Blood Vessel Branching: Beyond the Standard Calculus Problem

@article{Adam2011BloodVB,
  title={Blood Vessel Branching: Beyond the Standard Calculus Problem},
  author={John A. Adam},
  journal={Mathematics Magazine},
  year={2011},
  volume={84},
  pages={196 - 207}
}
  • J. Adam
  • Published 1 June 2011
  • Mathematics
  • Mathematics Magazine
Summary Calculating the optimal angle for blood vessel branching is a standard calculus problem. However, optimality in that setting is judged by a cost functional that turns out not to give realistic results. We study a sequence of improvements to the cost functional, finally arriving at one that passes an important modeling test: From this last functional, we derive three empirical laws of blood vessel branching, originally proposed by German zoologist Wilhelm Roux. 
Do Vascular Networks Branch Optimally or Randomly across Spatial Scales?
TLDR
It is found that material-cost optimizations are the strongest predictor of vascular branching in the human head and torso, whereas locally or intermediately constrained random branching is comparable to material- cost optimizations for the mouse lung.
Branch order regression for modeling brain vasculature
TLDR
A novel parametrization preserves an important aspect of tree structure, namely its branch order, and is amenable to standard methods of analysis, like generalized linear/additive models.
Fluid Flow at Branching Junctions
The flow of fluids at branching junctions plays important kinematic and dynamic roles in most biological and industrial flow systems. The present paper highlights some key issues related to the flow
Predicting bifurcation angle effect on blood flow in the microvasculature.
TLDR
This work suggests a new paradigm in microvascular blood flow properties, that vessel bifurcation itself, regardless of its angle, holds considerable influence on blood viscosity, and this phenomenon will help to develop new predictive tools inmicrovascular research.
Computational feasibility of simulating changes in blood flow through whole-organ vascular networks from radiation injury.
TLDR
It is demonstrated, for the first time, that it is computationally feasible to calculate radiation dose deposition in whole-organ vascular networks and the resulting change in blood flow.
Solute dispersion in bifurcating networks
Abstract Advective–diffusive transport of passive scalars in confined environments (e.g. vessels and channels) within a network is of fundamental importance in a plethora of biological and
Computational feasibility of simulating whole-organ vascular networks.
TLDR
It is demonstrated, for the first time, that it is feasible to computationally model the vasculature of the whole human brain using a fractal-based algorithm.
Computational feasibility of calculating the steady-state blood flow rate through the vasculature of the entire human body.
TLDR
This study designed and implemented a two-step algorithm to calculate the blood flow rates through a vasculature equal in size to that of the human body using principles of steady-state fluid dynamics.
A theoretical framework for determining cerebral vascular function and heterogeneity from dynamic susceptibility contrast MRI
  • I. Digernes, A. Bjørnerud, +5 authors K. Emblem
  • Physics, Medicine
    Journal of cerebral blood flow and metabolism : official journal of the International Society of Cerebral Blood Flow and Metabolism
  • 2017
TLDR
A theoretical framework for determining cerebral vascular function and heterogeneity from dynamic susceptibility contrast magnetic resonance imaging (MRI), which covers realistic structural architectures for vessel branching and orientations, as well as a range of hemodynamic scenarios for blood flow, capillary transit times and oxygenation is presented.
Analysis of Coronary Contrast Agent Transport in Bolus-Based Quantitative Myocardial Perfusion MRI Measurements with Computational Fluid Dynamics Simulations
TLDR
The analysis of contrast agent dispersion in bolus-based quantitative myocardial perfusion MRI measurements finds multi-faceted influences on bolus shape from length of traversed vessels to bifurcation angles and vessel curvature.
...
1
2
...

References

SHOWING 1-10 OF 14 REFERENCES
Segment analysis of human coronary arteries.
TLDR
The new concept of vessel segment and the method of analysis are proposed as a means for an accurate description of the branching characteristics of the coronary network, comparing the network with others in the cardiovascular system and comparing the vasculature in the normal versus that in the diseased heart.
Mathematics in Nature : Modeling Patterns in the Natural World
And while I have sought to shew the naturalist how a few mathematical concepts and dynamical principles may help and guide him, I have tried to shew the mathematician a field for his labour—a field
Introduction to the Calculus of Variations
This is a self-contained paper which introduces a fundamental problem in the calculus of variations, the problem of finding extreme values of functionals. The reader should have a solid background in
Calculus, Early Transcendentals
Chapter 0 Before Calculus 0.1 Functions 0.2 New Functions from Old 0.4 Families of Functions 0.5 Inverse Functions Inverse Trigonometric Functions 0.6 Exponential and Logarithmic Functions Chapter 1
On Growth and Form
TLDR
This book is an application of some of the concepts of physical science and sundry mathematical methods to the study of organic form and is like one of Darwin's books, well-considered, patiently wrought-out, learned, and cautious.
Guesstimation: Solving the World's Problems on the Back of a Cocktail Napkin
Guesstimation is a book that unlocks the power of approximation--it's popular mathematics rounded to the nearest power of ten! The ability to estimate is an important skill in daily life. More and
Mathematical Biophysics, Vol
  • 2, Chapter XXVII, Dover, New York,
  • 1960
The Branching Structure of Arterial Trees
  • Comments on Theoretical Biology
  • 1988
Modern Fluid Dynamics, Vol
  • 1, Van Nostrand Reinhold, Wokingham, England,
  • 1968
Modern fluid dynamics
...
1
2
...