Published in: Medical Imaging 1994: Physiology and Function from Multidimensional Images, Eric A. Hoffman, Raj S. Acharya, Editors, Proc. SPIE 2168, 380-392 (1994).




Posture Dependent Spatial Correlation: Similarity of Multiple CT-Derived
Pulmonary Structural and Functional Parameters

Jehangir K. Tajik, Collin L. Olson, Gopal Sundaramoorthy and Eric A. Hoffman

Department of Radiology
University of Iowa College of Medicine
Iowa City, IA 52242

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Table of Contents

ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
CONCLUSION
ACKNOWLEDGEMENTS
REFERENCES

ABSTRACT

To help characterize the determinants of the spatial distribution of regional pulmonary structure and function and to characterize a spatial autocorrelation (SAC) approach, we have applied SAC statistics to our pulmonary cine x-ray CT data of regional pulmonary blood flow and to various computer derived models (cubes and pyramids, 3-D wedges, and lung shapes in which pure "flow" gradients in either the x, y, or z directions were applied). To generate graphs of correlation vs. distance, we bin the data according to distance into a user specified number of groupings and then autocorrelate the data within each bin. Only regions of pulmonary parenchyma within the same lobe were used. We present the results of our analysis which show that several regional parameters exhibit a similar negative sloping correlation vs. distance relationship. Results from our computer models show: 1) there is an order dependence of the spatial correlation algorithm when each region pair is used only once, 2) that the negative correlation vs. distance relationship can come about solely through the effects of gravity, and 3) that the effect of an object's shape on the spatial autocorrelation relationship is minimal. Regional air content follows a spatial distribution pattern consistent with purely gravitational effects.The similarity of the SAC values for other pulmonary functional values such as pulmonary blood flow, mean transit time, pulmonary blood volume, etc. may relate to their partial dependence on regional alveolar expansion. This may be a likely driving force behind the similar spatial distribution of several other parameters. SAC statistics provide a unique tool for demonstrating the existence of underlying patterns to distribution of pulmonary function. However, because several regional distribution inhomogeneities can cause the same negative sloping relationship with a zero cross, it is not fully clear how well SAC uniquely defines the driving force describing the spatial distribution of a function. Care needs to be taken in finding appropriate normalization parameters in comparing data as the normalization factor itself can lend structure to spatial autocorrelation. Finally, shape of the spatial autocorrelation function may be as important or more important than slope in defining underlying structure.

2. INTRODUCTION

The primary function of the lung is to exchange oxygen and carbon dioxide between inspired air and pulmonary arterial blood at the level of the alveolar/capillary interface. Because matching of pulmonary blood flow and ventilation is Posture Dependent Spatial Correlation: Similarity of Multiple CT-Derived of paramount importance, and because the two parameters have been shown not to be distributed evenly throughout the lung, the determinants of this uneven spatial distribution has been the topic of much research in the field of pulmonary physiology over the last several decades or more. Regional ventilation and/or regional lung density changes have been shown to be greater in the lower or dependent lung region in the upright human1 or supine animal2, 3. However, it has been shown that this relationship does not simply reverse itself in the prone body posture, and gradients are apparent in the isogravimetric sagittal plane of animals in the right or left lateral decubitus body postures. Data has suggested that support of the heart may be an important parameter in determining the prone versus supine differences in regional ventilation: supine, the lung supports the heart and prone, the sternum supports the heart. Likewise, we and others4, 5, 6 have previously demonstrated that regional differences in pulmonary perfusion (blood flow to the lung parenchyma) are only partially determined by gravity, and evidence has been put forward to suggest that vascular geometry (conductance) may play a role in distribution of blood flow. Glenny and Robertson7 have applied measurement techniques which suggest that blood flow distribution fits a fractal distribution pattern, and again feel that gravity plays a minimal role in the perfusion heterogeneity. Hakim and colleagues have used SPECT and radiolabeling based techniques to suggest that pulmonary perfusion is actually radially distributed, reflective of a vascular branching pattern as well. Glenny8 has applied a spatial autocorrelation statistic9 to the pattern of pulmonary blood flow found using radioactive microspheres to tag pulmonary parenchymal blood flow in intact dogs with subsequent excision of the lungs for quantitation of microsphere distribution. The observed negative sloping relationship with a zero crossing and negative correlation coefficients between regions of large distances was felt to be reflective of the underlying branching pattern of the pulmonary arterial tree. There may, however, be correlations within other pulmonary parameters implying an overall pattern to pulmonary structure and function. In addition, gravity and/or other factors such as heart position may influence spatial distribution.

As shown schematically in Figure 1, spatial autocorrelation statistics are used to measure how the value of a parameter at one point (voxel) in space is related to the same parameter at varying spatial distances. Spatial autocorrelation statistics are calculated by pairing each sampled voxel or group of voxels successively with all other voxel locations and recording the parameter value associated with each location pair along with the distance between the two samples to generate a list of data in the form distance, value1, value2. After each spatial location has been paired with all other locations, data with similar distances between samples are grouped together. The data within each bin are treated as paired observations and linearly correlated, producing a correlation coefficient for that "bin" or "distance."

In our previous studies of pulmonary blood flow distribution, we found that the spatial location along the gravity vector defines only 50% +10%(SD)of the spatial heterogeneity supine and 40% +10%(SD) prone and the coefficient of variation measuring spatial heterogeneity was 31% across all body postures. We were unable to demonstrate a radial gradient close to the magnitude found by Hakim. If one invokes a role for vascular conductance and vascular geometry in determining pulmonary blood flow distribution, the minimal presence of a radial distribution pattern is not surprising: vascular branching, particularly in species other than humans, is not radially oriented. From these observations along with our observations regarding regional lung air content and regional differences in alveolar expansion with lung inflation, we hypothesized that regional air content and regional pulmonary perfusion have two non-overlapping determinants: 1) gravity and support of the heart, and 2) vascular conductance respectively. An interesting additional finding from our data to date has been that there is a strong and similar correlation between regional lung air content and the regional blood content of the lung, thus implying either a causal relationship between these two parameters or a common determinant. With an interest in evaluating the utility of the spatial autocorrelation construct in furthering our hypothesis testing, we have undertaken an evaluation of the spatial autocorrelation statistic, and have applied it to the various regional lung function parameters which are derived from time-intensity curves obtained via use of Electron Beam x-ray CT (EBCT: Imatron Corp, South San Francisco) scanning10.

3. METHODS

3.1 Evaluation of regional pulmonary function via Electron Beam x-ray CT:

We have previously described a method of image acquisition using EBCT11 scanning and image post-processing to simultaneously quantitate several regional structural and functional pulmonary parameters12. The Imatron C-100 scanner was used in the high temporal resolution mode to follow an injected bolus of contrast agent as it passed through the lungs. Image data were acquired from six anesthetized dogs using either eight spatially distributed levels sampled at ten time points each, or using six levels sampled at thirteen time points each (memory constraints of the scanner currently limit image acquisition to a maximum of 80 slices). Image data sets were acquired gated to the QRS complex with lung volume held at functional residual capacity (0 cmH2O airway pressure) in both the supine and prone body positions. Analysis of the regional time-intensity (gray level) changes within the lung field due to contrast passage allowed calculation of the following parameters: regional air, tissue and blood percentages; regional raw blood flow, blood flow normalized to regional air content, and blood flow normalized to regional tissue content; regional mean transit, peak and arrival times; peak opacification. Empirically derived selection criteria based on the calculated values were used to accept only regions of pulmonary parenchyma, eliminating regions containing major arteries, veins or airways. The selection criteria shown in Table 1 were used for all dogs in both body postures [note: it has been found that the appropriate criteria are not constant across species]. All parameters were calculated by a customized software module within VIDA13 and saved to a text file along with each accepted region's three dimensional coordinates.

3.2 Implementation of Spatial Autocorrelation:

We developed a computer program to bin data by distance into a user specified number of groupings and then calculate spatial autocorrelation as a function of distance for each of the calculated regional parameters and for our computer generated models (see 3.3 and 4.1).The concept of spatial autocorrelation is demonstrated above in Figure 1. Using the voxel dimensions of the image data set, the real distances from each voxel to all other voxels are calculated and the maximum distance between pairs of points in the data set is stored. The maximum distance is then divided by the user specified number of groups (or bins) to calculate the binsize. The real distance from each voxel to all other voxels are calculated and the parameter value of interest (i.e. air content) is recorded from each of the paired regions to generate data in the form: distance, value1, value2. The distance between two samples is taken to be the 3-d Euclidean distance and initially each region pair was used only once. The distance-parameter data is then binned according to distance into the user specified number of groups. The values within each bin are treated as paired observations and linearly correlated as given by the linear correlation coefficient r (below). Our software's correlation calculations were validated by comparing the r value calculated by our program to that calculated by a commercially available statistics package (StatView II, Abacus Concepts) for a sample data file we created. The results were identical (data not shown).

The output of the program is a text file listing distance, correlation coefficient, and number of pairs for each non-empty bin. The distance reported is from the center of each non-empty bin. The correlation coefficient (r) reported for each bin/distance is used to generate graphs of correlation (r) vs. distance (cm) for each parameter from the supine and prone body positions and also for the computer generated models. For our CT-derived data, only regions within the same lobe (lower left lobe as approximated by user-interactive editing of the image data using VIDA) were correlated. The lower left lobe was used as this was the least challenging in terms of separating the lobe from other lobes using the 8mm thick EBCT high temporal resolution image data sets.

3.3 Computer generated models:

To test for the effects of shape, gravity, etc. on spatial correlation, several computer generated models were analyzed in a similar fashion to that described in 3.2. We first generated cubes (voxel arrays) with either a random distribution of grey scale assignments to the voxels (using a random number generator) or a pure grey scale gradient in either the x, y, or z direction. Gradients were simulated by assigning a voxel value identical to the voxel position in the gradient direction so that all points at the same location in the plane perpendicular to the gradient vector had identical values assigned to them. Pyramids and 3-d wedges with pure single axis gradients were also generated and analyzed. Additionally, we applied pure gradients to the three dimensional voxel coordinates of a supine and prone lower left lung lobe of one dog in our study. The simulated geometric computer models were constructed of 10x10x10 pixel blocks (1000 pixels).

4. RESULTS

4.1 Computer models/simulations:


Figure 2


Figure 3

Data from the cube model are shown in Figure 2. The cube populated with random values showed no correlation (r = 0) over all distance, as would be expected for a random spatial distribution (Figure 2, upper left). The cubes simulating pure gravity gradients in either the x, y, or z direction should exhibit results identical to each other. This was true for the y and z gradients as seen in the lower panels of Figure 2. However, the spatial autocorrelation data from the cube with an x gradient Figure 2, upper right are clearly different. In our initial analysis, we assumed that it was not necessary to use each region pair twice (i.e. 1,3 and 3,1). While investigating the cause for the different result in the x based gradient, results clearly indicated that use of only a single pair introduces an "order dependence" since the only difference between the three cubes was the ordering of the data. To rectify the problem, we re-ran the analysis keeping all pairs and using the same number of bins as before. The results in this case showed all the cubes to be identical as should be expected. Figure 3 demonstrates the spatial correlation plot for the x based gradient data using all pairs to generate the plot on the left and only one pair for the plot on the right. All data reported below, therefore, result from analyses where all pairs in the data set are included.

Compared to the random population cube, the pure gradient simulations exhibit a different correlation-distance relationship. The gradient cubes display a high positive correlation for nearby samples that decays as distance increases, with the correlation coefficient becoming increasingly negative as distance increases (Figures 2 & 3). This negative correlation at greater distances is similar to that observed by Glenny8.

To test for the effects of shape on the spatial autocorrelation pattern, we analyzed pyramids and 3-d wedges with either pure x, y, or z gradients. Though the correlation-distance data from the 3-d wedges and cubes are from two distinct structures, they both show a similar correlation-distance relationship (Figure 4). In the case of the 3-d wedge, two simulations (x and y gradients) yielded identical results while the third (z gradient) simulation shows a slightly altered relationship relative to the other two. Although there is greater scatter in the data, the pyramid models display similar correlation vs. distance relationships to the cubes and wedges.

We also applied pure x, y, or z gradients to the three dimensional voxel coordinates of the left lower lobe from the supine and prone body postures. Again, there are high, positive correlations for nearby regions that become increasingly negative as the distances between regions increase. The results are not identical for the three gradient directions within a given body posture, and the correlation-distance relationship is the most consistent between the prone and supine posture for the y gradients, which correspond to the direction of the gravitational field in our experimental data sets. While the lobe changes shape from the supine to prone positions, the spatial correlation patterns for a given applied gradient direction remain nearly identical for the two body postures. It appears, therefore, that an object's shape has little influence on the correlation vs. distance relationship. Moreover, the increasingly negative correlation over larger distances is generated by gravitational effects alone. The altered pattern for the z gradient in our lobe data set might be explained by the reduced resolution in the z axis due to the 8 mm slice thickness.

4.2 Pulmonary CT data:

We followed the computer model autocorrelation analysis by applying spatial autocorrelation statistics to our x-ray CT derived regional structural and functional parameters of the pulmonary system. Autocorrelation analysis was done for the following nine parameters from the left lower lobe in both the supine and prone postures: regional air, blood and tissue percentages; regional arrival, mean transit, and peak times; regional raw blood flow and regional blood flow normalized to tissue or air content. The results reported here are from the left lower lobes of six canines imaged in the supine and prone body postures. Dogs were anesthetized with sodium pentobarbital and Inovar. Guidelines governing the care and treatment of animals issued by the NIH were adhered to and study permission was obtained from the institutional animal care and use committee.


Figure 4

Regional air content autocorrelation plots show a steep negative correlation vs. distance relationship supine which is significantly reduced or eliminated (essentially remaining flat around zero correlation for all distance) in the prone posture (figure 5, upper left). The regional air content autocorrelation plots of data from the supine posture display a similar correlation vs. distance pattern to those from the pure gravity gradient cubes (see figure 2) and particularly to the simulated lobar gravity gradient shown in figure 4, center bottom panel, suggesting that regional air content may be influenced by gravitational effects (such as the lung's support of the heart) supine while gravitational effects are reduced or eliminated in the prone posture (such as by a shift in support of the heart weight from the lungs to the sternum).

For all dogs, the spatial correlation of regional tissue content follows a closely similar pattern across the supine and prone body postures. However, in the instances where the autocorrelation graphs for tissue content are not similar for the two body postures (one dog), the supine plot showed an enhanced negative correlation-distance relationship relative to the prone posture (figure 5, lower left). Similarly, regional blood content is either posture independent or shows an enhanced negative correlation-distance relationship supine relative to the prone body posture. This is similar to the findings related to regional air content (figure 5, upper right vs. upper left). Mean transit time appears to exhibit the least spatial correlation (negative and positive) over all distance in both the supine and prone postures (figure 5, lower right).


Figure 5

Contrary to regional air content, the correlation vs. distance relationship for regional pulmonary perfusion normalized to tissue content appears nearly identical supine vs. prone (figure 6). In one dog, however, there is an enhanced negative relationship prone vs. supine. (figure 6, right). The correlation-distance relationship of regional pulmonary perfusion/tissue is not as steep as seen with regional air content; and the slope is significantly reduced, with a flattened appearance. Regional peak time follows a pattern similar to perfusion/tissue for most (4/6) dogs (data not shown).


Figure 6

We noticed a significant difference in spatial correlation supine vs. prone when regional perfusion was normalized to air content rather than tissue content (figure 7). This normalization may be closer to the normalization factor utilized by Glenny and others who analyze their data using the excised air dried lung. Perfusion normalized to regional air content generates similar supine to prone differences across all dogs, reminiscent of the regional air content autocorrelation plots (see figure 5, upper left). Regional raw blood flow (not normalized to air or tissue content) is steeper supine vs. prone in nearly every dog, apparently following a spatial correlation pattern similar to air content.

Regional arrival time remains essentially uncorrelated over distance in all dogs and postures with the entire lobe maximally enhanced at the same point in time.

5. DISCUSSION

The model based studies have demonstrated several important concepts concerning the use of the spatial autocorrelation method of evaluating the underlying basis for regional functional heterogeneity in the lung. Firstly, in its implementation, the method has an inherent stumbling block related to potential order effects. The results from the cube simulations clearly show that using each pair once leads to a spatial correlation pattern that can provide erroneous interpretations (i.e dependent on the order in which the data set is sampled). Glenny avoids this problem by randomizing the order in which he feeds the pairwise comparisons into the correlation evaluation. (personal communication). We felt that our evaluation would be most consistent by keeping all paired observations.

The results from the computer generated models demonstrate that a negative correlation-distance relationship is not unique. Data sets of multiple shapes which show a spatial distribution of a given parameter along a single axis exhibit negative spatial


Figure 7

correlation as a function of distance. The correlation vs. distance slopes are steeper for the computer simulations as compared to our CT-derived experimental data. This may be due to our methodology in generating the model data. In the gradient simulations, we set "flow" equal to the voxel location (i.e. for y gradients, all voxels with identical y locations had identical values) rather than reflecting gradient magnitudes similar to the experimental data. The simulated gradients used are steeper than the experimental data, therefore, leading to a steeper correlation-distance relationship. Forcing the gradient range of the simulations to match the range of actual PBF gradients may alter the slope of the spatial correlation relationship to that more similar to the experimental data.

It should be noted that, in our studies of in vivo lung function, we may not have been completely successful in separating out, digitally, a single lung lobe. We chose to study the left lung because of the fewer lobulations, and we chose to study the lower lobe because of its large ventral-dorsal expanse in the region just above the diaphragm. However, it should be noted that in our previous studies, the left lung shows a greater variability in spatial relationships as compared with the right lung. We have, in the past again attributed this to the support of the heart by the lung. When apparent zone 4 conditions14 are present, resulting in decreased flow to the dependent lung regions, this almost always appears most prominent in the left lower lobe15. The variability demonstrated in the pair of graphs shown in figure 6 may very well be due to our selection of the left lower lobe.

Our model data, contrary to the actual flow data reported by Glenny, reflects a relationship best fit by an exponential such as:

Similarly, our air content data, as well as the blood content data, and pulmonary blood flow normalized to air content spatial autocorrelation relationships all were best fit to an exponential relationship in the supine body posture. An exponential fit has also been found by Rodarte and colleagues when using a spatial correlation statistic to evaluate CT density distribution16. Spatial autocorrelation data for the prone body posture regardless of parameter being evaluated, and regional tissue content and mean transit times both supine and prone come closer to fitting a power law relationship. Of particular interest is the fact that blood flow normalized to tissue content rather than air content comes closer to reflecting data presented by Glenny. Normalization is a significant unknown when comparing our finding with those of others. Our data sets are fairly unique in that the information is evaluated in vivo. By excision and air drying the lung at total lung capacity (25cmH2O), normalization under these conditions may yield quite a different relationship. Normalization itself can impose a structure to the assessed distribution of function, and this is an important area for further thought. In figure 8 we show our blood flow data from a representative dog where flow is normalized to regional air content. Superimposed on the supine and prone spatial autocorrelation relationship for this dog are the power law based fits for the dogs studied in Glenny's experiments. While the slope of the prone data from our study is considerably more shallow, the shape of the relationship comes closer to fitting with Glenny's data than is the supine relationship. Our supine data, normalized in this fashion is considerably at odds with that of Glenny. In this case, our data closer reflects the pure, single gradient artificially imposed upon the lower lobe coordinates and shown in the lower, central panel of figure 4.


Figure 8

We have previously demonstrated that, while there is a significant difference in the height dependence of both pulmonary blood flow and regional lung air content supine versus prone, the relationship between the regional air/blood ratio and lung height remains nearly constant across body postures15. This may be demonstrating that as alveoli expand, regional blood volume is reduced. The close similarity between the two plots at the top of figure 5 nicely demonstrate this relationship through the use of the spatial autocorrelation approach. The flattening of the spatial autocorrelation plot for tissue content seen in the lower left panel of figure 5 may reflect that tissue remains more constant across lung regions while air and blood remains tied to each other through an inverse relationship.

6. CONCLUSION

Since models with different shapes show a similar negative correlation-distance relationship when pure "gravity" gradients were applied, an object's shape appears to have minimal effects on the correlation-distance relationship. Negative spatial autocorrelation can occur at a distance in models of a single axis based gradient. While a negative sloping spatial correlation and a negative absolute value for the spatial correlation at a distance can reflect a pure gravitational effect as well as other underlying geometric relationships determining spatial distribution patterns of function, the shape of the spatial autocorrelation may add to the underlying pattern. Our pure gravity gradients in the models as well as some of the functional distributions evaluated could best be fit with an exponential relationship. On the other hand, not all relationships studied showed an exponential pattern in the spatial autocorrelation plots. For multiple parameters studied, there appears to be a uniquely different driving force prone versus supine and these forces seem to be linked. The common parameter governing them all may be, in part, the relationships which serve to determine the regional differences in alveolar expansion. While the spatial autocorrelation technique offers a useful paradigm to discriminate varying patterns of spatial distribution, it is clear that the properties of the tool are not yet well understood in the context of lung function, and the sensitivity of the tool under in vivo conditions are not well characterized. Finally, the differences we find in the spatial autocorrelation relationship when normalizing pulmonary blood flow to regional lung air or tissue content demonstrates that further thought is needed to understand how normalization itself may be effecting the spatial autocorrelation measurements. Data to date suggest an inverse relationship between regional lung air content and regional lung blood content with lung tissue showing significantly less spatial correlation.



7. ACKNOWLEDGMENTS

This study was supported in part through NIH RO1-HL-42672. Send correspondence to: Eric A. Hoffman, Ph.D., Department of Radiology, University of Iowa College of Medicine, 200 Hawkins Drive, Iowa City, IA 52242.



8. REFERENCES

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2. Hoffman, E.A, "Effect of body orientation on regional lung expansion: A computed tomographic approach," Journal of Applied Physiology, Vol. 59, pp.468-480, 1985.

3. Hoffman, E.A., and E.L. Ritman, "Effect of body orientation on regional lung expansion in dog and sloth," Journal of Applied Physiology, Vol. 59, pp. 481-491, 1985.

4. Larsen, R.L., C.R. Bridges, K.C. Beck, and E.A. Hoffman, "Regional Pulmonary Blood Flow Via Cine X-ray Computed Tomography," The FASEB Journal, Vol.4, pp. A1074, 1990.

5. Glenny, R.W., W.J. Lamm, R.K. Albert and H.T. Robertson, "Gravity is a minor determinant of pulmonary blood flow distribution," Journal of Applied Physiology, Vol. 71, pp. 620-629, 1991.

6. Beck, K.C., and K. Rehder, "Differences in regional vascular conductances in isolated dog lungs," Journal of Applied Physiology, Vol. 61, pp. 530-538, 1986.

7. Glenny, R.W., and H.T. Robertson, "Fractal properties of pulmonary blood flow: characterization of spatial heterogeneity," Journal of Applied Physiology, Vol. 69, pp. 532-545, 1990.

8. Glenny, R.W., "Spatial correlation of regional pulmonary perfusion," Journal of Applied Physiology, Vol. 72, pp. 2378-2386, 1992.

9. Griffith, D.A. Spatial Autocorrelation. A Primer. Commercial Printing, State College, PA, 1987.

10. Tajik J.K., S.D. Kugelmass, and E.A. Hoffman, "An automated method for relating regional pulmonary structure and function: integration of dynamic multislice CT and thin-slice high-resolution CT," SPIE Proceedings,Vol. 1905 , pp. 339-350, 1993.

11. Boyd, D.P., and M.J. Lipton, "Cardiac computed tomography," Proceedings of the IEEE, Vol.71, pp.298-307, 1983.

12. Hoffman, E.A., J.K. Tajik, and S.D. Kugelmass, "Matching pulmonary structure and perfusion via combined dynamic multislice CT and thin-slice high-resolution CT," Computerized Medical Imaging and Graphics, In Press, 1994.

13. Hoffman, E.A., D. Gnanaprakasam, K.B. Gupta, J.D. Hoford, S.D. Kugelmass, and R.S. Kulawiec, "VIDA: An environment for multidimensional image display and analysis," SPIE Proceedings, Vol. 1660, pp. 694-711, 1992.

14. West , J.B., Respiratory Physiology-the essentials (4th ed.). Williams and Wilkins, Baltimore, 1990.

15. Hoffman, E.A., and J.K. Tajik, "Dynamic and high resolution CT assessment of pulmonary blood flow distribution," American Review of Respiratory Disease, Vol. 147, No. 4, pp. A1021, 1993.

16. Rodarte, J.R., M. Chaniotakis, and T.A. Wilson, "Variability of parenchymal expansion measured by computed tomography," Journal of Applied Physiology, Vol. 67, pp. 226-231, 1989.





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