefully inspect the data and use conventional techniques to extract as much and correct information as possible. So, in order to test the method and the proposed algorithm we contrasted it with a time series of pseudo random numbers (Fig. 4) as long as the one which corresponds to the photometric time series and it was found to be white noise. This is a random process in which the values at each time are statistical independent from each other. There is a perfect correlation of the signal with itself; and there is no correlation at the entire signal with any shifted version. The process has no memory; past values have no impact in subsequent values as it is expected on a pseudo random time series.

Fig. (4). Correlation Coefficient for a pseudoaleatory time series. The length of the time series is equal to the photometric time series analyzed.

As a preliminary indication of the possible clinical utility of this technique we have included (Fig. 5) the linear-linear plots of the correlation coefficient when the diffractometric data belongs to healthy controls. This method was also applied to diabetic erythrocyte samples (Fig. 6). In Figs. (7) and (8) the corresponding grand averages are given.

Fig. (5). Correlation Coefficient corresponding to the ten healthy control samples.

Fig. (6). Correlation Coefficient corresponding to the ten diabetic samples.

Fig. (7). Correlation Coefficient Grand Average for healthy control samples (mean ± SE).

Fig. (8). Correlation Coefficient Grand Average for diabetic samples (mean ± SE).

The results were very different comparing healthy controls and diabetic samples. On healthy controls, it could be ordinary Brownian motion (oBm), where statistical properties such as invariance or range are not related at all. On the other hand on diabetic patients, it would be fractional Brownian motion (fBm), it is fractal in the sense that self-similar, in an statistical sense, that is statistical properties are related over different time scales by a way of a power law, in other words, the stress process gives us some special information of the relaxation one in a short time and the series exhibit a great sensitivity to initial conditions.

There are many descriptions of the typical activity that accompanies the deformation but few detailed or quantitative analysis. On the other hand, the quantification of the recorded signals gives the possibility of studying the interactions among different anatomical structures, and the viscoelastic properties of several million of shear elongated cells.

One important finding relates on the complexity of the erythrocytes dynamics, using complexity in a rather loose manner, meaning more noise-like instead less regular, is to get insight in the process. This study found that complexity decreased with disease. In other words, there is such thing as healthy variability. A decrease in this variability can indicate a decrease in health, and variability can endow a system with flexibility and hence the ability to respond and adapt to environmental stressors [21].

Whether this variability is random or chaotic was a key in this study. One of the hopes of the recent application of nonlinear dynamical methods to physiology is that they could provide a general mathematical framework that has been missing from this traditional rather qualitative field.

In this respect, we think that introducing quantifiers derived from nonlinear dynamics could help with the description of the different red cell networks that could put at stake the cytoplasm and membrane interactions.

Moreover, the present results open the possibility of applied novel techniques, like resent methodology based in Information Theory, the causality plane Entropy - Statistical Complexity for a quantitative characterization of these behaviors [22]. Works in this direction, and also further studies with larger and well defined patient populations are in process in order to attain a better validation of the method.