CHAPTER 19 



Mathematical treatment of uptake and release 

 of indicator substances in relation to flow 

 analysis in tissues and organs' 



JAMES S. ROBERTSON 



Medical Research Center, Brookhaven Xalional Laboratory, 

 Upton, Long Island, New York 



CHAPTER CONTENTS 



Relationship Between Exchange Rates and Blood Flow Rates 

 Purpose, Scope, and Limitations of This Chapter 

 Circulatory Features Affecting Mixing and Distribution 

 Anatomical 

 Dynamic 



Intravascular mixing 



Extravascular distribution 

 The Tracer Concept 

 History 

 Terminology 

 Basic Assumptions 



Kinetic identity of label and carrier 



Absence of disturbing effects 



Homogeneity within compartments 

 Mathematical Basis 



Synopsis of Mathematical Techniques 



Linear differential equations 



Matrices and determinants 



The Laplace transform 

 Notation 



The One-Compartment Open System and Two-Compart- 

 ment Closed Systems 

 Three-Compartment and Multicompartment Systems 



Equations 



Calculation of rate constants 



Calculation of exponential constants 



Calculation of coefficients 

 Electronic Computer Application 



Analog computer methods 



Digital computers 



' Research supported by the \J. S. Atomic Energy Com- 

 mission. 



Role of Radioisotopic Tracer in Determining Regional Flow 

 Rates 



Cerebral Flow 



Myocardial Flow 



Hepatic Flow 



Flow Rates Measured by Tissue Uptake of K^ 



Regional Flow Rates by Tissue Clearance 

 Summary 



THE GENERAL PRINCIPLES of the use of indicators, or 

 tracers, have long been familiar to circulation physi- 

 ologists. Measurement of blood volume, cardiac 

 output, and regional blood flow rates, and the diag- 

 nosis of certain circulatory defects represent some 

 of the major areas of application which have been 

 developed with the use of dyes, gases, and other 

 materials. In the preceding chapter, Zierler (74) dis- 

 cusses in detail the theory and historical development 

 of the indicator dilution methods for determining 

 blood flow. Several other methods such as mechanical 

 and thermal stromuhrs are also available for measur- 

 ing flow rates. 



At least in part, because of the multiplicity of 

 methods applicable to blood flow studies and to the 

 successful uses of the older methods in circulatory 

 studies, the relatively recent availability of isotopic 

 tracers has had less, or at least less obvious, impact 

 on circulatory physiology than it has had in some 

 other branches of physiology, where the concept of 

 the dynamic state of body constituents and numerous 



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