# Diameters and cross-sectional areas of branches in the human pulmonary arterial tree.

код для вставкиСкачатьTHE ANATOMICAL RECORD 223~245-251(1989) Diameters and Cross-Sectional Areas of Branches in the Human Pulmonary Arterial Tree KEITH HORSFIELD AND MICHAEL J. WOLDENBERG The Midhurst Medical Research Institute, The Surrey Research Park, Guildford, Surrey GU2 5YU: United Kingdom (K.H.) and Department of Geography, State University of New York, W) Buffalo, New York 14260 (M.J. ABSTRACT Measurements were made of the diameters of the three branches meeting at each of 1,937 bifurcations in the pulmonary arterial tree, using resin casts from two fully inflated human lungs. Cross-sectional areas of the parent branch and of the daughter branches were calculated and plotted on a log-log plot, which showed that mean cross-sectional area increases in a constant proportion of 1.0879 a t bifurcations. The mean value of the ratio of daughter branch diameters a t bifurcations was 0.7849. The mean value of the exponent z in the equation flow = k (diameter“) was found to be 2.3 k 0.1, which is equal to the optimal value for minimizing power and metabolic costs for fully developed turbulent flow. Although Reynolds number may exceed 2,000 in the larger branches of the pulmonary artery, turbulent flow probably does not occur, and in the peripheral branches Reynolds number is always low, excluding turbulent flow in these branches. This finding seems to be incompatible with the observed value of z. A possible explanation may be that other factors may need to be taken into account when calculating the theoretical optimum value of z for minimum power dissipation, such as the relatively short branches and the disturbances of flow occurring at bifurcations. Alternatively, higher arterial diameters reduce acceleration of the blood during systole, reduce turbulent flow, and increase the reservoir function of the larger arteries. These higher diameters result in a lower value of z. The lesser circulation, the vessels through which it flows, and the changing pressures within them continue to be of interest to pneumonologists and cardiologists alike (Harris and Heath, 1986). Dimensions and numbers of branches in the asymmetric pulmonary arterial tree of humans have been described by Elliott and Reid (1965), Woldenberg et al. (1970), Singhal et al. (19731, and Horsfield (1978). In addition, the concept of optimal “design” (i.e., evolutionary development) in the systemic and pulmonary arterial trees has been presented by many authors (Murray, 1926a,b; Zamir, 1976; Horsfield, 1977; Uylings, 1977; Sherman, 1981; Woldenberg and Horsfield, 1983,1986). Pressure and flow within the pulmonary arteries are closely related to their diameters and cross-sectional areas, so that the way in which cross-sectional area changes down the tree is of particular interest. The purpose of this study is to measure diameter and cross-sectional area over a wide range of bifurcations in the pulmonary arterial tree and to relate these measurements to various physiologic and morphometric parameters, in particular to the exponent z in the equation flow = k x diameter’. The branches of these trees can be classified in terms of order, whereby branches are counted from the periphery toward the main stem. The method of ordering that will be used is that described by Horsfield (Horsfield and Cumming, 1968; Horsfield, 1984). In this method two order 1 branches meet to form a n order 2 branch, two order 2 branches a n order 3, and so on. When branches of differing order meet, the next order is one more than the higher of the two meeting branches. It should be understood that the order of the stem is determined by the longest pathway, because order increases by 1 a t each bifurcation. The number of branches in successive orders down the tree increases by a constant factor, the branching ratio Rb. Similarly, the mean diameter of branches in successive orders up the tree also increases by a constant factor, the diameter ratio Rd (Singhal et al., 1973; Horsfield, 1978). Murray (1926a) studied the trade-off between metabolic cost and power dissipation in relation to arterial diameter. As diameter increases, resistance, and hence work of moving the blood, fall, while the metabolic cost of maintaining the blood and vessel wall increases because their volumes increase. Assuming laminar flow MATHEMATICAL CONCEPTS Both the bronchial and pulmonary arterial trees in man can be modeled in terms of a repeatedly dichotomously branching structure arising from a single main stem, without any rejoining of branches (Horsfield and Cumming, 1968; Singhal et al., 1973; Horsfield, 1978). 0 1989ALAN R. LISS, INC. ~~ Received March 23, 1987; accepted August 8, 1988. Address reprint requests to Dr. K. Horsfield, the Midhurst Medical Research Institute, 30 Occam Rd., The Surrey Research Park, Guildford, Surrey GU2 5YW, U.K. 246 K. HORSFIELD AND M.J. WOLDENBERG and the applicability of Poiseuille's equation, he found lated by substituting measurements of the three diameters into the equation of continuity (Suwa et al., 1963; an optimum when Woldenberg and Horsfield, 1983). Thus f = kD3 (1) fo = f1 + f2 where f is flow of blood in a branch of diameter D, and k is a constant. For a given flow, the dissipation of energy where the subscript 0 refers to the parent branch, 1 is minimal with a fully developed laminar flow regime. refers to the major daughter branch, and 2 refers to the With more disturbed flow regimes, more energy is dis- minor daughter branch. Therefore, substituting for f sipated, and the optimum value of the exponent is lower from equation 2 in the equation of continuity we obtain than 3. For fully developed turbulent flow the optimum is 2.333 (Uylings, 1977). Thus, according to the minimum power model, the value of the exponent (z) depends on the flow regime, and in general Knowing the three diameters, Do, D1, and D2, a t a f = kDZ (2) bifurcation, equation 3 can be solved iteratively to find the positive value of z when Do > D1 3 D2, and the Because the observed values of z may range both above negative value of z when D1 2 D2 > Do. If D1 2 Do 2 and below 3 (Woldenberg and Horsfield, 1983) the mini- D2, then z cannot be calculated. mum power model may not be a n appropriate explanaIn addition to the overall diameter ratio of the system, tion for the values of z observed in the pulmonary artery we can also speak of a diameter ratio at a bifurcation or other vascular networks. We require a more general explanation of why flow is related to diameter by a Rd = Do/D1 (4) power function. That flow in a branch is a function of its diameter is intuitively obvious. Moving peripherally, the number of Similarly, we can define a flow ratio at a bifurcation as branches increases while their diameters and the flow Rf = (Do/D1)" through each decrease. That flow should be a power function of diameter is less obvious, but the following argument will show why this is probable. The number and, by our previous assumption of branches in successive orders down the tree increases Rb E Rf (5) by a constant proportion,Rb. On average, the mean flow in branches in successive orders will decrease by a similar proportion, namely 1 R b . In terms of moving up Note that Rb is the increase in the number of branches from the periphery, mean flow in branches of successive in successive orders down the tree, and Rf is the increase orders increases by Rb and mean diameter by Rd. Thus in mean flow in successive orders up the tree. The ratio of daughter branch diameters, a , is a n both flow and diameter are logarithmically related to order and are therefore likely to be related to each other expression of asymmetry at a bifurcation with respect to diameters, and is given by by a power function. Several methods have been suggested for finding the (Y = DZ/D1 < 1.0 (6) value of z. Direct measurement of river flow versus width, depth, etc. to yield l/z was introduced by Leopold In the lung the summed cross-sectional area of both and Maddock (1953). Shoshenko et al. (1982) and Mayrovitz and Roy (1983) have derived z from direct mea- the airways and the arteries increases distally. Thus in surements of blood flow and radius. Some predict z from optimality equations (Murray, 1926a; Zamir, 1976; Uylings, 1977; Sherman, 1981; Roy and Woldenberg, 1982; Woldenberg and Horsfield, 1983). Others have derived z Cancelling the constants and calling the proportional from overall system morphometry established by order- change (usually a n increase) in cross-sectional area at a ing (Leopold and Miller, 1956; Woldenberg et al., 1970; junction p, we get Woldenberg, 1972; Horsfield and Thurlbeck, 1981).Murray (1927) and Hooper (1977) derived z from the slope of p Do2 Dl2 + D2' the regression of log diameter versus log weight of a tree branch distal to the diameter. Horsfield and Cum- But Do = RdDl and D2 = aD1, so by substitutionwe get ming (1968) demonstrated in human airways a linear relation between log diameter of a branch, and log numRd2Di2 = D12 + a2Di2 ber of end branches (E) supplied by it. McBride and Chuang (1985) noted a similar relationship, while Suwa p Rd2 = 1 + a' et al. (1963) found a n analogous relationship between arterial radius and number of glomeruli in the kidney. p=----1 a2 Woldenberg (1972) found that river width was a power Rd2 function of E. If flow is proportional to E, then flow is a power function of diameter, and the exponent z is the slope of log E on log diameter. The value of z at individual bifurcations can be calcu- + UlMENSlUNS Ul" HUMAN TABLE 1. Relation of flow, Reynolds number, and mean In addition, Roy and Woldenberg(l982) have shown that velocity to arterial branch diameter and the exponent z at a bifurcation' Equation 2 faD" 2=2 2=3 2=4 faD2 faD3 faD4 Equations 7, 8, and 9 are dimensionless. They can 14 Re a D"-l Re 0: D1 Re 0: D2 Re a D3 therefore be used with values of a, fi, and Rd, to predict 15 v a D"-2 vaDo vaD1 vaD2 one variable when the other two are known. If, flow; D, diameter of an artery; Re, Reynolds number; v, mean Determination of z allows the calculation of flows, velocity. velocities, and other hemodynamic parameters. For example, Reynolds number, Re, is given by plane of focus and were corrected for magnification of x Re = vDpIq (10) 20. The diameters ranged from 1.0 to 0.06 mm. The third source was new measurements, made using an eye-piece where v is mean flow velocity across the area of a tube, scale in a dissecting microscope, of three peripheral segp = density, and TJ = viscosity. Viscosity varies, with ments broken off from different regions of a resin cast. shear rate (Caro et al., 1978a), being about 0.03-0.04 This cast had been used to provide other material for poise a t 1,000 s-l, and may be lower in vessels below distal data in a previous study (Horsfield, 1978).The two 0.3 mm diameter (Fahraeus and Lindqvist, 1931). We casts were made from saline-filled lungs, fully inflated will use the value 0.03, but different values may easily via the trachea a t 25 cm H20, with resin injected into be inserted where appropriate. For the density of blood the pulmonary artery at a pressure of about 40 cm HZO. we take 1.0595. Mean velocity is given by flow divided Horsfield et al. (1971) studied the form of branches a t bifurcations and showed how the shape of the cross by area section changes. In the majority of cases, the cross secv = 4fID'a (11) tion reverts to circular soon after the bifurcation; for consistency the diameters were measured half-way along and substitutingfor v from equation 11into equation 10 the length of each branch. In the occasional example in we get which the cross section was observed not to be circular at this point, in which case it was almost always elliptical, diameter was measured in two directions a t right 4fD x 1.0595 Re = (12) angles, and the mean value was used. However, this was aD' x 0.03 obviously impossible for the data set obtained from the Substitutingfor f from equation 2 into equation 12 gives photographs, and in this case there would be a small random error, some branches being recorded as too large, and others as too small. The effect of this procedure Re = 44.97k D"-' (13) would be to increase the standard deviation of the error of the measurement by a n unknown quantity. The error Re a D"-l (14) of measurement of cross-sectional area made on branches Thus Reynolds number for a branch can be calculated of the cast was estimated from 22 repeated measurements made on selected branches on different days. For from its diameter if k and z are known. a branch with a mean estimate of cross-sectional area of Mean velocity is given by v = flarea v a DVD~ v cx Dz-2 (15) Thus when z = 2, v a Do; hence v is constant, and crosssectional area is constant a t the bifurcation. These relations are summarized in Table 1. MATERIALS AND METHODS The data consist of the parent and daughter branch diameters at pulmonary arterial bifurcations measured on resin casts of the human pulmonary arterial tree. The data were obtained from three sources. The first was the data set described by Singhal et al. (1973) for pulmonary arteries down to 0.8 mm diameter, called the proximal zone. Diameters were obtained by measuring a resin cast of a pulmonary arterial tree using calipers. The second source consisted of the photographs made for the study of the intermediate zone of the same cast (Singhal et al., 1973). Measurements of diameter were made a t dichotomies where all the branches lay in the 35.33 mm' the standard deviation (SD) of the measurement was 1.138 mm', and the coefficient of variation (SDImean x 100) was 3.2%. For a branch with a mean estimate of cross-sectional area of 2.197 mm2, the SD of the measurement was 0.2457 mm2, and the coefficient of variation was 11.2%. The error in the estimated value of z depends on the errors of three different diameter measurements in a rather complicated way, as suggested by equation 3, and the coefficient of variation for this has not been calculated. However, Horsfield and Thurlbeck (1981)showed that the calculated value of z is very sensitive to errors in the measurement of diameters, and may therefore be subject to considerable variability, which is shown by the wide range of values for z which we found, from 0.4 to 14.5. In such circumstances it is essential to make a large number of estimates in order that the mean value shall be a good estimate of the true value. The three data files were merged and a computer program written to calculate z using equation 3, alpha, and cross-sectional areas at each bifurcation. Equation 3 was solved by iteratively substituting values of z with decreasing increments, until a value was found that gave a n error of less than 0.0001. The equation for the 248 K. HORSFIELD AND M.J. WOLDENBERG - 2.4 N E -5 1.6 Ln 6 W 02 0.8 a ./ 1 0 .:*.: J- 0 . *-2.4U 0 . ., Fig. 1. Grouped data for plot of log sum area of daughter branches against log sum area of parent branch at 1,937 arterial bifurcations. Data grouped at log intervals of 0.2. 0 , 1-10 data points; 0, 11-30 data points; 31 -100 data points; 0 , more than 100 data points. regression of y = log summed daughter areas versus x = log parent area was calculated to see whether there was any change in the ratio of daughter areas to parent area as a function of the size of the parent branch. Logarithms were used because the data are log normally distributed, and a bivariate regression of untransformed data would be invalid. RESULTS Figure 1 shows grouped data for y = log sum area of daughter branches against x = log area of parent. Mean values and the regression equation were calculated from the data before grouping; the equation is y = 0.0367 1.0003x, r = 0.993, mean x = -0.2601, mean y = -0.2235 and n = 1,937. + The slope is not significantly different from 1.0, and we therefore consider it reasonable to interpret the line as being parallel to the line of identity. It also lies above the line of identity. These findings imply that mean cross-sectional area a t a bifurcation increases in constant proportion, independently of the size of the parent branch. Given mean x, mean y, and a slope of 1.0, the intercept can easily be calculated. It is 0.0366, and its antilog is the area ratio, p = 1.0879. Table 2 gives the 95% confidence limits for the regression line and the corresponding values of fl calculated from the antilogs of the given values of y and x. The range is wide (0.648 to 1.831) but differs little with varying values of x. Table 3 shows the range, mean values, standard deviation, and 95% confidence limits of the means of CY and log z, along with the corresponding values of z. Mean log z was used because the value of z showed a log normal distribution. Although both CY and z have wide ranges, the 95% confidence limits of the means are fairly narrow because of the large number of observations. Neither z nor CY was found to correlate with parent branch diameter. Mean diameter ratio can be calculated from equation 8, and substituting 0.7849 for a! and 1.0879 for 0, we get Rd = 1.2188. Considering a hypothetical idealized bifurcation in which the system average values of a and Rd apply, the value of z may be calculated. Let Do = 1.0; then D1= 1/ Rd = 0.8205. Similarly D2 = D ~ C=Y0.6440. From equation 3 0.8205" + 0.6440" = 1.0 which, when solved iteratively, gives z = 2.2925. Alternatively, the same answer can be obtained from equation 9. This answer compares with a mean value of z from the data of 2.3779, and a geometric mean of 2.1951. Thus the three estimates of mean z fall approximately in the range 2.3 k 0.1. Considering flow a t this same bifurcation, then from equation 2, f0 a 1.0", f i 0: 0.8205", and f2 a 0.6440". For unit flow in the parent branch, and using the idealized value of z = 2.925, we calculate the proportion of flow to each daughter branch TABLE 2. Confidence limits (95%)of the regression line of log sum daughter areas versus log parent area (Fig. 1) and the corresponding values of p' Best estimate x = log of y = log sum parent area daughter areas 95%Confidence limits of y t2.8 -0.2601 -3.0 2.6123-3.0627 -0.4479-0.0009 -3.1884-2.7390 '0 = +2.8375 -0.2235 -2.9642 Corresponding values of = (antilog y)/(antilog x) p 1.0902 (0.6491-1.8310) 1.0877 (0.6489-1.8238) 1.0859 (0.6480-1.8239) area ratio at a bifurcation. TABLE 3. Mean, range, and standard deviation of a and log z, with 95% confidence limits of the means' Variable cy log z (16) Mean Range 0.7849 0.3415 (2.1953) 0.1-1.0 -0.3979-1.1614 (0.4-14.5) Standard deviation 95%Confidence limits of the mean 0.2043 0.1686 0.7742-0.7956 0.3328-0.3504 (2.1518-2.2408 1 'Values in parentheses are the corresponding values of z. 249 DIMENSIONS OF HUMAN PULMONARY ARTERIES TABLE 4. Morphometric parameters in the human pulmonary arterial tree fo = 1.0 f i = 0.6354 f2 = 0.3646 Parameter' It will be seen that fo = fl + f2, a s required by continuity. The ratio of flow in the daughter branches is f2/f1 = 0.5739. Mean flow velocity v is given by fkross-sectional area, and using the idealized values for f and D given above, we obtain vz/vl = 0.9314. We have already shown that the ratio fi/fo s 1/Rb, so that the flow ratio, Rf, is given by fo/f1 = Rb. Since P a! Rd Z z Z W Rb E Rf How obtained Value Regression equation from data Mean value from data Calculated from (Y and Arithmetic mean from data Geometric mean from data Calculated from Rd and a Calculated from Rd Calculated from RdZ 1.0879 0.7849 1.2188 2.3779 2.1951 2.2925 40 1.574 area ratio at a bifurcation; a,diameter ratio of daughter branches; z,exponent relating flow to diameter, where flow a DZ;Rd, diameter ratio; Rb, branching ratio; Rf, flow ratio; w, order of main pulmonary artery. I@, Rb z Rf = (Do/D1)" = RdZ (17) and given the values of z = 2.2925 and Rd = 1.2188, then Rb G Rf = 1.574 (Woldenberg, 1972; Horsfield and Thurlbeck, 1981). The number of orders in the arterial tree can be estimated from the diameter ratio, thus D, = D1 x RdW-' (18) where w is the order of the main pulmonary artery. Given Dw = 30 mm, the mean diameter of a n order 1 branch, D1 = 0.013 mm, and Rd = 1.2188, then w = 40. Note that the number of generations (counting bifurcations down from the main pulmonary artery) is equal to the number of orders on the longest pathway. Table 4 gives a summary of the values obtained for the more important morphometric parameters in the pulmonary arterial tree. In order to calculate Reynolds number from equation 13, the value of k is required. This of course depends on the total flow (cardiac output), and as a n example this will be taken to be 5.0 literdmin (83.33 ml/s) in the main pulmonary artery of 3.0 cm diameter. From equation 2 k = f/D" = 83.33/3.02.2925 k = 6.7143 (19) Substituting for k from equation 19 into equation 13 gives Re = 44.97 x 6.7143 Dzpl Re = 302 D1.2925 (20) Table 5 gives some examples of Re calculated from equation 20. Since viscosity is reduced in vessels less than 0.3 mm in diameter, perhaps by as much a s 50% (Fahraeus and Lindqvist, 1931), calculated Re for such vessels may be too high by the same factor. These calculations were based on the assumption of steady flow throughout the cardiac cycle, but mean systolic or peak systolic flow could be used instead as required. DISCUSSION The diameter data were obtained from two pulmonary artery casts that were measured in three different ways, so it was reassuring that the log area data showed a linear correlation with r = 0.993. Apart from a few trichotomies, all the available data down to 0.8 mm diameter were utilized. In the case of the photographs and the distal cast, samples were taken from three separate sites in the lung. Within these samples, all available data were utilized. Thus sampling bias could only arise from the choice of sites for each of the two sets of three samples. These sites were deliberately taken from widely separate parts of the casts, and no indication of the results they might give could be obtained until after they had been measured. We therefore do not consider it likely that sampling bias was a serious problem. We consider a n ordinary least squares regression to be a reasonable method to use when r2 = 0.986. Furthermore, because the measurement errors in y (log summed daughter areas) were assumed to be about twice the errors in x (log parent area), the regression line would give a good estimate of the functional relation (Davies and Goldsmith, 1980). More important is the fact that the data were obtained from measurements of casts. With the lung inflated a t 25 cm H2O and the resin injected at a pressure of about 40 cm HzO, the vessels are probably near or a t their maximum diameters, and thus may well have been smaller during life. The data must therefore be taken as applying to maximum arterial diameters. However, if vessels maintain the same relative dimensions that they have a t lower pressures, then the estimated values of z, diameter ratio, and area ratio would still be applicable, although k would be too small. An implicit assumption has been that the values of z and k are the same in the three branches forming a bifurcation, and as the daughter branches become the parent branches of the next bifurcations, it is also implied that the values are constant all the way down the tree. Alternatively, both k and z might vary together in such a way that equation 2 holds true for each individual bifurcation. We plotted the value of z calculated at each bifurcation against the diameter of the parent branch and found no change in mean z with parent branch diameter. Thus there is no evidence of a systematic change in z down the tree. However, considerable local variations in the value of z were observed. If at junction (i) the diameters of the three branches and the flow in the parent branch are known, and equation 3 TABLE 5. Diameter of arterial branches and Reynolds number calculated from equation 20 with cardiac output of 5.0 litedmin Diameter (cm) 3.0000 1.0000 0.1000 0.0100 0.0013 Reynolds number 1,179.000 285.000 14.500 0.740 0.053 250 K. HORSFIELD AND M.J. WOLDENBERG holds exactly (except when D1 2 Do > Dz), then k(i) and the flows in the daughters can be found. Should a daughter of junction (i) be also the parent of junction (j), then the values of z(j) and k(j) can similarly be calculated. Thus, if f(c)is the flow in the common branch, then [k(j)/k(i)] Dz(i)-z(i) = 1 dissipation during laminar flow, when z = 3, other factors that are optimized a t higher diameters are also operative, with the combined result that z has a lower observed value of 2.3. ACKNOWLEDGMENTS Part of this work was supported by Collaborative Research Grant 376184 from the Senior Scientists Program (21) of N.A.T.O. Equation 2 is only true in a statistical sense for the tree as a whole; there can never be perfect agreement at each individual junction, if only because of the errors in the measurement. The optimum value of z, calculated by Uylings (1977) to minimize the total of metabolic power losses (proportional to volume) plus frictional power losses for fully turbulent flow, is 2.333. A similar calculation by Murray (1926a) gave the optimum value for z for fully developed laminar flow as 3.0. Our mean values therefore fall on the lower limit of the predicted range of 2.3-3.0. Turbulent flow probably does occur in the proximal aorta (Caro et al., 1978b), in which Reynolds number exceeds 2,000. However, even though the main pulmonary artery has the same flow and a similar diameter, turbulent flow probably does not occur in the pulmonary artery ) because it has more elastic (Caro et al., 1 9 7 8 ~partly walls. As Table 5 indicates, Re falls off rapidly in the smaller branches, and turbulent flow in these is out of the question. The finding of such a relatively low value of z throughout the pulmonary arterial tree is therefore unexpected, although low values have previously been noted (Woldenberg and Horsfield, 1983).Optimum values of z for flow in a branching tubular structure with short segments has not yet been calculated, and these features might help to account for the low value. Alternatively, z may be optimized for other variables in addition to minimum power dissipation. A value of z lower than 3 suggests the possibility that, for equation 3 to hold true, diameters are greater than those appropriatefor minimum power loss with laminar flow. This suggestion raises the question of what might be the explanation €or such a n increase in the arterial diameters. First, a n increase in diameter lowers mean velocity and hence reduces the work of accelerating the blood during systole, a factor that we have ignored in our calculations. Second, the reduced velocity diminishes the likelihood of turbulent flow, which dissipates more energy than laminar flow. Third, the larger vessels have a reservoir function, which, along with the elasticity of their walls, enables much of the blood ejected during systole to be accommodated, and then forwarded during diastole, thus distributingflow more equally over the cardiac cycle. Finally, because we used a pressure of 40 cm water for the resin when making the casts, the arterial diameters would have been a t or near their maximum. Murray (1926a) showed that a t the optimum for minimum power dissipation with laminar flow, when z = 3, the work of overcoming friction during arterial blood flow is half the metabolic cost of the arterial blood volume. When, as a result of the requirement for increased arterial diameters, and hence volumes, the optimum moves towards more metabolic work, the value of z decreases. Thus, in summary, while the pulmonary arterial tree may be optimized partly for minimum power LITERATURE CITED Caxo, C.G., T.J. Pedley, R.C. Schroter, and W.A. Seed 1978 The Mechanics of the Circulation. Oxford University Press, Oxford. (a) p. 176; (b) p. 232; (c) p. 504. Davies, O.L. and P.L. Goldsmith 1980 Statistical Methods in Research and Production. Longman, London, pp. 203-210. Elliott, F.M. and L. 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