Assessment of the indicator-dilution technique in
nonstationary flow
Citation for published version (APA):
van Reth, E. A. (1984). Assessment of the indicator-dilution technique in nonstationary flow. Technische
Hogeschool Eindhoven. https://doi.org/10.6100/IR9372
DOI:
10.6100/IR9372
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Assessment of the indica.or-dilution technique in nonstationary flow
Proefschrift
Ter verkrijging van de graad van Doctor in de Technische Wetenschappen aan de Technische Hogeschool Eindhoven, op ge.:ag van de Rector Magnificus, Prof. Dr. S.T.M. Ackermans,
voor een commissie aangewe~en door het College van Dekanen in hel openbaar Ie verdedigen op vrijdag 2 maart 1984
Ie 16.00 uur
door
Eric AI bert von Reth geboren te Eindhoven
Prof. Dr. J.A. P()ulis en
Prof. Dr. A. Versprille
Aan:
CONTENTS.
ABSTIlACT.LIST OF SYMIlOLS.
1 INTRODUCTION.
1.1 The indicator-dilution technique. 1.2 Physiological backeround.
1.2.1 Errect~ of normal breathing and artificial ventilation on cardiac output.
1.2.2 E~timation or mean cardiac output in ~luctuating blood flow.
1.3 The purpo~e and ~cope of the present investigation. 1.4 Referenoe".
2 DESCRIPTION OF r~ INDICATOR-Dl~UTION CURV£.
2.1 Introduction.
2,2 Model simulation of indicator dilution curves. .2.2.1 Oi~tributed models. 2.2.2 Compartmental model~. 7 9 Ll 15 15 18 18 24 25 26 30 30 30 30 32
2.3 Compari~on of the compartmental and distribution models. 34
2.3.1 Description of investigation". 34 2.3.2 Fitti~s of theoretical function" to measured curves. 34 2.3.3 CompSri"on of the theoretical model function". 36 2.4 Di,,c1.105,,ion.
2. ;. References.
3 STUDIES ON THE INDICATOR-DILUTION METHOD IN MODELS OF VAIlIAI)J"E FLOW.
3.1 Introduction. 3.2 Theoretical analy.,ie,
3.3 Model experiments ~n modulated flow. 3.3.1 Experiment~l set-up.
3.3.2 Experimental procedllr;'e and data proceesing. 3.3.3 Ranp,e of eKPeriments. 38 40 44 44 4~ 47 47 48 50
3.4 Reeult".
3.4.1 Model experiments in consti'lnt fl<)w.
3.4.;> Model experiments in 5inusoicially modulated 3.5 Discussion.
3.6 References.
4 ERIlOR IlF;DUCT~ON WITH A WEIGHTING FUNCTION DERIVED FROM ARTERIAL PRESSURF-.
4.1 Introduction.
4.2 Approximations of the flow variation funcion. 4.3 Weighting functi,ms from pulse-contol)r re-lations. 4.4 Model experiments in modulated pulsating flow.
flow. 50 50 53 56 50 63 63 63 66 69 4.4.1 E~perimental set-up. 69 4.4.2 Experimentsl procedure. 70
4.4.3 Results from the experiments in the physical model. 73
4.5 Di:!;l-cUBsion. 4.6 ReferenC2s.
79
82
5 INSTANTANEOUS FLOW ESTIMATE FROM T~E INDICATOR-DILUTION CURVE. 84 5.1 Introduction.
5.2 Theory. 5.3 Method.
5.3.1 Data processing.
5.3.2 Testing with theoretical curves. 5.3.3 ExperimentaA set-up. 5.4 Results. 5.5 Discussion. 5.6 References. SUMMARY. SAMENVAtTING. NAWOORD. CURRrCULUM V11A£. 84 84 86 A6 86 88 1:l9 97 100 101 103 105 106
ABstRACT.
The investigations descr~bed in this thesis were intended to assess the indicator-dilution method with regard to its
application for estimat~ng the mean flow of the blood circulation. Earlier studies indicated that, especially during artificial ventilation, large errors could occur due to the nonstationsry character of blood flow.
9
The present study, comprises both theoretical analyses and model studies in hydrOdynamic set-ups. After an introduction in which the indicator-dilution method and the physiological background are described, a comparative study is given of different theoretical representations for indicator-dilution curves. Based on the results a compartmental approach was used to describe the effect of nonstationary flow on the mean flow estimate by the
indicator-dilution technique. The parameters for determining the deviation of the estimated from the actual mean flow were derived and the influence of each parameter was established. By meBnS of experiments w~th physical models, the theoretical results were confirmed. From these findings improvements to the accuracy of mean flow estimates with averaging techniques have been euggested. furthermore, an accurate estimate could be obtained from a single indicator-dilution curve by correctjng it for the flow variations. Since these variations can not be measured directly in practice, a related function was determined from the r~corded pressure signals. Finally, a method was considered for obtaining instantaneous flow information from the shape of the
LIST
OF SYMBOLS.
A relative amplitude a constant A relative amplitude of Q(t) q b constantc
concent.ation CO ca.diac outputCPPV oontinuous positive pressure ventilation concentration of the injected indicator concentration behind the k-th compartment concentration of the basic fluid
oonoentration behind two unequal mixing chambers effective diffusion coefficient
How
f flow v~iation
FPT first passage times F a Fc Fe Fm p w p pa F t Fm,k Fv •k IPPV
amplitude of flow variation conatant flow
estimated mean flow mean flow
estimated mean flow after weighting blood flow in pulmonary artery air flow in the traohea mean flow during the k-th beat flow variation during the k-th beat intermittent positive pressure ventilation LDRW local density random walk
M'IT mean transit time
mass of indicator injected number of compartments
PEEP positive end expiratory pressure Pe Peclet number
pressure of the dicrotic notch end diastolic pressure mean pressure m 11m3 11m3 11m3 11m3 cm2/5 mlls ml/s mlls mlls mlls mlls mIls mlls mlls mlls kg kPa kPa kPa 11
Pmd Pra P t Pth Ptm Pven Q q" maximal pressure mean diastolic pressure ri~ht atri~l pressure pressure in the trachea thoracic pressure
transmural right atrial pressure venous Pressure ~stimate of flow mean value of q(t) qA quotient of areas r RD ROm~x ROlli j quotient of MTT-valuee radius
place coordinate in radial direction relative deviation
maximal value of RD(~ap)
relative deviation after weighting with pulse-contour relation denoted with j
RMS root mean square SO standard deviation Sdi S sy S 1 T t u v
area of a pressure pulee during diastole area of a pressure pulse dur~ng sy6to~e part of S above p
ed sy
time constant time
time interval between injection and appearance time interval of diastole
total pulse time
passage time of the indicator time interv~l of systole start of pulse veloci ty volume of a compartment v relative flow VR venous return V inj Vk V m injected volume
stroke volume of the k-th beat
mean stroke volume over 3 modulation cycle
kPa kPa kPa kPa kPa kPa kPa !/l; lis m m kPa s kPa B kPa 13 s 13 s s mls ::l m ml ml ml
w.
J x x o ZEEPme~n relative flow during the k-th be~t variation of relatjve flow during the k-th beat weighting fUnction
weighting fUnction derived from the pulse-contour relation denoted with j
place
distance between injection and sampling points zero end expiratory press~re
~ scale factor
y dimensionless conoentration
Tk dimensionless concentration behind the k-th compartment A skewness parameter
'"
qo
frequenoy frequency of Q(t)-variation dimensionless frequenoy'j,k quantity related to stroke volume of the k-th beat derived with pulse-contour method denoted by j 'j,m mean of ~j.k for one modulation oyole • phase
+ap phase at the moment of appearance +inj phase at the moment of injection
'q
phase of Q{t)e dimensionless time variable
Sap dimensionless time interval between injection and ~ppearanoe of indicator
median transit time
13
m
l/s 1/13
1.
INTRODUCTION.
1.1 The indicator-dilution technique.
In intensive care medicinei ofteni pressure monitoring in the pulmonary artery is performeo by means of a multi-purpose (Swan-Ganz catheter). Beside~ pressure measurements, this catheter allows the sampling of pulmonary arterial blood and the estimation of the mean cardiac output with the indicator-oilution method. Flow determination with the indicator-dilution technique. however. can produce considerable errors when applied during variable flow conditions that exist in clinical situations due to the combination of the heart action and the influence of the ventilation on blood flow. This thesis presents the results of a model study conoerning these sOurces of error for the
indicator-dilution method and sug~esta ways to reduce them in clinical applications.
15
In 1897, Stewart described a method for measuring the flow through a system by injecting a fixed amount of indicator at the inlet and detecting the amount discharged at the outlet. The indicator-dilution method can be explained by considering a system that consi~ts of a mixing volum~ through which a fluid flows cont~jntng a COncentration of indicator, Co' The flow function is de~oted by P(t). An amou~t of indicator (m
inj) is injected at the inlet ~no compJ.ete mix),ng is as~umed t:lefore the samp~ing location. The i~dicato~ passing this point provides a coneentration versus time function, C(t), the indicator-dilution curve. A typical example of an ind~cator-diluUon curve for a constant flow measurement. after a delta-f~nction i~put of indicato~
(bolu~-injection), is giv~n in fig. 1.1. This proCsss can be described by the law of mass conservation for differences in the indjcator concentration with respect to the concentration of the t:lasic fhlid, assuming rotational symmetry and no loss of indioatorl
m . . lnJ
R '"
where R is the rad~us of the cross-section of the outlet. C(r,t) and u(r,t), respectively, are the concentration of indicator and the velocity of the fluid as R function of tim~ ( t l and place in the radial direction (r) in a cross-s~ction at the sampling point.
t
fig. 1.1, A typical example of an indicator-dilution curve (constant flow, bolus-injection).
Assuming a homog~neous cross-sectiona~ distrtbution of indicator gi.ves;
m = f~ Crt) [2~ fR u(r,t) r dr
l
dtinj •
Introducing the flow function defined by;
FIt) ~ 2~ [R U(r,t) r dr leads to, tn . . tnJ f~ F(t) Crt) dt ( 1.2) ( 1.3) ( 1.4)
In this study, a restriction will be made for a delta-function input (bolus-injection) of indicator with a fixed
17
Concentration, Ginj , because of its special clinical relevance. Other inje~tion functions were described by Cropp and Burton (1966). Lowe (1968) and Chamberlain (1975). ~or the bolus-injection. when a volume Vinj is injeoted, minj is determined by:
m. .
lnJ
The mean flow, ~m' defined by:
F m f lim 1. J F(t) dt t' .~ t ~ (1.5) (1.6)
~Bn be determined from the indicator-dilution curve and eq. (1.4):
F m where~ vItI m. . / ,- vItI Cit) dt lnJ F(t) ! F m
defines the relative flow function.
For ~onstant flow Fc (v(t) ~ 1) eq. (1.7) gives:
F c
which is ca1led the Stewart-Hamilton equation (FOX (1962)).
(1.7)
(1. 9)
The applioation of the indicator-dilution technique for physiologioal or c1inica~ purpo~es has been the subject of many investi~ation~ and it has become a routine clinical ob8~rvation technique. This techniquB is frequently used to obtain an estimate Fe of the mean t~ow using eQ. (1.9) under in-vivo conditions in spite of the nonstationarity of the flow due to heart action and ventilation:
F
e m . . InJ
i
~
For an extensive review of the history or the technique, reference is made to Dow (1956), Wood (1961), FoX (1962) and
ten Hoor (1969). The determination of error sources and improvement of the technique's accuracy were studied by many investigators (Swan and Beck (1960), Mohammed et al. (1963), Cropp and Burton (1966), Pavek at a1. (1970), Wessel et a1. (1971), Wilson et a1. (1972), fischer et al. (1978). Runciman et a1. (1981». Attention has been paid to the influences of variable flow on the method by Luthy and Ga1etti (1964), Cropp and Burton (1966), Bassingthwaighte et al. (1970), Solberg (1973). Scheuer-Leeser et al. (1977), Jansen at a1. (1979, 1981), Armengol at al. (19Bl), Snyder and
Pawner (1982) and von Reth et al. (1983).
1.2 Physiological background.
1.2.1 Effects of normal breathing and artificial ventilation on cardiac output.
Under steady state conditions, cardiac output is the result of an equilibrium between the pumping abiJity of the heart and the input of blood, the venous return (Guyton (1973)). The pumping ability can be expressed by the cardiac output as a function of the filling pressure Dr transmural right atrial pressure (p
tm), which is defined ~s the difference between the intra-atrial pressure (Pra) and the pre8sur~ in the thoracic cavity surroundjng the heart (Pth);
(1. 11)
When a slightly negative value is assumed for Pth of -0.5 kPa, whi<;h i" u!),lally the case fOr normal end-exp; ration, the cardIac output can be given as a f,metlon of atrial pressure (Pn,l with respect to atmospheriC pr'<::ssure by a transform~tlon
Dr
the horizontal axis. The relations between cardja" output and P["a and Ptm' respectively, are Illustrated in fiR. 1.2. Venous ["eturn jgnot feasible because venous return depends on the diff~rence between the pressure in the vei.ns outside the thorax and the pressure ineide the heart. Therefore. both pressures have to be related to the same reference value. atmospheric pressure. In fig. l.3. an example is shown to represent pumping ability and venous return under normal oonditions in manr The intersection of the curves determineS the values of venOus return and cardiac output in the state of equilibrium.
0
--0. 0 Pro (kP.)
.
•
-
--...
0 0,1> 1.!; P
lm (kP.)
fig. 1.2: Cardiac output (CO) curve as a function of Pra and P
tm' respectively (from Guyton (1973». Any factor that changee the thoracic pressure. such as breathing, artificial ventilation, or opening the thorax, will shift the cardiac output Curve as a function of P
tm along the Pra-axis (see fig. 1.2). This shift will influence the intercept of the venous return curve and the Cardiac output CUrVe.
During normal breathing, intra-thoracic pressure falls during inspiration and rises during expiration with the result that the cardiac owtpwt curve moves back and forth during each cycle
(f~g. 1.4). Th@ Venous peturn cUPVe ~tays almost the eame during normal breathing giving a per~odic variation of the cardiac output. In fact. there are sevepa~ interf<lring inter~ctionS
t
CO
CO
VR. 11UrMai
( Vminl
10fig. 1.3: Cardiac output (CO) curve and venouS return (VR)
curve under normal conditions ~n man (from Guyton (1973)). cO vR
t
( X,,;n
l 10 '0,5o
0.5fig. 1.4: Normal breathing cau8ing the cardiac output to mOve bet~een the points A and R
governing the cardiac output and venous return curves
(Guyton (1~7J)), but the back8round~ o~ the~e effect~ are beyond the scope o~ this study; only a de~cription o~ the basic idea is given.
21
During artificial ventilation, a statio lev~l of pressure in the airway and a eyclic fluctuation sUPerimposed on it, can be distinguished. Both influence flow and pressure in the circulation. The ventilation is called either intermittent positive pressure ventilation (rPPv) when a cyclic fluctuation ie applied and the end-expiratory pressure ie ~ero (ZEEP), o~
continuous positive pre~~ure ventilation (CPPV) when the cycliC fluctuation is superimpoeed on a positive end-expiratory pressure (P8eP)-level. Without going into the backgrounds of all types of ventilation, we will present an analysis of the e~~ecta o~ POsitive pressure ventilation.
Immediately after a positive pressure is applie~ by insufflation of the ventilatory volume, the normal cardiac output eurve shifts to the right due to the increase in right atrial pressure following the increase in thoracic p~essure. Cardiac output will be de~rea~ed from point A to B (fig. 1.5) when stationary conditions are obtained during in~ufflation. Howev~r, before stationary condition~ are e~tabli5hed. flow will be smaller than the value at point B, for the fir~t moments, due to the venous capacity whieh will be ri~led UP during pressure rise.
The oombined effect of PEEP and Superimposed insufflation during CPPV was studied by applying PEEP as a ramp, up to 1.5 kPa per 22.5 minutes, during e~periments with pigs (Schreuder et al. (1982)). In fig. 1.6. two individual examples of the
cardia~ output changes during this procedure are presented. Sohreuder et al. showed a corresponding, but Smoothed, response curve for the mean of 47 curves. A nonlinear decrease was observed with increasing PEEP which not only depended on the rise in central vanous pressure as indicated in fig. 1.5 but a150 on other control mechanisms.
(:0
VR
t
(1/. )
/"mIn 1
fig. 1.5: Analysis of the effect of positive pressure breathing on normal cardiac output
(from Guyton (1973)).
The effects of cycli c changes of airw('lY pre""ure during IPPV and cprv on t.he ('ImOlmt ot flow flu~ tuation through the right sj de of the heart, were r"port.ed hy V"rsprille et ,,1. (1982). One of their recordings, representing the Ch('ll"(\cteri",tic phenomena of flow ch8ngBS during IPPV, is Riven in fig. 1.7.
From these experiments it was shown that the cyc~ic fluctuationS
or
th" riRht ventricular stroke volume became more pronouncecl rel~ti vely when PRF.P wa!=: increased ~ hut insufflation volurr'1e w~s kept cunstant. Versprille et al. (1982) hypothesized that loading th~ venQ\.l;!7: C::;'73Pf!.C i ti~:s (juring i ntra-thQrac i<.: pr'1'l5~ure ri se wns D main mechanism explaining the fluctuations and th~t the I~v"l or mean flow to the riP.ht side of the heart at hiRhe!" PEEP-levels we>" one of the m.;\jo, conditions determining the relative "m()lmt of1
•
•
100 o •••
•
co
°DO.
OLl 0 • 0 ••
('/:.)••
0 0°
40o
-
.
~.II••
00•
00 DO.-••
•
• ••
•
[J • • • • • 0 0 C~"',", q" 0 -o DCDCCl'b." ()
0 0 o D CD PEEP (kPta) '-5....
fig. 1.6; Two i~dividual respo~ses of cardia~
output on PEEP. C<lrdi<lC OUtp\lt ie normal hIed
at the value at ZEEP (from Schreuder et ~l. (1982)).
fig. 1.7; Fluctuations of blood flow o~ring PEEP. After stabilization at Z££P, re~ordings
were mad~ at PEEP-levels of 0.5, 1.0 ano 1.5 kPa. Fpa volume flow (electro-magnetic) in the common pulmonary artery. Ft , volume flow of air in the trachea (I = insufflation. £ ~ expiration). Pt: airway presEure in the trachea
(from VerspriIIe et al. (1982)).
As was ~hown jn section 1.1, the estimation of the mean cardiac output with the indicator-dilution technique, ~~quires a Con$t~nt flow. In the preceding section it wa~ briefly explained that hlood flow will fluctuate during mechanical ventilation of patients. For physiological :applications of the indicator-dilution method the influence of nonstationarity of the flow is of interest.
In 1964, Luthy :atld Galetti published the results of a study in whiCh thermodilution measurements in artificia~~y ventilated dogs were performed. They observed a trend in the cardiac output e~timates related to the ventil:atory cycle. Although large v~riattons in the measurements prevented reliable conc~usjons being established, they found that the amplitude of the measured variation could be ~epetldent on the value o~ mean flow. Jansen et a1. (1979, 1981) performed experiments with arti fi.cially
ventilated p~gs. In order to estimate the errors in the
thermodilution technique ~ue to cyclic changes in c~rdi~c output caused by the ventilation, they measured c:ardiac output on both sides of the heart at increasing levels of PEEP. The l~tter vari~tion was used to change the cardiac output as well as th€ ~mplitude of the modul~tion. Under steady-~tate condition5, a th~rmodilution Measurement was taken after fifty joentical intervals of time in the venttlatory cycle. the inj~ctions were ", .. de in r"ndOM order. the individual readings gave a random error showing a maximal deviation between two measurements
or
the right ver1tt"iOular output of 70% of the mean flow value. The ratio of m~ximal and minimal va1\Je W~8 as high as two. PuttiTlg these meaSurements into sequence, with respect to the moment ofinje~tjon in the modulation cycle, yielded a cyclic modulation wi th only a "m" 11 random error superimposed on it. flurph singly,
the ~mplitude of th~ modulatjon in the esti~~tes of car-diae output W>J:; approxim"t~ly constant at th" clitferent levels of PF;RP, while djr~(':t flow r'ltJ'=..I~UrementR :!J.howerl an obvious iTlC'r'cHse of flow
fluctu~tjon8 {Versprille ct ~l. (19B?)). Also, Ar~engol ~t ~l. (\981) stuclied th~ ~ffBct of mechanical ventilation on the tt)erlnod'i llJt:ion techniqua. They injecte..1 ;at. only four instances
during the respira~ory cycle and ob~erved systemat~c d~fferences between the cardiac output estimates. Measurements at the end of
the inspiration showed the largest values with the smallest var~atlons. They concluded that timing the injection within the ventilatory cycle enhanced the reproducibility of cardiac output va'ues. This, however, does not imply a better e~timate of mean cardiac output. Snyder and Powner (1982) confirmed the
observations of Jansen et al. (1981) and r~ported thermodilution measurements in dogs and in one human patient showing a cyclic modulation in the estimates of cardiac output.
25
In summary, during mechanical ventilation, a modulation of blood flow exists, and it causes large errors in the estimates of mean cardiac output when applying the indicator-dilution technique randomly.
1.3 The purpose and scope of the present investigation.
The aim of this research project was to evaluate the errors in mean flow estimates due to the variable nature of the flow in physiological and clinical application~ for the indicator-dilution techn~que. Additionally, a reduction of these erro,~ was studied. Furth~rmore, it was investigated to ~ee whether information can be obtained about the instantaneous flow from an indlcator-dilution curve.
The investigations dcscribeo in this thesis can be summarized as follows: In chapter 2, some mathematical expressions for the mixing process between injec~ion and sampling poin~s are analysed. In chapter 3, experiments are described with various hydrodynamiC models. The simplest of these models allowed a study of the indicator-dilution technique with constant flow only. In a ~econd model, a sinusoidal flow mOdulation was Superimposed on a cOnstant flow. Experimentalty, the influence of the appropriate parameters could be established and compared with the results of the theoretical analyses. In chapter 4, a hydrodynamic model is presented which has a close resemblance to the ;n-vivo situation with regard to the indica~or-dilution measurement. In this chapter, the effects of a physiological flow pattern, consisting
of these pulsations resembling respiratory influences, are shown. Ways to reduce the ob~~rved e~rors were studied in a physical model. Finally, determination of the instantaneous flow from the indicator-di~ution curve is discussed in chapter ~.
Armengol. J •• Man. G. C. W., Balsys, A. J. and Wells, A. L.:
Effects of the respiratory cycle on cardiac output measurements: Reproducibility of data enhanced by timin~ the thermodilution injections in dogs. Crit. Care Med •• 1981.9.852-854.
Bassingthwaighte, J. B .• Knopp. T. J. and Anders~n, D. U.' Flow estimation by ind~cator-dilution (bolus injection); reduction of errors due to time-averaged sampling during unsteady flow. Cire. Res •• 1970. 27, 277-291.
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w.
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Scheuer-Leeser, M., Morquet, A., Reul, H. and Irnich, W.: Some aspects to the pulsation error in blood flow calc~lation5 by indicator dilution techniques. Med. BioI. Engng. Comput., 1977, 15. 118-123.
Schreuder, J. J., Jansen, J. R. C., Bogaard, J. M. and Versprille, A.: Hemodynamic effects of positive end-expiratory pressure applied as a ramp. J. Appl. Physiol.: Respirat. Environ. Exercise Physiol., 1982, 53, 1239-1247.
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Solberg, D. M.: Bestimmung der Volumenstrome in stationa~en und p~,sierenoen Strom~ngen ~nter epezie11er BerUeksichtigung der Thermodilutionsmethode fUr Messungen am Blutkreis~auf. Diss. Nr. 5200, 1973, ETH Zurich.
Stewart, G. N.: Researches on the circulation time in organs and the influences which affect it. IV: The output of the heart. J. Physiol., 1897, 22, 159-183.
Swan, H. J. C. and Beck~ W.~ VBnt~icular non-mixinR a5 a source of error i~ the estimati,on of ventrjC~lar volumes by th~ indic~tor dilution technique. eire. Res., 1960, 8, 989-998.
Versprille, A., Jansen, J. R. C. and Schreuder, J. J.: Dynamic aspects of the interaction between airway pressure and th~ circulation. In: Applied Physiology ~n C.inical Respiratory Care (Ed.: Prakash, 0.), 1982, 447-463, Martinus Nijhoff. the Hague.
Wessel, HTI Paul, M. Hot J~me~, G. W. and Grahn, A. R.~ Limitations of thermal dilution curves for cardiac output determ~nation. J. Appl. Physiol., 1971, 30, 643-652.
'-'ill son , E. M., Ranier-i, A. J., Updike, O. L. and Dammann, J. 1'.: An evaluation of therma. di~ution for obtaining serial
measurements of cardiac output. Med. BioI. Engng. Comput. 1972, 10, 179-191.
Wood, E. H.: Some past, pr-esent and futur-e aspects of indicator-dilution technics. Physiologist, 1961, 4, 62-73.
2.
DESCRIPTION OF THE INDICATOR-DILUTION CURVE.
2.1 Introduction.Valentinuzzi et a1. (l972) pointed out that a mathematical description allowes a measured indicator-d~lution curve to be represented by a set of parameters, from which the area or the curve can be estimated quickly. Hamil~on et al. (1932) used the matheical description for the elimination of recirculation effects. Furthermore, a mathematical analysis can be useful in studies concerning the indicator-dilution technique in variable flow (ijassingthwaighte et a1. (1970)).
Mathematical simulations of tracer transport are well known in chemical engineeri",.. ror the study of flow processes j.n chemical reactors. The two major groups of models used ror this purpose are distribution models and compartmental models (~even6piel (1972), Lighthill (1966). For simulation of the indicator transport in physiological systems, these models were adopted; in addition, purely mathematical simulations of the indicator-dilution curve were used, e.B. the gamma~function and the lognormal-function
(Spie~kerman and Brettschneider (1968»). A stUdy of the influen~es
of variable flow on the indicator-dilution te~hniQue restricts the choice or model to the distribution and the compartmental models. In section 2.2, these two models are described and, in
section 2.3, a comparative ~tudy of them is g~ven.
2.2 Model simulation of indicator-dilution curves.
2,2.1 O~stribution models.
Sheppard and Savagp. (1951) Were the f;r~t to suggest the use of random walk, or di.ffusion with drift functions, for the
interpretation of indicator-dilution C1)rvp.~. i'hen, tran5Port. of t.he indicator through part of the circul .. tory System can be reg"riled ~s a dj spers i on su~erl mpo:;;ed Orl a 11 "ear drift of the
31
applied to indicator-dilution methods, is the diffusion-with-drift eq~ation (Wi~e (1950», when a pe~rect c~o~~-~ectionaL mi~ing i~ a~sulTJed:
o
6'C(x,t) _ u 6C(X,t)h!
-6-x--... here;
C(x,t) is th~ tim~ and axial plaoe-dependent oonoent~ation;
o
is the effective long~tudina~ diffus~on coefficient; u is the mean flow velocity over the cross-section.(2.1)
In complicated transport eyeteme. euch ae olood circulation, 0 is determined by the contribution of various physical mechanisms, as Taylor diffusion, turbulent mixing and Brownian motion
(Wiee (1966). Bogaard (1980». For a bolue-injection of indicator, eq. (2.1) gives the 'looal density random walk' (LDRW)
dist~ibution when the ooncent~ation ie coneidered to be measured in a small crosB-sectional element perpendicul~r to the dire~tiQn of flow:
(2,2)
where l/a is a scale factor for the probability density functions of the transit times. The median traneit time ~ is def~ned by ~ = xJu, when Xo is the distance between injection and sampling points. The parameter • is a skewnese factor related to the aeymmetry of the curve which decreases with increasing 8symmetry. The tim~t is related to the ze~o time for the
distribution. Th@ first moment of the probability density run~tion gives the mean transit time (MTT, Bogaard (1860)) 50 that
eq. (2.2) givee:
(2.3)
Another ~olution for the diffusion-with-drift equation (eq. (2.~») can be obtained when a (virtual) abso~bing surface is assumed to be at the outlet, so that all the indicator particles will pass the measuring point only once. This solution, celled 'First
passage times' (FPT) distribution, is described by Sheppard (1962) and Wise (t966) and reads:
cit)
FOr this distribution, the m~an transit time is equal to the median transit timeI
)J
In the distribution expressions (LDRW and FPT) , there are two parameters describing the mixing system, namely ~ and A. Bogaard (1980) showed that A is proportional to the Peclet number (Pe) which is defined as the relative contribution of convection and dispersion to the indicator transport (Sl.ttel (1968)):
Pe 2 ~ .. U x./D
(2.4)
(2.5)
(2.6)
The parameter ~ can be derjved easily from the shape of the curve according to wise (1966).
2.2.2 Compartmental models.
Thp. compartmental approach to the study of tracer behaviour in chemical reactor engineering was described by Levenspiel (1962). For jndicator~dilution method'; the compartmental model has been proposed by Newman et al. (19~1). ~pp'cjal applications of this model have included the quick estimation of curve areas
(Schlossmacher et a1. (lR67), VRlp.ntinu~zj et a1. (1972). Its us~
for diagnostic purposes (Hill et ~l. (1973)), the deRcription of v"riabl" no", "rr"ct" (Sche\le)"-ceeser et al. (1977)) and the determination of ventricular volumes by JRrlov Bnd Mygind (l979) "nd J~rlov ana L"r~~n (1981) are other examples. Th" model b
b8~Ad on th~ t'"1~~~l.lrT'(.lt\on that the system between injection. And ""mpl;nl? pojt"1Vc
'''HI
l!e: divided into parts: (1) thp. ideal mixing ch~mhers in ~hic.} the lncticstor mjxe~ instantaneously with thefluid and (2) the pss5aHe tubes in which no mixing occurs ano th~
indicator is only transported. The equation describing the mixing process in an ideal mixing chamber ~s g~ven by the rollowing indicator balance:
33
v
dC(t)dt - F(t) [e(t) - C.(t») (2.7)
where V is the volume of the mixing chamber and Co(t) the concentration of the fluid entering. When the system between the injection and sampling points is assumed to be a series of N equal ideal mixing chambers, the mixing process is described by N
different~al equations of the first order. each representing the indicator balance in a m~x~ng chamber:
dC~(t) ' 1
-dt
~ .. 1..N (2.8)
where Ck(t) is the time-dependent concentration in the k-th m~x~ng chamber, V is the volume of each mixing chamber. For
bolus-injection of indicator at the entrance of the first mixing chamber at time t = 0, the boundary and initial conditions are:
c.(t) " 0 and Cl(O) = mjnj ! v (2.9)
For constant flow, F
c' eq. (2.8) with the conditions at eq. (2.9) can be solved analytically according to Schlossmacher
et a1. (1967): t _ minj __ 1 _
eN ( ) -
F (N-l)~ C N-l t t7
exp(-r)
(2.10)where T ~ V/fc is the tim~ constant to~ each mixinp, chamber. The parameters of the ~nct~on describing the mixing system in a model composed of equal mixing chambers in series 8~e N (intep,er) and V. When the mixing chambers are not equal, the number of parameters inc~eases with the number of compartments. The differential equations describing such a system are analoguous to eqns. (2.8),
but with unequal coefficients on the lefthand side of the equation. For two unequal mixine chambers, the $olutiQn for
constant ~low conditions and atter ~olus-injection o~ indicator is given ~y:
(2.11)
where V
1 and V2 are the volumes of the compartments.
2.3 Comparison ot the compartmental and di$tri~ut~on models.
2.3.1 Desoription of investigations.
In order to make a choice between the compartmental and the distribution model ~or indicator-dilution curves measured in varia~le flow, both models were compared in an experimental and mathematical study. The ~irst psrt of this study consists of fitting the N-oompartment function (eq. (2.10), the
two-compartment funct~on (eq. (2.11) and distri~ution model functions (eqns. (2.2) and (2.4) to indicator-dilution ourves measured in-vivo and to curves measured in B hydrodynam~c set-up with constant flow. The second part of the comparativ~ study comprises a direct comparison between the two models. We studied the suitability of fitting compartmental functions to the indicator-dilution curves represented by distribution tunctions, The ohoice of the latter functions as a referen~e for this oomparison is based on the ability to repre~ent the ~hape or asymmetry of a curve with a sinp;le par.j!meter and to establish a wide range of curve shapes by varying this para~ete~ (von Reth and Bogaard (1983)). In the cOMparative study, X ~ill refer to the skewness parameter o~ the (best-fit) LDRw-function.
2.3.2 Fitting of theoretical functions to measured curves.
In th~ comparative study, the ~our mOdel functions were fitted to constant flow curves (, < 6) obtained experimentally wjth a
hydrodynamic model as described in chapter 3. Also,
indicator-dilution curves obtained from anima. experimen~e are analysed. When only one side of ~he heart was involved the curves were rather skew with ~-values from 1 ~o 3. Indica~or-dilution curves obtained by injection on the venous side of the systemic circu.ation and sampling in the arterial part. allowing the indicator to paBB both sides of ~he hear~ and the pulmonary circulation in between, are much more symmetrical showing '-values from 3 to 6 (von Reth and Bogaard (1963». From the animal
experiments. curves are used with injection and sampling on the same side of the circulation. They were ob~ained in experiments performed with Yorkshire pigs (weight 7-10 kg) breathing spontaneously during the experiment. Detai.s on the sur~ical procedure and oon~rol measurements are presented elsewhere (Jansen et al. (1981».
With the four functions (eqns. (2.2), (2.4), (~.lO) and (2.11» measured indicator-dilution curves are described by the system parameters and the injection parameters. being the mass injected and the moment of injection which defines the translation with respect to the c~osen zero point. Two parameters define the scale for time and concentration respectively, ~he fourth parameter determines the shape of the indicator-dilution curve. The method of fitting was an automated, unweighted, numerical least squares fitting procedure which was available as a subroutine on a OEC!LSI-l1 microcomputer. This fitting procedure minimized the RMS-value of the sampled points of the measured curve (scaled with its maximum at one) and the func~ion by adjusting the four parameter values. Together with the RMS-va1Ue, the quotient (qA) for the area of the fitted curve and its measured value and the Quot~ent (QMTT) fDr ~he mean transit time of the fitted function and the measured value were calculated.
The results of fitting the four model functions to the measured curves are shown in figs. 2.1. 2.2 and 2.3.
0
•
0 -2 8,10...
t
¥
('0.1(0"", RM~ 2 LORW compo• •
FPTt
• •
0•
I:l.w
f
~
•
•
...
f
...
0 ~ 0 0 7 ~ Afig, 2.1: The RMS-value as a fun~tion of the skewness parameter of the best-fit LORW-function. The maximum of the curve is scaled at one. With increasing symmetry (A ~ 3), all models provided fairly
...
0
t
accurate representations, except ~or the two-compartment model with unequal compartments. This model gave the best
representations for skew ~urves (0.8 ~ ~ ~ 1.2).
2.3.3. Comparison of the theoretical model fun~tions.
When compar~ng the compartmental functions and the distribution functions theoretically. the compartmental and FPT-functions were fitted to LDRW-functions of varying shape. LDRW-~nctions were chosen as referen~e functions ~ith ~-values 0.5. 1. 2. 3. 5 and 10 respectively. The fittinp' procedure was the same as that explained in section 2.3.2. The results of the theoretical comparison between compartment models and distribution models is presented graphically jn figs. 2.4. 2.5, 2.6. The skewness-parameter of the reference LDRW-function i~ snown on the ho~izontal axis. On the ver:-ticaJ. "'lis the RMS-values for the fit. (fig. 2.4), the
ratio (gA) of the areas of the fitted compartmental curve and the reference cUrve (fig. 2.5) and the ratio (qMTT) of the MTT-values for the~e curves (fig. 2.6) are p,iven. Very accurate
1-2 Nco",," l comp
t
LDRW ~ f"T 'l"~75
.i
0•
~ 0-•
.0
1(0-i
.e.
..
•
•
•
0.8fig. 2.2: The quotient (G
A) of the areas of the fitted curve and the measured curve as a function
1.2
of the skewness parameter of the best-fit LDRW-function. Nco~ 2 comp
...
0
t
7...
0
t
fPT lDRWV
'I 0.IJ
•
0 11!
~1I "j•
If
0...
•
.r,t..
7.-
A...
~ fiB•
fig. 2.3: The quotient (G
MTT) of th~ MTT-v~lues of the fitted cwrve as a function of the skewness parameter of the best-fit LDRW-function.
can be obtained from the N-compartment fun~tjon. An acceptable accuracy for the FPT-fun~tjon restri~ted th~ skewness range to ;\ ; 3 and, for the two-~ompartment function with unequal compartments, to A ~ 3 •
. ,
5.10 \ 'I" R M~o~~~~~~~~~~
o 5 10..
fi~. 2.4, The RMS-values plotted as a fUnction of the akewness-parameter of th~ ref~rence LDRW-function, 2.4 Discussion
After f\tt~ng of the four model functions to measured curves, it was shown that I fot' curves in the ;\ -range ;\ :> 3, only the
two-compartm~nt model with unequal volumes fail~ to give accurate representations. In an ~arlier study within th)s project (von Reth and Bogaard (1983)), it was ~oncluded that only by the addition of more compartment$ I'ccurate representations or symmetrical curves could be obtained. This however, implIes an increase of the number of parameters in the model function. For skewed curves (. < 5),
th", accuracy fo, the FPT-model lessens and in this range the compartment mOO'll", a,\d the LDRW-model will be preferred. ThE'! preference for
the LDRW-model over the FPT-model was indicated by Bogaard (19~O). Tn the t'anp,e 1 ( A < 3, all model functione give accurate fits.
Oy choosing theot'etical LDRW-functirm'3 a~ a t'efel'ence 1"0" the comparison, :=3. wide range of asymmetry can 'be covered. Besides,
~ 1.1
1
~h ~~ -~-~ 0 $ W 2, A N~~ 3 QS1
M, ,,.,
fig. 2.5: The ratio Q
A of the areae ~nder the fitted and reference curves as a function of the
~T
,
,
,
,
skewness parameter of the reference LDRW-function.
,
~
0
fig. 2.5: The ratio qMTTof the MTT-values for the fitted curve and the reference LDRW-curve,
shown as a function of the ekewneee parameter of the distribution function.
with theoretical curves, ~o noise ~tIects c~n influence the Comp~rative study. The LDRW-function was chosen as a reference because of the proven fitting ability (Bogaard (1980), Msrinus et al. (1984)). A comparison of theoretical functions confirmed the results of the fits on measured curves, For very skewed curves, the FPT-function is less accurate, while for symmetrical curves the two-compartment function fails to p,ive accurate estimates. It w~s concluded that when covering a wide range for the shape parameter A, the LDRW and N~compartment functions ahould be preferred as representationa of indicator-dilution curves.
Cassot et al. (1978) used an LDRW-model and from the presented identification technique cardiac output could be estimated from the parameters of the fitted functiOn. From their ~t~dy with several flow- ~nd injection-functions they concluded that the LDRW-function gave good representations of the measured curves.
Fro~ the studies of Baasin~thwai~hte et al. (1970) and
Solberg (1973, 1975), it wes indiCated that with a compartmental approach the effects of variable flow could be easi,y incorporated in a description of the mixing behaviour, The comparison of model functions indicated no essential preference for LDRW- Or N-compartment models. Based on the simple model description, the N-compartment model will be used in this stu~y.
:;:.5 References,
9assingthwaighte, J. B., Knopp, T. J. and Andersoll, D. U.; Flow estimation by indicator~dilution (bolus injection); reduction of errors due to time-averaged sampling durinF, unsteady flow. rirc. Res., 1970, 27. 277-291.
Bagas-rd, J. M,: Interpretation of indicator-dilution curves with a random walk model. niss, Abstr. r~t.; s"ction C, 1980; thesis F.rasmus University Rotterdam, the Netherlands.
Rogaard, J. M., Wise, M. E., Smith, S. J. alld Versprille, A.: Some cons=.ideH .... F:lti one.; On the phy:siologi~f.Ll information obtainable from the skewness of indicator-d; l\,tion cu\"ves as obtained with the
double indicator-dilution method ~or the estimation of
extravascular lungwater. Bull. Europ. Phyaiopath. Resp., 1980, 15 37p-39p.
41
Cassot. F •• Saadjian. A. and Me Kay, C.: Heat and maSS trans~er of a thermal indicatcr in pulsatile flow through the cardio-pulmonary system;
r.
Modelling. Mad. Progr. Technol., 1978, 5, 203-214. Cas sot , F., Saadjian. A. and Mc Kay. C.: Heat and masa transfer of a thermal indicator in pulsatile flow through the cardio-pulmonary system: II. Identification of cardiac output. Med. Progr.Technol., 197$, 5, 215-222.
Guyton, A. C., Jones, C. R. and Coleman, T. G.: Circulatory Physiology: Cardiac output and its regulation, 1973,
w.
B. Saunders Co •• Philadelphia.Hamilton, W. F., Moore, J. W., Kinsman, J. M. and Spurling,
a.
G.: Studies on the circulation; lV. Further analysis of the injection method and of changes in haemodynamics under physiological conditions. Am. J. Physiol., 1932, 99, 534-551.Hill. D. W •• Valentinuzzi • M. E .• Pate. T. and Thompson. F. D.: The use of a compartmental hypothesis for the estimation of cardiac output from dye dilution curves and the analysis of ~adiocardiograms. Med. BioI. Engng. 1973, 11, 43-53.
Ja~sen, J. R. C., Schreuder, J. J., Bogaardt J. M.~ van Rooijen,
W. and Versprille, A.: The thermodilution technique for the measu~ement of cardiac output during artificial ventilation.
J. Appl. Physiol.: Respirat. Environ. Exercj,se Physiol •• 1981. 51. 584-591.
Jarlov, A. and Myginci, t.: Ventricular volumes determined (ram indicator-dilution curves. Med. BioI. Rngng. Comput •• 1979. 17. 31-37.
.I~rlov, A, and Larsen, B.: Stroke volumes and ventricular volumes determined from indicator-ddution curves. Med. BioI. Engng. Comput., 19R1, 19, 4C,7-4r,O.
Levenspiel, 0.: Chemical reaction engineering. John Wiley and Sons, New York, 1972.
Lighthill, M. J.: Initial development o~ di~fusion in ?oiseuille flow. J. Inst. Math. Applies., 1966, 2, 97-108.
Marinus, J. L. M., Bogaard, J. 101., Massen, C. H., von Reth, ~. A ••
Jansen, J. R. C. and Versprille, A.: Interpretation of circulatory "hunt di 1uti.on cUrves as bimodal distribution functions. Med. Biol. Engng. Comput., 1984, accepted.
Newman. E. V •• Merrel, 101., Genecin, A., Monge, C., Milnor, W. R.
and MCKeefer. M. p,: The dye dilution method for describing the central circulation; an analysis of factors shaping the time concentration curves. Circulation, 1951, 0, 1~5-746.
Norwich, K. H. and Zelin, S.: The dispersion of indicator in the cardio-pulmonary system. Bull. Math. Biol., 1970, 32, 25-43.
von ~eth, E. A, and Bogaard, J. 101.: Comparison between a two-compartment mooe1 and di~tributed models for
indicator-dilution stU(He'L "led, Blo1. Engng, Comput., 1983, 21, 453-459.
Scheuer-Lee~er, "I., MorQuet, A., Reul. H. and Irnich, W.: Some aspects to the pulsation error in blood ~low calculations by indicator dilution techniques, Med. Biol. F.ngng. Comput., 1977, 15, 118~123.
Schlossmacher. E. L., Weinstein, H., Lochaya, S. ano Schaffer. A, B.: Perfect mixer~ in ~eries mooel for fitting veno-arterial indicator-dilutiot1 curve~, J, Appl. Physiol., 1967, 22, 327-332.
43
Sneppard, C.
w.:
8aeic principle~ of the traoer method, 1~6~, John Wiley and Sone, New York.Sheppard, C.
w.
and Savage, L. J.: The random walk problem in relat~on to the physiology of circv1atory mixing. Phye. Rev., 1951, 83, 489-490.Sltte1, C. N •• Threadgill. W. D. and Schnelle. K. B.: Longitudinal diepersion for turbulent flow in pipes. Ind. Eng. Chem. Fund., 1968, 7, 39-43.
Solberg, D. M.: Bestimmung der Vo1umenetrome in et~tion~ren und pu1sierenden Stromungen unter spezieller BerUcksichtigung der Thermodilutionsmethod~ fu~ Me5sung~n am Blutk~eislauf. Di55. Hr. 5200, 1973, ETH Zurich.
Solberg. D. M.: Durchflussmessung mit Indikato~methoden, insbesondere mit lokaler ThermOdilution. 8~omed. Technol •• 1976. ~1, ~-9.
Spieckerman, P. G. and Brettechneider. H. J.: Vereinfachte quantitative Auswertung von !ndikato~ve~dUnnun&ekvrven. Archiv fUr Kreis1aufforschung, 1968, 55, 221-282.
Va1entin~zzi, M., Valentinuzzi, M. E. and Poscy, J. A.: Faet estimation of the dilution-curve area by a procedure based on a compa~tmenta1 hypoth~~i~. J. Ass. Adv. Med. Instr., 1972, 6, 335-343.
Wise, M. E.: Tracer dilution Qurves in cardiology and ~andom walk and log-norm~l di~tr~butions. Acta Physiol. Pharmacol. Neerl., 1950, 14. 175-204.
W~se, M. E.; Skew distributions in biomedicine including some with nep'ative powers of time. In: Ststi5ti~al distributions in
~cientific work, vol. II, Model building and model selecting, 1975, publ. Reidel, Boston.
3.
STUDIES ON THE INDICATOR-DILUTION METHOD IN MODELS
OF VARIAELE FLOW.
3.1 Introduction.
The general Stewart-Hamilton equation proved to be correct in the constal'lt flow si t\Jation. It ~s shown il'l section 1.1
theoretically and described in section 1.2 for in-vivo experiments that this equation is not vaiio when the flow is nonstationary. The accuracy of the ~tewart-~amilton equation when used to
estimete the mean flow has beel'l the subject of several model analyses. Cropp and Burton (1966) described theoretical considerations and model experiments of the indicator-dilution technique in variable flow. Particularly for the constant-infusion method, they
indicated possible sources of error. They concluded that where the method Of col'lstant infusion of indicator was used and sampling WaS possible with adequate mixing close to the injection point, accurate calculations were possible for the mean flow.
Bassingthwaighte et aI, (1970) performed a detailed mathematical analysis in order to describe the sources and magnit~des of errors of the indicator-dilution method in nonstationary flow, From theBe analyses, it was predicted that the error will be roughly
proportional to the amplitude of the variation in flow and that it will be maximal when the period of flow vaI"iatiol'l is s;mil,ar to the passage time of the dilution curve, The resulting error varies with the injection phase. Err6rs caused by variations with cardiac frequencies proved to be minimal. Their theoretical results were not verified experimentelly. Solberg (1973, 1976) p,)bli",hed a study in which the influence of a sinusoidal flow and a flow simulating the heart-action variation wa", investigated
theoreti~ally and experimentally. He based his theoretical analysis on ~ mixing chamber model and calculated the errors of the IOCDl indicator-dilution te~hnique for several injection
function~. Hi", observ8t ions were Inade fOI" or.ly thl"ee diffcl"eTlt inject.ion moments in the cycle of flow variation. With this small numher, tho> effect of tI.e ph<lsc in th" modulation cycle at the
45
moment of injection could not be ~naly~ed. The theoretical analysis was linked to the experimental data by model parameters derived from the geometrical dimensions of the system between injection and detection points and from a di~putable number of equal mixing chamberS, numbering three or six in these
calculations. The be~t model was, as Solberg reported, 'in fundamental agreement' with the experimental results. However, differences between model and experiments were as high as 70%. Scheuer-Leeser et al. (1977) performed a numerical and an experimental study of the effects of flow modulations caused by the heart action on the estimation of cardiac output. They concluded that for practical purposes like in-vivo, little loss of accuracy was caused by these flow variations. Their theoretical analyses were based on a compartmental approach to the behaviour of the mixing system.
we performed experiments and numerical calculations for testing the feasibility of the compartmental model in order to describe the influence of nonstationary flow and to quantify the influence of hea~t action and ventilation on the accuracy of the mean flow estimate (von Reth et a,. (l~8J)). The effects of heart actlon and ventil~tion were studied separately by choosing a flow function for both influences that consisted of a constant flow with a sinuaoidal modulation superimposed on it.
3.2 Theoretical analysl6.
Under the condition of nonstationary but periodic flow the flow function can be described &enerally by;
F(t) F + F f(tl
m a (3.1)
where Fm ~s the mean f,ow value. The variation function f(t) is a period~c function with Zero mean Value ~nd a maximum absolute value equal to unity. Fa represents the arnp,itude of the variation. Using sq. (3.1) into eq. (2.8) for a system described by an N-compartment model Eives,
k = 1..N (3.2) with T ~ V/f
m• A ~ fa/fm.
D~finlnR dimenBionl~~e p~r~m~ter~ e t/(N T) ~no r C
vi
m, _1nJ eQ. (3,2) c~n be rewritten as:
k 1. .N (3.3) The initial and boundary conditions, given in eq. (2.9), can be expressed in dimensionless form as:
o and "1(0)
The dimensionless differential equations can be solved numerically. The solution YN(S) represents the dimensionless indicator-dilution curve at the outl~t of the ml~ing system. The mean flow estimate Fe c~ be calculated from eQ. (1.10) and can be written. as:
F
..
F
III
(3.5)
The relative oeviation (RD) between the real mean flow Fm and the estimated mean flow f e can be calculated from:
f - f
flD e m
f 100% (3.6)
m
The flow function used in this section, both experjmentally and in the theoretical analyses, defined the variation function as:
f(t) (3.7)
where w is the angular frequency of the variation ~n(l ~ j,s the inj phase of the modu13tion cycle at the momet'\t of injection. Using the (ljmensionlcss parameters. i t produces:
wi th f2 = 1;,1 NT.
The 601ut~on of eqn~. (3.3) ~ep~e~~~t~ the di~~~sionless indicato,-dilution curve YN(9). The RD-value obtained with eq. (3.6) is dependent on N, A, 11 and~ . . ' Theoretically the
lnJ
moment of injection i~ the sam~ as the moment of first appearance of indicator. Experimentally, there mu~t be a time delay between these instance~. Thi~ delay, t , causes a phase shift according
ap to:
47
¢>inj - 'ap II
e
ap (3.9)whe,e ¢> t
IN
T. The value of t can be determinedap ap ap
expe~imentally by measuring the time between the mom~nt of injection and the first appearance of indicato, at the sampling point.
3.3 Model experiments in modulated flow.
3.3.1 Expe~imental set-up.
The experiments were performed in a system consisting of two pumps. a mixing chambe, and connecting tubes (fig. 3.1).
The constant flow produced by the first pump (Pl; Verder 114-1M-OOO) can be modulated sinusoidally with a plunger pump (P2). The actual flow as a function of time is reco,ded by means of an
electromagnetic (EM) flowmeter (Transflow 600. carrier
frequency 600 Hz., upper frequency r~sponse 100 Mz-3dB). A salt solution in water (0.1 gr%) is used as the circulating fluid with a tempe~ature of 18°C thermostatically controlled by a buffer volume (70 1; Tamson TCV70). Due to the size of th~ buffer volume no recirculation effects were noticeable. The volume of the mixing compartment (60 ml) is partially filled with glass beads to ensure good mixing. The tubes have an internal diameter of 15 mm and a total length of about 3 m.
The distance between the points of injection and sampling 8j,tes is kept to a minimum. The injection unit cOnsists of a pneumatically
driven syr'lrlge (Kuhnke 36.291), Bnd an electrically-controlled air-pressure valve (Kuhnke 44.250.01). Behind the mixinfl rep,ion, two electrodes are inserted, each consisting of a stainless-steel ring (1 .om. thick, 1 mm. apart and <l mm. in diameter). The impedance he tween the e~ectrodes is measured with an A.C.-Wheatstone hridge (carrier frequency 5 kHz and upper f~equency re~ponBe 1 kHz-3dB). For a~tual mea~urement~, the
tM
fig. 3.1: The hydrodynamic model
PI, constant flow pump; P2: plunger pump;
E M: electromagnetiC flowmeter; M: mixing region; C: computer;
i; injection point; s: sampl in" POi,nt,
passage of a small amount (1 mI.) of a 0.5 gI"% salt solution, injected instantaneously. was recorded. The conductj,vi ty ver",u", time cUrves and the flow function measured with the flowmeter were sampled hy a computer (DEC-LSI!ll. sample frequency 125 liz). The
indicator inject~on W~8 computer-controlled.
3.3.~ ~xperjmental procedure and data processing.
F.xperiments were performed under constant flow conditions in order' to cal ibratc the flowmeter :syste:m, to determine the 1 ineari ty Hnd the reproduc ibj, 1 i ty of the cond\Jctj vi, ty measurement
49
and to evaluate the mixing process in the region oetween injection and detection points. The reciprocal values of the areas under th~ curve~ measured at di~~erent flow values were used to che~k the linearity of the conductivity measurement. These experiments were performed with flow~ in a range of 10 to 125 m~/s. The mixing proces~ wa~ characterized by assuminp, that the mixing region was composed of N equal, ideal mixing compartments in series. The model function describing the mi~ina in such a system (eq. (2.10)) was fitted to the curve for a constant flow with value F . The
m
least-squares fitting procedure was used for a fixed number (N) of compartments, giving a best-fit value for the time con~tant T. This procedure was performed until a minimum value for the sum of the square~ of the deviations waS found. Next, a sinusoidal modulation was superimposed on the ~onstant flow with variable frequency and amplitude, generated by means of a plUnger pump. Th@ resulting flow function was measured with the flowmeter. UsinS a
least-squares fitting procedure, a theoretical ~inusoidal flow function was fitted to the sampl~d flow data points and this resulted in an experimental mean flow value, Fm, a relative amplitude. A. and a dirnenBionl~~~ modulation frequency, Q. The pha~e range was d~vided into 25 equidi~tant phase points .inj'
ea~h oeing used once to trigger off the inje~tion system. The phases were chosen at ~andorn in order to avoid systematic error~. Using eq. (3.9). the phases .ap' corrected for the e~perimental transit times, were determined. As opposed to ~inj' the phases ~ap were not equidi~tant. The indicator-dilution CUrVES obtained after the injection of 0.5 g~% salt solution at the phases ~ap were integrated numeri~ally. Using the Stewart-Hamilton ~quation (eq. (1.9)). an estimate of the mean flo~ could be made and the relative deviation (liD) between this mean flow estirnate Fe and the real mean flow value F m could be cal~ulated I)y eQ. (3. f:i) •
As shown in section 3.1. RD depends on the mixing system (N, T) and on the flow parameters (A. Q and ~ap)' The influence of the
flow parameters was ~tudied by determining RD versu,; ~ap fo~ variable A and constant Q and by determining RD versus ~ap for variable n and constant A.
