TEMPORAL FLUCTUATIONS IN REGIONAL RED BLOOD CELL FLUX IN THE RAT BRAIN CORTEX IS A FRACTAL PROCESS

1. INTRODUCTION

Temporal fluctuations ever since it was observed and recognized as an elementary feature of blood flow in the microcirculation by Krogh in 1929 (Krogh, 1929) became the essence of concepts termed “vasomotion” and “flow motion”. Such fluctuations in flow are common in various microcirculatory beds including that of the brain and can be continuously monitored by laser-Doppler flowmetry (Stern, 1975; Nilsson et al., 1980; Rosenblum et al., 1987; Fasano et al., 1988; Hudetz et al., 1992; Morita-Tsuzuki et al., 1992). When challenged, they often show slow, (6–12 cycles/minute) oscillatory pattern, which allow for a relatively simple characterization if the higher frequencies are eliminated from the signal (Hudetz et al., 1992; Morita-Tsuzuki et al., 1992). Temporal variation in micro-circulatory flow is however multifactorial; and the interactions among these factors can manifest in a complex structuring of the time series recorded by high-resolution laser-Doppler flowmetrx (LDF). With no apparent dominating frequency present, they are much too complex to be analyzed in specific terms by conventional descriptive statistics, amplitude and frequency measures. Thus we have used fractal methods genuinely holistic in nature to gain statistical insight into random signals of this kind and to determine if under unchallenged control conditions they represent disorganized behavior or show long-range correlations (West and Goldberger, 1987; Bassingthwaighte, 1988; West and Shlesinger, 1990; Weibel, 1991; Bassingthwaighte et al., 1994).

2. METHODS

2.1. Fractals and Time Series

Fractals describe objects and events which are characterized by consistencies in the degree of correlation between neighboring structures or events (Bassingthwaighte, 1988; Bassingthwaighte et al., 1995). The correlation may be positive or negative and is given by

$$r_1=2^{2H-1}-1$$

where H is the Hurst coefficient (Hurst, 1951). H is a measure of “roughness” of the structure or the signal, H near 1.0 indicating a high degree of smoothness and strong positive correlation, H near 0 indicating a high degree of roughness and strong negative correlation. The Hurst coefficient is independent of the Euclidean dimension E, in which the object is embedded and relates to the fractal dimension, D as

$$\mathrm{D}=\mathrm{E}+\mathrm{H}-1$$

Geometrical fractals are characterized by self-similarity of their structure viewed at varying scales of observation (Mandelbrot, 1983). Because the units in which a variable’s change is measured and the units of time in which these changes are recorded are different, fractal time series are not self-similar but self-affine.

When a signal has self-affine structuring, then this is reflected in its Fourier power spectrum

$${\left|A\right|}^2\approx \frac{1}{f^{\mathrm{\beta }}}$$

If the signal is composed of random independent events (random Gaussian noise, rGn), then β = 0, β= 2H - 1, H = 0.5 and r₁ = 0. When rGn is integrated to random Brownian motion (rBm) its β = 2, B = 2H + 1, H = 0.5 and r₁ = 0. Around the rGn and rBm, signals composed of correlated events are found, fractional Gaussian noises (fGn’s) (−1<β <1) and fractional Brownian motions (fBm’s) (1<B<3) (Mandelbrot and Ness, 1968).

2.2. Laser-Doppler Flowmetry in the Rat Brain Cortex

Urethane in a dose of 130 mg/100 g body weight was given i.p. for general anesthesia. The animals (n=8) were artificially ventilated by a gas mixture of 30% O₂ and 70% N₂. Ventilation was adjusted so that pO₂ in arterial blood samples be as close as possible to 100 mmHg. Core temperature was maintained by a Yellow Spring Body Temperature Controller switching an infra heating lamp at the threshold value of the rectal temperature of 37.5 °C. Mean arterial blood pressure (MABP) was measured by a Statham pressure transducer via a catheter introduced into the common carotid artery toward the heart. MABP and electrical activity of the brain cortex (ECoG) was recorded by a Grass polygraph. Red blood cell (RBC) flux was continuously measured with a time constant of 0.1 second under control conditions by a laser-Doppler Flowmeter (Moor Instruments, MBF3D) in the brain cortex of anesthetized rats through the thinned (closed) calvarium (Figure 1).

Figure 1.

Experimental arrangement for laser-Doppler flometry (LDF) and Near Infrared Spectrophotometry (NIRS) in the brain cortex of the anesthetized rat.

In our polished skull preparation (Fig. 2) using an illumination fiber of 0.4 mm in diameter the laser light penetrates across the brain cortex and beyond into the subcortical white matter, and even beyond reaching as deep as the subcortical nuclei and the thalamus. Scattering of laser light is seen in all directions. Subcortical white matter due to its high scattering coefficient allows little intensity to penetrate thus appears as a darker band splitting the photon diffusion sphere. Photons would have equal difficulty in escaping and reentering the cortex across this boundary. Hence the volume actually sampled by LDF is sandwiched between the pial surface and the subcortical white matter. It has a disk shape with the thickness of the brain cortex (2.5 mm) and 5–6 mm in diameter. These dimensions correspond well with the theoretical model of Bonner and Nossal (Bonner and Nossal, 1981).

Figure 2.

Assessment of the volume of measurement of laser-Doppler flometry in the rat brain cortex. Panel a: side view of a mid-line sagittal section of the rat brain in white light. Note the piece of the skull thinned for the RBC-flux measurement being interposed between the LDF-probe and the brain cortex. Panel b: Same view as in panel a with the Laser light being turned on and the white light turned off. The outline of the calvarium being trans-illuminated is shown.

2.3. Creating RBC-flux Time Series

RBC-flux time series were created in a length of 2¹⁷ data points by sampling the signal output at 200 Hz@12 bit, representing 655 seconds each. All time series collected in the 8 animals were proven fractional Brownian motions (β>1, Fig. 3 and Table I.) thus successive differences of the RBC-flux signal were taken to generate signals in fractional Gaussian noise format for analysis by the dispersional method for the Hurst coefficient H (Fig. 4.). The analyses were carried out on Sun workstations with a software developed by the group of Bassingthwaighte at the Center for Bioengineering, University of Washington (Bassingthwaighte and Raymond, 1995) operated via the Internet from Budapest. Additional analysis and data processing were carried out in Budapest on Macintosh Quadra 650 and 660AV computers by programs developed locally. Oxygen saturation and hemoglobin content of the volume of cerebro-cortical tissue under laser-Doppler measurement were also carried out by a Near Infrared Spectrophotometry (data are not shown and analyzed in this study). The latter did not show any interference with the laser-Doppler measurements.

Figure 3.

Schematics of the initial steps of data analysis. Upper panel: A record of RBC-flux in its original form. According to our signal classification, it is a fractional Brownian motion (fBm). Middle panel: Probability density funtion generated from the signal shown at the top. Lower panel: Fourier power-spectrum analysis. Power is plotted as a function of frequency on a log-log scale calculated from the time series shown at the top. The line of regression through the data is shown, which relates to the spectral index of the signal, β as β = −slope. Note, that the signal is fBm with β>l.

Table 1.

Characterization of RBC-flux time series with descriptive statistical and fractal tools

Time series		n	Mean	±SD	Skewness	Kurtosis
ts.1		131072	2488.843	32.638	−0.611	4.480
ts.2		131072	2461.350	15.306	0.187	3.003
ts.3		131072	2735.590	21.311	−0.174	3.649
ts.4		131072	2501.470	16.537	0.025	−2.941
ts.5		131072	2445.365	32.537	−0.247	2.710
ts.6		131072	2387.799	23.654	−0.011	2.900
ts.7		131072	2388.845	16.205	0.191	3.202
ts.8		131072	2456.693	22.038	0.675	4.113
Mean			2483.244	22.528	0.004	3.375
±SD			109.988	6.901	0.376	0.640

		Fourier (done on fBm)				Dispersion (done on fGn)
Time series		β	H	r1		r	H	r1
ts.1		1.494	0.247	−0.296		−0.995	0.254	−0.289
ts.2		1.466	0.233	−0.309		−0.998	0.163	−0.373
ts.3		1.477	0.239	−0.304		−0.997	0.158	−0.378
ts.4		1.452	0.226	−0.316		−0.997	0.139	−0.394
ts.5		1.510	0.255	−0.288		−0.999	0.220	−0.322
ts.6		1.507	0.254	−0.289		−0.999	0.213	−0.328
ts.7		1.458	0.229	−0.313		−0.996	0.145	−0.389
ts.8		1.480	0.240	−0.303		−0.997	0.162	−0.374
Mean		1.481	0.240	−0.302		−0.997	0.182	−0.356
±SD		0.022	0.011	0.011		0.001	0.042	0.038

Figure 4.

Steps of dispersional analysis. First, successive differences of RBC-flux signal shown at the top of Fig. 3 is taken (upper left) because the dispersional analysis can only be applied to fractional Gaussian noises (fGn). Then, neighbouring elements are lumped in succession resulting in series of decreasing length (2¹⁷ to 2⁹ datapoints) and relative dispersion. The Hurst coefficient is found by a linear regression analysis on the log-log plot of relative dispersion and window size according to Eq. 4.

2.4. Fractal Analysis of RBC-flux Time Series

In the present study the dispersional analysis was chosen because of its good overall performance under a variety of signal quality (Bassingthwaighte and Raymond, 1995). This method is based on finding the variability in the average signal over windows m elements long. In an iterative process it finds the Hurst coefficient from the regression analysis of log RD(m) versus log m/m_o (Fig. 4.), where RD is relative dispersion (SD/mean) and m_o is a reference window size, thus

$$\text{log RD}(\mathrm{m})=(\mathrm{H}-1)\text{log}(\mathrm{m}/{\mathrm{m}}_{\mathrm{o}})+\text{log RD}({\mathrm{m}}_{\mathrm{o}})$$

Dispersional method requires signals be subjected to analysis in noise format. Thus the signals need first be tested by the Fourier spectral analysis on their β before further processing and if they prove fBm, successive differences need to be taken to generate the fGn signal format (Fig. 3).

3. RESULTS

At a mean blood pressure of 101.25 ± 14.58 mm Hg the mean level of the LDF signal was 2483.244 ± 109.99 (mean ± SD in arbitrary units), the mean amplitude of LDF fluctuations within the time window of 655 seconds were 8.18 ± 2.55 % of the mean LDF signal. The mean skewness of the probability density functions (as in Fig. 3.) was 0.004 ± 0.376 indicating asymmetry of the distributions. Kurtosis>3 indicating high peakedness relative to Gaussian distribution were found in 5 of the 8 cases with one being marginal (3.375 ± 0.64). Thus probability density functions of the RBC-flux time series did not show a typical Gaussian distribution.

RBC-flux time series proved fractional Brownian motions (β>1, Fig. 3 and Table I.). H of 0.182 ± 0.042 was found by dispersional analyis in the 8 rats studied. This value of H implies that the correlation r₁ between neighbouring elements in the series is negative, r₁ = −0.356 (Table I.). These results are also supported by H of 0.24 ± 0.011 given by the Fourier analyis. In none of the studied cases had the fractal model represented by Eq. 4 to be rejected because of a poor fit (see r in Table I).

4. DISCUSSION

These findings indicate that RBC-flux in the rat cerebro-cortical microvessels is a fractal anticorrelated process. The basis appears to be chaotic dynamical fluctuations in diameters of arteries in series along the flow path, with fractal interpeak intervals and amplitudes. The disparity in timings of resistance changes in nearby vessels can cause local “steal” whereby an increase in flow in one vessel occurs at the expense of a reduction in flow in a nearby vessel. Owing to the modular and hexagonal organization of the vasculature in the neocortex of the rat (Bär, 1980) the monitored hemispherical volume contains a few hundred perforating arterial units of different sizes. Each of these can exhibit vaso-and flowmotion of a given character. Their manifestation in the temporal structuring of the overall RBC flux time series, as our findings indicate, is not independent of each other. On a more local level, hanging up of red blood cells at vessel bifurcations can create a flip-flop effectively alternating flow in between competing pathways. Frequency components of our RBC-flux time series are higher than that of potential interfering sources (cardiac and pulmonary). The frequency components of our RBC-flux time series should reflect local rheology since frequencies of this magnitude would significantly attenuate over distances beyond the dimensions of the laser-Doppler measurement.