Abstract
This paper is the second of a pair dealing with some mathematical properties of metabolic steady state. An investigator wishing to compute the rate of appearance and/or disappearance of a metabolite in steady state within an intact biological system will usually appeal to a method involving radioactive tracers. It is shown that while the investigator’s choice of the mode of tracer administration (constant infusion or single injection) is largely arbitrary, the mathematical interpretation of the results may depend upon the presence or absence of gradients in certain of the variables of the system. The latter will be the case if the system is sampled at a point within the distribution space of the metabolite which is not a source point but is otherwise arbitrary. In order to deduce a formula which gives the required rate, he must have knowledge of the gradient of concentration of the traced substance, and sometimes of the gradient of specific activity.
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Abbreviations
- \(a(\overrightarrow r ,t)\) :
-
specific activity; effectively the ratio of tracer to tracee concentrations
- \(C(\overrightarrow r ,t)\) :
-
concentration of tracee
- \(C*(\overrightarrow r ,t)\) :
-
concentration of tracer
- M *0 :
-
mass of tracer injected
- M *(t):
-
mass of tracer present in the system
- \(\overrightarrow r\) :
-
position vector
- R a (t):
-
total rate of appearance of tracee within the system [mass·time−1]
- \(R_{d,\upsilon } (\overrightarrow r ,t)\) :
-
rate of disappearance of tracee per unit volume [mass·vol−1·time−1]
- R d (t):
-
total rate of disappearance of tracee within the system [mass·time−1]
- t :
-
time
- \(\overrightarrow u\) :
-
velocity of a tracer particle
- \(\overrightarrow \upsilon\) :
-
velocity of a fluid particle
- V :
-
volume of distribution
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Norwich, K.H., Hetenyi, G. Basic studies on metabolic steady state. Incompletely mixed systems. Bulletin of Mathematical Biophysics 33, 403–412 (1971). https://doi.org/10.1007/BF02476782
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DOI: https://doi.org/10.1007/BF02476782