LIBRARY

Michigan Sula
University '

 

This is to certify that the

thesis entitled

MATHEMATICAL MODELLING OF ISOTOPE
EXTRACTION FROM SKELETAL
MUSCLE VASCULATURE

presented by

Mark Alan Holmes

has been accepted towards fulfillment
of the requirements for

M.S. , Chemical Engineering
degree in

 

 

 

Major professor

November 10, 1978
Date

0-7639

MATHEMATICAL MODELLING OF ISOTOPE
EXTRACTION FROM SKELETAL

MUSCLE VASCULATURE

BY

Mark Alan Holmes

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Chemical Engineering

1978

ABSTRACT

MATHEMATICAL MODELLING OF ISOTOPE
EXTRACTION FROM SKELETAL
MUSCLE VASCULATURE

BY

Mark Alan Holmes

The selection of parameter estimates and bounds
in a mathematical modelling technique was examined. A
technique was developed to analyze washout curves from
extraction studies on the canine gracilius muscle. The
mathematical models emnployed simulated vascular beds
with and without shunt channels. With the proper selec—
tion of initial parameter estimates and bounds, the
simulation technique adequately represented the experi-
mental data.

The two-path model (with shunt) describes the
experimental data more accurately than the single path
model (no shunt). However, from the analysis of the
data available to date, it is not clear if the skeletal

muscle vasculature contains shunt vessels.

TO THE LORD,

who got it done, and

TO KAREN,

who helped.

ii

ACKNOWLEDGEMENTS

The author would like to express his appreciation
to his academic advisor, Dr. Donald K. Anderson, for his
guidance and assistance. He would also like to express
his gratitude to Dr. Jerry B. Scott, Michael L. Goodnight,
and Gregory A. Goslow for their advice and cooperation
during the course of this research. Finally, he would
like to thank his wife, Karen, for all of her support
and assistance.

The financial support of the National Institute
of Health and the Michigan Heart Association is gratefully

acknowledged.

iii

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . .
LIST OF FIGURES O O O O O O C O O O

NOWNCLATURE O O O O O O O O O O 0

INTRODUCTION . . . . . . . . . . .
BAC KG ROUN D O O C O O O O O I O O

The Capillary Bed . . . . . . .
Work of Other Investigators . . . .
Skeletal Muscle Studies . . . .
Population Balance Studies . . .

MODELS, EQUATIONS, AND COMPUTER PROGRAMS .

Age-Distribution Functions . . . .
Internal-Age-Distribution Function
Exit—Age-Distribution Function . .
Intensity Function . . . . . .

The Models and Equations . . . . .
Renkin Models . . . . . . .
Friedman Models . . . . . . .

Computer Program Modifications . . .

METHODS AND RESULTS 0 O O O O O O 0

Surgical Procedure . . . . . . .
Experimental Procedure . . . . .
Results . . . . . . . . . .

METHODS OF ANALYS I S O O O O O O O 0
Preliminary Calculations . . . . .
Calculation of Tracer Recovery . .

Mean Residence Time Calcuation . .
Data and Weighting Factors . . .

iv

Page
vi
vii

ix

Parameter Estimations
Plug Flow Volume Calculations
Tank Number Parameters
Fractional Flow Parameter .
Ratio of Tank Volumes
Diffusible Model Parameters

The Modelled Data

DISCUSSION .

Experimental Procedure .
Equipment Modifications
Tracers
Plug Flow Tubing Assumption

Modelling Techniques
Curve Fitting Capabilites . .
Other Age-Distribution

Conclusions
Theory Comparison
Suggestions

APPENDIX .

BIBLIOGRAPHY

Functions

50

50
50
SO
51
54
54
55
55
55
56

57

68

LIST OF TABLES

Table Page
1 Experimental Parameters . . . . . . . . 25
2 Data Set 10 - C - D . . . . . . . . . 26
3 Data Set 10 - C - s . . . . . . . . . 27
4 Data Set 10 - St - D . . . . . . . . . 28
5 Data Set 11 - C - D . . . . . . . . . 29
6 Data Set 24 - C - D . . . . . . . . . 30
7 Computer Derived Parameter Values . . . . . 38

8 Calculated Physical Parameters . . . . . . 39

vi

Figure

l
2

10

11

12

13

LIST OF FIGURES

Schematic diagram of vascular bed . . . .
Schematic diagram of a capillary network with
change in flow direction. Adapted from

Borghi (6, p. 2) . . . . . . . . .

Schematic diagram of the one-channel, non-
diffusible tracer model (Renkin) . . .

Schematic diagram of the one-channel, diffu-
sible tracer model (Renkin) . . . . .

Schematic diagram of the two-channel, non-
diffusible tracer model (Friedman) . . .

Schematic diagram of the two-channel, diffu-
sible tracer model (Friedman) . . . .

Schematic diagram of experimental proce-
dure O O O O O O O O O O O O 0

Comparison of one- and two-channel models to
data set 11 - C - D . . . . . . . .

Comparison of computer derived curve and
data set 10 - C - D . . . . . . . .

Comparison of computer derived curves and
data set 10 - C - S . . . . . . . .

Comparison of computer derived curve and
data set 10 - St - D o o o o o o 0

Comparison of computer derived curve and
data set 11 - C - D . . . . . . . .

Comparison of computer derived curve and
data set 24 - C - D . . . . . . . .

vii

Page

13

15

17

19

22

35

4O

42

44

46

48

Figure Page

14 Comparison of normalized washout data from
tubing and muscle (data set 24 - C - D)

and tubing only . . . . . . . . . 52

15 Schematic diagram of the two-channel, non-
diffusible tracer model with stirred
tanks (ST) replacing the plug flow
section . . . . . . . . . . . . 53

viii

E(t)

E(9)

I(t)

1(6)

NOMENCLATURE

In one-path model, fraction of the total system
volume in the main flow stream, dimensionless.
In two-path model, the fraction of stream 1
volume in the main stream, dimensionless.

In the one-path model, ratio of tracer exchange
rate to the main stream flow, dimensionless. In
two-path model, ratio of tracer exchange rate to
stream 1 flow, dimensionless.

Concentration, mc/ml.

Computer parameter, equivalent to B

Exit-age-distribution function as a function of
time, sec'l.

Exit-age-distribution function as a function of
dimensionless time, dimensionless.

In two-path model, fraction of total flow going
to stream 1, dimensionless.

Constant, see equation 35, dimensionless.

Internal—a e-distribution function as a function
time, sec" .

Internal-age-distribution function as a function
of dimensionless time, dimensionless.

Constant, see equation 15, dimensionless.

In two-path model, number of stirred tanks in
stream 2.

In one-path model, number of stirred tanks in
series. In two-path model, number of stirred
tanks in stream 1.

Constant, see equation 16, dimensionless.

Constant, see equation 15, dimensionless.

ix

t Time, minutes; seconds.

E Mean residence time, minutes; seconds.

v Blood flow rate, ml/min; m1/min/100 gm.

V Volume, m1; ml/lOO gm.

a In two-path model, the ratio of the mean resi-
dence time in stream 2 to the mean residence time
in stream 1, dimensionless.

B Constant, see equation 21, dimensionless.

y Computer parameter equivalent to "a".

6 Fraction of total volume in the plug flow section,
dimensionless.

n In the two-path model, the ratio of the volume of
a tank in stream 2 to the volume of a tank in
stream 1, dimensionless.

6 Dimensionless time.

6 Variable, see equation 12, dimensionless.

A(t) Intensity function as a function of time, sec-1.

A(e) Intensity function as a function of dimensionless
time, dimensionless.

¢ Fraction of tracer recovered, dimensionless.

Subscripts

B Blood

D Diffusible

I Injected

ND Nondiffusible

T Total system

tubing tubing

1 Function 1 or stream 1

2 Function 2 or stream 2

X

INTRODUCTION

The goal of this research was to continue a study
on the vasculature of skeletal muscle. In particular,
this work involved further development of computer
modelling techniques, initially presented by M. R.
Borghi (6), to be used in the analysis of radioisotope
pulse injection data obtained from canine gracilis
muscles.

The equations used in the computer simulations
were derived from modelling techniques used extensively
in the engineering field. A discussion of the work of
other investigators, in both the engineering and physio-
logical areas, is presented in the Background section.
A description of the models and equations used is also
presented.

The data which were analyzed were obtained from
experimental procedures described in detail by M. L.
Goodnight (12). In brief, the gracilis muscle of a
mongrel dog was isolated in situ, leaving only one
artery leading to the muscle and one vein draining it.
Radiolabelled compounds were injected as a bolus into
the artery feeding the muscle, and samples of venous

blood were taken serially and analyzed.

The experimental data from the washout studies
were converted to age-distribution curves and analyzed.
The primary analytical method consisted of fitting the
experimental results with computer generated curves of
mathematical models.

Discussion of the results and conclusions of this

research is contained in the final section.

BACKGROUND

The Capillary Bed

 

Flow through a vascular bed can best be described
using the schematic diagram in Figure 1. Blood enters
through the artery (A) which branches into arterioles
(a) which branch into capillaries (c). Exchange of
nutrients and metabolic wastes between the blood tissue
occurs at the capillary level. The capillaries drain
into venules (v) which, in turn, empty into veins (V).

Vessel diameters range from 600 - 1,000 microns
for terminal arteries and veins, 20 - 30 microns for the
arterioles and venules, and 8 - 12 microns for true
capillaries. True capillaries can also be distinguished
from the larger vessels by their walls, since they con-
tain only endothelial cells, while the walls of the
larger vessels also contain smooth muscle and collagen
fibers.

Actually, the capillary network is far more
complex than is indicated in Figure 1. For example,
flow in a given capillary can, and does, change direc—
tions as indicated in Figure 2. A more detailed des-
cription of the capillary and its function can be found

in Mountcastle (20, pp. 984-991).

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WOrk of Other Investigators

 

The work done by other investigators can roughly
be divided into two groups: 1) those studying skeletal
blood flow and, 2) those developing the modelling

techniques.

Skeletal Muscle Studies

 

Two important contributors to research on blood
flow through skeletal muscle are J. J. Friedman (8-11)
and E. M. Renkin (22-25). Both have conducted indicator
dilution studies on skeletal muscles, and each has pro-
posed a theory to explain his observations.

Friedman supports the hypothesis that skeletal
muscle blood flow is best described as a flow limited,
two-path system, where one path (referred to as the
nutritive path) supplies oxygen and nutrients to the
tissue, and the second path (or nonnutritive path)
shunts excess blood around or past the capillary bed,
as in an arteriovenous anastomosis. No anatomical
anastomoses have been identified in skeletal muscle.
However, a physiological shunt, consisting of preferen-
tial pathways, may exist (see 6, Figure 1). In this
system, the preferential pathways would act to shunt
excess blood flow by the capillary bed, while the true
capillaries would supply the nutritive needs of the

muscle.

According to Renkin's theory, nutrient supply to
the skeletal muscle is permeability limited, with the
possibility of one or more paths. In this study, only
the one-path model was considered. The use of this
model is consistent with Renkin's definition, since the
additional stream(s) serve(s) as a source of additional
capillaries for increased blood flow (22, 23).

A third investigator, C. H. Baker (1-4, 19), has
also been working in the analysis of blood flow through
skeletal muscle vascular beds. Utilizing the parallel
path model, he has attempted to calculate the volumes
associated with radio-labelled red blood cell, albumin,

and rubidium flow paths.

POpulation Balance Studies

 

In the engineering field, the work of Danckwerts
(7) in 1953 is considered the forerunner of present
tracer washout modelling techniques. In the area of
physiology, indicator dilution studies were used as
early as 1928 by Hamilton (13, 14), with detailed
derivations by Zierler (18, 26-29). It is interesting
to note that Zierler went through an analysis of vascular
beds very similar to the one Danckwerts did with reactor
beds one year earlier. The work in the physiological
area has been further developed by investigators such

as Martin and Yudelivich (17), while Levenspiel (16),

Bischoff (5), and others (see citations in 15) have
developed the modelling techniques formalized by
Danckwerts in the engineering field.

The research described inthis thesis involves

applying engineering models to the study of the skeletal

muscle vasculature.

MODELS, EQUATIONS, AND COMPUTER PROGRAMS

Age-Distribution Functions

The equations used in this study are from a group
of functions called age-distribution functions, which
have been described previously (7, 21). The equations,
or functions, describe the residence time of fluid
elements in a theoretical, idealized system, which can
hopefully be related to the physical system being
examined. In engineering applications, this would
include reaction mixtures flowing through reactor beds.
In this biological study the equations are used to
describe the residence time of blood in the muscle
capillary bed. Three of the age-distribution functions

are described below.

Internal-Age-Distribution Function

The internal-age-distribution function, I(t), is
the age distribution inside the system, and has the
units: fraction of ages per unit time. The "age" of
any element is defined as the time, t, the element has
spent in the system. From this, the fraction of elements
in the vessel with ages between t and t+dt is given by

I(t)dt. The function is normalized, which leads to

10

/”I(t)dt = 1 <1)

Exit-Age-Distribution Function

The exit-age—distribution function, E(t), is
defined as the residence time distribution of the stream
leaving the system. It is the distribution, or fre-
quency, of fluid elements of a given age in the exit
stream. It has the same units as I(t), and the fraction
of the exit stream with ages between t and t+dt is given
by E(t)dt. This function is also normalized which leads

to
°f°°E<t>dt = 1 (2)

The exit-age-distribution function is related to the

internal-age-distribution function by

_ -- é__
E(t) — wt dt I(t) (3)

Intensity Function

 

The intensity function, A, is defined as the
fraction of fluid in the system of age t that will leave
at a time between t and t+dt. It is related to the other

two functions by

m
E

A(t) = (4)

wind
H
£1

11

The functions can be expressed in dimensionless

form, where e, the dimensionless time, is defined as

(5)

CD
u
«Inf

and t, the mean resident time, is defined as

E = °I°°tE(t)dt (6)

The dimensionless age—distribution functions then become

E(e) = EE(t) (7)
1(9) = EI(t) (8)
me) = E (t) = §{%-} ‘9)

More detailed derivations of the functions can
be found in the texts by Levenspiel (16) or Himmelblau

and Bischoff (15).

The Models and Pguations

 

Borghi (6, pp. 11-57) develOped four models to
describe the age-distribution functions for diffusible
and non-diffusible tracers expected from the theories
proposed by Renkin and Friedman. The models used con-
sist of arrangements of ideal stirred and/or ideal plug
flow vessels. In an ideal stirred tank, the entire

contents of the vessel are completely mixed and uniform.

12

An ideal plug flow reactor is one in which there is
complete mixing perpendicular to the direction of flow,

but no mixing or diffusion along the direction of flow.

Renkin Models

 

As mentioned earlier, the Renkin hypothesis
concerning blood flow is represented by a single pathway.
For the non-diffusible tracer, the model consists of
a plug flow vessel, (PF) followed by a series of equal
volumed stirred tanks (ST) as shown in Figure 3. The

age-distribution function is given by

 

 

 

O for 8 5 e
E(e) = 1
N N-l (10)
N 9 exp (-N O) for 6 3 e
\ F(N)(l-e)
where
V
e=V§-F- (11)
T
0 - e
o—l_€ (12)

The model for the diffusible tracer is similar
to the model for the nondiffusible tracer. The diffusion
model consists of the same plug flow tank and series of

stirred tanks with the addition of a diffusion tank (DT).

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mm

 

 

 

14

The diffusion tank is used to represent the diffusion of
tracer in and out of the interstitial space, and consists
of a stirred tank which receives flow from, and sends
flow to, the last stirred tank in the series as indicated
in Figure 4. The dimensionless age-distribution func-

tion for the single-path diffusion model is give by

 

 

 

 

 

 

 

K Q '- _ N-i-l
E(e) = (1 exp (-Q O)N£l [31+aR1)Qé] -exp (aQRlO)
l-e) ._
1‘1 (N-i-l)!
K Q _ N-i-l
_ (2 ) exp (-Q O)N£1 [31+aR2)Qé] -exp (aQRZO)
1-e ._
1'1 (N-i-l)!
(13)
> <
for 6- e and O for 6- e, where
- _ _ 4aB(1-a)
R12. = 1 a+B li/ l —1—+—B-—2' (l4)
2a(l-a) ( -a )
1 R1 + B/(l - a)
K2 = 2 (15)
(l + aRi’)N-1 (R2 - R1)
Q = aN - a + l (16)

a

15

IX
A \

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xcmu 30am msam 1 mm

 

 

 

 

 

 

 

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16

Friedman Models

 

The Friedman model is represented by a two-
channel flow path. The nondiffusible tracer path is
represented by a plug flow vessel followed by two
parallel streams of stirred tanks, designated as streams
l and 2. Stream 1 has N tanks, each having a volume V1,
and stream 2 has M tanks, each having volume V2. A
schematic diagram of the model is presented in Figure 5.
The dimensionless exit-age-distribution function for

this model is given by

 

 

(NB)N ON 1
E(O) = f exp (- N80)
T (N) (l-E)
(MB/d)M 9M 1 ms
+ (l-f) exp - —— O
P(M) (l-E) a (17)

for 6 2 e, and 0 for 6 s e, where

VPF

e = __ (18)
VT

0 = 9:2 (19)

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I
m

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xcme pmnuflum I am
xcma 30am msam I mm

N Emouum
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H Ecmuum
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.HmccmnOIozu

 

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18

fMVz

_ (20)
a ’ (1-f)Nv1
B = f+(1—f)a (21)

For the diffusible tracer, a diffusion tank (DT) is added

to stream 1, as indicated in Figure 6.

r‘ N-i-l
£11138 N-l QB(l+aR])€->
E(O) = T—[exp (-069) .21 N_1_1 , :l - eXP(aQR189):|
1:

 

 

N-i-l

f B N-l QB(1+ )e

'QKZ T—y|:exp (-QBG) .21 [(N_i:lR)2, :1— - eXP(aQR280):I
l:

(1- f) 91“ MB M MB (22)

TI-e) (M-l)! [3'] exp '[?9]

for 6 2 e and O for e E e, where

fame!

a-f) (aN-a+1)V1 ‘23 ’

 

and s, G, R%, Ki, 8, and Q are as defined earlier.

Computer Program Modifications

 

The computer programs used in this research were
versions of Fortran IV programs written by Borghi (6, p.
100). Four major modifications were made in the pro-
grams. First, subroutine Area was eliminated because

the values of t calculated in the subroutine were

19

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same cohmsmmhn I as
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# Em II.|.II.I Em

     

 

 

 

 

 

 

- I H Emmuum
Em aEm II I.II
—mm——

 

20

unreasonable, often greater than 200 seconds. Secondly,
the channel volume calculations were found to be incor-
rect and were replaced.

Thirdly, nonzero values of E(e) were given by the
model subroutines for values of 9 5 e, which is incor-
rect. Alterations were made to account for the lower
values. Finally, the main program and model subroutines
were modified so the subroutines could be called up by
calling either Model (used by the curve fitting program)
or by calling the individual subroutine name (Series,
Renkin, or Friedman; used with other programs). This
eliminated the need to alter the subroutines' names each
time they were used with a different program.

The modified programs are presented in the
Appendix. The optimization portion of the program (sub-
routine Opt) and the plotting subroutine (Plotter) were

not altered, and are not presented.

METHODS AND RESULTS

Surgical Procedure

 

Mongrel dogs weighing approximately 20 kg were
anesthetized with sodium pentobarbitol (30 mg/kg). The
gracilis muscle (M, see Figure'N‘was surgically approached
from the medial aspect of the hind limb, and all vessels
except the primary artery and vein (indicated by A and V,
respectively) were ligated. An essentially nonpulsatile,
perfusion pump (P in Figure 7) was used to pump blood
from the contralateral femoral artery to the primary
artery. Venous blood from the muscle flowed through a
sampling valve (SV), which, when activated, flushed
intermittent, serial blood samples of equal volume into
collection vials (CV).

During the washout studies, it was desirable
that the blood not be returned to the dog in order to
eliminate the possibility of tracer recirculation.
Therefore, during the washout periods, the blood leaving
the sampling value was discarded, and dextran-70 was
infused into the contralateral femoral vein at approxi-
mately the same rate as blood was lost. Between studies,
the blood was returned to the dog via the contralateral

femoral vein.

21

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23

The muscle was tied at its origin and insertion
to eliminate any collateral flow, and the obturator
nerve (O) was cut and connected to an electronic stimu-
lator (S). Systemic arterial, perfusion, and venous
pressures were monitored using pressure transducers,
and recorded on a strip chart. Timing signals from the

sampling value controller were also recorded.

Experimental Procedure

Sucrose, labelled with C14, and H3 tagged

 

dextran (MW 82,400) were used as the diffusible and
nondiffusible tracers, respectively. One bolus of a
mixture of the tracers in saline solution was injected
into the primary artery during each of the three wash-
out studies performed on each dog. As a control, the
first study was done without stimulation. The second
injection was made immediately after electrical stimu-
lation of the muscle (6v, 6 cps, 1.6 msec, for 30
seconds). The third injection was made after five
minutes of histamine infusion (10 mg/ml at 0.5 ml/min).
The histamine infusion continued throughout the sampling
period.

The blood samples were bleached with percholoric
acid and hydrogen peroxide, then counted in a liquid
scintillation counter. The counts obtained from the

scintillation counter were converted to tracer

24

concentrations and the concentrations were then con-

verted to exit-age parameters utilizing the formula

C v
E(t) = B (24)

CIVI

 

where C is the tracer concentration in the blood, v is

B
the blood flow rate, CI is the tracer concentration
injected, and V is the volume of tracer solution

I
injected. A more detailed explanation of the experi-

mental procedure is presented by Goodnight (12).

Results

Much of the experimental data could not be fully
analyzed due to low tracer recoveries. In many instances,
dextran recovery at the completion of a washout study
was less than 70%. This was especially true for the
stimulation experiments. Further complications arose
from the need to use dextran washout parameters in
analyzing the corresponding sucrose data.‘ The poor
quality of the dextran data, therefore, often led to the
discarding of otherwise usable sucrose data.

Five sets of experimental data were selected for
analysis: Four dextran washout studies and one sucrose
washout study. Four of the sets of data are from
control experiments, and one is from a stimulation ex-
periment. Values for the experimental parameters are

given in Table 1, and the .data is presented in Tables 2-6.

25

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Table

26

2.-—Data Set 10 - C - D

 

Time

E(t)

Time

E(t)

 

(seconds) (seconds- ) (seconds) (seconds-1)
2.00 .00000 102.48 .00103
8.28 .00000 108.76 .00087

14.56 .00003 115.04 .00084
20.84 .00288 121.32 .00076
27.12 .03385 127.60 .00064
33.40 .03982 133.88 .00059
39.68 .02563 146.44 .00045
45.96 .01460 152.72 .00039
52.24 .00986 159.00 .00035
58.52 .00585 165.28 .00028
64.80 .00401 171.56 .00029
71.08 .00303 177.84 .00028
77.36 .00249 184.12 .00025
83.64 .00179 190.40 .00023
89.92 .00137 196.68 .00021
96.20 .00122 202.96 .00019

 

27

Table 3.--Data Set 10 - C - S

 

 

Time E(t) _ Time E(t) _
(seconds) (seconds ) (seconds) (seconds )
2.00 .00000 102.48 .00132
8.28 .00000 108.76 .00116
14.56 .00003 115.04 .00106
20.84 .00243 121.32 .00096
27.12 .02684 127.60 .00089
33.40 .03069 133.88 .00082
39.68 .01827 146.44 .00068
45.96 .01065 152.72 .00065
52.24 .00713 159.00 .00062
58.52 .00500 165.28 .00061
64.80 .00373 171.56 .00057
71.08 .00299 177.84 .00055
77.36 .00245 184.12 .00052
83.64 .00202 190.40 .00047
89.92 .00172 196.68 .00046
96.20 .00152 202.96 .00045

 

28

Table 4.--Data Set 10 — St - D

 

 

Time E(t) _ Time E(t) _
(seconds) (seconds ) (seconds) (seconds )
2.00 .00000 108.76 .00108
8.28 .00003 115.04 .00095
14.56 .00007 121.32 .00081
20.84 .00193 127.60 .00064
27.12 .02188 133.88 .00058
33.40 .04498 140.16 .00049
39.68 .02953 146.44 .00053
45.96 .01502 152.72 .00036
58.52 .00579 159.00 .00030
64.80 .00416 165.28 .00027
71.08 .00291 171.56 .00025
77.36 .00217 177.84 .00021
83.64 .00173 184.12 .00019
89.92 .00170 190.40 .00017
96.20 .00136 196.68 .00016

102.48 .00127

 

29

Table 5.--Data Set 11 - C - D

 

 

Time E(t) _1 Time E(t) _1

(seconds) (seconds ) (seconds) (seconds )
4.00 - .00000 172.00 .00027
9.60 .00000 177.60 .00023
15.20 .01684 183.20 .00023
20.80 .05498 188.80 .00019
26.40 .03658 194.40 .00020
32.00 .01348 200.00 .00024
37.60 .00871 205.60 .00018
43.20 .00531 211.20 .00019
48.80 .00294 214.20 .00019
54.40 .00259 219.80 .00020
60.00 .00201 225.40 .00015
65.60 .00149 231.00 .00015
71.20 .00113 236.60 .00017
76.80 .00099 242.20 .00014
82.40 .00092 247.80 .00013
88.00 .00088 253.40 .00011
93.60 .00083 259.00 .00012
99.20 .00063 264.60 .00013
104.80 .00055 270.20 .00012
110.40 .00049 275.80 .00012
116.00 .00039 281.40 .00012
121.60 .00037 287.00 .00018
127.20 .00042 292.60 .00010
132.80 .00028 298.20 .00010
138.40 .00031 303.80 .00009
144.00 .00025 309.40 .00009
149.60 .00024 315.00 .00009
155.20 .00022 320.60 .00009
160.80 .00024 326.20 .00010
166.40 .00019 331.80 .00014

 

30

Table 6.--Data Set 24 - C - D

 

 

Time E(t) _ Time E(t) _
(seconds) (seconds ) (seconds) (seconds )

2.68 .00000 45.13 .00440
4.75 .00000 47.25 .00411
6.88 .00000 49.38 .00373
9.00 .00211 51.50 .00303
11.13 .01451 53.63 .00273
13.25 .05200 55.75 .00309
15.38 .07486 57.88 .00298
17.50 .08238 60.00 .00232
19.63 .04792 62.13 .00220
21.75 .04520 64.25 .00195
23.88 .02928 66.38 .00176
26.00 .02354 68.50 .00211
28.13 .01774 70.63 .00179
30.25 .01726 72.75 .00164
32.38 .01218 74.88 .00176
34.50 .00913 77.00 .00152
36.63 .00865 79.13 .00154
38.75 .00719 81.25 .00151
40.88 .00655 83.38 .00146

43.00 .00529

 

METHODS OF ANALYSIS

Preliminary Calculations

 

Calculation of Tracer Recovery

 

The tracer recovery, I), is obtained by integration
of the exit-age-distribution function. In equation form,

the fraction of tracer recovered to time t is given by
t
¢(t) =°f (E(t)dt (25)

Since all of the tracer should eventually be washed out

to the tissue,
I(w) =.r” E(t)dt = 1 (26)

Estimation of the tracer recovered during an
experiment was made using Euler's estimation of the
integral. The dextran tracer concentration in the venous
blood was sufficiently low by the end of most experiments
to justify the conclusion that all of the tracer that
could be washed out of the tissue had been extracted.
Extrapolation of the curves usually accounted for less
than an additional 5-10% of the tracer recovery. There-
fore, dextran data sets with less than 90% recovery by

the last data point were not analyzed. The dextran

31

32

data set from the control experiment of dog number 11 was
included, however, to illustrate the problem encountered

with low tracer recovery.

Mean Residence Time Calculation

The equation for the mean residence time, E,
was presented in Section II. (p. 11). The integration
of the exit-age-distribution data was done graphically,
as numerical techniques often gave unreasonable values
due to the methods used to estimate the tail portion of
the curve. The mean residence times for the sucrose
data are only approximations. The tail portion of that
data set made a more exact determination of t impossible.

However, the data are adequate for illustration of

the analytical techniques.

Data and Weighting Factors

 

For curve fitting purposes, the data beyond
6 = 2 or 3 could not be considered reliable (15, p. 72),
and were not used in the analysis. To compensate for
the large number of data points in the tail portion of
the washout curves, the peak data point was weighted
10 times heavier than the other points. Once the
initial optimization was completed, any necessary alter-
ations in the curve fit could be made by using a
weighting scheme to emphasize the portion of the curve

where the improvement was desired.

33

Parameter Estimations
Plug Flow Volume Calculations
In order to obtain an estimate of the plug

flow volume, V blood flow in the tubing from the

PF'
injection site to the muscle, and from the muscle to
the sampling valve, was assumed to be plug flow in
nature. The tubing volume used in the experiments, as
calculated from the tubing diamter and length, was

1.9 i .4 ml.

The total system volume, V , is related to E

and blood flow, v, by
V = vt (27)

From this, a, the fraction of the total volume in the

plug flow section,is given by

8 = VPF : vtubing :: 1._9 (28)
VT VT vt

The value of 6 could not be defined precisely
since the primary artery and vein might also show plug
flow characteristics. However, the calculated value
was used as the first estimate of e in the curve fitting

Optimization.

34

Tank Number Parameters

 

Borghi (6) found that the two-stream model gave
a better curve fit than the one-stream model, since the
one-stream model could not simultaneously fit both the
peak and the tail portions of the data adequately.
Similar results were found in this study, as shown in
Figure 8. When the two-path model was used, the lower
limit on the number of tanks in each stream was set at
3.0 to avoid anomalies in the mathematics, and the
upper limit set at 50.0. Initial estimates for N and M

were taken as 3.0 and 15.0, respectively.

Fractional Flow Parameter

The fraction of the total fluid flow in stream
1, f, was allowed to range from 0.0 to 1.0. The initial
parameter estimate was defined as 0.4, as the optimized

values were usually less than 0.5.

Ratio of Tank Volumes

 

The ratio of a tank volume from stream 2 to a
tank volume from stream 1, n, was allowed to range from
0.0 to 10.0, with an initial estimate of 0.2. The final,

optimized values for this parameter were less than 0.5.

Diffusible Model Parameters
For the two-path diffusible model, the values

for e, f, N, M, and n were defined by the corresponding

, sec

E(t)

35

0.06 r

0.05? KEY

0 data '
--- single flow channel ( Renkin )
0,04 - —1wof|ow channal ( Friedman )

0.02 -

 

0.0l '-

Jn °.

0 20 40 so 8 IOO I20 I40 I60I80
I,sec

 

 

 

 

 

Figure 8. Comparison of one- and two-channel models to
data set 11 - C - D.

36

nondiffusible optimization. The values for f, N, M, and
n were the same in the diffusible case as in the non-

diffusible case, and SD was defined by

It should be noted that stream 1 was defined as
the nutritive (diffusion) stream, and stream 2 was
defined as the shunt (nondiffusion) stream when the two-
channel, diffusible tracer model was used. Therefore,
with this model, f was the fraction of the total blood
flow going to the nutritive stream, and N was the number
of tanks in the nutritive stream. Likewise, n was the
ratio of the volume of a tank in the shunt stream to
the volume of a tankiJIthe nutritive stream.

The value of "a" (referred to as Y in the compu-
ter programs), the fractionof the total stream 1 volume
which was present in the main stream, was calculated

using the equation

 

 

Gt
a = -ND -
- +
(G l)tND tD .
where
G _ 1 - ND
14.51).

37

B (referred to as D in the computer programs)
was defined as the ratio of flow to and from the dif-
fusion tank to the main stream flow. The value of B
was allowed to range from 0.0 to 100.0, with an initial
estimate of 0.01. Since this was the only parameter
being optimized in this model, the initial estimate was
not as critical as in the case of the nondiffusibile
tracer model parameters. Curve fitting of the sucrose
data was done in which each of the streams from the
corresponding dextran optimization was defined as the
nutritive stream. The best fits for each configuration

were compared, and the optimum fit was selected.

The Modelled Data

 

The mean residence times, optimized parameter
values, and the sum of the squared residuals for each
of the sets of modelled data are presented in Table 7.
The calculated values of a and B, from equations 20
and 21, respectively, for the nondiffusible tracer
model are also included in Table 7 to facilitate the use
of equation (d) from Himmelblau‘s text (15, p. 74). The
calculated physical parameters are presented in Table 8.
Comparisons of the computer derived curves and the

experimental data are presented in Figures 9-13.

38

 

 

cowumcmamxa Ham H manna mama
>5. mm. vma. mHH. o.NH o.m mm. ham. mm.- o I U I «N
mm. as. moa. mma. v.5H o.m we. mHH. 0.5m o I U I oa
mo. he. mmo. mna. o.mH o.m mm. omm. o.ov I um I OH
Hmm. omn.a ma. om. o.m o.v on. Hum. o.om m I U I oa
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Ne. em. mmm. mom. o.¢ o.m «m. 5mm. o.m¢ o I U I ca
Aoamv
m a mamnpflmmm c a > z z u w u
pmumsvm I
no Sam mumumEmumm «pom mumo

 

mmsam> Hmuwemnmm pm>wnmo HODSQEOUII.h OHQMB

39

o.m

o.om u u .Hmeos umomuu manflmsOMHa
m u u .macoe Hooch» manflm5mmwpcoz

N

manmohaamm “Hm.
coflumcmamxm you a canoe mama

 

 

m.m m.H m.m m.m ~.~ a I o I «N
o.HH m.m m.oa m.¢ H.H a I o I Ha
m.m m.m m.e H.e ~.~ I um I OH
H.m o.m me. s.m e.~ o.v m I o I oa
o.m H.~ m.¢ m.m H.v m I o I ca
~.m e.~ H.m m.m o.v a I o I oH
am coaxcns\as so ooa\as em coaxas em ooa\efls\as em ooH\Hs am ooa\as
30am mEdHo> madao> 3on mesao>
pooam mnmaaflmmu macaw UOOHm mumHaflmmo
HmhuhumuwucH mssao>
«aAEmaHum ucsnmv «aafimmuum coflmnmmwpv 30am
m ammuum a ammuum made I now mama

 

mnmuafimnmm Hmowmhnm pmumHsoamoII.m magma

40

Figure 9.--Comparison of computer derived curve and
data set 10 - C - D.

E (6)

LS "

|.O-

OSr

 

0.0

 

 

 

1...!

 

41

42

Figure 10.--Comparison of computer derived curves and
data set 10 - C - S.

.03 P-

O
N
T

E ( 1 )(seconds")

.Ol-

 

.00

\-

 

43

KEY

— nondiffusible model,
I = 55 seconds

--- diffusible model.
1 = soseconds

 

1 (seconds)

44

Figure 11.--Comparison of computer derived curve and
data set 10 - St - D.

 

45

 

 

 

 

2.0 "

L5-

0.5 "

46

Figure 12.--Comparison of computer derived curve and
data set 11 - C - D.

 

47

 

 

 

 

L5"

L0-

Amvm

0.5 -

48

Figure l3.--Comparison of computer derived curve and
data set 24 - C - D.

 

49

 

 

 

 

 

 

L5 l'

|.0 -

on m

0.5 -

000.0

DISCUSSION

Experimental Procedure

 

Equipment Modifications

 

Equipment modifications improved the quality of
the data obtained from the experiments. The use of a
chromatographic sample injection valve (12) eliminated
the dead space problems encountered by Borghi (6). The
use of a new timing device decreased the sampling inter-
val from 4-6 seconds to 1-3 seconds. This increased
the number of data points on the peak portion of the
washout curve, and decreased the possibility of missing
the peak portion of the curve, which occasionally occur-

red with the previous timer.

Tracers

The primary difficulty encountered in this study
was the poor quality of the dextran washout data. Over
60% of the control and stimulation experiments had
dextran recoveries less than 70%, and in almost 40% of
the experiments, dextran recovery was less than 50%. In
contrast, only 20% of the control and stimulation sucrose
recoveries were less than 70%. The sampling period for

each experiment exceeded 250 seconds, allowing adequate

50

51

time for the complete extraction of the nondiffusible
tracer (the blood recirculation time in the normal canine
is approximatley 30 seconds).

As can be seen in Figure 12, the low tracer
recovery in data sets 11 - C - D made adequate fitting
of the data impossible. The computer derived curve is
above the data at all points. This problem also occurred
when attempts were made to model the data from experiment
10 - C - S (recovery 85.5%).

The low level of detected dextran recovery is
believed to be due to the loss of the tritium through an
exchange reaction. The possibility of replacing the

125

dextran tracer with I labelled albumin is being

examined in the continuation of the project.

Plug Flow Tubing Assumption

 

One washout study was conducted on blood flowing
through tubing, without the muscle. The normalized data
collected in that experiment is compared to the non-
diffusible tracer data from dog number 24 in Figure 14.
Additional washout studies on blood flow through the
tubing should be conducted to test the assumption of
plug flow in the tubing. If that assumption is incorrect,
it would be necessary to replace the plug flow section

of the model with a series of stirred tanks (Figure 15).

2.0 *-
L5
255
LIJ
I.0
0.5

 

52

fl KEY

o data from tubing and
muscle
a data from tubing only

 

 

 

 

 

|.0 2.0 3.0

Figure l4.--Comparison of normalized washout data from

tubing and muscle (data set 24 — C - D) and
tubing only.

53

.coHuowm 30am mafia as» mcflomammu Aamv mxcmu pmunwum nufl3
Hapos umomuu manwm:MMApcoc .HmccoSOIozu an» MC Emummep aflumaacomII.mH anamam

fi~£.....fl
E II..- a
1... 2.63.5

54

Modelling Techniques

 

Curve Fitting Capabilities

 

With the exceptions of data sets 10 - C - S
and 11 - C - D, the modelling techniques adequately
represented the data (see Figures 9, 11 and 13). The
primary requirements for the optimization techniques
were high tracer recovery and the selection of reason-
able initial parameter estimates and bounds.

With the use of randomly selected parameters,
the curve fitting program often "optimized" to curves
which did not reasonably represent the data. The problem
was especially pronounced when the data peak was not
preferentially weighted. The large number of data
points in the tail overemphasized that portion of the
curve. In addition, if multiple maxima were present in
the optimization scheme, random initial parameter
selection may have caused the program to optimize around
the incorrect maximum.

Because of the uncertainty of the mean residence
time calculations for the sucrose data set, that set
was optimized using several different values for E.

The best fits for the nondiffusible and diffusible
models were presented (see Figure 10). As mentioned
earlier, the low tracer recovery for this data set
hindered the optimization efforts. However, since only

one parameter is optimized with the diffusible tracer

55

model, the optimization technique is expected to perform

adequately with more reliable data.

Other Age-Distribution Functions

 

The possibility of replacing the exit-age-
distribution function, E(t), in the optimization tech-
nique with one of the other-age distribution functions
was considered. The information obtained from the
internal—age-distribution function, I(t), is similar to
that obtained from E(t), and there was no significant
advantage to using the former. The intensity function,
A(t), is helpful for detecting bypassing or stagnation
in a flow system. The appearance of two or more maxima
in the A(t) curve indicates the presence of nonuniform
flow. However, the intensity function is not as useful
as the other two functions in quantifying the effect

(15, p. 73).

Conclusions

 

Theory Comparison

 

It is not possible to draw any definite con-
clusions from the data available at this time concerning
which of the two hypotheses (Renkin's or Friedman's) is
correct. The need for a two-path system to represent
the data supports Friedman's theory, as does the high
sucrose exchange rate which is inconsistent with the

permeability limited system described in Renkin's theory.

56

However, analysis of more reliable data than are pre-

sently available will be required to resolve the question.
With the proper selection of initial parameter

estimates and bounds, the modelling techniques described

above can be expected to perform adequately.

Suggestions

 

The utilization of the sampling valve and control
timer, developed by other investigators working on the
project, has significantly improved the quality of the
data. Replacing the radiolabelled dextran tracer and
testing the assumption of plug flow in the tubing are
areas which should be addressed in the continuation of

the skeletal muscle microcirculation research.

APPENDIX

COMPUTER PROGRAMS

57

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B FITT 0
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EXPERIMENT.
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ED
T
TO BE PERFORMED BY 18.

T0 E
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Oktuhhum<<ztmnt OU<WI<O<
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m UDXH ZZZ HJJKQC K C

ARE THE LOUER BOUND. INITIAL GUESS AND UPPER BOUND
I
V
E

58

AND PRINT OUT DATA.

READ IN

UUUUUUUUUUUUUUUUUUUUUUUUUUUU

QFPPQGMQELDW

QIPUNQITRACERgFPNQGH

O.ICN
P 9TRUNQITRACER

p
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THE DATA IN DINENSIONLESS F0

A
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CHANGE DATA To

C

59

A

2

¢

>

Z

O O

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a“ of
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mm 0
P.— A A
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SURROUTINE.

CALL OPTIMIZATION

UUU

CALL OPT(NOE,Y,NPA°oBQDELgCHPAXqBNDLUQBNDUPoREDAQRSSTOLoITHAX,
ISIS)

Fl

LL:L.L

LIBOFQNOBQNPARI

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Cp.XEMQ1,
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pLOT VALUES OF EXPERIMENTAL AND CALCULATED DATA POINTS.

UL' UULL.’

CALL PLOTTFR(696095.?Q-N089109IX9IXCNDBQII)

CTOR VOLUMES USING TFAR IN SEC AND FLOH IN

EA
CNS.

60

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61

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VII v-I

B IBLIOGRAPHY

68

BIBLIOGRAPHY

Baker, C. H. "Blood Flow and Volume Distribution
with Elevated Venous Pressure." American Journal
of Physiology, 218(3):674-680 (March, 1970).

 

Baker, C. H., and Menninger, R. P. "Histamine-
induced Peripheral Volume and Flow Changes."
American Journal of Physiology, 226(3):731-737
(March, 1974)}

Baker, C. H., and Davis, D. L. "Isolated Skeletal
Muscle Blood Flow and Volume Changes during
Contractile Activity." Blood Vessels, 11:32-44
(1974).

 

Baker, C. H.: Menninger, R. P.: Schoen, R. E.; and
Sutton, E. T. "Skeletal Muscle Vascular Volume
Changes with Increased Venous Pressure." Blood
Vessels, 13:222-237 (1976).

Bischoff, K. B., and Dedrick, R. L. "Generalized
Solution to Linear, Two-compartment, Open Model
for Drug Distribution." Journal of Theoretical
Biologx, 29:63-83 (1970).

Borghi, M. R. "Mathematical Simulations of Isotope
Extractions in Nutritional and Nonnutritional
Flow Channels in Vascular Beds." Master's
thesis, Michigan State University, 1976.

Danckwerts, P. V. "Continuous Flow Systems."
Chemical Engineering Science, 2(1):1-13 (Feb.,
1953.

 

Friedman, J. J. "Microvascular Flow Distribution
and Rubidium Extraction." Federation Proceed-
ings, 24:1099-1103 (1965).

 

Friedman, J. J. "Total, Non-Nutritional, and
Nutritional Blood Volume in Isolated Dog Hind-

limb." American Journal of Physiology, 214(1):
151-156 (1966).

 

69

70

Friedman, J. J. "Muscle Blood Flow and 86Rb Extrac-

tion: 86Rb As a Capillary Flow Indicator."
American Journal of Physiology, 214(3):488-493
(I968).

Friedman, J. J. "Single-passage Extraction of 86Rb
from the Circulation of Skeletal Muscle."
American Journal of Physiology, 216(3):460-466
(1969).

Goodnight, M. L. Master's thesis, to be published,
Michigan State University, 1978.

Hamilton, W. F.; Moore, J. W.; Kinsman, J. M.;
and Spurling, R. G. "Simultaneous Determination
of the Pulmonary and Systemic Circulation Times
in Man and of a Figure Related to the Cardiac
Output." American Journal of Physiology, 84:
338-344 (1928).

 

Hamilton, W. F.; Moore, J. W.; Kinsman, J. M.; and
Spurling, R. G. "Further Analysis of the Injec-
tion Method, and of Changes in Hemodynamics Under
Physiological and Pathological Conditions."
American Journal of Physiology, 99:534-551 (1932).

Himmelblau, D. M., and Bischoff, K. B. Process
Analysis and Simulation. New York: Jo n
Wiley and’Sons, 1968.

Levenspiel, 0. Chemical Reaction Engineering. New
York: John Wiley and Sons, 1972.

 

Martin, P., and Yudilevich, D. "A Theory for the
Quantification of Transcapillary Exchange by
Tracer-dilution Curves." American Journal of
Physiology, 207(1):l62-168 (1964).

 

Meier, P., and Zierler, K. L. "On the Theory of
the Indicator-dilution Method for Measurement
of Blood Flow and Volume." Journal of Applied
Physiology, 6(12):731-744 (June, 1951).

 

Moore, J. C., and Baker, C. H. "Red Cell and
Albumin Flow Circuits During Skeletal Muscle
Reactive Hyperemia." American Journal of
Physiology, 220(5);121311219 (1971).

 

Mountcastle, V. 8., ed. Medical Physiology. St.
Louis: The C. V. Mosby Company, 1974.

 

21.

22.

23.

24.

25.

26.

27.

28.

29.

71

Naor, P., and Shinnar, R. "Representation and
Evaluation of Residence Time Distributions."
Industrial andvgngineering Chemistry, Funda-
mentals, 2(4):278-286 (1963).

Renkin, E. M. "Exchangeability of Tissue Potassium
in Skeletal Muscle." American Journal of
Physiology, 197(6):1211-I215 (1959).

 

 

Renkin, E. M. "Transport of Potassium-42 from
Blood to Tissue in Isolated Mammalian Skeletal
Muscles." American Journal of Physiology,
197(6):1205-1210 (1959).

 

Renkin, E. M., Hudlicka, 0., and Sheehan, R. M.
"Influence of Metabolic Vasodilatation on Blood-
tissue Diffusion in Skeletal Muscle." American
Journal of Physiology, 211(1):87-98 (1966).

Renkin, E. M. "Blood Flow and Transcapillary Ex-
change in Skeletal and Cardiac Muscle." Int.
Symp. on the Coronary Circulation and Energetics
of the Myocardium, Milan 1966, pp. 18-30, New
York: Karger, Basel, 1967.

Zierler, K. L. "Circulation Time and the Theory of
Indicator-dilution Methods for Determining
Blood Flow and Volume." In Handbook of Physio-
logy, Sec. 2 Circulation, Vol 1., Washington,

D. C., Am. Physiol. Soc., 1962.

 

Zierler, K. L., M.D. "Theory of Use of Indicators
to Measure Blood Flow and Extracellular Volume
and Calculation of Transcapillary Movement of
Tracers." Circulation Research, 12:464-471
(May, 1963).

 

Zierler, K. L., M.D. "Equations for Measuring
Blood Flow by External Monitoring of Radioiso-
topes." Circulation Research, 16:309-321
(April, 1965).

 

Zierler, K. L. "Tracer-dilution Techniques in
the Study of Microvascular Behavior." Federation
Proceedings, 24:1085-1091 (1965).