Detection of early central circulatory transits in patients with cirrhosis by gamma variate fit of indicator dilution profiles
Jens H. Henriksen,
Søren Møller,
Stefan Fuglsang, and
Flemming Bendtsen
Departments of Clinical Physiology and Gastroenterology, Hvidovre Hospital, University of Copenhagen, Copenhagen, Denmark
Submitted 6 May 2004
; accepted in final form 13 November 2004
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ABSTRACT
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Patients with cirrhosis have hyperdynamic circulation with abnormally distributed blood volume and widespread arteriovenous communications. We aimed to detect possible very early (i.e., before 4 s) and early (i.e., after 4 s) central circulatory transits and their potential influence on determination of central and arterial blood volume (CBV). Thirty-six cirrhotic patients and nineteen controls without liver disease undergoing hemodynamic catheterization were given central bolus injections of albumin with different labels. Exponential and gamma variate fits were applied to the indicator dilution curves, and the relations between flow, circulation times, and volumes were established according to kinetic principles. No significant very early central circulatory transits were identified. In contrast, early (i.e., 4 s to maximal) transits corresponding to a mean of 5.1% (vs. 0.8% in controls; P < 0.005) of cardiac output (equivalent to 0.36 vs. 0.05 l/min; P < 0.01) were found in cirrhotic patients. These early transits averaged 7.7 vs. 12.7 and 17.2 s of ordinary central transits of cirrhotic patients and controls, respectively (P < 0.001). Early transits were directly correlated to the alveolar-arterial oxygen difference in the cirrhotic patients (r = 0.46, P < 0.01) but not in controls (r = 0.04; not significant). There was good agreement between the CBV determined by the conventional indicator dilution method and that determined by separation of early and ordinary transits by the gamma variate fit method (1.51 vs. 1.53 liter; not significant). In conclusion, no very early central circulatory transits were identified in cirrhotic patients. A significant part of the cardiac output undergoes an early transit, probably through pulmonary shunts or areas with low ventilation-perfusion ratios in cirrhotic patients. Composite determination of CBV by the gamma variate fit method is in close agreement with established kinetic methods. The study provides further evidence of abnormal central circulation in cirrhosis.
central blood volume; exponential fit; indicator dilution technique; kinetic principles; pulmonary shunts; ventilation-perfusion ratio
BESIDES PORTAL VENOUS HYPERTENSION, patients with cirrhosis have a hyperdynamic systemic circulation with increased heart rate and cardiac output, low arterial blood pressure, and decreased systemic vascular resistance (12, 37). A number of these patients have fluid retention with expansion of the blood volume, but the circulating medium is abnormally distributed with central and arterial hypovolemia and splanchnic pooling of blood (15, 26, 32). The combination of hyperdynamic systemic circulation and central hypovolemia suggests that these patients have short central circulation times (CCTs) (1, 15, 29, 32). Evaluation by single-photon emission computerized tomography and with the indicator dilution technique obtained by central injection and arterial sampling (1, 5, 15, 32) indicates that patients with cirrhosis have a substantially reduced mean CCT [i.e., reduced central and arterial blood volume (CBV) relative to cardiac output]. The short CCT bears a significant relation to poorer survival (29). Moreover, it has been reported that the existence of ultrafast circulatory transits are caused by pulmonary shunting and that pulmonary hyperperfusion may invalidate the concept of central volume evaluation and transit-time determination (30).
The present study was therefore undertaken to detect possible very early and early central circulatory transits in patients with cirrhosis. We applied a gamma variate fit to indicator dilution curves after central bolus injections of radiolabeled serum albumin in patients with cirrhosis undergoing a hemodynamic examination. The influence of early transits on the determination of CBV was also analyzed.
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METHODS
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Study Population
The study population comprised 36 patients with cirrhosis who had been referred for hemodynamic investigation to diagnose and quantify portal hypertension. The age range was 3771 yr, mean age 50 yr. The diagnosis was established by liver biopsy or, if this was contraindicated, on well-established clinical, biochemical, and ultrasonographic criteria. None of the patients had experienced recent gastrointestinal bleeding or had encephalopathy above grade I. All had abstained from alcohol for at least 1 wk, and no signs of withdrawal symptoms were present at the time of the study. None of the cirrhotic patients had signs of other major diseases, including primary heart disease, pulmonary disease, organic renal disease, and cancer. Patients were divided into three groups according to the modified Child-Turcotte criteria (11). Eight patients belonged to class A, 17 to class B, and 11 to class C. Ultrasonography showed ascites in 17 patients. Patients with ascites were prescribed a diet containing 40 mmol of sodium per day, and in addition they received diuretics, but none received
-adrenergic blocking or vasodilating agents. Clinical and biochemical characteristics are summarized in Table 1.
Nineteen patients without liver disease served as controls. The age range was 3879 yr, mean age 59 yr. The hemodynamic investigation was performed to exclude mesenterial ischemia and renal vascular hypertension with unilateral overflow of renin. The diagnoses were essential arterial hypertension (n = 6), hypertension and renal disease (n = 1), suspected mesenteric angina with irritable bowel syndrome (n = 10), and chronic pancreatitis (n = 2). None of these patients had signs of congestive heart failure, pulmonary disease, cancer, or other major disease. Seven patients received diuretics. Clinical and biochemical data are summarized in Table 1.
All subjects consented to take part in the study, which was approved by the Ethics Committee for Medical Research in Copenhagen and was carried out in accordance with the guidelines established in the Helsinki Declaration II. No complications or side effects were encountered during the study.
Catheterization
Catheterization was performed in the morning after an overnight fast for at least 1 h in the supine position, as described elsewhere (18). In brief, a Swan-Ganz catheter (7 Fr) was guided to the hepatic veins and right atrium through the femoral route under fluoroscopic control with the patient under local analgesia (15, 18, 32). A small indwelling polyethylene catheter (5 Fr) was introduced into the femoral artery by the Seldinger technique and was placed with the tip at the aortic bifurcation.
Measurements of pressures, cardiac output, and CCTs have been described elsewhere (15, 17, 18, 32). Cardiac output was determined by the indicator dilution technique after a bolus injection of 150 KBq of 125I-labeled human serum albumin (IFE IT.20S; Institute of Energy Technique, Kjeller, Norway) into the right atrium followed by arterial sampling. In addition, cardiac output was determined independently by a quantitative injection of 0.5 MBq 99mTc-labeled human serum albumin (Vasculocis; CIS Biointernational, Griffe-sur-Yvette, France) from a catheter depositing directly into the right atrium followed by automatic arterial sampling for 60 s, as described recently (32). In this way the technetium indicator was injected straight away, immediately followed by the iodine indicator. Separate control experiments disclosed an interval (i.e., that between the mean time of the catheter outflow profiles of the two indicators) of 0.6 s. Arterial blood samples and standards were counted in a well-type scintillation detector (Compugamma 1210 Wallac; LKB, Helsinki, Finland). 99mTc was counted after sufficient decay to avoid overflow and dead-time problems in the scintillation counter. After an interval of at least 4 days, the samples were recounted in the 125I spectrum. At least 10,000 counts were recorded and were corrected for background activity and decay. Simultaneous collection of counts in the 99mTc spectrum showed no activity above the background here.
Arterial compliance was determined from pulse pressure and stroke volume as described elsewhere (17). Systemic vascular resistance was assessed as 80 x (mean arterial pressure right atrial pressure)/cardiac output.
The CCT (mean indicator transit time), which represents the mean indicator sojourn in the central vascular bed (i.e., heart cavities, lung vasculature, and central arterial tree up to points that are temporarily equidistant to the aortic bifurcation), was determined from the indicator dilution curve as the time-weighted average of outflow at the aortic bifurcation (16, 22, 31) (see Determination of CTTs and Initial Short Circulation Transits). All transit times were corrected for catheter transit time by the formula: tcath = catheter volume/catheter flow (9.6811.88 s).
CBV was assessed in accordance with the kinetic theory as cardiac output multiplied by CCT (15, 25, 32).
Determination of CCTs and Initial Short Circulation Transits
Indicator dilution curves were determined at sample intervals of 1 s. At least 10,000 counts were obtained in each sample (coefficient of variation < 1%). A characteristic curve is shown in Fig. 1. Both equidistant and semilogarithmic presentations were available. After reaching a peak radioactive blood concentration, but before recirculation, several samples showed a linear fall in the semilogarithmic presentation. This constitutes the basis of exponential extrapolation of the part of the indicator dilution curve affected by recirculation (25). The area under the extrapolated part of the total curve constituted <12% of the total curve area. The area under the curve and the time-weighted area were determined by the combination of numerical analysis and exponential integration, as previously described (15, 32). All curves were inspected thoroughly for the presence of very early (i.e., before 4 s) and early (i.e., 4 s to maximum) transit times, reflecting fast anatomic shunts before the upstroke of the indicator dilution curve.
In addition, correction of recirculation was performed with a gamma variate fit that included curve points from two samples before peaking until recirculation (i.e., exclusion of the upstroke) (27, 28, 34) (see APPENDIX). The difference in area between the gamma variate fit and the upstroke of the indicator dilution curve was taken to represent possible early circulatory transits, and the flow through these was estimated as the relative area multiplied by the cardiac output. The radioactivity-time integral of the transits (area under the curve) and the mean transit time of this curve were determined and taken as the fast fraction of cardiac output and early mean CCT, respectively. Flow, flow fraction, and circulation time of early transits were related to the alveolar-arterial oxygen difference.
Measurement of Arterial Blood Samples
Arterial oxygen tension (PO2), carbon dioxide tension (PCO2), and pH were measured by an ABL-300 blood gas analyzer, and arterial oxygen saturation (SO2) was measured by an OSM-2 hemoximeter (both from Radiometer, Copenhagen, Denmark). Coefficients of variation of PO2, SO2, and PCO2 were determined from duplicate arterial samples taken at an interval of <15 min: 0.7%, 2.1%, and 3.1%, respectively (33). The alveolar-arterial oxygen gradient (AaPO2) was calculated from the alveolar gas equation
 | (1) |
where FIO2 is the inspiratory O2 fraction, PB is the barometric pressure, and PACO2 is the alveolar carbon dioxide tension, assumed to be equal to PCO2. R is the respiratory exchange ratio, set at 0.80 as found in other studies of cirrhotic patients (35).
Statistical Evaluation
Comparisons between multiple data were performed by one-way ANOVA with Turkey's test for multiple comparisons or by the Kruskal-Wallis test with Dunn's correction for multiple comparisons. Bivariate paired or unpaired data were compared by Student's paired and unpaired t-tests or by the Wilcoxon and Mann-Whitney tests, respectively. The nonparametric tests were applied in the cases of nonnormality. Correlations between independent variables were performed by the least squares method (Pearson). P < 0.05 was considered significant.
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RESULTS
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Physiological Variables in Patients with Cirrhosis and Controls
Hemodynamics.
Cardiac output, stroke volume, heart rate, and arterial compliance were significantly higher in the patients with cirrhosis than in the controls (Table 2). Systemic vascular resistance, arterial blood pressure, CCT and CBV were either significantly reduced or equal to that of the controls.
Arterial blood gas analysis.
A characteristic pattern with slightly decreased oxygen saturation and tension, and significantly reduced PCO2 owing to hyperventilation, was seen in the cirrhotic patients (Table 3).
Analyses of Indicator Dilution Curves
An indicator dilution curve with exponential and gamma variate correction of recirculation is illustrated in Fig. 1. A close relation was found between the cardiac output determined by the two indicators (r = 0.96, P < 0.001) and cardiac output determined by the exponential extrapolation technique and the gamma variate fit technique (r = 0.83, P < 0.001). Likewise, CCTs determined by the two indicators and the two different techniques in each indicator were closely related (Fig. 2). CCT with 99mTc label was 0.49 ± 0.16 s (P < 0.001) shorter than CCT with 125I label, owing to catheter deposit of the technetium-labeled tracer. A slight but statistically significant difference was found between the mean values (gamma variate CCT = 11.6 and exponential extrapolation CCT = 12.7 s, difference 1.06 ± 0.17 s; P < 0.001), owing to the presence of early transits. Correlations of CCT, cardiac output, and CBV determinations are summarized in Table 4.
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Table 4. Correlations between variables obtained by ordinary correction for recirculation (exponential) and by the gamma variate method
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Detection of Possible Very Early and Early Transit Times
A close inspection of the initial part of the indicator dilution curve showed that neither 125I-albumin nor 99mTc albumin (which was shifted
0.5 s) disclosed any very early peaks, which would indicate the existence of very early transits (see Fig. 1). When analytical error is taken into account, any very early transits would be <0.4% of the cardiac output.
Early transits, as reflected by the difference between the measured indicator dilution upstroke and the gamma variate retropolation, amounted to an average of 5.1 ± 0.89% in patients with cirrhosis and 0.8 ± 1.5% in controls (P < 0.005) (Fig. 3). These values correspond to 0.36 ± 0.065 l/min and 0.05 ± 0.08 l/min, respectively (P < 0.01) (Table 5). A Bland-Altman analysis indicates no relation to the level of cardiac output (not shown). There was no significant difference in the values obtained by 125I-albumin and 99mTc-albumin. A significant relation was found between the fraction of early transits and early transit flow and the estimated alveolar-arterial gradient in patients with cirrhosis (Fig. 4) but not in controls.

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Fig. 3. Indicator dilution curves from patients with cirrhosis. A: no evidence of early transits (gamma variate upstroke is identical to measured upstroke). B: small fraction of early transits (small area between gamma variate fit and measured upstroke). C: significant early transits (substantial area between gamma variate fit and measured upstroke).
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Fig. 4. Relation between early transits and alveolar-arterial oxygen difference (AaPO2, kPa). A: AaPo2 and early transit fraction in patients with cirrhosis (r = 0.46, P < 0.01). B: AaPO2 and early transit flow in patients with cirrhosis (r = 0.47, P < 0.01). AaPO2 and early transits in controls (r = 0.040.08; not significant) are not shown.
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"Composite" Determination of the CBV and Arterial Blood Volume
The CBV and arterial blood volume determined by the exponential and gamma variate techniques were directly correlated (Fig. 5A). However, a closer look at the individual patients revealed differences between 0.27 and 0.95 in 10 patients, which may be explained by early transits.
In the patients with cirrhosis, the mean time of the early transits was on average 7.7 ± 0.77 s. When CBV was determined as the sum of early transits (mean early transit time multiplied by its volume flow) and the CCT as obtained by the gamma variate function multiplied by cardiac output less early flow components, there was a good agreement to the CBV obtained in the conventional way (Fig. 5B and Table 6).
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DISCUSSION
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The present study shows that 1) extrapolation of indicator dilution curves for correction of recirculation by the exponential technique and the gamma variate fit technique give very similar results in patients with cirrhosis; 2) no very early transits (i.e., before 34 s) could be detected; 3) a significant contingent of early transits was identified in patients with cirrhosis, which may represent pulmonary vascular shunts or vascular areas with low ventilation-perfusion ratios, related to the size of the alveolar-arterial oxygen gradient; and 4) CBV determined from the sum of a shunt component and a nonshunt component show good agreement with the conventional determination of CBV.
It has been claimed on experimental and theoretical grounds that the gamma variate fit of indicator dilution curves would be better to determine cardiac output, owing to a better correction of recirculation (8, 14, 21, 40). However, any extrapolation procedure may underestimate the very long transits and may thereby lead to an erroneously low area under the late parts of the indicator dilution curve and thus to some reduction in the mean transit time compared with a "true" value (24). In keeping with this point of view, cardiac output, as determined by the Steward-Henriques-Hamilton technique, is often a few percent above that determined from oxygen uptake by the Fick principle (13). When the gamma variate fit technique is applied, the shortcomings of the indicator technique may be somewhat reduced (8, 14, 21). However, from the present results in patients with cirrhosis we found no substantial difference between the traditional exponential (or semilogarithmic) extrapolation and the more sophisticated gamma variate technique. One reason could be that patients with cirrhosis, even when they are resting in the supine position, are clearly hyperdynamic with short central and noncentral circulation times. Besides the cardiac output (which is simply reflected by dose relative to the area under the indicator dilution curve), the CCTs determined by the exponential technique and the gamma variate technique were also similar, and consequently almost identical CBV values were found. In addition, the present results confirm earlier observations in patients with cirrhosis of either normal or contracted CBV (15, 16, 32) and the presence of a low systemic vascular resistance and high arterial compliance (7, 9, 17).
Fitting of the gamma variate function to the indicator dilution curve can be performed with confidence, as described in the APPENDIX. Laplace transformation of this function discloses a number of convoluted exponential functions, which may represent an allusion to several serial compartments, and the gamma variate function has been widely used to model indicator dilution curves from the heart (8, 14, 21, 23, 25, 38, 40). However, no specific model is related to the concept of determination of flow and mean transit time from outlet registration (25). Other types of functions like the shifted random walk function or the lagged normal density function may also be fitted to the present data (23, 38). We tried out these functions on some of our patient data. The shifted random walk function and the gamma variate fit gave very similar results. (This could be expected, as the random walk function in some aspects is close to a gamma variate function). However, the lagged normal density function gave values of cardiac output and mean transit time that were 69% higher than that of the other two functions. The far greater experience with and ease in setting the starting conditions of the gamma variate function (see APPENDIX) are the main reasons for choosing this function in the present study.
We found no significant very early transits in our patients with cirrhosis. Very early transits are seen in patients with, for instance, intracardiac right/left shunts and persistent arterial duct and pulmonary malformations with anatomic shunts. Portopulmonary shunts have been described in some patients with cirrhosis (19). A portopulmonary shunt may reduce the arterial oxygen saturation somewhat but will not be detected by the present technique. Theoretically, a shunt could have a very short temporal dispersion and might thereby escape blood sampling at an interval of 1 s. However, in the present study we used two indicators that were shifted in time by about half a second. In this way, the temporal solution was increased, and from the present measurements any very early transit is insignificant, i.e., <0.4% of the cardiac output in patients with cirrhosis.
Estimated on the values of oxygen saturation in arterial blood and mixed venous and portal venous blood, right-to-left shunts of up to 20% of cardiac output may be present in patients with cirrhosis (33). According to the present study, a certain number of early transits may be present in patients with cirrhosis. The fraction was
5%, corresponding to a flow rate of 0.36 l/min. These early transits did not show a significant relation to the level of cardiac output. The fact that they showed a direct relation to the alveolar-arterial oxygen gradient (Fig. 4) may suggest that they represent either anatomic shunts or areas with low ventilation-perfusion ratios.
Transit time analysis of vascular indications has been applied increasingly in the diagnosis and classification of cirrhosis. Thus Blomley and colleagues (1, 3, 4) have used a noninvasive technique with intravenous injection of ultrasound microbubble contrast agents and subsequent determination of appearance time in hepatic veins and the carotid artery. By this technique they found hepatic vein and carotid transit times consistently shorter with worsening cirrhosis, in agreement with the present findings and earlier reports on prognostic information of short transit times (29).
However, this noninvasive technique requires a change in the intensity of the baseline signal of 10% (4), rendering this method less suitable for detecting minor early circulatory transits. Moreover, at present the microbubble ultrasound technique is semiquantitative with focus on vascular appearance times. But with further development, quantitative data on transit curves can probably be obtained by a noninvasive method.
Studies have shown that the incidence of a hepatopulmonary syndrome may vary considerably among study populations with cirrhosis (6, 10, 36). The hepatopulmonary syndrome is defined as a condition with dilation of pulmonary vessels, the presence of pulmonary shunts, and arterial oxygen desaturation. Only a few patients in the present study population meet these criteria in a strict sense, in accordance with the clinical selection of our patients. But several patients with cirrhosis showed signs of early transits, which may be identified as originating from areas with low ventilation-perfusion ratios or even minor shunts. Thus there may be a gradual transition from minor pulmonary abnormalities to a full-blown hepatopulmonary syndrome. The present technique may prove useful for the study of these patients in greater detail, for instance in combination with other techniques such as contrast echocardiography, ventilation-perfusion isotope scintigraphies, and multi-indicator inert gas studies (2, 20, 35). However, this is a topic for future investigations.
The present study showed CBV values in patients with cirrhosis that were below or equal to those found in controls, as reported previously. Values were significantly reduced in the men, but in the women they were equal to control values, a result which is in keeping with a recent report on other patients (32). Independent determination of CBV by the classic technique and by a composition of early transits and late transits gave closely comparable results (Fig. 5 and Table 6). Lastly, the present results indicate that the assumptions set out in the kinetic theory are also met in patients with cirrhosis with respect to determination of CBV (15, 16). It could be expected that an underestimation of early or late transits would lead to an erroneous determination of the size of the CBV. However, application of the gamma variate technique, which is especially useful to correct for recirculation with conservation of long transits, did not show a significantly different CBV from that of the classic technique.
In conclusion, no very early central circulatory transits were identified in patients with cirrhosis. A significant part of the cardiac output undergoes an early transit, probably through pulmonary shunts or areas with low ventilation-perfusion ratios in patients with cirrhosis. Composite determination of CBV by the gamma variate fit method is in close agreement with established kinetic method. This study presents further evidence of an abnormal central circulation in cirrhosis.
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APPENDIX
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Fitting the Gamma Variate Function to Indicator Dilution Curves
The gamma variate function is defined as
 | (A1) |
It can be shown (25) that this expression can be written as
 | (A2) |
or, if the function begins in t = t0, 0 < t0 < tmax:
 | (A3) |
with A = ymaxtmax
exp(
) and
= tmax/
. Here
and A are eliminated and instead the parameters ymax and tmax are introduced, which are relatively easy to determine from the data since ymax is the maximum value and tmax is the time for maximum. It is possible, with a logarithmic transform of Eq. A3, to estimate
by linear regression, but this requires determination of t0 from the data, which is not generally possible. A better approach is to estimate
, tmax, ymax, and t0 by a multidimensional minimization algorithm. We have chosen to use the downhill simplex method developed by Nelder and Mead (34), which is implemented in the mathematical program MATLAB (v. 5 for Windows 95/NT; The MathWorks, Natick, MA) as the function fmins. This requires provision of a function to be minimized, in this case an expression for the error between the measured curve and the gamma variate function. We use the square root of the sum of squared differences between the values, divided by the maximum value of the data, that is
 | (A4) |
where the subscripts d and
refer to the measured data and the gamma variate function, respectively.
In the case of early transit times, the gamma variate upward slope lags behind the upward slope of the measured data, so we only used curve points from two samples before peaking until recirculation in the fitting process.
When the parameters are estimated, the difference between the data and the gamma variate function is calculated. The mean transit time of early transits is the center-of-mass of the first nonnegative part of the resulting curve.
Determination of Mean Transit Time
For determination of the mean transit time, which is defined as
 | (A5) |
we integrate the gamma variate function and the first moment curve from 0 to
(for t0 = 0)
 | (A6) |
 | (A7) |
where the gamma function is defined as
 | (A8) |
By dividing Eq. A7 by Eq. A6 and using the fact that
(x + 1) = x
(x), the transit time can be expressed as
 | (A9) |
or for t0 > 0
 | (A10) |
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GRANTS
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This work was supported by grants from The John and Birthe Meyer Foundation and Savværksejer Jeppe Juhl and Hustru Ovita Juhl's Memorial Foundation. Jens H. Henriksen was awarded The Niels A. Lassen Prize for 2003.
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ACKNOWLEDGMENTS
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We thank Hanne Hansen and Rosemary Sørensen for skilful assistance.
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FOOTNOTES
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Address for reprint requests and other correspondence: J. H. Henriksen, Dept. of Clinical Physiology and Nuclear Medicine, 239, Hvidovre Hospital, DK-2650 Hvidovre, Denmark (E-mail: jens.h.henriksen{at}hh.hosp.dk)
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
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