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Am J Physiol Heart Circ Physiol 284: H654-H667, 2003. First published October 3, 2002; doi:10.1152/ajpheart.00594.2002
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Vol. 284, Issue 2, H654-H667, February 2003

Flow heterogeneity following global no-flow ischemia in isolated rabbit heart

Robert C. Marshall1,2, Patricia Powers-Risius1, Bryan W. Reutter1, Amy M. Schustz1, Chaincy Kuo1, Michelle K. Huesman1, and Ronald H. Huesman1

1 Department of Nuclear Medicine and Functional Imaging, Ernest Orlando Lawrence Berkeley National Laboratory, University of California, Berkeley 94720-8119; and 2 Martinez Veterans Affairs, Northern California Health Care System, Martinez, California 95616


    ABSTRACT
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

The purpose of this study was to evaluate flow heterogeneity and impaired reflow during reperfusion after 60-min global no-flow ischemia in the isolated rabbit heart. Radiolabeled microspheres were used to measure relative flow in small left ventricular (LV) segments in five ischemia + reperfused hearts and in five nonischemic controls. Relative flow heterogeneity was expressed as relative dispersion (RD) and computed as standard deviation/mean. In postischemic vs. preischemic hearts, RD was increased for the whole LV (0.92 ± 0.41 vs. 0.37 ± 0.07, P < 0.05) as well as the subendocardium (Endo) and subepicardium considered separately (1.28 ± 0.74 vs. 0.30 ± 0.09 and 0.69 ± 0.22 vs. 0.38 ± 0.08; P < 0.05 for both comparisons, respectively) during early reperfusion. During late reperfusion, the increased RD for the whole LV and Endo remained significant (0.70 ± 0.22 vs. 0.37 ± 0.07 and 1.06 ± 0.55 vs. 0.30 ± 0.09; P < 0.05 for both comparisons, respectively). In addition to the increase in postischemic flow heterogeneity, there were some regions demonstrating severely impaired reflow, indicating that regional ischemia can persist despite restoration of normal global flow. Also, the relationship between regional and global flow was altered by the increased postischemic flow heterogeneity, substantially reducing the significance of measured global LV reflow. These observations emphasize the need to quantify regional flow during reperfusion after sustained no-flow ischemia in the isolated rabbit heart.

myocardium; microspheres; reperfusion


    INTRODUCTION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

COMPARED WITH IN VIVO HEARTS, isolated hearts have several advantages for the study of myocardial injury during ischemia and reperfusion. First, flow deprivation is uniform during global no-flow ischemia in in vitro hearts because collateral flow is not possible. Second, the presence and amount of physiological and pharmacological substances in the perfusate that might alter the response of the myocardium to ischemia and reperfusion can be controlled. Third, because of the systemic effects of a low cardiac output in vivo, experimental perturbations that depress cardiac function are more easily studied under sustained steady-state conditions in isolated hearts. Fourth, the ability to control and measure tracer delivery and prevent recirculation facilitates development and application of tracer methodology to assess flow and metabolism during and following myocardial ischemia. Although extrapolation of results to in vivo hearts requires caution, these advantages make the isolated heart an important experimental model to investigate mechanisms that contribute to ischemic and postischemic injury as well as interventions to reduce the severity of their damage (29, 32, 37, 45).

One problem in the use of isolated hearts is that quantification of regional blood flow distribution and heterogeneity during reperfusion following global ischemia has received little investigative attention. It has been established that regional blood flow to normoxic myocardium is spatially heterogeneous (3, 5, 7, 8, 14, 19, 27). Similarly, blood flow during regional ischemia and reperfusion is spatially heterogeneous (12, 17, 30). However, blood flow heterogeneity during reperfusion following global, no-flow ischemia is not known. Also, there are conflicting reports about the presence of subendocardial (Endo) no- or impaired-reflow zones following prolonged complete flow deprivation in isolated hearts (1, 24, 25). If there are zones with impaired reflow following sustained no-flow ischemia, then regionally inadequate postischemic blood flow could be a limiting factor determining myocardial salvage in isolated hearts.

The purpose of this investigation was to quantify the effect of 60 min of global no-flow ischemia on blood flow distribution and heterogeneity during reperfusion. The experimental preparation was the isolated, isovolumic rabbit heart perfused retrograde with a buffer containing erythrocytes and albumin. Microspheres labeled with different radionuclides were used to quantify regional blood flow before ischemia and during reperfusion. Blood flow heterogeneity was assessed by the measurement of microsphere deposition in small left ventricular (LV) segments. Because accurate measurement of regionally impaired reflow depends on microsphere content, a statistical model was developed to determine the precision with which very low flows could be measured in small myocardial segments containing few microspheres.

Our results indicate that during reperfusion in the isolated rabbit heart: 1) flow heterogeneity is increased relative to preischemia in the left ventricle as a whole and in the subendocardium and subepicardium considered separately; 2) impaired reflow develops primarily in the subendocardium; 3) reflow is directed away from the subendocardium into the subepicardium; and 4) the extent of reflow heterogeneity and impaired reflow varies between hearts and correlates with the intensity of contracture.

The increase in postischemic flow heterogeneity and the development of impaired reflow decrease the efficacy with which reperfusion relieves ischemia. In addition, a measured value for global postischemic LV flow does not provide information about regional flow distribution or the source of venous outflow. These observations emphasize the need to quantify regional coronary reflow in isolated rabbit hearts after sustained no-flow ischemia.


    METHODS
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

Experimental Preparation

All procedures were performed in accordance with institutional guidelines for animal research. The preparation of isovolumic retrograde erythrocyte- and albumin-perfused rabbit hearts was similar to previous reports (32, 33). Hearts were obtained from male New Zealand rabbits (R&R Rabbitry; Stanwood, WA) weighing 3.5-4.5 kg. They were given 4,000 units of heparin sodium (Upjohn; Kalamazoo, MI) and 250 mg of pentobarbital sodium (Abbott; North Chicago, IL) via an ear vein. The heart was immediately excised through a median sternotomy, arrested in ice-cold saline, and rapidly attached to a cannula to allow retrograde perfusion. After an apical drain was inserted into the left ventricle, a fluid-filled latex balloon connected to a pressure transducer (P23ID, Gould; Oxnard, CA) was inserted across the mitral valve into the LV cavity. Perfusion pressure and systolic and diastolic ventricular pressures were recorded continuously on a Graphtec Linearecorder (Western Graphtec; Irvine, CA). A coronary venous sampling catheter and a needle thermistor (Omega Engineering; Stamford, CT) were inserted into the right ventricle (RV). The vena cavae and pulmonary artery were ligated so all coronary venous drainage flowed out of the sampling catheter. The atrioventricular node was crushed to allow controlled stimulation using 4-V, 4-ms stimuli from a Grass SD44 stimulator. Temperature was maintained between 36° and 38°C with a water-jacketed heating coil and heart chamber. Coronary flow was held constant with a peristaltic pump (Rainin Instruments; Woburn, MA). Coronary blood flow rate was measured by timed collection from the venous sampling catheter. The perfusate was not recirculated.

During the 40-to-50 min interval required for surgical preparation, coronary flow was held at 65-75% of control values. After surgical preparation was complete, electrodes were placed against the LV and RV, and hearts were initially paced at 80-90 min-1. Approximately 0.1 ml of distilled water was placed in the LV balloon. Over the next 10-15 min, coronary flow, stimulus rate, and LV balloon volume were incrementally increased to the experimental values used in control hearts. Control coronary flow was ~1.4 ml · min-1 · g LV wet wt-1. Stimulus rate was 180 min-1. Balloon volume was increased until developed pressure was >60 mmHg with a diastolic pressure <= 10 mmHg.

The perfusate buffer was a modified Tyrode solution that included 5 mM glucose and 2 mM pyruvate, 22 g/l dialyzed, filtered BSA (Fraction V, fatty-acid free, Roche Diagnostics; Indianapolis, IN), and oxygenated bovine erythrocytes [red blood cells (RBCs)] adjusted to a hematocrit of 18-20%. The plasma and white blood cell fractions of bovine blood were separated from the RBCs by centrifugation. RBCs were oxygenated by multiple spins in perfusate equilibrated with 100% oxygen. The specific electrolyte concentrations of the perfusate buffer were (in mmol/l) 110 NaCl, 2.5 CaCl2, 6 KCl, 1 MgCl2, 0.435 NaH2PO4, and 28 NaHCO3. The pH and PO2 of the RBC + albumin perfusate were measured with a blood gas analyzer (IRMA, Diametrics Medical; St. Paul, MN). The means ± SD pH value was 7.39 ± 0.05 and the PO2 was 429 ± 105 mmHg. To maintain oxygenation and a stable pH, the surface of the RBC and albumin containing perfusate was equilibrated with a mixture of 98% O2-2% CO2 during the experiment.

Technical problems that excluded hearts from this study included aortic valve incompetence, air emboli, and inadequate systolic performance. These problems were always evident during surgical preparation or the subsequent adjustment of perfusion and function parameters to control values. Fewer than 20% of the excised hearts were excluded from this study.

Regional Myocardial Blood Flow

Regional coronary blood flow was measured with radiolabeled microspheres. Microspheres (mean diameter 15.5 ± 0.1 µm, New England Nuclear Life Science; Boston, MA) labeled with 57Co, 95Nb, 103Ru, 113Sn, and 153Gd were suspended in 0.01% Tween 80. The stock solution was prepared by sonicating for about 20 min then vortexing before removing an aliquot for dilution in perfusate buffer. The diluted suspension was vortexed, and 0.2 ml was immediately injected over 30 s into an arterial port 12 cm above the aortic cannula. After the last microsphere injection, the heart was removed from the apparatus and the LV was dissected free, weighed, and plunged briefly into liquid nitrogen. The frozen LV was cut into five rings parallel to the atrioventricular groove. Each ring was cut radially into eight sections (the apical ring into four) using the junction of RV and LV septum for the first section of each ring. Each section was cut into inner (subendocardium) and outer (subepicardium) segments for a total of 72 pieces (92 ± 12 mg average weight). The location of each piece was recorded, and the pieces were weighed in tared vials. The vials were counted on a gamma counter (TM Analytic; Brandon, FL) enhanced with an automated measurement system (MICRAD; Knoxville, TN) (33). The computer-based multichannel analyzer simultaneously quantified and recorded the spectrum from all isotopes in the sample. Reference standards for each of the five isotopes were also counted. Each sample spectrum was decomposed into a linear combination of the spectra of the reference standards using a weighted least-squares fit.

Experimental Protocols

After the heart was prepared, an equilibration period of at least 15 min preceded all experimental interventions. Hearts that generated a systolic developed pressure (peak systolic minus diastolic) >60 mmHg and were stable during equilibration were considered acceptable for use. Developed pressure, diastolic pressure, coronary blood flow rate, and perfusion pressure were recorded after equilibration and before each microsphere injection. Intraventricular balloon volume was maintained constant during all experiments. By clamping the tubing just above the aortic cannula, global no-flow ischemia was produced. The temperature of the heart during ischemia was maintained between 36° and 38°C by wrapping the water-jacketed heart cup and bubbling nitrogen into a saline solution just beneath the heart. At the end of 60 min of ischemia, the tubing was unclamped and flow was restarted and returned to preischemic levels over a 10- to 12-min interval.

Protocol 1

Reproducibility of microsphere technique. The number of microspheres in myocardial segments with very low flows is expected to be considerably less than the "400 microsphere per piece" rule (11). Because reperfusion after 60 min of no-flow ischemia was associated with impaired regional blood flow, the reproducibility of the microsphere technique was tested in two nonischemic and five postischemic hearts by injecting five different radiolabeled sets of microspheres simultaneously (57Co, 95Nb, 103Ru, 113Sn, and 153Gd). In the two nonischemic hearts, microspheres were injected 15 min after equilibration. In the five postischemic hearts, microspheres were injected 18 ± 4 min (~20 min) after the start of reperfusion.

Protocol 2

Distribution of microspheres in normoxic myocardium. The spatial heterogeneity and temporal stability of regional myocardial blood flow in normoxic myocardium were assessed by injecting different radiolabeled microspheres at 15 min, 1 h, and 2 h after equilibration in five control hearts. Although coronary flow was constant in each heart during the 2-h experiment, flow rates in individual hearts ranged from 0.8 to 1.7 ml · min-1 · g LV wet wt-1. The average flow for all five hearts at each injection time is listed in Table 1.

                              
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Table 1.   LV function and perfusion values at time of microsphere introduction

Protocol 3

Distribution of microspheres in pre- and postischemic myocardium. Regional blood flow was measured in five hearts with radiolabeled microspheres just before 60 min of no-flow ischemia and 20 ± 2 min (~20 min or "early" postischemia) and 47 ± 4 min (~50 min or "late" postischemia) after the start of reperfusion. After initiation of ischemia, the stimulator was turned off when contraction ceased. To reduce the incidence of ventricular fibrillation during reperfusion, the stimulator was not restarted until spontaneous contractions appeared (the two hearts that developed ventricular fibrillation were successfully cardioverted in 2-3 min).

Data Analysis

Calculation of regional deposition density. The deposition density (DD) for each piece of tissue was computed from the concentration of a radiolabeled microsphere in that piece divided by the mean concentration of that microsphere over the entire left ventricle
d<SUP>j</SUP><SUB>i</SUB>=<FR><NU>c<SUP>j</SUP><SUB>i</SUB>/m<SUP><IT>j</IT></SUP></NU><DE><IT>C<SUB>i</SUB>/</IT>M</DE></FR> (1)
where d<UP><SUB><IT>i</IT></SUB><SUP><IT>j</IT></SUP></UP> is the DD of isotope i in tissue piece j, c<UP><SUB><IT>i</IT></SUB><SUP><IT>j</IT></SUP></UP> is the number of disintegrations of isotope i counted from tissue piece j, mj is the mass of tissue piece j, Ci = Sigma <UP><SUB><IT>j</IT> = 1</SUB><SUP><IT>J</IT></SUP></UP> c<UP><SUB><IT>i</IT></SUB><SUP><IT>j</IT></SUP></UP> is the total number of disintegrations of isotope i counted from the LV, M = Sigma <UP><SUB><IT>j</IT> = 1</SUB><SUP><IT>J</IT></SUP></UP> mj is the total LV mass, and J is the number of tissue pieces. DD is the equivalent of relative regional blood flow after mean flow has been normalized to one. The absolute blood flow per unit mass in each sample is equal to the product of DD and the mean coronary blood flow to the whole LV. In this study, flow is reported as DD instead of absolute flow to allow quantitative comparisons between nonischemic and postischemic hearts perfused at different flow rates.

To display regional DD in histogram form, the individual DD values were grouped into bins centered on bk of width Delta b (8)
w<SUB><IT>ik</IT></SUB><IT>=</IT><FR><NU><LIM><OP>∑</OP></LIM><SUB><IT>j</IT></SUB> <FENCE>m<SUP><IT>j</IT></SUP><IT> : b<SUB>k</SUB>−</IT><FR><NU><IT>&Dgr;b</IT></NU><DE>2</DE></FR><IT>≤d</IT><SUP><IT>j</IT></SUP><SUB><IT>i</IT></SUB><IT><b<SUB>k</SUB>+</IT><FR><NU><IT>&Dgr;b</IT></NU><DE>2</DE></FR></FENCE></NU><DE>M</DE></FR> (2)
where wik is the fractional mass in histogram bin k for isotope i.

Spatial heterogeneity was expressed as the relative dispersion (RD) and calculated as the weighted sample standard deviation of DD in the tissue pieces, normalized by the weighted mean DD
r<SUB>i</SUB>=<FR><NU><RAD><RCD><LIM><OP>∑</OP></LIM><SUP>72</SUP><SUB>j=1</SUB> (d<SUP>j</SUP><SUB>i</SUB>−<A><AC>d</AC><AC>&cjs1171;</AC></A><SUB>i</SUB>)<SUP>2</SUP>(m<SUP><IT>j</IT></SUP><IT>/</IT>M)</RCD></RAD></NU><DE><IT><A><AC>d</AC><AC>&cjs1171;</AC></A><SUB>i</SUB></IT></DE></FR> (3)
where ri is the RD for isotope i and
<A><AC>d</AC><AC>&cjs1171;</AC></A><SUB>i</SUB>=<LIM><OP>∑</OP></LIM><SUP>72</SUP><SUB>j=1</SUB> d<SUP>j</SUP><SUB>i</SUB>(m<SUP><IT>j</IT></SUP><IT>/</IT>M)
which is the weighted mean DD. When computing RD for subendocardium and subepicardium, the weighted mean DD for each layer was used. Comparison of Endo and subepicardial (Epi) blood flow was computed using the ratio (Endo/Epi) and the difference (Endo-Epi) in control hearts. In the postischemic hearts, only the difference was computed because a few Epi pieces had very low DD values.

Precision of flow measurements. Sources of statistical fluctuation in microsphere activity measurements include the probabilistic natures of microsphere entrapment and radioactive disintegration. With the use of statistical considerations described by Buckberg et al. (11), the number of microspheres trapped in a segment can be assumed to follow a Poisson distribution. The fluctuations in the number of disintegrations detected from a single radiolabeled microsphere obey a second Poisson distribution. However, when the expected number of detected disintegrations is large enough, the variance in counts observed from a segment is due primarily to statistical fluctuations in microsphere entrapment (39).

Because of the Poisson characteristic of microsphere entrapment, the DD measurement variance is proportional to the expected value of DD. As a result, absolute precision increases as flow decreases (Fig. 1). For example, if the standard deviation is 0.05 for an expected DD measurement of 1, then the standard deviation will be only 0.025 for an expected DD measurement of 0.25 (i.e., decreasing flow by a factor of four reduces uncertainty by a factor of two). Thus smaller absolute differences can be discriminated at low flow compared with high flow. This may be somewhat counterintuitive to the "400 microsphere per piece" rule (11), which is useful for setting the baseline relative precision of nonischemic flow measurements.


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Fig. 1.   Depiction of increase in absolute precision with decreasing flow. Poisson probability distributions are shown for measurements involving 400, 100, and 25 expected microspheres, corresponding to deposition densities (DD) of 1 ± 0.05, 0.25 ± 0.025, and 0.0625 ± 0.0125, respectively. Decreasing flow by a factor of 4 reduces the standard deviation by a factor of 2. E(DD), expected deposition density.

We developed a scaled Poisson statistical model to determine the precision with which regional low flows can be measured with radiolabeled microspheres (see APPENDIX). This model was tested by comparing measured data from the seven protocol 1 hearts with simulated data derived from Monte Carlo-generated independent Poisson random variables. The model was then used to determine the confidence with which microsphere flow measurements could be used to indicate impaired reflow.

Criteria for impaired reflow. In postischemic hearts, many segments had activities that were much lower than those observed in nonischemic or preischemic myocardium. By combining measurements from control hearts with preischemic measurements from the ischemic reperfused hearts and by using only one isotope from the two nonischemic hearts in which five differently labeled sets of microspheres were injected simultaneously, there were 1,584 regional blood flow measurements in normoxic myocardium. Seven segments had a DD <0.1 (0.5%). On the basis of these observations in nonischemic myocardium, all segments with DDs <0.1 were considered to have an abnormally low DD in postischemic myocardium. When postischemic segments were analyzed for abnormally low DD, the scaled Poisson statistical model developed in the APPENDIX was used to determine the confidence with which microsphere DD measurements indicated relative flow values <0.1. As described in the APPENDIX, this information was then used to calculate the uncertainty in the number of segments having relative flow <0.1.

Description of statistics. Data are expressed as means ± SD and regression analyses were performed using KaleidaGraph (Synergy Software) statistical software. Comparisons of functional performance, RD, and relative blood flow data were performed with Student's two-tailed t-test for paired and unpaired observations with the use of StatView statistical software (Abacus Concepts). P < 0.05 was considered statistically significant.


    RESULTS
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

LV Function and Perfusion

Table 1 lists the mean values for developed pressure, rest pressure, coronary blood flow, perfusion pressure, and coronary vascular resistance for each experimental group. There were no significant differences in any of these parameters when comparing initial values in nonischemic control hearts to preischemic values in the ischemia-reperfusion group (protocol 3) (P > 0.2 for all comparisons). During the 120-min experimental period in control hearts, developed pressure, rest pressure, and coronary flow did not change (P > 0.2) while perfusion pressure (P < 0.03) increased and coronary vascular resistance (P > 0.05) increased insignificantly.

In comparing early reperfusion to preischemia (protocol 3), developed pressure was severely depressed (P < 0.001), whereas rest pressure (P < 0.05) and perfusion pressure (P < 0.05) were increased. Coronary blood flow (P > 0.2) and coronary vascular resistance (P > 0.06) were not significantly different during early reperfusion. In comparing early and late reperfusion, developed pressure recovered from 32% of preischemic values during early reperfusion to 51% in late reperfusion (P < 0.02). Rest pressure (P > 0.6) and perfusion pressure (P > 0.6) did not change significantly during reperfusion. Coronary flow (P < 0.03) decreased and coronary vascular resistance (P > 0.05) tended to increase.

Precision of Flow Estimates with Microspheres

Protocol 1. In accordance with the scaled Poisson statistical model developed in the APPENDIX, the observed variance of measurements involving simultaneous injection of five differently radiolabeled sets of microspheres decreased with decreasing DD. The observed standard deviations of the measurements exceeded those predicted by the model by an average of ~27%. Possible explanations for the increased statistical fluctuation include errors in average activity per microsphere reported by the manufacturer; deviations in activity from sphere to sphere; errors in our measurements of gamma counter efficiency, due in part to errors in reference standard activities reported by the manufacturer; and sphere aggregation. With the use of protocol 1 heart data, the counter efficiency factors were adjusted as described in the APPENDIX so that the average observed standard deviations were within 0.01% of those predicted by the model. Data from other protocols were then analyzed using the adjusted counter efficiency factors.

Flow Heterogeneity and Distribution in Nonischemic Control Hearts

Protocol 2. Table 2 shows the mean RD values for the whole LV, subendocardium, and subepicardium as well as relative Endo and Epi blood flow at three different times during 120 min of perfusion in control hearts. RD for the whole LV did not change over 120 min of perfusion (P > 0.03). Heterogeneity was insignificantly greater in the subepicardium than the subendocardium at all three time points (P > 0.1 for each comparison). Comparing 15 and 120 min values, RD declined 9.1% (P > 0.1) for the subendocardium and 20.9% for the subepicardium (P > 0.1). Endo DD was increased compared with the subepicardium, possibly reflecting the presence of an incompressible balloon in the LV cavity. This difference increased with time (P < 0.03).

                              
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Table 2.   Relative dispersion and blood flow distribution for nonischemic control hearts

Figure 2 displays in histogram format the mean DD values derived from preischemic and nonischemic hearts in which microspheres were injected 15 min after equilibration. Bin width is in 0.1 µm increments of DD and the myocardial mass in each bin is expressed as the percent mass for the whole LV (Fig. 2A), subendocardium (Fig. 2B), and subepicardium (Fig. 2C). The distribution of DD (i.e., percent myocardial mass) is a frequency function of relative blood flow that is centered around the mean normalized value of one. Although there is variation, all three curves are skewed to the right (skewness of 0.73, 1.20, and 0.89 for LV, subendocardium, and subepicardium, respectively). The LV and subepicardium curves have somewhat flattened peaks (kurtosis of -1.11 and -0.58, respectively), whereas the peak of the subendocardium curve has a nearly Gaussian shape (kurtosis of -0.04).


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Fig. 2.   Binned values of DD expressed as the percentage of whole left ventricular (LV) mass (A), subendocardium mass (B), and subepicardium mass (C). The data are from the preischemia microsphere (MS) distributions of the 60-min ischemia + 50-min reperfusion experiments (n = 5), the first MS distribution of the control experiments (n = 5), and the nonischemia 5 MS experiments (n = 2). For the 5 MS experiments, only 113Sn-labeled MS were included. Of the 864 segments included in this illustration, six had deposition densities <0.1.

Flow Heterogeneity and Distribution in Ischemic-Reperfused Hearts

Protocol 3. Table 3 shows the values for pre- and postischemia RD and Endo-Epi relative blood flow for the five hearts subjected to 60 min of ischemia, followed by ~50 min of reperfusion. The preischemic values for RD and Endo-Epi relative blood flow in these hearts were comparable to the initial values in the control hearts (Table 2) (P > 0.5 for each comparison). Comparing early reperfusion to preischemia, there was a marked increase in RD for the whole LV (P < 0.03) as well as the subendocardium and subepicardium analyzed separately (P < 0.05 for both comparisons). Comparing late reperfusion to preischemia, the increased RD for the whole LV and the subendocardium remained significant (P < 0.05 for both comparisons) while the subepicardium was no longer significantly different (P > 0.1). The high postischemic standard deviations are due to variability in RD values between the five hearts.

                              
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Table 3.   Relative dispersion and relative blood flow for 60-min ischemia hearts

Relative blood flow was redistributed from the subendocardium into the subepicardium during early reperfusion (P < 0.05). During late reperfusion, this difference was not significant (P > 0.05). Comparing late versus early reperfusion, there was a small reduction in the redistribution of blood flow from subendocardium to subepicardium (P > 0.1). The negative values for the difference between subendocardium and subepicardium indicate that, in contrast to preischemia, Epi DD exceeded Endo DD during reperfusion.

Figure 3 shows a DD histogram obtained during early (Fig. 3, A, C, and E) and late (Fig. 3, B, D, and F) reperfusion from one heart after 60 min of no-flow ischemia. The RD for the LV of this heart was 0.79 and 0.74 during early and late reperfusion, respectively. Compared with nonischemic hearts, DD is not unimodal about the mean value. Instead, DD is bimodal with markedly reduced reflow to a large fraction of the subendocardium while reflow to the remaining myocardium is widely dispersed with higher peak flows than nonischemic myocardium (Fig. 2). During early reperfusion, 40.0 ± 3.9% of Endo segments had impaired reflow. During late reperfusion, 25.3 ± 1.9% had impaired reflow. Epi flow was widely dispersed. There was only one Epi segment with reduced reflow during early reperfusion.


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Fig. 3.   The sum of the section weights in 0.1 increments of DD expressed as the percent of the total LV, subendocardium, or subepicardium mass. A: early postischemia, whole LV. B: late postischemia, whole LV. C: early postischemia, subendocardium. D: late postischemia, subendocardium. E: early postischemia, subepicardium. F: late postischemia, subepicardium. In C-F, the DDs have been renormalized so that the mean DD in the subendocardium and subepicardium considered individually is each 1.

All five postischemic hearts that were reperfused for ~50 min had qualitatively similar changes. However, quantitatively, there was a wide range of RD values. Compared with the heart shown in Fig. 3, two hearts had flow that was more heterogeneous (average LV RD of 1.35 and 0.91 during early and late reperfusion, respectively) and two had flow that was less heterogeneous (average LV RD of 0.57 and 0.48 during early and late reperfusion, respectively). For the five hearts combined, reflow heterogeneity tended to decrease as the duration of reperfusion increased.

Similar to RD, the number of segments with impaired reflow varied between hearts. The extent of impaired reflow and increased RD were closely correlated in the subendocardium during early and late reperfusion (Fig. 4, A and B). In the subepicardium, there were fewer segments with impaired reflow and there was no significant correlation between flow heterogeneity and impaired reflow. For the five hearts combined, 37.5 ± 1.3% of Endo segments and 6.0 ± 0.9% of Epi segments had impaired reflow during early reperfusion and 27.7 ± 0.7% and 0.8 ± 0.3% had impaired reflow during late reperfusion. Despite the considerable variability between hearts, the average extent of impaired reflow was similar to the heart illustrated in Fig. 3. Thus the bimodal distribution of flow seen in Fig. 3 is representative of the results observed for the five ischemia reperfused hearts.


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Fig. 4.   Relative dispersion vs. the number of subendocardial (, solid line) and subepicardial (open circle , dotted line) segments with impaired flow after 60-min ischemia (5 experiments). A: early reperfusion. B: late reperfusion. The equations for the linear regression lines are shown in the panels.

The relationships between flow heterogeneity and impaired reflow versus intensity of ischemic contracture during early reperfusion were examined in protocols 1 and 3 hearts (Fig. 5, A and B). In protocol 1 hearts, only one isotope (113Sn) was used to compute RD and number of segments with impaired reflow. There was a positive correlation between intensity of contracture and both flow heterogeneity and impaired reflow.


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Fig. 5.   The relationship between end-ischemic rest pressure and relative dispersion (A) and the number of segments with impaired flow (B). Data are from protocols 1 and 3 hearts (see Table 1). The relative dispersion values and sections with impaired reflow in protocol 1 hearts were computed with the use of a single isotope (113Sn). The equations for the linear regressions are shown in each panel.


    DISCUSSION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

Microspheres were used to quantify blood flow distribution and heterogeneity during reperfusion after 60 min of global no-flow ischemia in the isolated rabbit heart. The four most salient findings are the following: 1) although flow deprivation was homogeneous and absolute, regional flow was more heterogeneous during reperfusion than before ischemia; 2) zones with impaired reflow developed primarily in the subendocardium; 3) reflow was directed away from the subendocardium into the subepicardium; and 4) the severity of flow heterogeneity and reduced reflow varied between hearts and correlated with the intensity of contracture.

These observations are relevant to the use of the isolated rabbit heart as an experimental model to study ischemia and reperfusion. The abnormal postischemic flow distribution adversely affects the efficacy with which reperfusion relieves ischemia. The development of zones with impaired reflow indicates that regional ischemia can persist after reperfusion relieves global ischemia. Also, it is possible that some regions with very high relative flows might be ischemic because the high flows could represent arterial-venous shunting. Because persistent regional ischemia presumably results in continued tissue damage, investigations that use the isolated rabbit heart to study sustained global no-flow ischemia and reperfusion need to quantify postischemic regional reflow to assess the efficacy with which reperfusion relieves ischemia.

In addition to persistent regional ischemia, the markedly increased postischemic flow heterogeneity changes the relationship between global and regional LV reflow. (Because regional flow has been expressed as DD in this study, global LV flow per unit mass corresponds to the mean DD value of one.) The altered relationship between global and regional flow during reperfusion poses two problems. The first is that measured postischemic global LV flow provides no information about regional flow distribution. During reperfusion, regional flow distribution tends to be bimodal and not unimodal and centered about the mean as in preischemic hearts (see Figs. 2 and 3). Second, because global LV venous outflow is primarily derived from regions with maintained or increased flow, arterial venous concentration differences do not provide accurate information about either global or regional substrate or tracer extraction. In postischemic hearts with heterogeneous regional reflow, accurate measurement of either global or regional tracer distribution and tissue metabolism can only be achieved by techniques that directly interrogate the myocardium on a region-by-region basis.

Methodological Considerations

The stochastic nature of microsphere entrapment imposes uncertainty on the relationship between measured microsphere content and expected microsphere content based on true regional flow. Some investigators have characterized this uncertainty using relative error, which is computed as standard deviation/mean (4, 11). Because relative error increases as the number of microspheres decrease, the major emphasis of these studies was to define a minimum number of microspheres allowing accurate flow quantification. In an early study, Buckberg et al. (11) reported that 384 microspheres were required for 95% confidence that measured flow was within 10% of true flow. On the basis of this work, the familiar "400 microsphere per piece" rule was established as a prerequisite for accurate flow quantification. Subsequent studies, however, have shown that fewer microspheres are needed. Nose et al. (39) reported that it was possible to quantify absolute flow with 95% confidence in myocardium containing as few as 49 microspheres as long as the reference sample contained >400 microspheres. Bassingthwaighte et al. (7, 8) made directionally similar observations. Recently, Polissar et al. (42) reported that global parameters of flow heterogeneity and correlation coefficients between different sets of microspheres could be accurately measured with <400 microspheres per sample.

All of these studies evaluated flow quantification with microspheres in normoxic myocardium. In the present study, there were myocardial regions in postischemic hearts with impaired reflow and far fewer microspheres than 400. Nonetheless, microsphere measurements were reproducible and had absolute error that decreased with decreasing flow, in accordance with a Poisson model for microsphere entrapment. As discussed in the APPENDIX, an accurate estimate of the total number of microspheres trapped in the heart is needed to estimate measurement uncertainty. The present study suggests that it is beneficial to perform calibration experiments, in which differently radiolabeled sets of microspheres are injected simultaneously. Calibration factors can then be checked and adjusted, if need be, for use in estimating uncertainty in subsequent experiments in which only one set is injected at a time.

In small arteries, microspheres tend to enter branches with higher rather than lower flows (7, 8). Microsphere biasing into high-flow regions is due to their particulate nature and the greater pressure drop in the direction of higher flow when the vessel diameter approximates the size of the microsphere (15). This bias might have contributed to the increased reflow heterogeneity observed in postischemic hearts. Also, intense vasoconstriction or external compression of the microvasculature following ischemia might have exaggerated biasing because the size of the surrounding vessels might have decreased relative to microspheres. Other work has suggested that microsphere biasing is responsible for relatively small errors in normoxic myocardium (7, 8). However, the role that biasing played in the heterogeneous reperfusion in postischemic hearts is uncertain: accurate evaluation of microsphere biasing requires comparison to a "molecular microsphere" that is completely extracted and retained in myocytes under both normoxic and postischemic conditions (9). A molecular microsphere fulfilling these requirements has not been identified.

Instead of absolute flow, we assessed regional blood flow as DD, which is the equivalent of relative regional blood flow after normalizing mean flow to one. Reporting regional blood flow as DD has the advantage of allowing quantitative comparisons between hearts studied at different LV perfusion rates. In addition, the values for RD and the Endo/Epi ratios are unaffected with the use of DD instead of absolute flow. Because postischemic global LV flow tended to be ~20% lower than preischemia, DD overestimated absolute regional flow during reperfusion relative to preischemia. As a result, the fraction of LV mass with reduced reflow and redistribution of flow away from the subendocardium into the subepicardium was slightly underestimated. These errors were small and did not affect the conclusions of this investigation.

Possible Mechanism(s) for Reflow Heterogeneity

In normal hearts, autoregulation controls the rate of oxygen delivery and regional differences in oxygen consumption are thought to be responsible for heterogeneous regional blood flow (5, 6, 44, 51). However, in postischemic hearts, regionally impaired reflow is not consistent with effective vasomotor control of oxygen delivery, indicating that autoregulation is not the only factor controlling regional blood flow. Ischemic or postischemic injury could also affect regional flow distribution with severely injured hearts displaying the most reflow heterogeneity and regionally reduced reflow. Two observations support this contention: 1) flow heterogeneity and reduced reflow were increased in the subendocardium relative to the subepicardium, consistent with the known vulnerability of the subendocardium to ischemia; and 2) increased flow heterogeneity and reduced reflow correlated with the intensity of ischemic contracture, an index of ischemic injury.

Ischemic or postischemic injury could alter flow distribution during reperfusion by directly injuring the microvasculature. Although originally thought to be a late phenomenon that occurred after myocardial necrosis (28), recent evidence suggests that microvascular injury can occur after only 15 min of ischemia when myocardial injury is still reversible (35). In crystalloid-perfused isolated hearts, microvascular injury appears to involve primarily the endothelium with most of the damage developing after the reintroduction of oxygen during reperfusion (34, 35, 48, 49, 52). Postischemic microvasculature injury can increase vascular resistance (18, 49), depress vasodilatory reserve (36, 38), increase capillary permeability to ions (46) and macromolecules (47), alter blood flow distribution (40, 41), and produce endothelial structural changes with bleb formation capable of impeding or obstructing blood flow (16, 20). There is no information to suggest which of these abnormalities (or combination of abnormalities) might have altered postischemic flow distribution and heterogeneity in the present study.

Ischemic or postischemic injury could also change postischemic flow distribution by external compression of the microvasculature due to osmotic swelling or contracture of the surrounding myocardium. Expansion of the extravascular space by osmotic swelling follows a severe ischemic insult and requires at least transient reperfusion. Although the ischemic myocyte does accumulate water due to the osmotic load of intracellular metabolic catabolites (26), extravascular volume does not change or might even decrease during permanent no-flow ischemia (13). Expansion of the extravascular space requires an outside source of water that is supplied with reperfusion. If extravascular expansion is sufficient to increase intramyocardial pressure, compression of capillaries or postcapillary venules (50) could increase resistance or back pressure to regional reflow and alter postischemic flow distribution.

Contracture also develops after severe ischemic injury (2). By analogy with systolic contraction (21, 43), contracture could increase intramyocardial pressure and compress the microvasculature. Similar to osmotic swelling, if compression of the microvascular is sufficient to increase resistance or back pressure to regional reflow, flow distribution during reperfusion could be altered (1, 22-24). The positive correlation we observed between contracture and reflow heterogeneity suggests that these two processes might be related. However, a cause and effect association was not established and the nature of that relationship is unknown.

To account for heterogeneous reflow, ischemic injury and microvasculature dysfunction have to be heterogeneous. Because flow deprivation was both homogeneous and absolute, heterogeneous injury can only be accounted for by assuming that vulnerability of the LV to ischemia is heterogeneous. However, the presence and potential mechanism for heterogeneous vulnerability has received little investigative attention. Although it is well known that the subendocardium is more vulnerable to ischemic injury than the subepicardium, increased flow heterogeneity was observed in each of these layers considered separately, indicating that differences between the subendocardium and the subepicardium are not sufficient to account for the present results. Ghaleh et al. (17) observed that heterogeneous infarction was related to preischemic, normoxic flow heterogeneity with higher flow regions suffering more damage than lower flow regions. However, another group of investigators observed that metabolic indices of ischemia were not related to preischemic flow heterogeneity (12). Although myocardial strain is known to be heterogeneous (10), contractile activity stopped at least 10 min before any detectable increase in diastolic pressure in all but one heart, suggesting that heterogeneous contraction is not the sole cause of heterogeneous ischemic injury. Other potential contributors, such as myocyte stress or ion flux and homeostasis, are difficult to measure on a regional basis at a spatial resolution adequate to assess flow heterogeneity. Further exploration of potential mechanisms that could account for heterogeneous injury and heterogeneous reflow following sustained global no-flow ischemia in isolated hearts is needed, in part because the presence and distribution of both ischemic injury and postischemic reflow will determine myocardial salvage.

Reflow Heterogeneity, Reduced Reflow, and Ischemia

On the basis of measurements in nonischemic myocardium, segments were identified as having abnormally reduced relative blood flow when DD was <0.1. It is possible that some segments with DD <0.1 might not be ischemic or infarcted. Reduced relative blood flow in nonischemic segments could reflect severe relative regional contractile dysfunction or altered ion homeostasis with reduced energy consumption following sustained ischemia. Therefore, although reduced reflow does not necessarily identify ischemic or infarcted myocardium, abnormally low relative flow after prolonged absolute flow deprivation does indicate the presence of myocardium that is severely dysfunctional as a result of ischemic damage sustained during no-flow ischemia.

Despite the uniform duration of no-flow ischemia, the intensity of ischemic contracture was variable when comparing individual hearts. To determine whether preischemic function or perfusion might be related to the variable ischemic contracture, preischemic coronary flow rate, perfusion pressure, coronary vascular resistance, and rest and developed pressure were correlated with end-ischemic contracture. There was an insignificant trend for preischemic coronary flow to be inversely related to ischemic contracture (R = -0.61, P > 0.07). The other parameters were not correlated. Although it is possible that hearts perfused at lower flow rates might have been under-perfused before the initiation of no-flow ischemia, previous work (31) has shown that lower flows than those encountered here were not associated with detectable high-energy phosphate depletion or lactate accumulation, suggesting that additional unrecognized factor(s) probably contributed to ischemic injury. Although the reason(s) are not apparent, the variability of ischemic contracture did not adversely affect the conclusions of this investigation. Instead, the correlation between contracture, reflow heterogeneity, and regionally reduced reflow suggests that the severity of ischemic injury might have contributed to the abnormal postischemic flow distribution.

Humphrey et al. (24) reported that the presence of an inflated balloon in the LV cavity during ischemia, followed by deflation before reperfusion allowed the return of normal Endo blood flow during the first 20 s of reperfusion. However, later studies (1, 25) showed that Endo no reflow developed if reperfusion was continued for longer periods of time (e.g., 5 min). The results of the present study are more consistent with these later studies.

In conclusion, reperfusion after sustained global, no-flow ischemia results in increased regional flow heterogeneity relative to preischemia and the development impaired reflow zones. The severity of abnormal reflow distribution is variable between hearts and correlates with the intensity of ischemic contracture. These observations affect the use of the isolated rabbit heart as an experimental model to study ischemia and reperfusion because 1) restoration of global flow after sustained no-flow ischemia can be associated with persistent regional ischemia, and 2) measured global flow provides limited information about regional flow distribution and the source of venous outflow. Although our observations were made in the isolated rabbit heart, it is possible that similar findings could be detected in hearts from other species.


    APPENDIX
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES

A statistical model was developed to determine the precision with which very low flows can be measured in small tissue segments containing few radiolabeled microspheres. With the use of statistical considerations described by Buckberg et al. (11), the number of microspheres trapped in a segment can be assumed to follow a Poisson distribution. Fluctuations in the number of disintegrations detected from a single radiolabeled microsphere obey a second Poisson distribution. However, when the expected number of detected disintegrations is large enough, the variance in counts observed from a segment is due primarily to statistical fluctuations in microsphere entrapment (39).

Entrapment Fraction Calculation

The Poisson statistical model for microsphere entrapment was tested in two nonischemic and five postischemic hearts. Five sets of differently radiolabeled microspheres were injected simultaneously in each heart (see METHODS). Each heart was divided into 72 segments, which were counted in a gamma counter for 5 min so that the number of counts detected from each microsphere was large enough so that variability due to statistical fluctuations in the number of disintegrations could be ignored. For each isotope i the gamma counter efficiency factor varepsilon i was measured using a reference standard whose activity level was specified by the isotope manufacturer. The expected number of disintegrations per microsphere eta i was calculated using data from the microsphere manufacturer. For each segment j the number of counts from each type of microsphere c<UP><SUB><IT>i</IT></SUB><SUP><IT>j</IT></SUP></UP> was determined by decomposing the observed spectrum into a linear combination of spectra of reference isotope standards using a weighted least-squares fit.

Combining this information, the number of microspheres labeled with isotope i trapped in segment j, n<UP><SUB><IT>i</IT></SUB><SUP><IT>j</IT></SUP></UP> was calculated
n<SUP>j</SUP><SUB>i</SUB>=<FR><NU>c<SUP>j</SUP><SUB>i</SUB></NU><DE>ϵ<SUB>i</SUB>&eegr;<SUB>i</SUB></DE></FR> (A1)
The fraction of microspheres trapped p<UP><SUB><IT>i</IT></SUB><SUP><IT>j</IT></SUP></UP> was calculated, also
p<SUP>j</SUP><SUB>i</SUB>=<FR><NU>c<SUP>j</SUP><SUB>i</SUB></NU><DE>C<SUB>i</SUB></DE></FR>=<FR><NU>n<SUP>j</SUP><SUB>i</SUB></NU><DE>N<SUB>i</SUB></DE></FR> (A2)
where Ci = Sigma <UP><SUB><IT>j</IT> = 1</SUB><SUP><IT>J</IT></SUP></UP> c<UP><SUB><IT>i</IT></SUB><SUP><IT>j</IT></SUP></UP> was the total number of counts from isotope i for the entire LV, Ni = Sigma <UP><SUB><IT>j</IT> = 1</SUB><SUP><IT>J</IT></SUP></UP> n<UP><SUB><IT>i</IT></SUB><SUP><IT>j</IT></SUP></UP> was the estimated total number of microspheres labeled with isotope i trapped in the LV, and J was the number of segments (i.e., J = 72).

Statistical Modeling

Assuming that the expected value of the entrapment fraction for segment j is the same for each isotope i and that the error in estimating the total number of trapped microspheres can be neglected, the expected value of the entrapment fraction µj is defined as follows
&mgr;<SUP>j</SUP> ≜ <IT>E</IT>(<IT>p</IT><SUP><IT>j</IT></SUP><SUB><IT>i</IT></SUB>)<IT>=</IT><FR><NU><IT>E</IT>(<IT>n</IT><SUP><IT>j</IT></SUP><SUB><IT>i</IT></SUB>)</NU><DE><IT>N<SUB>i</SUB></IT></DE></FR> (A3)
where E(n<UP><SUB><IT>i</IT></SUB><SUP><IT>j</IT></SUP></UP>) is the expected number of microspheres labeled with isotope i trapped in segment j. With the use of a Poisson model for microsphere entrapment, the variance of the entrapment fraction for isotope i in segment j is
Var(<IT>p</IT><SUP><IT>j</IT></SUP><SUB><IT>i</IT></SUB>) = <FR><NU>Var(<IT>n</IT><SUP><IT>j</IT></SUP><SUB><IT>i</IT></SUB>)</NU><DE><IT>N</IT><SUP>2</SUP><SUB><IT>i</IT></SUB></DE></FR><IT>=</IT><FR><NU><IT>E</IT>(<IT>n</IT><SUP><IT>j</IT></SUP><SUB><IT>i</IT></SUB>)</NU><DE><IT>N</IT><SUP>2</SUP><SUB><IT>i</IT></SUB></DE></FR><IT>=</IT><FR><NU><IT>&mgr;<SUP>j</SUP></IT></NU><DE><IT>N<SUB>i</SUB></IT></DE></FR> (A4)
Given models for the mean and variance of the entrapment fraction for each isotope i, an estimate of the expected entrapment fraction for segment j is obtained by defining a weighted least-squares criterion in which squared differences between the observed and modeled entrapment fractions are inversely weighted by the modeled variances
<FENCE>X<SUP>j</SUP></FENCE><SUP>2</SUP>=<LIM><OP>∑</OP><LL>i=1</LL><UL>I</UL></LIM> <FR><NU><FENCE>p<SUP>j</SUP><SUB>i</SUB>−&mgr;<SUP>j</SUP></FENCE><SUP>2</SUP>N<SUB>i</SUB></NU><DE>&mgr;<SUP>j</SUP></DE></FR>=<LIM><OP>∑</OP><LL>i=1</LL><UL>I</UL></LIM> <FENCE>x<SUP>j</SUP><SUB>i</SUB></FENCE><SUP>2</SUP> (A5)
where (x<UP><SUB><IT>i</IT></SUB><SUP><IT>j</IT></SUP></UP>)2 = (p<UP><SUB><IT>i</IT></SUB><SUP><IT>j</IT></SUP></UP> - µj)2 Nij and I is the number of isotopes (i.e., I = 5). Minimizing this criterion with respect to the model parameter µj yields an estimate of the expected entrapment fraction <A><AC>&mgr;</AC><AC>ˆ</AC></A>j
<A><AC>&mgr;</AC><AC>ˆ</AC></A><SUP>j</SUP>=<RAD><RCD><FR><NU><LIM><OP>∑</OP></LIM><SUP>J</SUP><SUB>i=1</SUB> <FENCE>p<SUP>j</SUP><SUB>i</SUB></FENCE><SUP>2</SUP>N<SUB>i</SUB></NU><DE><LIM><OP>∑</OP></LIM><SUP>I</SUP><SUB>i=1</SUB> N<SUB>i</SUB></DE></FR></RCD></RAD> (A6)

Statistical Model Validation

With the use of simulated Poisson microsphere entrapment data, it was verified that the criterion given by Eq. A5 is distributed approximately as a chi 2 random variable with I - 1 degrees of freedom when <A><AC>&mgr;</AC><AC>ˆ</AC></A>j estimated by Eq. A6 is substituted for µj in Eq. A5 to obtain the following statistic
(<A><AC>X</AC><AC>ˆ</AC></A><SUP>j</SUP>)<SUP>2</SUP>=<LIM><OP>∑</OP><LL>i=1</LL><UL>I</UL></LIM> <FR><NU><FENCE>p<SUP>j</SUP><SUB>i</SUB>−<A><AC>&mgr;</AC><AC>ˆ</AC></A><SUP>j</SUP></FENCE><SUP>2</SUP>N<SUB>i</SUB></NU><DE><A><AC>&mgr;</AC><AC>ˆ</AC></A><SUP>j</SUP></DE></FR>=<LIM><OP>∑</OP><LL>i=1</LL><UL>I</UL></LIM> <FENCE><A><AC>x</AC><AC>ˆ</AC></A><SUP>j</SUP><SUB>i</SUB></FENCE><SUP>2</SUP> (A7)
For the case where a DD of 1 corresponds to 400 microspheres, 1.25 million independent Poisson random variables with mean values of 40 were generated to simulate the trapping of I = 5 types of microspheres in J = 250,000 heart sections having an expected DD of 0.1. For each heart section, j as the expected value of the entrapment fraction was estimated using Eq. A6 and substituted into Eq. A5 to yield Eq. A7. The sample mean for each (x<UP><SUB><IT>i</IT></SUB><SUP><IT>j</IT></SUP></UP>)2 term in the sum in Eq. A7 was ~0.8, as expected given that their sum was distributed approximately as a chi 2 random variable with I - 1 = 4 degrees of freedom.

Similar calculations were then made using measured microsphere data obtained from 486 segments of the 7 hearts injected with 5 sets of microspheres. Sections were selected using the criterion that there should be evidence of flow such that the estimated activity for at least one of the isotopes was at least three times greater than its estimated uncertainty. Sections meeting this criterion had DD values ranging down to ~0.001 in postischemic hearts. On the basis of initial estimates of the total numbers of microspheres trapped in the LV (the Ni), the observed normalized deviations of the measurements (the x<UP><SUB><IT>i</IT></SUB><SUP><IT>j</IT></SUP></UP> in Eq. A7) exceeded those predicted by the model by ~27%. Possible explanations for the increased statistical fluctuation include errors in the average activity per microsphere eta i reported by the manufacturer; deviations in activity from sphere to sphere; errors in our measurements of the gamma counter efficiency factors varepsilon i, due in part to errors in reference standard activities reported by the manufacturer; and sphere aggregation.

The estimates of the total numbers of microspheres trapped in the LV were then adjusted so that the average observed deviations were within 0.01% of those predicted by the model. Figure 6A contains a histogram (depicted as a bar graph), which shows the resulting distribution of normalized squared deviations [the term (x<UP><SUB><IT>i</IT></SUB><SUP><IT>j</IT></SUP></UP>) in Eq. A7] for 113Sn-labeled microspheres. Figure 6A also contains a histogram (depicted using solid circles and error bars), which shows the shape of the distribution of normalized squared deviations obtained from the simulated Poisson entrapment data. There is good agreement between the measured and modeled distributions after adjustment of the estimates of the total numbers of microspheres trapped in the LV. Figure 6, B-E, shows the distributions of normalized squared deviations for 153Gd, 57Co, 103Ru, and 95Nb-labeled microspheres, which are also in good agreement with the model.


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Fig. 6.   Measured and modeled distributions for normalized squared deviations of entrapment fractions for microspheres labeled with (A) 113Sn; (B) 153Gd; (C) 57Co; (D) 103Ru; and (E) 95Nb. Bar graphs depict the measured distributions and solid circles depict the modeled distribution obtained from simulated Poisson entrapment data. Error bars depict a range corresponding to plus or minus one standard deviation, where the standard deviation has been calculated as the square root of the number of simulated segments in the model histogram bin. The horizontal axis has been truncated at a value beyond which lies only ~1% of the simulated segments.

Confidence of DD Measurements

The variance (Var) of a DD measurement for isotope i in heart segment j is obtained from Eqs. 1, A2, and A4 as
Var<FENCE><IT>d</IT><SUP><IT>j</IT></SUP><SUB><IT>i</IT></SUB></FENCE><IT>=</IT><FENCE><FR><NU>M</NU><DE>m<SUP><IT>j</IT></SUP><IT>N<SUB>i</SUB></IT></DE></FR></FENCE><SUP>2</SUP> Var<FENCE><IT>n</IT><SUP><IT>j</IT></SUP><SUB><IT>i</IT></SUB></FENCE><IT>=</IT><FENCE><FR><NU>M</NU><DE>m<SUP><IT>j</IT></SUP><IT>N<SUB>i</SUB></IT></DE></FR></FENCE><SUP>2</SUP><IT> E</IT><FENCE><IT>n</IT><SUP><IT>j</IT></SUP><SUB><IT>i</IT></SUB></FENCE> (A8)

<IT>=</IT><FR><NU>M</NU><DE>m<SUP><IT>j</IT></SUP><IT>N<SUB>i</SUB></IT></DE></FR><IT> E</IT><FENCE><IT>d</IT><SUP><IT>j</IT></SUP><SUB><IT>i</IT></SUB></FENCE>
where it has been assumed that error in measuring mj can be neglected. With the use of this scaled Poisson model, the statistical significance of the difference between a DD measurement and a value of interest can be assessed. In particular, a number between 0 and 1 can be assigned that quantifies the confidence that relative flow is above or below a certain level.

Approximating the scaled Poisson model with a Gaussian, the confidence that a DD measurement indicates a relative flow of at least alpha  is given by the function
&rgr;<SUP>j</SUP><SUB>i</SUB>(&agr;)=&PHgr;<FENCE><FR><NU>d<SUP>j</SUP><SUB>i</SUB>−&agr;</NU><DE>(M<IT>d</IT><SUP><IT>j</IT></SUP><SUB><IT>i</IT></SUB><IT>/</IT>m<SUP><IT>j</IT></SUP><IT>N<SUB>i</SUB></IT>)<SUP>1<IT>/</IT>2</SUP></DE></FR></FENCE> (A9)
where
&PHgr;(x)=<LIM><OP>∫</OP><LL>−∞</LL><UL>x</UL></LIM> <FR><NU>1</NU><DE><RAD><RCD>2&pgr;</RCD></RAD></DE></FR> exp<FENCE>−<FR><NU><IT>u</IT><SUP>2</SUP></NU><DE>2</DE></FR></FENCE><IT>du</IT>
is the cumulative distribution function for a Gaussian with zero mean and unit variance. Conversely, the confidence that relative flow is less than alpha  is 1 - rho <UP><SUB><IT>i</IT></SUB><SUP><IT>j</IT></SUP></UP>(alpha ). On the basis of this single measurement, a Bernoulli random variable B<UP><SUB><IT>i</IT></SUB><SUP><IT>j</IT></SUP></UP> can be used to model the outcomes of repeatedly measuring DD in a heart segment for which the expected value is d<UP><SUB><IT>i</IT></SUB><SUP><IT>j</IT></SUP></UP>. The random variable B<UP><SUB><IT>i</IT></SUB><SUP><IT>j</IT></SUP></UP> either takes a value of 1 with probability 1 - rho <UP><SUB><IT>i</IT></SUB><SUP><IT>j</IT></SUP></UP>(alpha ) to indicate that DD is less than alpha , or takes a value of 0 with probability rho <UP><SUB><IT>i</IT></SUB><SUP><IT>j</IT></SUP></UP>(alpha ) to indicate that DD is at least alpha . Thus the expected total number of segments for which DD is less than alpha  is