238.

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Page 88
1.5.8—
Example Where the Variance Does Not Exist:
Blood Flow in the Heart
Bassingthwaighte et al. measured blood flow in the heart by adding radioactive tracers to the blood. These tracers leak out of the blood vessels and become trapped in the muscle of the heart. The amount of tracer that leaks out is proportional to the blood flow in that region. Thus they determined the blood flow by measuring the amount of radioactivity in different pieces of the heart.
They determined the relative dispersion, RD, which is equal to the standard deviation divided by the average of the blood flow measured in a piece of weight w. They measured the blood flow in pieces of different weight. The weight of the pieces set the spatial resolution of the measurement. They found that the relative dispersion had the power law scaling relationship that RD was proportional to w1-d, where d is the fractal dimension.
Thus blood does not flow evenly throughout the heart. The flow of blood through the heart is fractal. There are regions of higher than average blood flow and regions of lower than average blood flow. The pattern of blood flow is self-similar. There are ever smaller regions of higher and lower than average blood flow. The dimension d is a measure of the correlation between regions with different amounts of blood flow.
They showed that if the large blood vessels connect to slightly unequal smaller branches, then the relative dispersion would increase after each branching. This would explain why the relative dispersion increases as it is measured from pieces of smaller weight that correspond to the finer spatial resolution.

 
[Cover] [Abbreviated Contents] [Contents] [Index]


Fractals and Chaos Simplified for the Life Sciences
Fractals and Chaos Simplified for the Life Sciences
ISBN: 0195120248
EAN: 2147483647
Year: 2005
Pages: 261

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