Data Acquisition and Processing in Biology and Medicine. Proceedings of the 1966 Rochester Conference

362 Pages·1968·8.574 MB·English

by  Kurt Enslein (Eds.)

DATA ACQUISITION
AND PROCESSING
IN BIOLOGY AND MEDICINE
VOLUME 5
PROCEEDINGS OF THE
1966 ROCHESTER CONFERENCE
Edited by
KURT ENSLEIN
ROCHESTER, N.Y.
THE QUEEN'S AWARD
TO INDUSTRY 1 ·•·
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Copyright © 1968
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First edition 1968
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CONFERENCE COMMITTEE
EUGENE AGALIDES THOMAS H. KEEN AN
General Dynamics/Electronics Computing Center
1400 North Goodman Street University of Rochester
Rochester, New York 14609 Rochester, New York 14627
FORBES H. NORRIS, JR.
GERALD H. COHEN
Department of Medicine
Dept. of Electrical Engineering
University of Rochester
University of Rochester
Rochester, New York 14627
Rochester, New York 14627
ROBERT B. SMITH
KURT ENSLEIN Research Laboratory
Eastman Kodak Company
42 East Avenue
Kodak Park
Rochester, New York 14604
Rochester, New York 14613
JOSEPH IZZO ROLAND ZINSMEISTER
Department of Medicine General Dynamics/Electronics
University of Rochester 1400 North Goodman Street
Rochester, New York 14627 Rochester, New York 14609
V
AN ANALOGUE COMPUTER SOLUTION OF A
SET OF NON-LINEAR DIFFERENTIAL
EQUATIONS OF MOTION OF THE DOG AORTA
I. THE EFFECTS OF HYPERTENSION, AGING
AND BLOOD INFLOW FORCINGS ON
PRESSURE CURVES
JULIA T. APTER* and LESTER S. SKAGGS
Mathematical Biology and Radiology
University of Chicago, Chicago, Illinois
INTRODUCTION
THE purpose of this study is to show that a model based on the visco-elastic
behavior of the aorta and arterioles can generate aortic pressure curves
resembling real ones. Because the source of irregularities in the model-
generated curves is known, it is possible that similar irregularities in real
curves can be assigned to similar sources, pending experimental verification.
In fact, the model has called attention to certain details of real curves
previously overlooked or attributed to artefacts.
The visco-elastic behavior modelled is stress-relaxation, that is, when an
arterial segment ligated at both ends is stretched by a step-function increase
in volume, the pressure rises to a peak, then drops along an essentially
exponential course to a constant level within 2 sec. 1,2 This behavior can be
formalized3 as a combination of conservative (or elastic) elements and
dissipative (or viscous) elements (Fig. 1).
PROCEDURES AND RESULTS
The behavior of circumferential aortic strips 1 resembles the behavior of
a number n of elastic rods R (length l elastic constant a) in parallel with
x x i9 x
a number n of elastic rods R (length l, elastic constant a) each of which
2 2 2 2
* Present address: Presbyterian-St. Luke's Hospital, Division of Surgery.
1
2 JULIA T. APTER AND LESTER S. SKAGGS
is in series with a viscous rod G (length / , viscous constant b). All R and
3 Y
R + G are curved to encircle a lumen of radius
2
r = pli (1)
where ρ = 1/2π. Tangential force F results from elongation of R and R
t 2
1 -
ttoooooooooooooooSWïiftWffiïr
1
JMUSCLE
ELASTIN
COLLAGEN
MUSCLE
ELASTIN
FIG. 1. 1. A combination of springs and dashpot. By joining Β to B' and D to D'
to form a circle and then combining many such circles to form a tube, a model
artery will be built which will show stress relaxation resembling real arteries.
2. Anatomical counterparts of the model. It simplifies to model 1.3. Experiments4
show that this is more nearly accurate for the aorta than is model 2.
past rest lengths / and l . In a tube of unit length and wall thickness, D,
lo 2o
pressure
ρ F M/l-/lo) + *2(/2-/2o)]£
= = (2)
r pl
x
Let the tube represent an aorta of length L exposed to a blood inflow, W(t).
Take the blood outflow Q{t) to be laminar against a peripheral resistance, 2
maximal at R during systole and early diastole, and R(t) at a time Δ, from
0
the onset of systole; a time which occurs later in diastole, where
R(t) = R-yP' and Q = -. (3)
0
Κ
P' is the pressure above the threshold pressure of the carotid sinus mechanism ;
DIFFERENTIAL EQUATIONS OF MOTION OF THE DOG AORTA 3
such pressures exist during systole and are approximated, here, by half a sine
wave. 7 is a constant dependent on the responsiveness of the peripheral
resistance and the excitability of the carotid sinus if we assume that R(t)
because of carotid sinus response to pressures existing during systole. It is
also possible that R(t) because of stress relaxation in peripheral arterioles
whose muscle tension is responsible for R.
This aorta has a rate of volume change, V:
V = pLll. (4)
1l
Set
h = h + r =pl,
0 lo
a' = aD\ a" = aD; (5)
x 2
η = bD.
Then material balance gives
V = W -Q (6)
or3
plR
x
and
h = / + / . (8)
2 3
Several non-linearities are incorporated into the model. 1 For example:
a' = 0 for /i < / .
lo
a" = oo for l < l - (9)
2 2o
Then the strain rate of G is / , or
3
/ = +-(/ -/ )when/ >0 (10)
3 2 2o 3
n
and
h = --(/i-/i )when/ <0. (11)
0 3
Ά
Steady state requires that / at the end of diastole equal / at the beginning
3 3
of the previous systole. Therefore,
U
= o
at some time t* during the cardiac cycle and / is discontinuous at /*.
3
* R{t) implies R is a function of /.
4 JULIA T. APTER AND LESTER S. SKAGGS
Equations (7), (8), (10) and (11) were programmed for the general purpose
Beckman EASE (2132) analogue computer (Fig. 2). Plots of P, l /, /,
u 2 3
R(t) and W(t) permitted analysis of events in the cardio-vascular system so
modelled. This study was divided into three parts: (1) to analyze the effects
FIG. 2. Program for Beckman EASE (2132) analogue computer which solves
equations (7), (8), (10), and (11). C = a'(l-l) + a"(l-l); C = fl'(/i-/i)
l lo 2 2o 0
H- T71 / I. K05 driven by master (M) clock reset at end of each cardiac cycle was
A
put in to prevent an artefact from retrace of function generator reset. V indicates
100 volts. R(t) driven by slave (S) clock with delay time, Δ.
induced on P, l l, and / by changing separately R , R(t), W(f) (approxi-
l9 2 3 0
mated by a function commensurate with clinical records) and the viscous
and elastic constants; (2) to combine such changes in / , R, R(t), W(t),
lo 0
a\ a" and η as are consistent with hypertension or aging and following the
effects on Ρ and Ι (circumference); (3) to show the response of the system
γ
to a sinusoidal wave for normal and hypertensive conditions to demonstrate
the non-linearities of the response of P.
DIFFERENTIAL EQUATIONS OF MOTION OF THE DOG AORTA 5
1. The effects of individual parameter changes are displayed in Figs. 3,
4, 5, 6, 7. These seldom change individually in vivo. However, these figures
help identify the cause for particular events in the computer-generated curves
and therefore help assign causes for similar events in real pressure curves.
2. Figure 8 shows the influence on the model of an injection of a cate-
cholamine which raises the tone of vascular smooth muscle in aorta and
arterioles and has a positive inotropic action on cardiac muscle.
I40h
I20h
lOOh
80h
60h
x * χ
40h
20h
_J I I I—
4 6 8 10
1
a cm Hg
FIG. 3. P and P versus a'. P and P-P rise with a'; P rises only for small
s d s s d d
increase in a'\ at high a\ high P drives out blood fast, reducing P.
s d
As the dose of drug increased, the following constants would also increase:
R (due to constriction of arterioles); a\ a\ η (due to contraction of aortic
0
smooth muscle); and W{t) amplitude (due to positive inotropic action on
cardiac muscle). The following constants are assumed to decrease: R(t)
(due to resetting of the carotid sinus nerves which regulate arteriolar muscle
tone); / (due to contraction of aortic smooth muscle). Actual values for
lo
a'» a"> ^i> R, and W(t) were obtained from dogs (Apter et α/. 4) and the
0 0
computer-generated pressure curves were to be compared with dog aortic
pressure curves.
The responses of the model to these changes, individually and in various
combinations, are summarized in Figs. 3 to 10. Systolic pressure P and
s
6 JULIA T. APTER AND LESTER S. SKAGGS
140
120
:
100
FIG. 4. A plot of steady state P
s
80 and P versus R. A marked
d Q
increase in P and P, but a slight
s d
decrease in P—P accompanies
s d
60 an increase in R0, due to peri-
pheral vasoconstriction.
40
20
.25 .3 .35
J
Ro cm Hg sec. cm
120
100
80
60
40
20
_L _L
350 370 390 410 430 450 470 490 510 530
-1
w(t) cc sec.
FIG. 5. An increase in cardiac output generated by increased inotropic action of
the heart is reflected as an increase in W{t). P and P increase markedly; P-P
s d s d
very slightly. Inset shows how W(t) is varied.
DIFFERENTIAL EQUATIONS OF MOTION OF THE DOG AORTA 7
diastolic pressure P, all increase with an increase in a\ R , W(t) and with
d 0
a decrease in / . The pulse pressure (P-P) increases also, but more
lo s d
markedly with a change in a' and l .P does not reflect a change in a" or η,
u s
FIG. 6. R(t), as potentiometer settings [R scaled as (400 R)], versus systolic and
diastolic pressures for two values of R, (0.4 and 0.2). Insets show form of R(t)
0
and steady state pressure curves for the 8 R(t) settings. Lowest Ρ curve means
greatest R(t) change.
but P decreases when η increases. There is evidence, in clinical hyper-
d
tension,2'5 that the pressure does drop at the end of diastole as these computer
curves do. The circumference, l of the aorta mirrors the pressure changes
u
except those due to η, while l (t) and l(t) are markedly influenced by η and
2 3
a".
Β