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Brigham Young University Brigham Young University
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Theses and Dissertations
1970-6
A Feasibility Study of the Application of Bondgraph Modeling and A Feasibility Study of the Application of Bondgraph Modeling and
Computerized Nonlinear Model Parameter Identification Computerized Nonlinear Model Parameter Identification
Techniques to the Cardiovascular system Techniques to the Cardiovascular system
Randall L. Taylor Brigham Young University - Provo
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BYU ScholarsArchive Citation BYU ScholarsArchive Citation Taylor, Randall L., "A Feasibility Study of the Application of Bondgraph Modeling and Computerized Nonlinear Model Parameter Identification Techniques to the Cardiovascular system" (1970). Theses and Dissertations. 7199. https://scholarsarchive.byu.edu/etd/7199
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D O
A FEASIBILITY STUDY OF THE APPLICATION OF BONDGRAPH MODELING
AND COMPUTERIZED NONLINEAR MODEL PARAMETER
IDENTIFICATION TECHNIQUES TO THE
CARDIOVASCULAR SYSTEM I
A Thesis
P resented to the
D epartm ent of M echanical Engineering
Brigham Young U niversity
In P artial Fulfillm ent
of the R equirem ents fo r the Degree
M aster of Science
by
Randall L . Taylor
June 1970
A FEASIBILITY STUDY OF THE APPLICATION OF BONDGRAPH MODELING
AND COMPUTERIZED NONLINEAR MODEL PARAMETER
IDENTIFICATION TECHNIQUES TO THE
CARDIOVASCULAR SYSTEM
A Thesis
P resented to the
D epartm ent of M echanical Engineering
Brigham Young U niversity
In P artia l Fulfillm ent
of the Requirem ents fo r the Degree
M aster of Science
by
Randall L . Taylor
June 1970
This th esis , by Randall L . T aylor, is accepted in its p re sen t form
by the D epartm ent of M echanical Engineering of Brigham Young U niversity
a s satisfy ing the th esis requ irem ent for the degree of M aster of Science.
Date / ’ /
Typed by Katherine Shepherd
ACKNOWLEDGMENTS
my gratitude to Tau Beta Pi national engineering honor society fo r seeing fit to
aw ard me the 1969-70 Honeywell Fellowship sponsored by the Honeywell C o r
pora tion . Without th is financial help it would not have been possible fo r me to
continue my graduate studies and accom plish th is w ork.
I am especially indebted to D r. Weldon B. Jolley of Loma Linda U niver
sity and D r. Karl R . Kelly Nicholes of the Mayo Clinic fo r th e ir unselfish w ill
ingness to offer suggestions and help in connection with th is study.
The patient advice, counsel, and instruction given by my graduate
advisory com m ittee m em bers, D r. Joseph C . F ree and D r. Henry J . N icholes,
was m ost valuable to m e . My opportunity to associate with them on th is
p ro jec t w ill be long rem em bered as a highlight in my life .
The excellent physiological drawings p resen ted h ere in a re a lso the
re su lt of D r. Henry J. N icholes' w ork and I am certa in ly grateful to him for
th is h e lp . I am likewise thankful for the help rendered by Les Dorrough in
p reparing some of the o ther figures used in th is p ap er.
Perhaps m ost of a ll I am gratefu l to my wife Sondra fo r h e r unequalled
love, insp iration , and devotion to me and our children over these y ears of
fo rm al education. She indeed m akes it a ll w orthw hile.
iii
I would certa in ly be rem iss if I did not take th is opportunity to ex p ress
TABLE OF CONTENTS
ACKNOWLEDGMENTS..........................................................................................
LIST OF TABLES......................................................................................................
LIST OF FIG U R ES........................................................................... . . . . . . . .
C hapterI . INTRODUCTION...............................................................................................
Selection of a system Method of investigation
MODELING THE SYSTEM .......................... ............................... ...............
Basic physiology of the heart Derivation of the heart model Derivation of the model equations Experim ental data
THEORY OF PARAMETER IDENTIFICATION................. . . . . . . .
Developing a generalized e r r o r function Developing a perform ance c rite rio n Application to the h ea rt model equation The com puter program Sim ulating the model
COMPUTER R E S U L T S ....................... ..........................................................
Introduction T est re su lts
DISCUSSION OF RESULTS AND CONCLUSIONS...................................
Identification of I3Identified constitutive re la tionships fo r R3 Sim ulating the identified m odels Conclusions
IV
4
19
28
44
iii
vi
vii
1
II.
III.
IV.
V.
REFERENCES 49
L ite ra tu re CitedR eferences Consulted but not Cited
APPENDIXES................................................. .. ...............................
APPENDIX A. The Com puter Program APPENDDC B. Davidon’s M inimization Method
v
53
LIST OF TABLES
Bondgraph symbol re p re se n ta tio n s .......................................................
Vital s ta tis tic s fo r the lin ea r c a s e ................................ ..
Vital s ta tis tic s fo r the PLIN 2 c a s e .....................................................
Vital s ta tis tic s fo r the PLIN 2 c a s e .....................................................
Vital s ta tis tic s fo r the PLIN 3 c a s e .....................................................
Vital s ta tis tic s fo r the PLIN 3 c a s e .............................................. . .
Vital s ta tis tic s fo r the APOL case ........................................................
Vital s ta tis tic s fo r the APOL c a s e .......................................................
vi
10
30
33
34
35
35
40
40
PageTable
1.
2 .
3 a .
3b.
4 a .
4b.
5 a .
5b.
LIST OF FIGURES
Diagram to show the four cham bers of the human h e a r t ...............
(a) A m ore accura te represen tation of the right h e a r t , (b) A m ore accura te rep resen ta tion of the left h e a r t. ...........................
(a) Three views (somewhat schem atic) of the left h ea rt only.(b) Bondgraph rep resen ta tion of the left h e a r t .............................
D iastolic bondgraph m o d e l ......................................................................
Systolic bondgraph model rep resen ta tion : (a) a tr ia l portion,(b) v en tricu la r p o rtio n ............................................................................
Sample problem s y s te m ............................................................................
Bondgraph model of the sam ple p ro b le m ............................................
L inear R3 ......................................................................................................
Flow v e rsu s tim e fo r the lin ea r c a s e ....................... ..........................
R3 for PLIN 2 ................................................................................................
Flow v e rsu s tim e fo r the PLIN 2 c a s e ...................................... ..
R3 for PLIN 3 ................................................................................................
Flow versu s tim e fo r the PLIN 3 c a s e ..................................................
R3 fo r the APOL c a s e .......................... ....................................................
Flow v e rsu s tim e fo r the APOL case ..................................................
Com parison of constitutive re la tio n sh ip s. .........................................
G ross overview of the en tire p r o g r a m .................... .............. ..
vii
6
6
8
12
12
19
20
31
32
34
36
38
39
41
43
46
60
PageFigure
1.
2 .
3.
4 .
5.
6 .
7.
8 .
9.
10.
11 .
12.
13.
14.
15.
16.
17.
CHAPTER I
INTRODUCTION
Recent years have w itnessed an increasing trend fo r people in the
m edical and engineering professions to couple th e ir ta len ts in a m ore
united effort to b e tte r understand the functioning of the human body. This
union of technologies has m eant the introduction of new viewpoints and te ch
niques in the field of physiological re s e a rc h . The eng in eers ' basic tool,
m athem atical modeling, has perhaps been one of the m ost noticeable add i
tions . There has been a significant amount of w ork done in recen t y ears
along these lines of deriv ing m athem atical m odels to ex p ress the functioning
of a subsystem of the body, and then, using analog, d ig ital, or hybrid com
p u te rs , analyzing the system through its model rep resen ta tio n . Apparently,
how ever, the la te st modeling techniques, nam ely bondgraph concepts and
com puterized model p a ram ete r identification, have not been utilized in this
physiological re sea rch effo rt.
The purpose of th is work is to investigate the feasib ility of apply
ing bondgraph and com puterized model p a ra m e te r identification techniques
to physiological re se a rc h .
1
Selection of a system
The card iovascu lar system was felt to be p a rticu la rly well suited
2
fo r use in th is study fo r severa l re a so n s . Study of the card iovascu lar system
is certa in ly tim ely and therefo re an investigation into the applicability of
these techniques to it would be of in te re s t to m any. Also from a p rac tica l
standpoint, the card iovascu lar system is perhaps the physiological system
m ost easily tran sla ted into the rea lm of understanding of one schooled in the
m echanical engineering d isc ip line . In addition, co rre la ted tim e h is to rie s of
the state variab les p re sen t in a system need to be m easured and recorded
before com puterized model p a ram ete r identification techniques can be applied.
F o r purposes of th is study, how ever, it was deemed unnecessary to p rocure
such data firsthand since th is w ork is intended to be a feasib ility study
ra th e r than an attem pt to gain new knowledge about the p a rtic u la r physiolog
ica l system used . It was therefo re n ecessa ry to se lec t a system whose state
va riab les had been previously recorded and presen ted in technical l i te ra tu re .
This kind of inform ation is available in the lite ra tu re for portions of the
card iovascu lar system . It was therefo re decided that the h ea rt with portions
of the m ajor veins and a r te r ie s connected to it would be a good system to
consider in th is study.
Method of investigation
Having selected the p a rtic u la r physiological system to be investigated,
the next step was to construct a bondgraph rep resen ta tion of that sy stem .
This requ ired considerable background re sea rch into the physiology of the
h ea rt in o rd e r to make the function of the model resem ble the function of
the re a l system as closely as p o ssib le . The m athem atical equations
rep resen tin g the functioning of that model w ere then d erived . Thus having
the model and its rep resen ta tive equations, the system was analyzed to see
3
what state v a riab les needed to be m onitored en vivo in o rd e r to provide the
experim ental data requ ired to successfully identify the model p a ra m e te rs .
This experim ental data and inform ation re la ting to the type of model r e p re
sentation being used was then operated on in a digital com puter by a g en e ra l
ized model p a ram ete r identification p ro g ram . In sim ple te rm s , th is p ro
gram v a rie s the values of the model p a ram e te rs until the difference between
the response of the model and the experim ental data is a m inim um . The
validity of the model, with its com puter identified p a ra m e te rs , w as then
checked by exciting the model represen tation with the same driving force
the rea l system experiences and com paring the model response, a s ind i
cated by its state v a riab les , to the rea l sy stem 's resp o n se . If the responses
w ere not close enough, the model was varied as the new inform ation in d i
cated and the procedure repeated until good correspondence between the
response of the model and the re a l system was achieved.
CHAPTER II
The h ea rt, in sim plest te rm s , is a device for pumping the blood
throughout the body’s c ircu la to ry system . The h ea rt consists of four v a r i
able volume cham bers, and ac ts as two distinct pumps attached to g e th er.
(Refer to F igure 1, page 6.) The two upper cham bers a re called a tr ia and
the two low er a re called v e n tric le s . The adjectives "righ t" and "left" re fe r
to th e ir re la tive position in the body as viewed by the person in whose body
the h ea rt l ie s . The right side of the h eart receives venous blood from the
body and pumps it to the lungs. The left side receives oxygen-rich blood
from the lungs and pumps it throughout a ll the body. During diasto le (that
portion of the h e a rt cycle in which the h ea rt re laxes) the valve between each
a trium and its respective ven tric le is open and the valves leading from the
ven tric les a re c lo sed . The blood flows into the a tr ia and then d irec tly into
th e ir respective v e n tric le s . Just before v en tricu la r systole (that portion of
the h ea rt cycle in which the h e a rt ven tric les contract), both a tria contract
(called a tr ia l systole), forcing one la s t charge of blood into the v e n tric le s .
As both ven tric les begin to contract in v en tricu la r systo le, the valve between
each a trium and its ven tric le is pushed shut. The p re s su re rapidly r is e s in
the ven tric les and the valves leading from each ven tric le to its respective4
MODELING THE SYSTEM
Basic physiology of the h ea rt
5
a r te ry is pushed open. These valves a re en tire ly passive and sim ply open
and shut as the p re s su re gradient d ic ta te s . During v en tricu la r systole blood
continues to flow into the a tr ia , which act as receiving cham bers for the blood.
It is in te resting to note that there a re no valves between the a tria and the
veins emptying into them in h igher anim als such a s m an .
The basic h ea rt rhythm is m aintained within the h eart wall itse lf
(see F igure 2). The heartbeat orig inates in a bundle of specialized tissue
ce lls located in the wall of the right a tr iu m . This is called the s in o -a tria l
(S-A) node. This signal to contract sp reads concentrically from the S-A
node a c ro ss both a tr ia , but it does not spread into the v en tric les because of
a connective tissu e b a r r ie r . A s im ila r bundle of specialized h ea rt m uscle
cell is located in the wall of the right a trium and is called the a tr io v e n tric
u la r (A-V) node. As the signal to con tract reaches the A-V node it is d e
layed for about .1 second, and then it is tran sm itted very rapidly through
the Purkinje f ib e rs , illu s tra ted in F igure 2, to the v e n tric le s . These s ig
nals a re e lec tro -chem ica l signals of a nature s im ila r to that of nerve im
pulses .
form s w ill be iden tical. The model derivation p resen ted below w ill speak
Derivation of the h e a rt model
The previous d iscussion of h eart physiology indicates that fo r
model purposes, the left and right halves of the h eart a re the sam e in
s tru c tu re and function. The p a ram e te rs of the model will change depending
upon w hether the right o r left heart is being investigated , but the model
to left lung from right
ven tric le
6
from right lung to left a trium
ao rta - to all p a rts of the body - from the left ven tric le
from left lung back to the left a trium
aortic valve (left sem ilunar valve) from left v en tric le into aorta
m itra l valve (left a -v valve) from left a triu m into left ven tric le
therighta trium
Figure 1 . --D iagram to show the four cham bers of the human h e a r t . (This diagram is schem atic . It is im possible to show the th ree-d im ensional h ea rt in a two-dim ensional d iagram accu ra te ly . See below .) The situation is d ia s to le .
blood back to righ t atrivuti
blood to left lung
from righ t lung
to left a trium
Pur kin je j . fib e rs f
blood back to right a trium(a)
blood to a ll p a rts of the bodyfrom the left ven tric le
" X
blood back from left lung
to left a trium
(b)
F igure 2 . --(a) A m ore accu ra te rep resen ta tion of the right h e a r t. (Stippled a rea rep re sen ts septum common to right and le ft a t r ia .) The s itu a tion is d iasto le , (b) A m ore accura te rep resen ta tion of the left h e a r t. Blood flows back to left a trium from right and left lungs. F rom left a trium blood goes into left ven tric le . The situation is d iasto le .
Position, of s -a I node
a-vnode
from a ll p a rts of body I back to a
torigh tlung.'
If we consider the short lengths of pulm onary vein and ao rta included
in the model as variab le volume cham bers (they have e lastic w alls), then,
from a lumped p a ram ete r modeling viewpoint, the left h ea r t can be envisioned
a s a s e r ie s of four cham bers connected respective ly by an o rifice and two
v a lv es . These cham bers a re excited by an e lec tro -chem ica l pacem aker
s ignal. Although very little is understood about the coupling m echanism
between th is e lec tro -chem ica l pacem aker signal and h e a rt m uscle co n tra c
tion, th is portion of the model can sym bolically be rep resen ted a s an e lec tric
voltage signal "e" which activates a switch controlling h ea rt con traction .
In addition, a f i r s t o rd e r l /b 's + l) te rm can be included to rep resen t the
frequency dependency of the strength of m uscle con traction . This signal
can be rep resen ted as following two different paths, one leading to the
a trium and the o ther to the v e n tr ic le . The v en tricu la r signal path a lso
needs an additional tim e delay function as previously d iscussed in the section
on basic h ea rt physiology. These two symbolic con tractile m echanism
paths serve as flow sources respectively to the a trium and to the v en tric le .
The bondgraph rep resen ta tion of th is model may be found in Figure 3.
In bondgraph notation the "0" elem ent is called a zero junction and
re p re sen ts a point in the system w here, in fluid system term inology, the
p re s su re is common to a ll model elem ents p re sen t at that po in t. The "1"
elem ent is called a one junction and rep re sen ts a point in the system where
to the left h ea r t only, with the understanding that it could a lso be applied to
the righ t h e a rt.
7
8
ao rta contracting because of e lastic ity
pulm onary vein
midway during ven - tr ic u la r d iastole a tr ia l systole
(final filling)
(a)
ao rta fully expanded again
Figure 3 .--(a )T h re e views (somewhat schem atic) of the left h ea rt only. At the re a d e r 's left a continuous flow is shown through the ao rta as the e lastic w alls of the ao rta con tract, forcing out into the b ranches of the ao rta the blood pushed into the ao rta during the previous v en tricu la r systole . In the m eantim e blood flows steadily through the left a trium (from the lungs) into the left ven tr ic le through the m itra l valve . Just before the next v en tricu la r systole an a tr ia l systole w ill occur (middle p ic tu re) causing a b rie f final filling of the ventr ic le . (b) Bondgraph rep resen ta tion of the left h e a r t.
v en tricu la r systole
SWITCH TIME DELAY
SWITCH
l/(Ts + 1]f<e W
e
f2 P3 f3 P4 faP2f lP1fpv
1- 0 -I
C1
— 1----/ \
R1
0 -I
C2 *2 R2
1 0
C3
1
*3 R3 C4
0 1
Pulm onaryVein
O rificeto A trium
LeftAt.
A-VValve
(b)
LeftVent.
AorticValve
Aorta
a p a rticu la r fluid flow ra te is common to all model elem ents p re sen t at that
point. (F or a m ore thorough discussion of bondgraph techniques, please
re fe r to re fe ren ces 1 and 2 .) Each ze ro junction in the h e a rt model above
corresponds to one of the four cham bers included in the m odel. Going from
left to right they correspond respectively to a segm ent of pulm onary vein,
the left a triu m , the left ven tric le , and a segm ent of the a o rta . S im ilarly ,
the one junctions correspond respectively to some point upstream of the left
a trium in a pulm onary vein, the o rifice between the pulm onary vein and the
left a triu m , the A-V valve between the a trium and the ven tric le , the valve
leading into the ao rta , and finally, some point dow nstream of the ao rtic valve
in the a o r ta . The "R," "C, " anu "I" elem ents correspond respectively co
re s is tan ce , capacitance, and inertance as used in o ther modeling techniques.
Table I indicates that portion of the re a l system to which the symbols used
in the bondgraph rep resen ta tion re fe r .
As previously d iscussed in the section on the physiology of the
h e a rt, during diastole fg=0 and Rg = o° since the ao rtic valve is shu t. L ike
w ise, during systole fg=0 and Rg= ® since the A-V valve is shut. This
m eans that the bondgraph model rep resen ta tion p resen ted above can be f u r
th e r sim plified into m odels rep resen ting the h e a rt during diasto le and s y s
tole resp ec tive ly . Using the sam e symbol rep resen ta tio n s , the d iastolic
model would be as shown in F igure 4, page 12 . S im ilarly , the systolic
model can be broken into two segm ents, as shown in F igure 5, page 12.
TABLE 1
Symbol R epresents
e E lectro -chem ical pacem aker signal at the S-A node
fpv Flow in the pulm onary vein before the a trium
P1 P ressu re in the pulm onary vein
f l Flow into the left a trium
^ e )as A trial flow source (a function of e)
P2 P ressu re in the left a trium
f2 Flow into the left ven tric le
f(e )vs V entricu lar flow source (a function of e)
P3 P ressu re in the left ventric le
f3 Flow into the ao rta
P4 P ressu re in the aorta
^a Flow in the ao rta dow nstream of the aortic valve
C 1 Capacitance of the pulm onary vein segm ent
h Inertance of the o rifice between the pulm onary vein and atrium
» i R esistance associated with the sam e orifice
c 2 Capacitance of the left a trium
i2 Inertance of the A-V valve
*2 R esistance of the A-V valve
10
BONDGRAPH SYMBOL REPRESENTATIONS
e
fpv
P1
^ e )as
P2
f2
f(e )vs
P3
f3
P4
C 1
h
c 2
h
*2
^ e )as
^a
f(e )vs
fxpv
11
Symbol R epresents
C3 Capacitance of the left ven tric le
h Inertance of the ao rtic valve
*3 R esistance of the ao rtic valve
c 4 Capacitance of the ao rtic segm ent
TABLE 1 - -Continued
12
Figure 4 . --D iasto lic bondgraph model
F igure 5 . --Systolic bondgraph model rep resen ta tion : (a) a tr ia l division o r portion , (b) v en tricu la r p o rtio n .
D erivation of the model equations
13
In d iscussing bondgraph model rep resen ta tions, the ze ro and one
junctions with th e ir attached re s is tan ce , capacitance, and inertance model
elem ents a re som etim es re fe rre d to as n o d es . In o rd e r to derive the equa
tions rep resen ting the en tire model, the equations rep resen ting each node
a re w ritten and then these nodal equations a re combined in m atrix fo rm .
The nodal equations a re derived by applying a few sim ple m athe
m atical re la tionships defined in the theory of bondgraphs. The basic r e la
tionships for zero junctions a re that the p re ssu re is constant a t that junction
and that the sum of the flows into that junction a re z e ro . In equation form ;
tionships p resen ted in th e ir linear derivative form a re :
P = Constant
Z f { = 0
The basic relationships fo r a one junction a re ju s t the opposite; i . e . , the
flow is constant and the sum of the p re ssu re s to which that junction is
exposed a re equal to z e ro . In equation form :
f z Constant
iP i - 0
The re s is tan ce , capacitance, and inertance elem ents a lso have basic defin
ing m athem atical re la tionsh ips called constitutive re la tio n sh ip s. These re la -
f = C dP/dt
To illu s tra te the derivation of a nodal equation, le t us derive the
equation fo r the second one junction in the d iastolic bondgraph m odel. At
th is junction f j is constant and the sum of P^, ?2 , P jp and P-^^ equals z e ro .
But P jj = I j df-^/dt and Pp^ = R jf i . T herefore the summation of p re s su re s
a t th is node is:
S im ilarly , the nodal equations for the P j, P2, P3, and f2 nodes can be d e
rived . They a re :
dPi/dt - 1/C 1(fpv - fj)
dP2/d t = 1/C 2(f1 - f2 * f(e)ag)
dP3/d t - 1/C 3(f2 * f(e )v s )
df2/d t = 1 /I2 (P2 - P3 - R2f2)
These five nodal equations can now be combined into one m atrix
P =Rf
P = I d f/d t
14
Pr l i d f j/d t - R ]fx - P2 = 0
Solving fo r the f ir s t o rd e r te rm we get:
d f j/d t -- 1 /IX (Px - R jfi - P2)
15
equation of the form X = AX + BU. This m atrix equation as p resen ted below
desc rib es m athem atically the bondgraph model of the h eart during diastole .
In th is model the variab les P^, P2 , P3, f j , f2 , and fpV a re called the state
v a riab les of the system . The f(e)ag and f(e)vs te rm s a re the forcing func
tions applied to the system .
Applying identical p rocedures to the a tr ia l and v en tricu la r portions
of the systolic bondgraph yields m atrix equations of the sam e fo rm . The
equation rep resen ting the a tr ia l systolic model is :
The equation rep resen tin g the v en tricu la r systolic model is likewise:
dt
ddt
0
0
0
0
0
0
0
0
0
l / I l - l f t i
l / I l -1 /I20
p l
P2
P3
h
f2
P 1
p2
P3
h
*2
i / c 1
1/C 2
1/C 3
0
0
fpv
f(e>as
f(e )vs
0
0
-1 /C j
1/C 2
0
- R i / i i
0 -R2/I2
0
0
- i / c 2
1/C3
d_dt
P1
P2
f l
0
0
0
0
0 • 0
i /C 2 • f(e)as
1 /C X fpv
i / l x -1/I1 -R1/I1
- l / C i
1/C2
P1
P2
h m
P3
P4
f3
0
0
0
0
i / I 3 -V I3 0 0
»
0
1/C3
-1 /C 4
f(e )vsP3
P4
_f3 ,
-V C 3
41/C 4
- r 3/ i 3
Experim ental data
16
In o rd e r to successfully identify a ll of the re s is tan ce , capacitance,
and inertance p a ram e te rs in any of the m atrix equations p resen ted above,
the dynamic values of a ll the state variab les as functions of tim e need to be
known as well as the forcing functions a s functions of tim e . T herefore , the
ing bondgraph and com puterized model p a ra m e te r identification techniques
Even though the full m atrix equation could not be used, it w as fe lt
that th is s ca la r equation, though certa in ly not as com plex, s till se rv es to
m eet the purposes of th is w ork, viz . , to investigate the feasib ility of apply-
df3/d t : I/I3 (P3 -P4) - ( R 3/ I 3) f 3
could possib ly be worked with was that rep resen tin g the ven tric le during
systole . However, the forcing function f(e)yS and the state variab le fa a re
not known so even the full m atrix equation for th is model cannot be worked
w ith . The only equation left to be used is the sc a la r equation:
With only these th ree state variab les available, the only model that
lite ra tu re specifies only the state v a riab les P3 , P4 , and f3 . Two a r tic le s ,
one by Kern W ildenthal, Donald S. M ierzwiak, and Jere H . M itchell (3) and
one by M ark I. M. Noble (4) both presen ted en vivo sim ultaneous record ings
of P3 , P4 , and fg v e rsu s tim e daken in dogs.
Unfortunately, the only usable experim ental data available in the
known in o rd er to identify a ll the model p a ram e te rs in the th re e -h e a rt m odels.
values of P l5 P2 , P3 , P4 , fpV, f ( e )a s> f 1, f2 , f3 , and fa need to be^ eV s ’
17
to physiological re s e a rc h . The p a ram e te rs I3 and R3, rep resen ting the fluid
inertance and re s is tan ce associated with flow through the ao rtic valve, w ill
then be those model p a ra m e te rs identified in th is w ork .
In o rd e r to fu rth er dem onstrate the flexibility of the techniques
being investigated, the p a ram ete r R3 w ill be identified both as a lin ea r and
as a nonlinear p a ra m e te r. In the nonlinear case it w ill be identified both as
a piecew ise lin ea r and an a rb itra ry polynomial p a ra m e te r.
Being m ore specific, th is m eans that the res is tan ce R3 will be
identified in the lin ea r form of its constitutive re la tionship a s presen ted
b e fo re .
In graphical form this relationship appears as:
In the piecew ise lin ea r form , the constitutive re la tionship becom es
a group of n equations rep resen ting the n segm ents of the piecew ise lin ea r
non linearity . In graphical form th is becom es, fo r a two segm ent nonlinearity :
PR3 = R3f3
18
The polynomial can be extended o r shortened as d esired and the exponents
a , b, c , e tc . , can be a rb itra r ily specified .
p R3 = R3f3 for f3 < f
PR3 = R3,f3 * Po fo r f3 > f
PR3 = ^ * Bf3b * Cf3C * • • •
In equation form th is becom es the two equations;
and
w here P0 is the y -ax is in tercep t of the extended second lin ea r segm ent.
In the a rb itra ry polynomial form , the constitutive relationship
becom es a polynomial equation:
CHAPTER III
THEORY OF PARAMETER IDENTIFICATION
Basically the technique of p a ra m e te r identification re lie s on v a ry
ing the values of the model p a ram ete rs until the e r r o r between the model
response and the tru e system response , as indicated by the state va riab les ,
is a m inim um . A generalized m easure of th is e r r o r , let us call it T(t),
m ust therefo re be derived . Perhaps the b est way to illu s tra te the fo rm ula
tion of a generalized e r r o r function would be to derive it in a basic sam ple
p ro b lem .
19
Figure 6 . - -Sample problem system
Let us take the c lass ic m echanical v ibration problem modeled as
a lumped m ass , suspended from the ground by a spring and dashpot in p a ra l
le l, and under the influence of a forcing function F . This model is i l lu s
tra ted in F igure 6 below .
Developing a generalized e r r o r function
20
The bondgraph rep resen ta tion of th is system is sim ply a one junction with
the M, K, and C elem ents attached to it and with the forcing function F a c t
ing upon i t .
If th is model equation tru ly rep resen ts the rea l system and if the
model p a ram e te rs M, C, and K a re co rrec tly identified, then the instan tan
eous e r r o r „f(t) w ill indeed be identically ze ro when the forcing function and
the state variab le X and its derivatives a re applied to the equation. M odels,
how ever, a re rep resen ta tions and never exact, so /( t ) w ill have some v a lue .
What the p a ram ete r identification technique seeks to do, th e re fo re , is to
Figure 7 . - -Bondgraph model of the sam ple problem
m inim ize </ ( t) .
MX + CX + KX - F = cT(t)
MX + CX + KX = F
The instantaneous e r ro r , / ( t ) , fo r th is model is then defined by the equation:
As d iscussed e a r l ie r , the nodal equation rep resen ting th is model is derived
by applying the lin ea r constitutive re la tionships to the one junction sum m a
tion equation. This yields the linear equation:
21
The defining equation for f ( t) may have to be fu rth er modified in
each p a rtic u la r application of these methods depending on what state v a r i
ab les may be experim entally determ ined from the re a l sy stem . Suppose,
fo r exam ple, that in th is sam ple problem only the variab les X and X can be
m easured in the rea l sy stem . The defining equation for £(t) th e re fo re needs
to be modified so that it only has the X and X te rm s in i t . This can be done
by m ultiplying the equation through by X and then in tegrating with respec t
to tim e . This modified equation then becom es the defining equation fo r the
e r r o r m easu re , £ (t) .
MXXdt + ^ CXXdt + \ KXXdt - \ FXdt = £(t) o o m o
Some of these in tegrals m ay be perform ed explicitly yielding the equation:
,t 9 ftMX2/ 2 *■ C \X 2dt + KX2/2 - \ FXdt = £(t)
This resu ltan t equation rep resen ts the generalized e r r o r equation. It can
be rep resen ted in equivalent form as:
ociqi * oc2q2 * a3q3 + <*4q4 = £(t)
w here
ofj = M
«2 " C
°f3 = K
« 4 = 1
q x = X2/ 2
- S *X2dt
q3 = X2/2
q4 z FXdt
T herefore the generalized e r r o r equation can be w ritten as:
22
nz« i< k - £ (t)i = l
This rela tionsh ip holds because the q 's depend only on the experim ental
data and a re , th e re fo re , independent of the cc’s . The «-’s a re independent
because they rep resen t the individual p a ram e te rs of the system m odel.
Now, the p a rtia l derivative of the perform ance c rite rio n , E, with
* £ ( t ) _hcci qi
<*lQl + cc2q2 t oc3q3 + « 4q4 = & t )
with re sp ec t to each oc gives us
Since £ (t) is a function of the p a ram e te rs cq then obviously E is a lso a func
tion of the p a ram ete rs The function E will therefo re tend to zero as the
model p a ram e te rs approach the system p a ram e te rs .
Taking the p a rtia l derivative of the generalized e r r o r equation:
Developing a perform ance c rite rio n
In o rd e r to make the e r r o r te rm m ore sensitive and useful, a non-
negative perform ance c rite rio n is defined using the e r r o r m easure £ (t).
The sim plest is the in tegral of the e r r o r squared .
E r ( £ (t)2dt jo
In th is expression , iL, as usual, rep re sen ts the f ir s t derivative of fg with
then becom es:
*3% + R3f3 = P3 ' P4
It would now be of in te re s t to apply the theory of model p a ram e te r
identification described above to the equation being used in th is study. That
equation is p resen ted again here below.
Application to the h ea rt model equation
With th is inform ation a g radient search technique may be used to
find the oc's which m inim ize the perform ance c r ite r io n . Thus the p a ram ete rs
a re identified.
F o r a given set of cc’s and q 's this expression may be explicitly
ca lcu la ted . It is therefo re possib le to calculate both the perform ance c r i t e r
ion, E, and its gradient with resp ec t to the unknown p a ram e te rs <x at any
point in the p a ram ete r space .
/"t£(t)q dt
;o
Applying the expression ju s t derived fo r the p a rtia l of £ (t) with resp ec t
to oc to th is equation yields:
re sp ec t to each is:
- A l _ = \ 2 £(t) - A £ (t) dt<* *i ^ c c i
23
resp ec t to tim e . F o r convenience le t P3 - p 4 be called DP. The equation
X3 3 f R3f3 = DP
24
The instantaneous e r ro r , / (t), is therefo re defined as:
I3f3 + R3f3 - DP = «T(t)
In th is p a rtic u la r system , only the state variab le f3 and the forcing
function DP a re available in experim ental da ta . The defining equation fo r
/ ( t ) th e re fo re needs to be modified so that it only contains the f3 and DP
te r m s . This can be accom plished by m ultiplying the equation through by f^
and then in tegrating with resp ec t to tim e .
The f ir s t te rm of th is equation can be explicitly in tegrated which yields:
E xpressing th is generalized e r r o r equation in equivalent form we get:
I3 \ f 3f3d t * R3 i f32d t - \ DPf3d t = £<‘> 0 0 0
✓ t A.I3f32 + R3 \ f g Ot - ^ DPf3dt = £(t)
o o
<xl ql + *2*2 + % q3 = £(t)
w here <x3 = I3 q l : f32
.t
0C3 - 1
<*2 ” R3 q2 - \ f32dto
q3 = C Spfgd to
Having the model equation in th is generalized e r r o r fo rm , we a re
able to evaluate the q 's from the experim ental data, the perform ance
25
The com puter p rogram used in th is study is a very generalized p ro
gram called Model Identification Algorith (M. I . A .) (5). It is a very g e n e r
alized program with a g rea t deal of flexibility built into i t . It has the cap a
bility of identifying model p a ram e te rs with lin ea r, p iece-w ise lin ea r, and
a rb itra ry polynomial constitutive re la tionsh ips in e ith e r sca la r o r m atrix
equations. Since such a generalized p rogram req u ires a g rea t amount of
com puter storage, it is w ritten as a linked p ro g ram . A linked program
sim ply m eans a la rge p rogram which has been divided into sm a lle r sec tions.
Thus the com puter can work on ju s t one section of the program a t a t im e .
When those calculations requ ired by that section a re finished, the com puter
need only save those quantities needed by subsequent sections and can e ra se
the re s t of the inform ation it had sto red to make storage locations available
fo r the next section . At p resen t the program used consists of five links plus
a sixth link added for the specific purpose of sim ulating the response of the
model being worked with in th is p a rtic u la r study once its p a ram e te rs have
been identified . A m ore thorough discussion of what model sim ulation m eans
in th is p a rtic u la r study is p resen ted below in the section entitled "Simulating
the m od e l." The en tire program is w ritten using F o rtran IV.
In some w ays, applying such a la rge and generalized com puter program
c rite rio n , E, and its g rad ien t. This is sufficient inform ation to identify the
model p a ram ete rs using a m inim ization technique re fe rre d to as Davidon’s
M ethod. (Refer to Appendix B .)
The com puter program
26
applying the p a ram e te r identification techniques to th is equation with this
experim ental data, we a re able to identify the values of the model p a ram ete rs
Ig and Rg , Now, in o rd e r to check the valid ity of the model with its id en ti
fied p a ra m e te rs , we m ust re v e rse the p ro cess ju s t gone through. In o ther
w ords, we assum e we know what Ig, Rg, and DP a re , and then solve the
model equation for fg as a function of t im e . If the model is an exact r e p re
sentation of the rea l system , the calculated fg will be identical to the fg
m easured experim entally in the rea l sy s te m . The re la tive degree of
V S + R3f3 = DP
Sim ulating the model
In identifying the model p a ram ete rs Ig and Rg in the model equa
tion
to the sim ple model equation used in th is study is analogous to killing
in sec ts with a sledge-ham m er. The work could be done with something
le ss (indeed, the lin ea r p a ram ete r identification w as originally done in this
study using a much sim pler program ), but the end re su lt is the sam e using
the heav ier too l. In a deeper sense, how ever, the use of the generalized
p rogram was requ ired to dem onstrate what tools a re available and also to
identify the nonlinear case of the p a ram e te rs in question.
A m ore detailed discussion of the com puter p rogram may be found
in Appendixes A and B .
we know what and DP a re as functions of tim e in the rea l system . Byf3
27
s im ila rity between the two fg’s is therefo re a m easure of the validity of the
m odel. This, in essence , is what is m eant by sim ulation of the m odel.
CHAPTER IV
COMPUTER RESULTS
Hie analog data p resen ted by Kern W ildenthal, e t a l . (3), and M ark
I. M. Noble (4) was manually digitized in o rder to use it on the d igital com
p u te r . Since the equation being worked with was taken from the model r e p r e
senting v en tricu la r systo le, the data was divided into as many equal tim e
steps a s possib le beginning at the point in the cycle w here the ao rtic valve
just begins to open and ending on the n e a re s t full tim e step before the flow
through the ao rtic valve, f3, re v e rse s in the p ro cess of shutting the valve.
Both papers p resen ted the data in such reduced physical dim ensions tha t the
maximum num ber of data points which could be obtained over th is in terval
with any degree of accuracy was th ir teen . The tim e step which resu lted in
the data from W ildenthal, et a l . , was .012 seconds and in the data from
Noble w as .014 seconds. T hirteen data points a re too few data points to
effectively identify higher o rd e r nonlinear p a ra m e te rs , a s w ill subsequently
be dem onstrated , so A itken's algorithm (6) o r ite ra ted lin ea r interpolation
was applied to the th irteen data points and they w ere expanded to 121 and
241 data point v e rs io n s .
Since the accuracy of the a lgorithm used in sim ulating the model
depends upon the size of the tim e step used, i t is im portant to note that the
28
Introduction
only difference between the application of these methods to the 13 and 121
data point versions was in the identification of the model p a ra m e te rs . I te r
ated lin ea r interpolation was applied during the sim ulation phase of the p ro
gram fo r the th irteen data point version so that the sim ulation of the model
used the sam e tim e increm ent in both the 13 and 121 data point v e rs io n s .
However, the 121 and 241 data point versions d iffer both in the model p a ra m
e te r identification phase and in the model sim ulation phase of the p ro g ram .
These com puterized model p a ra m e te r identification techniques w ere
f i r s t applied to the data from both re fe ren ces in the lin ea r and piecew ise
lin ea r cases of R g. The num erical re su lts w ere quite s im ila r so it was
decided to use only one of the data sources fo r the bulk of the t e s t s . The
data p resen ted by W ildenthal, et a l . (3), was the e a s ie s t to read and d ig itize .
It was therefo re selected as the one to be used . The re su lts p resen ted below
w ere obtained using the data obtained from th is refe rence .
The p a ram e te r Ig was identified in each case as a lin ea r p a ram ete r
only. The p a ram ete r Rg, however, was identified in the lin ea r, two segm ent
piecew ise lin ea r, th ree segm ent piecew ise lin ea r, and th ird o rd e r polynomial
fo rm s of its constitutive relationship as d iscussed in C hapter II. These
p a ram e te r identifications w ere , in addition, perform ed in each of the 13, 121,
and 241 data point v e rs io n s .
ica l fo rm . It would be well to explain some of the notation which w ill be u se d .
29
T est re su lts
The te s t re su lts w ill be p resen ted below in both tabu lar and graph-
The tabu lar heading "V ersion" re fe rs to the 13, 121, o r 241 data point v e r
s ions. "No. M ins." re fe rs to the num ber of one dim ensional m inim izations
requ ired to be taken in o rd e r to find the absolute minimum of the perform ance
c rite rio n E . The m easure of e r r o r p resen ted re fe rs to the e r ro r between the
sim ulated f^ ve rsu s tim e and the re a l fg v e rsu s tim e taken a t those data
points corresponding in tim e to the th irteen orig inal data po in ts . The "Ave.
Abs. E r r o r ” column therefo re re fe rs to the sum of the absolute values of
TABLE 2
VITAL STATISTICS FOR THE LINEAR CASE
V ersion No. M ins. R3 J3 A ve. Abs No.
E rro r%
XEC
13 4 .07047 .002354 26.5 19.6 1121 10 .07149 .002506 23.5 17.4 3241 4 .07148 .002507 22.7 16.8 4
L inear c a s e . The v ita l s ta tis tic s associated with the identification
of the lin ea r case of R3 a re p resen ted in Table 2 .
30
re fe rs to the percentage of the maximum flow, found in the orig inal th irteen
data points, that that average e r r o r r e p re s e n ts . That maximum flow was
135 m l/s e c . The "XEC" column re fe rs to the com puter tim e, in m inutes,
requ ired in the execution of the p ro g ram . The graphical re su lts p resen ted
below a re taken from the 241 data point v e rsions in each c ase .
the e r ro r s divided by th ir te e n . The column under th is e r r o r heading%Ttft
As is the case with a ll the te s ts , the value of the average e r r o r d ecreases
31
and the com puter tim e requ ired in c rease s with increasing num bers of data
p o in ts .
Two segm ent piecew ise lin ea r (PLIN 2) c a s e .- - In identifying p ie ce -
w ise lin e a r constitutive re la tionsh ips, the p rogram does not d irec tly solve
fo r Rg . Instead, the one using the p rogram judiciously picks values of fg
Figure 9 p resen ts a graphical com parison between the re a l and
sim ulated values of fg v e rsu s tim e . Note how in th is lin ea r case of R3 the
sim ulated response does little m ore than approxim ate the general wave form
of the re a l fg _
Figure 8. - -L inear Rg
The lin ea r constitutive relationship rep resen ted by the identified
value of Rg is p resen ted in graphical form in Figure 8.
Figure 9 . --Flow versus time for the linear case
This then m eans that the coordinates of the end points of the lin ea r segm ents
a re specified and thus the constitutive relationship is identified . If th ree
b reak point values of fg a re specified, then a two segm ent piecew ise linear
relationship w ill be identified . If four b reak point values of fg a re specified,
then a th ree segm ent piecew ise lin ea r constitutive relationship will be id en ti
fied, and so on. The break point values fo r th is two segm ent piecew ise linear
case w ere chosen to be fg = 0, 66, and 135. Table 3 p re sen ts the v ital s ta t is
tic s associated with th is PLIN 2 c a s e . The num bers p resen ted under the "0, "
V ersion N o. m in . h0
Break Valu 66
es135
13 8 .001945 -1.061 .9373 11.88121 12 .002127 2.016 .9316 13.01241 15 .002127 2.011 .9330 13.01
The constitutive re la tionship fo r Rg which re su lts from th is end
point coordinate specification is p resen ted in F igure 10 a lso found on the
following page.
TABLE 3a
VITAL STATISTICS FOR THE PLIN 2 CASE
a a
(called b reak point values) and then the com puter p rogram identifies the value
o£pR3 which corresponds to each of the specified b reak point values off 3 *
"6 6 ," and "135" columns rep resen t the values of identified by the com
pu ter p rogram to ex ist a t those corresponding b reak point values of fg .
PR3
TABLE 3b
34
VITAL STATISTICS FO R THE PLIN 2 CASE
V ersion A ve. A bs. No.
E r ro r%
XEC
13 22.2 16.4 2121 16.7 12.4 5241 16.3 12.1 9
Figure 10 . --R g fo r PLIN 2
Note that in rea lity the coordinates of the f i r s t end point should be
(0 ,0 ). Since so little inform ation is p resen ted by the data in that low flow
region, the com puter program let the value of Pr s tray a t that point andO
identified its coordinates as (0 ,2 ). The sim ulated response does not heavily
depend on the identification at th is point so th is does not seriously effect the
end re su lt .
The graph ical com parison between the re a l and sim ulated values of
35
fg v e rsu s tim e fo r th is PLIN 2 case is p resen ted in Figure 11. Allowing the
constitutive relationship for Rg to assum e th is sim ple nonlinear form sign ifi
cantly im proved the sim ulation of the decreasing portion of the fg v ersu s tim e
curve while degrading the sim ulation of the increasing p o rtio n . The average
absolute e r r o r , how ever, shows considerable im provem ent over the lin ea r
c a s e .
TABLE 4a
VITAL STATISTICS FOR THE PLIN 3 CASE
Version No. M ins. %0
Break Point 18
Value40 144
13 21 .001243 73.49 9.700 -4.878 13.33121 21 .001795 100.2 2.076 -2.878 12.77241 17 .001795 100.2 2.006 -2.877 12.77
TABLE 4b
VITAL STATISTICS FOR THE PLIN 3 CASE
V ersion A ve. A bs. No.
E r ro r%
XEC
13 490 364 2121 24.0 17.7 5241 19.3 14.3 10
T hree segm ent piecew ise lin ea r (PLIN 3) c a s e .--T h e break point
values for th is PLIN 3 case w ere chosen to be fg = 0, 18, 40, and 144. The
v ita l s ta tis tic s fo r th is case a re p resen ted in Table 4 .
36
Figure 11. --Flow v e rsu s tim e fo r the PLIN 2 case
37
In try ing to identify th is h igher o rd er nonlinearity , the th irteen data
points a re com pletely inadequate as illu s tra ted by the average e r r o r f ig u res .
So few data points sim ply do not provide enough inform ation for identifying so
many p a ra m e te rs .
The constitutive relationship for Rg which the com puter p rogram
identified in the 121 and 241 data point cases is p resen ted in F igure 12.
Once again, the coordinates of the f i r s t end point should be (0,0), but, fo r
the sam e reasons d iscussed e a r lie r , the p rogram identified the coordinates
(0 ,100). This g ro ss deviation from what in theory should occur caused a
problem when sim ulating the m odel.
In o rd e r to sim ulate the m odel, one equation rep resen ting each of
the th ree lin ea r segm ents had to be used . If the sim ulated value of fg fe ll
between zero and eighteen, the f i r s t of these th ree equations was used, and
so on. When the tru e equation rep resen tin g th is f i r s t lin ea r segm ent was
used, the ex trem ely high value of the v e rtic a l axis in tercep t, 100, and the
accompanying high slope of the line so g ro ssly affected the in itia l sim ulation
that the en tire sim ulation, in essence , d iverged . To avoid th is p roblem ,
the equation rep resen ting the f ir s t segm ent was modified such that it assum ed
a v e rtic a l axis in tercep t of z e ro and m aintained the sam e slope. This e s s e n
tia lly invalidates the sim ulation between f3 = 0 and 18, but it p re se rv e s the
in tegrity of the rem aining m ajority of the sim ulation by controlling the a d
v e rse effect of the in itia l sim ulation. The discontinuity n ear the end of the
sim ulated fg v e rsu s tim e curve (see F igure 13) occurs w here the sim ulation
38
prog ram switches back into the range of the f i r s t equation. The data points
beyond there a re therefo re m ean ing less. This problem does not appear so
vividly on the in creasing side of the curve because on that side the flow is
le ss than 18 m l/se c fo r only a very short tim e .
T hird o rd e r a rb itra ry polynom ial (APOL) c a s e . - -P aram eter iden tifi-
cation in the a rb itra ry polynomial case is s im ila r to the piecew ise lin ea r
case in that R3 is not d irec tly solved fo r . An a rb itra ry polynom ial in f3 is
specified and then the com puter p rogram identifies the coefficients of that
the constitutive re la tionship was indeed opposite to the form identified in the
Figure 12. - -Rg fo r PLIN 3
polynom ial. In th is te s t a th ird o rd e r polynom ial
was specified . The com puter p rogram then identified the coefficients A, B,
and C . The v ita l s ta tis tic s for th is APOL case a re p resen ted in Table 5.
As was the case with the PLIN 3 te s t , th irteen data points w ere not
enough to successfu lly identify th is h igher o rd e r non linearity . The form of
pRs = Af33 + Bf32 + Cf3
39
Figure 1 3 .--Flow versus time for the PLIN 3 case
121 and 241 data point v e rs io n s . The constitutive relationship identified in
40
the 121 and 241 data point versions is presen ted in Figure 14.
TABLE 5a
VITAL STATISTICS FOR THE APOL CASE
V ersion No. M ins. h A B C
13 4 .000626 .0001018 -.02006 .9777121 16 .002330 -.00001636 .004124 -.1771241 14 .002331 -.00001639 .004130 -.1773
TABLE 5b
VITAL STATISTICS FOR THE APOL CASE
V ersion Ave. Abs. No.
E r ro r%
XEC
13 ____ _____ 1121 15.5 11.5 4241 14.8 11.0 6
In fitting a polynomial to the conditions specified by any set of data
points, one runs the r isk of having that polynomial fit over the range of values
defined by the specified data points but not fitting a t o ther possib le but not
specified po in ts. This hazard is illu s tra ted in th is polynomial identified as
the constitutive relationship fo r R g. The value of fg in the data used only
ranged between 0 and 144 m l/se c , as noted in F igure 14. The polynomial
F igure 14. —Rg fo r the APOL case
41
42
could therefo re be expected to be a good rep resen ta tion of the constitutive
relationship between these po in ts. Indeed the polynom ial, as illu s tra ted in
F igure 14, looks s im ila r to the piecew ise lin ea r constitutive re la tionships
over the sam e range . However, fo r fg g re a te r than 144 the polynomial
assum es a negative slope and drops off very rapidly from there . Once again,
th is type of thing could only be expected since the input data only covered the
range of fg from 0 to 144 m l /s e c . In sim ulating fg ve rsu s tim e using th is
constitutive re lationship , the value of fg overshot 144 m l/se c , fell into the
negative slope region, and th e rea fte r diverged from the rea l fg . Upon reco g
nizing th is problem , the constitutive relationship was modified as indicated
by the dotted line in F igure 14. This m eant that fo r any value of fg g re a te r
than 120 m l . / s e c . the constitutive re la tionship becam e lin ea r having the same
slope a s was p resen t at fg = 120 m l/se c . The resu ltin g com parison of the
sim ulated and re a l fg v e rsu s tim e curves is p resen ted in Figure 15.
The sim ulated fg v e rsu s tim e now very closely m atches the re a l
curve over m ost of the descending portion of that cu rv e . In addition, the
e r r o r p re sen t over the ascending portion of the curve is le ss than in e ither
the PLIN 2 o r PLIN 3 c a se s .
43
Figure 1 5 .--Flow versus time for the APOL case
CHAPTER V
One salien t s im ila rity between a ll of the sim ulated fg v e rsu s tim e
curves is that fg increased much fa s te r than in the re a l system . This d is
crepancy occurs because Ig was constrained to be lin ea r in a ll the cases
studied . In a fluid system where the state v a riab les used a re p re ssu re and
flow ra te , the inertance te rm is defined by the equation I = C l/A . In the
equation being studied in th is work, the inertance Ig is the inertance a s s o
ciated with a h ea rt valve which, in essence , acts like a variab le a rea o rif ice .
When speaking about the inertance associated with flow through an o rifice ,
the equivalent length, 1, in the defining equation fo r inertance is proportional
to the d iam eter of the o r if ic e . Since the a rea of the orifice is proportional to
the d iam eter squared, the inertance te rm fo r an o rifice is therefo re p ro p o r
tional to one over the d iam eter of the o rif ice . T herefo re , the constitutive
re la tionship for Ig is not in rea lity lin ea r since the orifice d iam eter is not
constant with t im e . As the valve begins to open, the o rifice d iam eter is
sm all and the inertance te rm is la rg e . When the valve is fully open, the
d iam eter is a maxim um and the inertance te rm a m inim um .
The ao rtic valve.opens very rapidly and then rem ains fully open
during the m ajority of v en tricu la r sy sto le . The ine rtia te rm , I3, is th e re -
Identification of I3
DISCUSSION OF RESULTS AND CONCLUSIONS
44
45
The constitutive re la tionships fo r Rg identified in each of the case s
tr ied a re presen ted together in F igure 16. It is obvious that they a re all
re la ted to one another in th e ir general shape. This indicates that the model
p a ram ete r identification technique is consisten t and is not sim ply identifying
random re la tionsh ips.
As previously d iscussed in C hapter IV, a ll the constitutive re la tio n
ships should begin at coordinates (0 ,0 ). A b e tte r sim ulation and p a ra m e te r
identification could be accom plished in the piecew ise lin ea r cases if the s ta r t
ing point w ere constrained to fall at (0,0) when the constitutive relationship
is specified to be an odd re la tionsh ip . This im provem ent is p resen tly being
im plem ented in the general model identification algorithm p ro g ra m .
The occurrence of a negative slope a t the beginning of each of the
constitutive re la tionsh ips, except the lin ea r case , is quite su rp ris in g and as
yet not fully explained. This could perhaps be a re su lt of getting the phase
Identified constitutive re la tionships for Rc»
fore a t its minimum value over m ost of the tim e involved in this study. When
that inertance te rm is constrained to have a lin ea r constitutive relationship
( i . e . , be a constant), the model p a ram e te r identification p rogram identifies
an average value fo r i t . The identified value of Ig is therefo re close to the
value which occurs when the valve is fully open and is sm alle r than the values
of Ig which occur during the tim e the valve is opening. Thus the model is
in itia lly sim ulated using a value fo r inertance which is lower than it should b e .
The model therefo re responds fa s te r than the re a l system .
46
Figure 1 6 .--C om parison of constitutive re la tionsh ips
re la tionship between fg and DP out of line in digitizing the data o r in the
reproduction of the analog d a ta . In fu ture applications of these techniques,
the ideal way to take the data would be to pass the analog data through an
analog to digital co n verter as it is taken . This w ill b e tte r insure p ro p er
phase re la tionships and avoid the e r ro r s inherent in in terpolating between
data p o in ts .
It is in te restin g to note that th e re is essen tia lly no difference between
the model p a ram ete rs identified in the 121 and 241 data point v e rsions of each
c a se . T here is an im provem ent over the 121 data point version in sim ulating
the 241 data point v e rs io n s . Com puter tim e could conceivably be saved and
the sam e re su lts as the 241 data point version be obtained by identifying the
p a ra m e te rs from the 121 data point version and then by applying ite ra ted
lin ea r in terpolation to the data points during the sim ulation, a s w as done in
the 13 data point c a s e .
ev er .
47
Sim ulating the identified models
The com puter p rogram used to sim ulate the model in th is study was
a specialized p rogram good only fo r sim ulating m odels rep resen ted by f ir s t
o rd e r d ifferen tia l equations. The theory fo r w riting a generalized p rog ram ,
using m atrix exponential techniques, capable of sim ulating h igher o rd e r
equations o r m atrix equations, has been developed. The actual F o rtran p ro -4
gram m ing of th is generalized sim ulation method has not yet been done, how-
Conclusions
48
Bondgraph and com puterized model p a ram ete r identification te ch
niques have been shown to be effective tools in modeling th is physiological
sy stem . They certain ly a re not an ultim ate end in and of them selves, for
th e re is s till considerable leeway fo r the person applying them to use h is
judgment and b e tte r the re su lts obtained. This is p a rticu la rly illu s tra ted ,
in th is study, in the judgment involved in choosing the b reak point values fo r
fg in the piecew ise lin ea r c a s e s . Significantly d ifferent re su lts can be had
by applying the sam e methods to different break points . Obtaining the best
model and c o rrec tly identifying its p a ra m e te rs is therefo re s till an ite ra tiv e
p ro cess even with these m ethods. It is obvious, how ever, that these m ethods,
particu la rly the com puterized model identification technique, do ra ise the
level of the problem of modeling system s above the level of sim ply applying
intuition and estim ating the values of the p a ra m e te rs until the m odel's s im u
lated response approxim ates well the re a l sy stem 's resp o n se . The tim e
involved in co rrec tly identifying the model p a ram e te rs in physiological s y s
tem s can therefo re be significantly d ecreased . These methods can therefo re
be effective tools in the m athem atical modeling of physiological sy s te m s .
REFERENCES
LITERATURE CITED
1. F ree , Joseph C . "Bondgraphs--A Flexible Modeling Concept fo r SystemDynamics and C o n tro l," Provo, Utah, 1968. (M im eographed.)
2 . Kamopp, Dean, and Rosenberg, Ronald C . Analysis and Simulation ofM ultiport S y stem s. C am bridge, M ass .: The M. I . T . P ress , 1968.
3 . W ildenthal, Kern; M ierzwiak, Donald S .; and M itchel, Jere H. "Effectof sudden changes in ao rtic p re s su re on left v en tricu la r dp /d t, " Am erican Journal of Physiology, V ol. 216, No. 1 (January, 1969),187.
4 . Noble, M ark I . M . "The Contribution of Blood Momentum to Left V en tric u la r Ejection in the Dog, " C irculation R esearch , XXIII (November, 1968), 666.
5 . F ree , Joseph C . "P rogress Report on Development of the Model Identification Algorithm (M .I .A .) ," unpublished rep o rt subm itted to Lawrence Radiation Laboratory , U niversity of C alifornia, Septem ber, 1969.
6 . Conte, S. D. E lem entary N um erical A nalysis . San F rancisco : M cGraw-H ill, 1965.
7 . F re e , Joseph C . "Sum m er Effort 1968," unpublished paper of work atLawrence Radiation Laboratory , U niversity of C alifornia, 1968.
8 . F le tch er, R . D ., and Powell, M. D. "A Rapidly Convergent D escentMethod fo r M inim ization," Com puter Journal, Vol. 6 (July, 1963), 163-168.
50
REFERENCES CONSULTED BUT NOT CITED
Physiology
Abildskov, J. A .; Eich, Robert H .; Harum i, Kenichi; and Smulyan, H arold . "O bservations on the Relation between V entricu lar Activation Sequence and the Hemodynamic S ta te ." C irculation R esearch , XVII (Septem ber, 1965), 236-47.
Brady, Allan J. "Excitation and Excitation-C ontraction Coupling in C ardiac M usc le ." Annual Review of Physiology, XXVI (1964), 341-56.
C arlsson , E rik . "Experim ental Studies of V en tricu lar M echanics in Dogs Using the Tantalum -labeled H e a r t ." Federation Proceedings,XXVIII, No. 4 (July-August, 1969), 1324-29.
Herndon, Caleb W .; Sagawa, K iichi. "Combined Effects of Aortic and Right A tria l P ressu res on Aortic F low ." A m erican Journal of Physiology, V ol. 217, No. 1 (July, 1969), 65-72.
Hinds, Joseph E .; Hawthorne, Edward W .; M ullins, C harles B .; and M itchell, Jere J . "Instantaneous Changes in the Left V entricu lar Lengths O ccurring in Dogs during the C ardiac C y cle ." Federation P roceedings, XXVIII, No. 4 (July-August, 1969), 1351-57.
Lynch, P e te r R ., and Bove, Alfred A. "Geom etry of the Left V entricle As Studied by a High-speed C ineradiographic Technique." Federation P roceedings, XXVIII, No. 4 (July-August, 1969), 1330-33.
M itchell, J e r e H .; W ildenthal, Kern; and M ullins, C harles B. "G eom etrical Studies of the Left V entricle U tilizing Biplane C inefluorography." Federation Proceedings, XXVIII, No. 4 (July-August, 1969), 1334-43.
Olson, Robert M. "A ortic Blood P ressu re and Velocity As a Function of Time and P osition ." Journal of Applied Physiology, XXIV, No. 4 (A pril, 1968), 563-69.
O 'Rourke, M ichael F . "Im pact P re ssu re , L a te ra l P ressu re , and Impedence in the Proxim al Aorta and Pulmonary A rte ry ." Journal of Applied Physiology, XXV, No. 5 (November, 1968), 533-41.
51
52
Sandler, Harold, and Ghista, D hanjooN . "M echanical and Dynamic Im plications of Dim ensional M easurem ents of the Left V en tric le ." F e d e ra tion P roceedings, XXVIII, No. 4 (July-August, 1969), 1344-50.
Spencer, M errill P ., and G re iss , F rank C . "Dynamics of V en tricu lar E jec tio n ." C irculation R esearch , X (M arch, 1962), 274-79.
T sa k iris , A nastasios G .; Donald, David E .; Sturm , Ralph E .; and Wood,E a rl H. "Volume, Ejection F raction , and Internal Dimensions of Left V entricle D eterm ined by Biplane V ideom etry ." Federation P ro ceedings, XXVIII, No. 4 (July-August, 1969), 1358-67.
Physiological System Modeling
Beneken, Jan E . W ., and Rideout, Vincent C . "The Use of Multiple Models in C ard iovascu lar System Studies: T ran sp o rt and Perturbation M ethods." I . E . E . E . T ransactions on Bio-M edical Engineering, V o l. BME-15, No. 4 (October, 1968), 281-89.
C hristensen , Burgess N .; W arner, Hom er R .; and P ryor, T . A llan. "Sim ulation in the Quantitative Study of C arotid Sinus B ehavior." Simula - tion, VIII, No. 2 (F ebruary , 1967), 89-93.
McLeod, John. "PHYSBE . . . A Physiological S im ulation." Sim ulation, VII, No. 6 (D ecem ber, 1966), 324-29.
________ . "PHYSBE . . . A Y ear L a te r ." Sim ulation, Vol. 10, No. 1(January, 1968), 37-45.
Topham, W. Sanford. "An Analog Model of the Control of C ardiac O utput.” Sim ulation, VIII, No. 1 (January, 1967), 49-53.
APPENDIXES
APPENDIX A
THE COMPUTER PROGRAM
THE COMPUTER PROGRAM
To help c larify the p rog ram , each link w ill be outlined here with a
b rie f descrip tion about its overall function. In addition the m ain subroutines
in each link w ill be lis ted and briefly d esc rib ed . F o r sim plicity , flow
ch arts of each link of the p rogram a re provided in F igures 17-21.
Main
The main link is the independent link and serves only to d rive the
dependent lin k s . It has only one m ain subroutine called ch a in .
C hain . - -Subroutine CHAIN'S only functions a re to estab lish the
common dim ension statem ents used by the dependent links and to ca ll each
dependent link in tu rn .
Link one
Link one reads in the input variab les which specify, among other
things, the num ber of equations to be worked with, what type of constitutive
relationship each h as , how many data points a re going to be subsequently
provided, and the magnitude of the tim e step between data p o in ts .
Input. - -This subroutine is the driving subroutine fo r the link . It
ca lls the o ther subroutines and then m anipulates the inform ation they p ro
vide to get it into the p ro p er form to be used by subsequent links.
55
56
A read. --AREAD is a subroutine designed to allow fo rm atless input
of e ith e r alphanum eric, in teg re r, o r floating point variab les . It is the sub
routine which actually reads the input v a ria b le s .
V ecout. - -VECOUT sim ply p rin ts out what AREAD has re a d . This
provides a check to insure that the input inform ation was read c o rre c tly .
U korkn. - -This is a function subroutine which determ ines w hether
each model p a ram ete r defined in the input inform ation is specified as a
known p a ram ete r o r an unknown p a ram e te r which is to be identified.
Typef. --TY PEF is a function subroutine which determ ines w hether
the unknown p a ram ete rs a re to be identified a s lin ea r, piecew ise lin ea r, o r
a rb itra ry polynomial functions.
S y m etr. --SYMETR is a function subroutine which determ ines w hether
the unknown p a ram ete rs a re to be a rb itra ry , even, o r odd functions.
Link two
Link two reads in the data ca rd s specifying the values of the state
v a riab les to be used in p a ram ete r identification.
XVF. --XVF is the driving p rogram fo r the link . It sim ply se ts up
the common storage fo r the link and ca lls DA TAG.
Da ta g . --DATAG is the main p rogram of the link . It ca lls the o ther
subroutines and then m anipulates the inform ation they provide to get it in
p roper form to be passed on to subsequent links.
5 7
A read . - -AREAD was previously d iscussed in connection with link
on e .
V ecout. --VECOUT was previously d iscussed in connection with link
one.
Link th ree
Link th ree takes the input control specifications provided by link
one and the experim ental data specifying the s ta te v a riab les as provided by
link two, and then generates the q^’s previously described in the discussion
of the generalized e r r o r equation.
G etq .--G ET Q is the subroutine in link th ree which actually does
the calculations ju s t d iscu ssed .
Link four
Link four is a buffer link which sim ply re -a rra n g e s some of the
storage locations for quantities calculated in previous links and a lso p rin ts
out the values of the qds fo r re fe rence p u rp o ses. Its only subroutine is
called QOUT.
Link five
Link five takes the inform ation provided by a ll the previous links
and does the actual p a ram ete r identification.
58
S ea rch . --T h is is the subroutine which im plem ents Davidon's Minima -
zation Method. It therefo re m anipulates the values of the <*'s until the g e n e ra l
ized e r r o r £(t) is m inim ized. A m ore thorough discussion of th is subroutine
may be found in Appendix B.
G ee.--G E E is the subroutine which calcu lates the gradient of the
perform ance c rite rio n E .
E fk .--E F K is the subroutine which evaluates the perform ance c r i
terion E .
E r r . - -ERR is a function subroutine which form s the summation of
products («jHqj) which is the generalized e r ro r £(t).
Link six
Link six is the link added to the generalized p rogram to sim ulate
the response of the h ea rt model being used in th is study once the model
p a ram e te rs have been identified. This then se rv es a s a check to see how
closely the model approxim ates the rea l system it re p re se n ts .
Sim u. --SIMU is the driving p rogram of link s ix . It re -a rra n g e s
some of the inform ation provided it by previous links and then ca lls the
subroutine EULER.
E u le r . - -This subroutine applies E u le r 's method for the solution
of o rd inary d ifferen tia l equations (6) to the model equation and solves fo r
59
fg . It a lso p lots out an X-Y plot of the re a l system s fg v e rsu s tim e with the
sim ulated fg v e rsu s tim e superim posed upon i t .
60
Figure 1 7 .--G ro ss overview of the en tire p rogram
Link One
61
Figure 1 8 .--L ink one
Figure 1 9 .--L inks two, th ree , and four
Link Two
Link Five
62
Figure 2 0 . --L ink five
Link Six
Figure 2 1 .--L ink six
APPENDIX B
DAVIDON’S MINIMIZATION METHOD*
APPENDIX B
*The en tire Appendix has been taken from "Sum m er E ffort 1968,” b y j . C . F ree , pp. 56-59. (R e fe re n c e ? .) No fu rth e r re fe ren ce will be made to the p ap e r .
64
Si = g (S i)
Given H j and oc, compute
DAVIDON'S MINIMIZATION METHOD*
This Appendix p resen ts the algorithm ic steps used in Davidon’s
m ethod. It a lso explains the one dim ensional m inim ization procedure used
a s one of the e ssen tia l steps in the m ethod. An application to a te s t function
is given.
The notation used is :
positive definite m a trix .
tinvecto r of n p a ra m e te rs a t the 1 stage in the m inim ization p ro cess .
the sca lo r function which is being m in im ized with resp ec t to 9^ .
the gradient of with resp ec t to gc .
H
S£i
H* i)
Si
The method req u ires an in itia l H which is positive defin ite . An
orig inal choice could be an H with diagonal elem ents equal to one if no b e tte r
inform ation is availab le .
The requ ired steps a re a s follows:
Take a step
65
Having the new H, repeat the steps described above by finding the
which m inim izes <j> along the line
£ = « i* l ” Ki t i Hi + 1 Si + 1
Find the minimum of <f> along the line
by doing a one dim ensional search on K j.
Then set
Compute the new grad ien t vecto r
and the change in g .
Update the H m atrix by
= -Ki Hjgi
c c . stj ' ^ ^ g i
oc. , = oc. -K- . H- er;—l 4-! ~ i ^ l m m n i a i
f U l = g < s i + l)
ASi = f i +i - g i
Hi + 1 =Hj *• (HiASi) (Hi/Vgi)T^ « iT&gi AgiTH iAgi
Ki + 1
and so on .
66
the step size is in c reased . Additional steps a re taken with continually in c re a s
ing step size until the function in c re a se s . When th is happens a parabolic fit
is made to the la s t th ree points and the minimum determ ined fo r that pa rab o la .
Although th is may be a coarse minimum, it could be refined by fitting another
parabola to the minimum point determ ined and the la st two points, e tc . As
a p a rt of Davidon’s technique it is probably sufficient to do the coarse m in i
m ization as was done in th is w ork.
Since the c h a rac te ris tic s of the <f surface a re unknown to begin with,
it is possib le fo r the in itia l step to in c rease <f> ra th e r than d ecrease it, by
stepping c lea r a c ro ss the valley of the m inim um . The f ir s t step is therefo re
adjusted until <f is reduced and account is taken of which side of the valley
the descent is proceeding, i . e . , w hether <j> is reduced by increasing or
decreasing K.
w ard d irection of the l in e . If the function being m inim ized is reduced then4
The one dim ensional m inim ization method used in th is work is a
basica lly sim ple technique which begins with an a rb itra ry step in the down-
Observe that the m inim ization of a function of n v a riab les has been
reduced to a num ber of m inim izations of that function with respec t to a s in
gle variab le K. F le tch er and Powell show that exactly n of these m in im iza
tions a re requ ired for a quadratic surface (8).
A FEASIBILITY STUDY OF THE APPLICATION OF BONDGRAPH MODELING
AND COMPUTERIZED NONLINEAR MODEL PARAMETER
IDENTIFICATION TECHNIQUES TO THE
CARDIOVASCULAR SYSTEM
Randall L . Taylor
D epartm ent of M echanical Engineering
M. S. D egree, June 1970
ABSTRACT
A sim ple model rep resen tin g the function of the left h ea rt and short segm ents of the m ajor v e sse ls connected to it was developed using bondgraph modeling concepts. The m athem atical equations rep resen ting that model w ere then derived from the m odel.
The num eric values of the model p a ram e te rs rep resen ting the r e s i s t ance to flow and the fluid inertance associated with flow through the ao rtic valve w ere then identified . The inertance p a ram e te r was evaluated only in the lin ea r form of its constitutive relationship while the re s is tan ce p a ram e te r was identified in its linear form as well as in two and th ree segm ent piecew ise lin ea r and a th ird o rd e r polynomial fo rm .
The resu lting model response sim ulations showed the model and its identified p a ram ete rs to be an adequate rep resen ta tion of the rea l system under the constra in ts imposed on the model and dem onstrated the u tility of applying these techniques to physiological sy s te m s .
COMMITTEE APPROVAL:
