Model combining hydrodynamics and fractal theory for analysis of in vivo peripheral pulmonary and systemic resistance of shunt cardiac defects

Self-Similar Functional Circuit Models of Arteries and Deterministic Fractal Operators: Theoretical Revelation for Biomimetic Materials

In this paper, the self-similar functional circuit models of arteries are proposed for bioinspired hemodynamic materials design. Based on the mechanical-electrical analogous method, the circuit model can be utilized to mimic the blood flow of arteries. The theoretical mechanism to quantitatively simulate realistic blood flow is developed by establishing a fractal circuit network with an infinite number of electrical components. We have found that the fractal admittance operator obtained from the minimum repeating unit of the fractal circuit can simply and directly determine the blood-flow regulation mechanism. Furthermore, according to the operator algebra, the fractal admittance operator on the aorta can be represented by Gaussian-type convolution kernel function. Similarly, the arteriolar operator can be described by Bessel-type function. Moreover, by the self-similar assembly pattern of the proposed model, biomimetic materials which contain self-similar circuits can be designed to mimic physiological or pathological states of blood flow. Studies show that the self-similar functional circuit model can efficiently describe the blood flow and provide an available and convenient structural theoretical revelation for the preparation of in vitro hemodynamic bionic materials.

Radius exponent in elastic and rigid arterial models optimized by the least energy principle

It was analyzed in normal physiological arteries whether the least energy principle would suffice to account for the radius exponent x. The mammalian arterial system was modeled as two types, the elastic or the rigid, to which Bernoulli's and Hagen-Poiseuille's equations were applied, respectively. We minimized the total energy function E, which was defined as the sum of kinetic, pressure, metabolic and thermal energies, and loss of each per unit time in a single artery transporting viscous incompressible blood. Assuming a scaling exponent α between the vessel radius (r) and length (l) to be 1.0, x resulted in 2.33 in the elastic model. The rigid model provided a continuously changing x from 2.33 to 3.0, which corresponded to Uylings' and Murray's theories, respectively, through a function combining Reynolds number with a proportional coefficient of the l - r relationship. These results were expanded to an asymmetric arterial fractal tree with the blood flow preservation rule. While x in the optimal elastic model accounted for around 2.3 in proximal systemic (r >1 mm) and whole pulmonary arteries (r ≥0.004 mm), optimal x in the rigid model explained 2.7 in elastic-muscular (0.1 < r ≤1 mm) and 3.0 in peripheral resistive systemic arteries (0.004 ≤ r ≤0.1 mm), in agreement with data obtained from angiographic, cast-morphometric, and in vivo experimental studies in the literature. The least energy principle on the total energy basis provides an alternate concept of optimality relating to mammalian arterial fractal dimensions under α = 1.0.

Quantifying left ventricular trabeculae function - application of image-based fractal analysis

The ventricular-blood interface is geometrically complex due to the presence of ventricular trabeculae carneae (VTC). We introduce a new image-based framework to quantify VTC function using high-resolution computed tomography (CT) imaging and offer new insights into the active role of VTCs during ejection. High-resolution Cine CT scans of a patient with normal cardiac function were acquired at a resolution of 0.77 mm per pixel at 10 phases of the cardiac cycle. The images were segmented and the VTC surface was obtained by triangulating the segmented data. Fractal dimension of the VTC surface was calculated for each cardiac phase as a function of scale size using the box-counting algorithm. The fractal dimension, D corresponding to VTCs ranged between 2.05 and 2.2 and varied as a function of time during the cardiac cycle. Fractal dimension is highest at diastole and lowest at peak systole with the change being significantly different (P < 0.003). This variation of D when plotted against stroke volume (i.e., D-V loop) revealed an active VTC role due to hysteresis in the loop. Physically the hysteresis in the D-V loop indicates a new mechanical function of VTCs as structures that provide mechanical leverage during early systolic ejection through contraction. VTC relaxation is noted to occur during late diastole at larger ventricular volume. D-V loop of VTCs quantifies VTC function. A new dynamic physical role of VTCs is suggested by way of mechanical leverage, as opposed to the traditionally accepted passive role.