### Abstract

Unidirectional extraction of a substrate S in the capillaries following the arterial injection of a bolus containing S and a reference tracer R is assumed to follow first-order kinetics. If C_{R} and C_{S} denote normalized venous effluent concentrations of R and S, respectively, let L(t)=ln[C_{R}(t){plus 45 degree rule}C_{S}(t)]. We derive a formula which expresses the experimental L(t) data in terms of the mean μ(t) and variance of the transit times of those capillaries which are contributing indicators at each sample time t. We examine the information thus contained in the L data about capillary and noncapillary transit times under several kinematic assumptions. We show that if the capillary and noncapillary transit times are stochastically independent with frequency functions h_{c}(t) and h_{av}(t), respectively, then the shapes of the graphs of L(t) and μ(t) depend on the variances and skewnesses of h_{c}(t) and h_{av}(t). Specifically, let r_{2} be the ratio of the variance of h_{c}(t) to the variance of h_{av}(t), and let r_{3} be the ratio of skewnesses in the same order. Then the graph of μ(t) is concave downward if r_{2}{plus 45 degree rule}r_{3} > 1, concave upward if r_{2}{plus 45 degree rule}r_{3}< 1, and linear if r_{2}{plus 45 degree rule}r_{3} = 1. If the fraction of S extracted is not too large, L(t) has nearly the same shape as μ(t), and therefore, L(t) contains information about h_{c}(t) and h_{av}(t).

Original language | English (US) |
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Pages (from-to) | 199-225 |

Number of pages | 27 |

Journal | Mathematical Biosciences |

Volume | 83 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 1987 |

### ASJC Scopus subject areas

- Agricultural and Biological Sciences(all)
- Ecology, Evolution, Behavior and Systematics

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## Cite this

*Mathematical Biosciences*,

*83*(2), 199-225. https://doi.org/10.1016/0025-5564(87)90112-X