Optimal branching structure of the vascular tree

Kamiya, Akira; Togawa, Tatsuo

doi:10.1007/bf02476705

Cited by 159 publications

(97 citation statements)

References 3 publications

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“…At each bifurcation and at each time, the law of matter preservation is veriÿed: the same blood quantity enters the bifurcation by the father vessel and leaves it by the two sons vessels. Another relation, extracted from experimental data, is used to relate the radius of the father vessel and the radii of its sons [15,16].…”

Section: Vesselsmentioning

confidence: 99%

Hepatic tumor enhancement in computed tomography: combined models of liver perfusion and dynamic imaging

Bézy-Wendling

Krȩtowski

Rolland

2003

Computers in Biology and Medicine

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The objective of this study is to show how computational modeling can be used to increase our understanding of liver enhancement in dynamic computer tomography. It relies on two models: (1) a vascular model, based on physiological rules, is used to generate the 3D hepatic vascular network; (2) the physical process of CT acquisition allows to synthesize timed-stamped series of images, aimed at tracking the propagation of a contrast material through the vessel network and the parenchyma. The coupled models are used to simulate the enhancement of a hyper-vascular tumor at di erent acquisition times, showing a maximum conspicuity during the arterial phase. ?

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Section: Vesselsmentioning

confidence: 99%

Hepatic tumor enhancement in computed tomography: combined models of liver perfusion and dynamic imaging

Bézy-Wendling

Krȩtowski

Rolland

2003

Computers in Biology and Medicine

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“…The Murray's law has been used for the analysis of blood vessel, tracheas, trees, and the design of circuit and fluid systems. [12][13][14] However, the formulae above have some limitations. It works only for circular tubes.…”

Section: A Biomimetic Design Of Microchannelsmentioning

confidence: 99%

“…It obeys the rule of the minimum energy loss in order to absorb more nutrition for photosynthesis, which has been widely applied in many areas including micro-electronics, space flight, and medicine and so on in last century. [10][11][12][13][14][15][16][17] For the microfluidic planar reactors, the minimum energy loss is also important. This is to minimize the effective power that is used to pump the fluids through the reactors.…”

Section: Introductionmentioning

confidence: 99%

Biomimetic microchannels of planar reactors for optimized photocatalytic efficiency of water purification

Liao

Wang

et al. 2016

Biomicrofluidics

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This paper reports a biomimetic design of microchannels in the planar reactors with the aim to optimize the photocatalytic efficiency of water purification. Inspired from biology, a bifurcated microchannel has been designed based on the Murray's law to connect to the reaction chamber for photocatalytic reaction. The microchannels are designed to have a constant depth of 50 lm but variable aspect ratios ranging from 0.015 to 0.125. To prove its effectiveness for photocatalytic water purification, the biomimetic planar reactors have been tested and compared with the non-biomimetic ones, showing an improvement of the degradation efficiency by 68%. By employing the finite element method, the flow process of the designed microchannel reactors has been simulated and analyzed. It is found that the biomimetic design owns a larger flow velocity fluctuation than that of the nonbiomimetic one, which in turn results in a faster photocatalytic reaction speed. Such a biomimetic design paves the way for the design of more efficient planar reactors and may also find applications in other microfluidic systems that involve the use of microchannels. V C 2016 AIP Publishing LLC.

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“…In the following a short review is given based on the article by Kamiya and Togawa (1973). As for the branching of blood vessels, Thompson (1917) gave some qualitative rules as follows, which are seen in most blood vessels:…”

Section: Branching Rules Of Blood Vesselsmentioning

confidence: 99%

Branching Structures in Nature and Human Societies

Takaki¹

2016

Forma

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expressions of human relations and communication networks. Some of these examples appear in this review article.As a mathematical framework developed since many years we have the method developed by a geologist Horton (1945), who found so-called Horton's law in the tree-type river branching structures, which is introduced briefly in Sec. 2. This method has a great advantage in a sense that the geometrical properties of tree-type structures can be expressed in terms a single parameter. On the other hand, the network-type branching systems have been often treated successfully by scientists from various fields, but they were not based on a simple method similar to that of Horton. The present author has once proposed a method to treat network-type system for leaf veins and road systems, which was included in a monograph by the present author (Takaki, 1978) and not published as a scientific paper. It is introduced in Sec. 4.Of course, the framework of analysis of branching systems is not limited to the Horton's method, and some remarkable examples are introduced in the following sections. In particular, an application of the topology (one of mathematical fields) is made by a pathological scientist Shimizu (1992) for analysis of 3-dimensional (3D) network-type structures of blood vessels in human liver along with a topological concept called "Betti number".Here, it is expected that introductions of various method and concepts would give a larger scope of branching systems, which would stimulate further development of studies in future. Horton's Law for River Structures and Its Derivations Horton's lawA river made of branching streams has a shape belonging to the category of tree-type. The tree structure of riv- Branching Structures in Nature and Human Societies Ryuji TakakiTokyo University of Agriculture and Technology (emeritus professor), 2-23-12 Yuigahama, Kamakura, Kanagawa 248-0014, Japan E-mail address: jr.takaki@iris.ocn.ne.jp (Received July 19, 2015; Accepted March 25, 2016) Several examples of branching structures in the nature, social structures and the human body are introduced, and results of their analyses are given. It is shown that the Horton's law, which is confirmed for river branching structures, is satisfied also in variety of branching structures in the nature and human societies. It is suggested that these structures are constructed owing to mechanisms similar to that for river structure. As for the branching structures in human body some trials are introduced to construct them numerically by the use of mathematical models. The object of this review article is to show that the analyses of branching forms are interesting topics as the science of forms.

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Optimal branching structure of the vascular tree

Cited by 159 publications

References 3 publications

Hepatic tumor enhancement in computed tomography: combined models of liver perfusion and dynamic imaging

Hepatic tumor enhancement in computed tomography: combined models of liver perfusion and dynamic imaging

Biomimetic microchannels of planar reactors for optimized photocatalytic efficiency of water purification

Branching Structures in Nature and Human Societies

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