Optimality, Cost Minimization and the Design of Arterial Networks

Theory of sleep/wake cycles affecting brain elastography

As elastography of the brain finds increasing clinical applications, fundamental questions remain about baseline viscoelastic properties of the brainin vivo. Furthermore, the underlying mechanisms of how and why elastographic measures can change over time are still not well understood. To study these issues, reverberant shear wave elastography using an optical coherence tomography scanner is implemented on a mouse model, both under awake conditions and in a sleep state where there are known changes in the glymphatic fluid flow system in the brain. We find that shear wave speed, a measure of stiffness, changes by approximately 12% between the two states, sleep versus awake, in the entire cortical brain imaging volume. Our microchannel flow model of biphasic (fluid plus solid) tissue provides a plausible rheological model based on the fractal branching vascular and perivascular system, plus a second parallel system representing the finer scale glymphatic fluid microchannels. By adjusting the glymphatic system fluid volume proportional to the known sleep/wake changes, we are able to approximately predict the measured shear wave speeds and their change with the state of the glymphatic system. The advantages of this model are that its main parameters are derived from anatomical measures and are linked to other major derivations of branching fluid structures including Murray's Law. The implications for clinical studies are that elastography of the brain is strongly influenced by the regulation or dysregulation of the vascular, perivascular, and glymphatic systems.

Image-based morphometric studies of human coronary artery bifurcations with/without coronary artery disease

It is of great clinical significance to study the relationship between coronary bifurcation's morphometrical feature change and coronary artery disease (CAD) lesion. The purpose of this study is to determine the morphological changes in patients with CAD lesion when compared with non-CAD subjects and to find indicators that may be used for cardiovascular disease diagnosis. Computed tomography angiography images from Southern Chinese populations were used to reconstruct three-dimensional coronary arterial trees. Murray's law was introduced to assess the level of deviation of the realistic vascular networks from their optimal state. The results showed CAD Left had the highest deviation values of ARR (0.2552 ± 0.0071 ) and DERR (0.5059 ± 0.0098 ), while non-CAD Right had the lowest values (ARR : 0.1892 ± 0.0066 and DERR : 0.3733 ± 0.0092 , respectively). Moreover, the slope values of the ratio between D m 3 and D s 3 + D l 3 for non-CAD Left, CAD Left, non-CAD Right, and CAD Right were 0.7428, 0.7004, 0.7628, and 0.7577, respectively. Theoretically, the slope value should equal to 1 when the bifurcation structure is in its optimal state. Therefore, these results indicated that coronary bifurcations with CAD lesion deviated from the optimal structure further than those without CAD lesion and coronary bifurcations in right were closer to the optimal structure than those in left. More importantly, the present study found that DERR and AER depended only on the diseased state, but not age, suggesting that DERR and AER were potentially used as two novel indicators for early CAD diagnosis.

Fractal analysis of concurrently prepared latex rubber casts of the bronchial and vascular systems of the human lung

Fractal geometry (FG) is a branch of mathematics that instructively characterizes structural complexity. Branched structures are ubiquitous in both the physical and the biological realms. Fractility has therefore been termed nature's design. The fractal properties of the bronchial (airway) system, the pulmonary artery and the pulmonary vein of the human lung generates large respiratory surface area that is crammed in the lung. Also, it permits the inhaled air to intimately approximate the pulmonary capillary blood across a very thin blood-gas barrier through which gas exchange to occur by diffusion. Here, the bronchial (airway) and vascular systems were simultaneously cast with latex rubber. After corrosion, the bronchial and vascular system casts were physically separated and cleared to expose the branches. The morphogenetic (Weibel's) ordering method was used to categorize the branches on which the diameters and the lengths, as well as the angles of bifurcation, were measured. The fractal dimensions (DF) were determined by plotting the total branch measurements against the mean branch diameters on double logarithmic coordinates (axes). The diameter-determined DF values were 2.714 for the bronchial system, 2.882 for the pulmonary artery and 2.334 for the pulmonary vein while the respective values from lengths were 3.098, 3.916 and 4.041. The diameters yielded DF values that were consistent with the properties of fractal structures (i.e. self-similarity and space-filling). The data obtained here compellingly suggest that the design of the bronchial system, the pulmonary artery and the pulmonary vein of the human lung functionally comply with the Hess-Murray law or 'the principle of minimum work'.

Fundamental constraints of vessels network architecture properties revealed by reconstruction of a rat brain vasculature

The studies of mammalian vasculature are an essential part of biomedical research, enabling the development of physiological understanding and forming the background of medical techniques and therapy. Despite the fact that the basic principles of vessel network description were established in the first quarter of the twentieth century, a digital model describing the vasculature in full accordance with experimental data has not yet been created. In the present study, we combine the determined structure design of basic arterial vessels with the stochastic creation of small vessel networks. By the example of rat brain arterial network model it was shown that the arterial blood volume and the magnitude of the blood flow impose a limitation on the network architecture. In particular, the bifurcation exponent (γ) should not be less than 2.7, and the optimal value of this parameter lies in the range of 2.9-3.0. Although the networks with a low γ appear as branched and complex, they do not fill out the phantom properly. Thus, the architecture of the vasculature is fundamentally determined by topological geometrical parameters.

Geometries of vasculature bifurcation can affect the level of trophic damage during formation of a brain ischemic lesion

Ischemic lesion is a common cause of various diseases in humans. Brain tissue is especially sensitive to this type of damage. A common reason for the appearance of an ischemic area is a stop in blood flow in some branch of the vasculature system. Then, a decreasing concentration gradient results in a low mean level of oxygen in surrounding tissues. After that, the biochemical ischemic cascade spreads. In this review, we examine these well-known events from a new angle. It is stressed that there is essential evidence to predict the formation of an ischemic micro-area at the base of vascular bifurcation geometries. Potential applications to improve neuroprotection are also discussed.

The fractal heart - embracing mathematics in the cardiology clinic

For clinicians grappling with quantifying the complex spatial and temporal patterns of cardiac structure and function (such as myocardial trabeculae, coronary microvascular anatomy, tissue perfusion, myocyte histology, electrical conduction, heart rate, and blood-pressure variability), fractal analysis is a powerful, but still underused, mathematical tool. In this Perspectives article, we explain some fundamental principles of fractal geometry and place it in a familiar medical setting. We summarize studies in the cardiovascular sciences in which fractal methods have successfully been used to investigate disease mechanisms, and suggest potential future clinical roles in cardiac imaging and time series measurements. We believe that clinical researchers can deploy innovative fractal solutions to common cardiac problems that might ultimately translate into advancements for patient care.