### Abstract

Resistance, r_{in}, follows a 3/4 allometric scaling law from a fractal architecture, but an explanation for biological variability is lacking. For an asymmetric fractal branching tree of Horsfield order N with degree of asymmetry, Δ = 0,1,2 ...; k orders of arteries with branching ratio, R_{b1}; N-k small arteries/capillaries with branching ratio R_{b2} ;bifurcation design a, where 1/a = 4/x - 1/λ, x is a scaling exponent in the diameter law d_{p}
^{x} = d_{1}
^{x} + d_{2}
^{x}, and λ in length l_{p}
^{λ} = l_{l}
^{λ} + l_{2}
^{λ}, r_{in} = r_{c}/R_{b2}
^{N}(R_{b2}
^{k}/R _{b1}
^{k/a}(1 - κ^{k}/1 - κ) + N - k) where κ = (Formula Presented) and 0 < κ < ∞. Unlike a symmetric tree, r_{in} is not bounded by κ ≤ 1, but varies continuously with a and is quantized according to Δ. From morphometric data and the slope of pressure-flow curves from dog, cat and human, this model demonstrates a conjugate mapping between structure and function, and a quantum/continuum variance in resistance influenced by individual differences in branching asymmetry and/or arterial design.

Original language | English (US) |
---|---|

Journal | FASEB Journal |

Volume | 12 |

Issue number | 5 |

State | Published - Mar 20 1998 |

### Fingerprint

### ASJC Scopus subject areas

- Agricultural and Biological Sciences (miscellaneous)
- Biochemistry, Genetics and Molecular Biology(all)
- Biochemistry
- Cell Biology

### Cite this

*FASEB Journal*,

*12*(5).

**Pulmonary vascular resistance variability/reactivity in terms of a quantized/continuum fractal network.** / Bennett, S. H.; Eldridge, M. W.; Milstein, Jay M; Goetzman, B. W.; Woldenberg, M. J.

Research output: Contribution to journal › Article

*FASEB Journal*, vol. 12, no. 5.

}

TY - JOUR

T1 - Pulmonary vascular resistance variability/reactivity in terms of a quantized/continuum fractal network

AU - Bennett, S. H.

AU - Eldridge, M. W.

AU - Milstein, Jay M

AU - Goetzman, B. W.

AU - Woldenberg, M. J.

PY - 1998/3/20

Y1 - 1998/3/20

N2 - Resistance, rin, follows a 3/4 allometric scaling law from a fractal architecture, but an explanation for biological variability is lacking. For an asymmetric fractal branching tree of Horsfield order N with degree of asymmetry, Δ = 0,1,2 ...; k orders of arteries with branching ratio, Rb1; N-k small arteries/capillaries with branching ratio Rb2 ;bifurcation design a, where 1/a = 4/x - 1/λ, x is a scaling exponent in the diameter law dp x = d1 x + d2 x, and λ in length lp λ = ll λ + l2 λ, rin = rc/Rb2 N(Rb2 k/R b1 k/a(1 - κk/1 - κ) + N - k) where κ = (Formula Presented) and 0 < κ < ∞. Unlike a symmetric tree, rin is not bounded by κ ≤ 1, but varies continuously with a and is quantized according to Δ. From morphometric data and the slope of pressure-flow curves from dog, cat and human, this model demonstrates a conjugate mapping between structure and function, and a quantum/continuum variance in resistance influenced by individual differences in branching asymmetry and/or arterial design.

AB - Resistance, rin, follows a 3/4 allometric scaling law from a fractal architecture, but an explanation for biological variability is lacking. For an asymmetric fractal branching tree of Horsfield order N with degree of asymmetry, Δ = 0,1,2 ...; k orders of arteries with branching ratio, Rb1; N-k small arteries/capillaries with branching ratio Rb2 ;bifurcation design a, where 1/a = 4/x - 1/λ, x is a scaling exponent in the diameter law dp x = d1 x + d2 x, and λ in length lp λ = ll λ + l2 λ, rin = rc/Rb2 N(Rb2 k/R b1 k/a(1 - κk/1 - κ) + N - k) where κ = (Formula Presented) and 0 < κ < ∞. Unlike a symmetric tree, rin is not bounded by κ ≤ 1, but varies continuously with a and is quantized according to Δ. From morphometric data and the slope of pressure-flow curves from dog, cat and human, this model demonstrates a conjugate mapping between structure and function, and a quantum/continuum variance in resistance influenced by individual differences in branching asymmetry and/or arterial design.

UR - http://www.scopus.com/inward/record.url?scp=33749350171&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33749350171&partnerID=8YFLogxK

M3 - Article

VL - 12

JO - FASEB Journal

JF - FASEB Journal

SN - 0892-6638

IS - 5

ER -