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Despite the fact that the concept of sheet-flow in the pulmonary microcirculation of mammals was introduced more than three decades ago, the capillary circulatory model still prevails in the physiological literature. Since cardiac output is identical in the systemic and in pulmonary circulations, it is noteworthy that in the former, the resulting arterial pressure is five times higher than that of the latter, which means that the corresponding microcirculations must be radically different. The present study addresses this problem from both morphological and physiological perspectives.

A wide spectrum of cyclic functions in terrestrial mammals of different size, from the 3-gram shrew to the 3-ton elephant, yields an allometric exponent around 0.25, which is correlated--as a kind of common denominator--with the specific metabolic rate. Furthermore, the applicability of these empirical findings could be extrapolated to chronological events in the sub-cellular realm. On the other hand, the succession of growth periods (T98%) until sexual maturity is reached also follows the 1/4 power rule. By means of Verhulst's logistic equation, it has been possible to simulate three different biological conditions, which means that by modifying the numerical value of only one parameter, revertible physiological and pathological states can be obtained, as for instance isostasis, homeostasis and heterostasis.

In contrast with classical physics, particularly with Sir Isaac Newton, where time is a continuous function, generally valid, eternally and evenly flowing as an absolute time dimension, in the biological sciences, time is in essence of cyclical nature (physiological periodicities), where future passes to past through an infinitely thin boundary, the present. In addition, the duration of the present (DP) leads to the so-called 'granulation of time' in living beings, so that by the fusion of two successive pictures of the world, which are not entirely similar, they attain the perception of 'movement,' both in the real world as well as in the sham-movement in the mass media (TV).

The purpose of this study was to characterize time-frequency behavior using the Continuous Wavelet Transform (CWT) and Fast Fourier Transform (FFT) to analyze ventricular and arterial pressure signals from anesthetized mongrel dogs. Both ventricular and arterial pressure pulsations were recorded using catheter-tip manometers and the CWT was applied to these signals to obtain module coefficients, associated contours, and the 3-D representation of these modules. FFT was applied to obtain the Fourier spectrum. The mathematical analysis of the cardiovascular pressure pulsations permitted the identification of the evolution of the frequency components for the aortic and pulmonary valve functions as well as the intra-ventricular and respiratory influences on the cardiovascular dynamics. The CWT is a very sensitive and reliable procedure for determining the three-dimensional (time-frequency-amplitude) of the oscillatory phenomena during each cardiac cycle, providing more, although complementary, information than the spectral analysis obtained with the FFT. Thanks to the FFT, exact values in Hz could be found for the different events produced in each cycle, and thus the information provided by CWT could be related to the information provided by FFT. The combination of both mathematical methodologies permitted identification of each component of the analyzed signals. The 3D representation allowed an easy comparison of the relative importance of the complex magnitudes in frequency for the different components of the pulsatile waves.

The applicability of dimensional analysis (DA) is discussed in relation to the metabolic scaling laws. The evolution of different theories of biological similarity has shown that the calculated reduced exponents (b) of Huxley's allometric equation are closely correlated with the numerical values obtained from the statistical analysis of empirical data. Body mass and body weight are not equivalent as biological reference systems, since in accordance to Newton's second law, the former has a dimension of a mass, while the latter should be dimensionally considered as a force (W = MLT-2). This distinction affects the coefficients of the mass exponent (alpha). This difference is of paramount importance in microgravity conditions (spaceflight) and of buoyancy during the fetal life in mammals. Furthermore, the coefficients (beta) of the length dimension, and (gamma) of the time dimension do not vary when mass or weight are utilized as reference systems. Consequently, the "specific metabolic time," that results from the ratio of basal oxygen consumption and body mass or body weight yields the "biological meaning" of the time dimension, which is of fractal nature.

In the present paper we have examined the applicability of dimensionless and invariant numbers (DN & IN) to the analysis of the cardiovascular system of mammals, whose functions were measured at standard metabolic conditions. The calculated IN did not change when we compared these figures with those obtained in dogs while they were submitted to graded exercise on a treadmill. In both instances, rest and exercise, the constancy of the IN prevailed, in accordance with Cannon's principle of "homeostasis" (1929). On the contrary, when dogs were examined during a standardized hypovolemic shock, we observed a breakdown of the IN, and the resulting DN evolved as a reliable index of the condition of "heterostasis" as defined by H. Selye. The robustness of the homeostatic regulations is based on high-gain integral feedback mechanisms, while "heterostasis" could be associated with low-gain integral feedback processes, when organisms are submitted to unitary step disturbances or to changes of the set-point at the entrance of the feedback loop.

The present allometric study deals with the duration of three electrocardiographic intervals (PQ, QRS, QT) and their relationships with the corresponding cardiac cycle length (R-R interval) in mammals across a wide body mass range. The numerical values of the different ECG intervals were obtained from Grauwiler's (1965) monograph on the subject. Because the corresponding body masses were not given by this author, Heusner's (1991) data on basal metabolic rate as function of body mass were used to establish the most likely body mass figure for each case, based on the taxonomic identity between the corresponding specimens. On the other hand, in a recent study we established the "duality" of physiological times (Günther & Morgado, 1996) and, therefore, we adopted this novel approach to investigate the ECG intervals and their relationships with the R-R interval (heart rate reciprocal). Considering that the anatomy and physiology of auricles and ventricles are different (spheroids versus quasi-cylinders), and that excitation (sino-atrial node and His-Purkinje's system) and contraction processes can be described either by Euclidean or fractal geometries, only a quantitative analysis of the different ECG waves could resolve the dilemma. From the present preliminary study we can conclude that fractal geometry is prevalent with regard to ECG intervals.

The aim of the present study was to differentiate two modalities of intrinsic time scales: i- the geometric or Euclidean modality, which is based on the constant speed of mass transport or of wave transmission in cylindrical structures (arteries, veins, nerves), whose allometric exponent (TE = aMb) is b = 0.33, where M is body mass (kg) and a the mass coefficient; ii- the fractal time scale (TF), which is characteristic of organs with self-similar branching structures and with volume-specific flows, whose allometric exponent is b = 0.25. The proposed dichotomy could be confirmed by means of the statistical analysis of empirical allometric exponents (b). Our findings demonstrate the need to separate the chronology of bulk transport at long distances (inter-organic) which follows an Euclidean geometry (cylinders), from the fractal time scale, which operates at short distances (intra-organic) and is represented by a self-similar branching system which determines both the morphometric and physiometric characteristics within each organ.

The aim of the present study is to emphasize the applicability and versatility of the allometric equation in the biological sciences. This equation (Y = a x Mb) was introduced by Huxley (1932) for intra- and interspecific comparisons of morphological, physiological and ecological variables (Y), when they are expressed as functions of body mass (M). The regression analysis of the experimental data, plotted in a double logarithmic scale, yields a straight line, which is equivalent to the logarithmic form of the above mentioned allometric equation [log Y = log(a) + (b) x log(M)]. Only the exponent (b) can be calculated a priori for a given function, based firstly on the corresponding dimensional analysis in accordance with the MLT-system of physics, and secondly on one of the theories of biological similarity, while parameter (a) is of empirical nature. A relevant feature of the allometric equations is that they can be treated algebraically to obtain allometric ratios, mass independent numbers (MIN), and even dimensionless numbers (M0L0T0), which are valid for all organisms pertaining to the same taxonomic classification.

In homeotherms, the standardized (basal) metabolic rate should not be expressed per kilogram of body weight (specific metabolic rate), nor per unit of body surface (square meters of body-ambient interface), since both mitochondrial thermogenesis and heat-loss mechanisms (radiation, conduction, convection, evaporation) are not uniform processes. On the contrary, each organism is an heterogeneous bioreactor, which is composed at least of two compartments: 1) a metabolically active volume (aV), where oxidative phosphorylation takes place; and 2) a metabolically inactive volume (iV), where oxygen consumption is negligible. The ratio (aV/iV) is not invariant, since iV increases disproportionately with the scaling up of body size, and as shown by us, when the three main components of iV, i.e., skeleton, fat deposits, and blood volume, are added together, a similar disproportionality is found. The aV was determined by subtracting the iV from the total volume (V) of an organism, or by estimating the volume occupied by all mitochondria, or mitochondrial volume (mtV). For this purpose two procedures are discussed: 1) the stereological or morphometric method; and 2) the oxygen consumption per unit time or physiometric method. The latter procedure is based on the equivalence between an VO2 = 3 ml O2.min-1 and a mtV of 1 ml, whose oxidative phosphorylation yields an approximate power output of 1 watt. The correspondence between oxygen consumption, heat production, and electron flux at the respiratory chain of the mitochondrial cristae, is discussed. From a physical point of view, the metabolic rate is a "power" function (P = M L2T-3), where M = mass, L = length, and T = time. The dimensional analysis and the statistical treatment of the corresponding numerical values of more than 200 allometric equations yields the 3/4 power, law established by Kleiber (1961), for the relationship between basal metabolism and body weight. Instead of expressing the metabolic rate per unit body weight (kg-1) or per unit body surface (m-2) structural and functional criteria should be taken into account as, for instance, the distinction between iV and aV, and particularly by emphasizing the paramount importance of the mtV where oxidative phosphorylation takes place. An allometric equation relating mtV and body weight (W) could be tentatively established for interspecies comparisons.

Twelve biological variables were submitted to dimensional analysis in accordance with the MLT-system of physics (M, mass; L, length; T, time). Each of these variables has a characteristic numerical value for the exponents alpha for mass, beta for length, and gamma for time. By means of Newton's reduction coefficient (chi), the three dimensions (MLT) can be expressed as power functions of body mass (Mb); the exponent (b) is the result of the combination of the three dimensional exponents (alpha, beta, gamma). By linear regression analysis of 203 allometric exponents (betaE) obtained from the literature, the following equation was found for the regression exponent (bR) (equation: see text). The estimated numerical coefficients (ki) for the three exponents (alpha, beta, gamma) of the basic dimensions (MLT) do not agree with those of the prevailing theories of biological similarity.