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The estimation of avian egg surface area. A comparison of different methodological approaches
Cortrp.Bioclrrr~~. Pk.rsicrl.Vol. 73A. No. 1. PP. 101to 103. 1982 Printed in Great Britain
03~.96~9~82~090~0l-03~3.00/0 0 1982Pergamon Press Lrd
THE ESTIMATION OF AVfAN ECG SURFACE AREA. A COMPARISON OF DIFFERENT METHODOLOGICAL APPROACHES M. GONZP;LEZ, P. ROCA, F. S&K? and M. ALEMANY Departament de Bioquimica, Facultat de Ciencies, Universitat de la Ciutat de Mallorca, Ciutat de Mallorca, Balears, Spain
Abstract--t.
Five methods for egg surface area determination have been presented and applied to house sparrow, rock dove. domestic fowl, goose and ostrich eggs. The data were compared with a direct measurement integration of small tronchoconic segments measured from the photograpb~~ egg profiles. 2. A very good fit was that of surface area estimation of the shell from its weight, density (thus volume was calculated) and thickness. The best method was a modification of this one corrected by egg volume and shell thiekness. 3. Simple mathematic models such as the sphere (with a volume equal to that of the egg) and a prolate spheroid gave poor fits for all samples studied. 4. The computation of the surfaces of two paraboloids joined at their bases calculated by a regression method from the photographic profiles of the eggs gave better results. but nevertheless inferior to those found with the shell parameters methods.
MATERIALS AND
R’iTRODUCTlON The avian egg is a singular physi~l~gi~ai enclosed system containing practically ail the energy to be used for the generation of a new bird. The egg wastes little in maintenance of thermal homeostasis: this is provided by external systems, conserving an important amount of resources without compromising the speed (dependent on tissue temperature) of the involved metabohc processes. In homeotherm animals there exists a direct relationship between basal metabolic rate and body surface area (Brady and Procter, 1932: Hemmingsen, 1960). Since animal body surface (BS) area is difficult to determine (Diack, 1930: Hong et ol., 1977), an exponential value of body weight, as (BW)O.“*’ (theoretical value) or (BW)“‘7h or (BW)*.” (experimental values, Brody et ul., 1932; Hemmingsen, 1960) is currently used for physiological comparisons. The egg is practically a perfect solid of revolution (ThoIll~son~ 1942). However. the different egg shapes vary considerably (Preston, 1953). thus complicating the determination of surface area. The egg surface area is an important factor in determination of gas and water interchange through the shell (WangenSteen & Rahn, 1970-1971: Ar et crl., 1974) and a number of procedures have been devised for its estimation. The egg surface area has been calculated from profiles (Besch ut nf.. t968), by adjusting the shape to the nlathematicai formulae and then calculating the surface of the solid of revolution (Preston. 1953: Paganelli rt ~1.. 1974; Hoyt, 1976). The available methods fall in two groups: (a) they ari: complicated and cumbersome, not easily applicable to large numbers of eggs or (b) they are too simplified and fail to give precise measurements essential for work on surface area dependent effects. Were we present a different experimental approach for estimation of egg surface area and compare it with other approaches.
METHODS
Unincu~t~ eggs of house sparrow (fusser ~om~sr~~~s), rock dove (Co&&r liria). domestic fowl--Sussex stock-(Gallrts [~~~e.sf~~us)and goose (Anser rrnser) as well as an ostrich (~t~~r~~~~ ~it#~~~~.~) egg were used. Their externai measures are presented in Table 1. Egg volumes were determined by differential weighing of the eggs suspended in a wire basket immersed in water. Water density was corrected for temperature using a thermistor couple. The eggs were dissected and the shell carefully peeled otTof the internal membranes. Shell weight was then measured, as well as its apparent density (determined by differential weighing in water of a large piece of shell in a controlled temperature bath). She11thickness was determined by measuriu~ with a micrometric screw in at least five different places. The volume of the ostrich egg was measured by estimating the volume of dry fine sand displaced by the egg. Three measurements were done for each egg and the mean was selected. Derermintrtion
ofegg sucfuce ctren
(1) ,4~~~ffsr~~~r to 61sphere.From the sole value of egg volume (EV) the hypothetical radius of a sphere with the same volume was determ~ued from: EV
=
4rri 3
as egg surface area (ES) is ES = 4nrZ: then ES =
3EV. f
(2) Adjustntcnr to (I prolatr spheroid. From the values of (I and h (semi-major and semi-minor axes, respectively) measured BS4 of maximal height and radius of equatorial girdle, the value of ES can be calculated: ES = 2nh’ + 2n ‘2 arcsin E: e where l
‘
M. GONZALEZ
102 (3) E.stimution
of meun shell surfuce ureu from
This method considers the egg shell volume from its weight SW and density SD:
she// o&es.
ESV calculated
ESV = g; the egg mean surface ES can be deduced from ESV because the mean shell thickness ST is known; then the mean surface area is calculated from these ESV = ES.ST:
SW ES = ___ SD.ST
(4) Estimution tf meun she// surfuce ureu from shell rulues corrected by thickness. This is a further refinement of method (3) that takes into account the differences between internal and external surface areas of the shell (IES and EES). In method (3) the mean IES + EES 2 egg shell surface area is obtained. This value corresponds to an ideal sphere (if for the sake of simplicity the values are corrected to those of a sphere) of radius r. Thus, from the above calculated data: ES = TDs:
the EES corresponds
ES = 4d:
to a sphere with radius
R where
R=r+y:
thus,
(5) Estimation of she// surfuce ureu by inteyrution puruboloids ofrecolution. Each egg was photographed
of two
so as to obtain sagittal projections. The figures were projected and drawn over cross section paper. The main axes were also traced along the maximal length and perpendicular to the equatorial circle. The shorter axis was used to separate the two parabolae (upper and lower halves of the egg) with a common longer axis. From the drawings, sets of points were used to fit parabolae by the method of least squares yielding a parabolic correlation coefficient >0.99. From the formulas and the length of the semiaxes a computer program integrated the successive tronchoconic segments into which the curves were divided. (6) Direct integrcttiou oj’ egg surfice uretr Jvm strgitttrl c’ross section projec~tions. This laborious method was used as a means to contrast the other methods presented. It was based on the same drawings as method (5). and the drawings were carefully transferred to millimetre paper. A series of parallel lines spaced exactly 0.5, 1 or 2mm (depending on the size of the egg) were used to determine at each level the width of the egg. A simple computing program was used to determine the tronchoconic surfaces of the segments obtained and, thus, the egg surface area was estimated. The method had a repeatability of more than 99.4yzC, on measurements done on projections (checked with goose eggs).
RESULTS
In Table
I the
mean
AND
of the same egg
DISCUSSION
values of experimental data of egg surface area are shown. The different approaches used to determine the egg surface areas are presented in Table 2 in which the correlation between the different fittings can be compared. used for the calculation
ef ul.
103
Avian egg surface area Table 2. Comparison of different methods used for the determination Species domesticus Columhu liuia Gtrllus domesticus Anser unser Struthio ctrmelus
Passer
of egg surface area
N*
Method (1)
Method (2)
Method (3)
Method (4)
Method (5)
Method (6)
4 5 5
9.18 + 0.18 31.05 + 2.02 71.14 + 1.73 131.7 + 0.8 591.3
11.17 + 0.28 37.33 + 2.83 75.17 + 2.87 141.6 f 0.5 664.8
11.93 * 0.47 28.62 + 2.04 70.26 + 1.24 145.5 + 0.7 589.0
12.07 f 0.47 28.99 + 2.06 71.33 + 1.26 148.0 + 0.7 606.3
12.17 f 0.52 28.79 + 2.11 72.51 k 3.23 144.0 * 1.2 613.4
12.15 + 0.52 28.33 + 1.91 69.92 k 2.05 145.6 k 1.3 613.4
5
1
All values are the mean f SEM surface areas expressed in cm’ from the same eggs indicated in Table 1 The number of the methods correspond to their description in the text.
The direct integration method (6) was selected as more representative of the real egg surface area because of the degree of precision obtained in the measurements as well as in the simple integrative process, which had a small residual error. However, it must be taken into account that the method used has some built-in bias towards slightly smaller values than the real egg, because of the conversion of small segments of arc into straight lines for calculation of trapezoid circular segments areas. The mean variations for each of the methods studied (data calculated from the means) expressed as a percentage ( f SEM) values of the deviation from the assumed -5.3 + 5.1 real (direct integration) data were: (method l), 4.9 + 7.2 (method 2), -0.9 + 0.9 (method 3), 0.8 + 0.7 (method 4) and 1.9 & 1.7 (method 5). It is directly apparent that methods (1) and (2) were poor fits, the first biased towards smaller surface areas than the actual values and the second with higher estimations; in both cases the degree of error as observed by the gross deviation (high SEM values) was extremely marked. The method presented here showed some bias towards higher values (method 4) or the reverse (method 3). However, they showed better fit to the calculated results of method (6) than the other methods presented, including that of the two parabolas, where despite a very good fit with both poles of the egg, the curves met at the egg equator creating a ridge that significantly altered the results. The best approximation to the direct integration results was, in the average, that of method (4), as it gave some positive bias (a situation that can be better tolerated in terms of real values than the reverse because of the estimated negative bias of the control method) with the lowest average deviation from the control method values. It also gave the lowest dispersion of data of the methods contrasted. This method (4) has the advantage that is self-correcting and does not require complex measurements and mathematical treatment of the data. The form of the egg is a factor that deeply influences the surface area measurement. depending on the method used, as most systems try to fit the complex form of the egg into the constraints of a given mathematical figure (Besch et al., 1968; Paganelli et al., 1974; Hoyt, 1976). The precision of the method used depends directly on the precision of the measurement done, being es-
pecially critical in two points: the estimation of shell thickness and the determination of shell apparent density. However, when these factors are adequately controlled and the repeatability is adequate, the method poses no problem, despite different egg shapes and varying curvatures (Preston, 1953: Hoyt, 1976), allowing for routine determinations of egg surface area in most studies. The main problem it presents is that it requires the destruction of the egg, a situation that is not always feasible; it also assumes a uniform shell thickness despite changes in curvature. Acknowledgement-Thanks are given to Dr. Ralph Mansfield for his help with some mathematical criteria and for reading and critical revision of the text. REFERENCES
BESCHE. L., SLUKAS. J. & SMITHA. H. (1968) Determination of surface area using profile recordings. Poultr) Sci. 47, 82-85. BRODYS. & PROCTERR. C. (1932) XXII. Relation between basal metabolism and body weight in laboratory animals: published data. Missouri ugric. Co//. exp. Stn. Rex Bull. 166, 83-101. BRODYS.. FUNK E. M. & KEMPSTER H. L. (1932) XIX. Relation between basal metabolism and body weight in the growing domestic fowl. Missouri trqric. Co/l. exp. Stn. Res. Bull. 166, 67-70. DIACKS. L. (1930) The determination of the surface area of the white rat. J. Nutr. 3, 289-296. HEMMINGSEN A. M. (1960) Energy metabolism as related to body size and respiratory surfaces, and its evolution (Part 11). Reporfs of the Steno Memorid Hospitul und Nordisk Insu/inlahorutorium, 9, l-l 10. HONG C. C., EDIGERR. D., RAETZ R. & DJURICKOVKS. (1977) Measurement of guinea pig body surface area. Ltrh. Animctl Sci. 27, 474476. HOYT D. F. (1976) The effect
of shape on the surfacevolume relationships of birds aggs. Condor 78. 343-349. PAGANELLIC. V., OLSZOWAA. & AR A. (1974) The avian eggs: surface area, volume and density. Condor 76. 319-325.
PRESTONF. W. (1953) The shapes of birds’ eggs.
Auk
70,
16&182.
THOMPSOND’A.W. (1942) On Growth und Form, pp. 934944. Cambridge University Press. WANGENSTEEN 0. D. & RAHN H. (197c-1971) Respiratory gas exchange by the avian embryo. Resp. Pkysiol. 11, 3145.
03~.96~9~82~090~0l-03~3.00/0 0 1982Pergamon Press Lrd
THE ESTIMATION OF AVfAN ECG SURFACE AREA. A COMPARISON OF DIFFERENT METHODOLOGICAL APPROACHES M. GONZP;LEZ, P. ROCA, F. S&K? and M. ALEMANY Departament de Bioquimica, Facultat de Ciencies, Universitat de la Ciutat de Mallorca, Ciutat de Mallorca, Balears, Spain
Abstract--t.
Five methods for egg surface area determination have been presented and applied to house sparrow, rock dove. domestic fowl, goose and ostrich eggs. The data were compared with a direct measurement integration of small tronchoconic segments measured from the photograpb~~ egg profiles. 2. A very good fit was that of surface area estimation of the shell from its weight, density (thus volume was calculated) and thickness. The best method was a modification of this one corrected by egg volume and shell thiekness. 3. Simple mathematic models such as the sphere (with a volume equal to that of the egg) and a prolate spheroid gave poor fits for all samples studied. 4. The computation of the surfaces of two paraboloids joined at their bases calculated by a regression method from the photographic profiles of the eggs gave better results. but nevertheless inferior to those found with the shell parameters methods.
MATERIALS AND
R’iTRODUCTlON The avian egg is a singular physi~l~gi~ai enclosed system containing practically ail the energy to be used for the generation of a new bird. The egg wastes little in maintenance of thermal homeostasis: this is provided by external systems, conserving an important amount of resources without compromising the speed (dependent on tissue temperature) of the involved metabohc processes. In homeotherm animals there exists a direct relationship between basal metabolic rate and body surface area (Brady and Procter, 1932: Hemmingsen, 1960). Since animal body surface (BS) area is difficult to determine (Diack, 1930: Hong et ol., 1977), an exponential value of body weight, as (BW)O.“*’ (theoretical value) or (BW)“‘7h or (BW)*.” (experimental values, Brody et ul., 1932; Hemmingsen, 1960) is currently used for physiological comparisons. The egg is practically a perfect solid of revolution (ThoIll~son~ 1942). However. the different egg shapes vary considerably (Preston, 1953). thus complicating the determination of surface area. The egg surface area is an important factor in determination of gas and water interchange through the shell (WangenSteen & Rahn, 1970-1971: Ar et crl., 1974) and a number of procedures have been devised for its estimation. The egg surface area has been calculated from profiles (Besch ut nf.. t968), by adjusting the shape to the nlathematicai formulae and then calculating the surface of the solid of revolution (Preston. 1953: Paganelli rt ~1.. 1974; Hoyt, 1976). The available methods fall in two groups: (a) they ari: complicated and cumbersome, not easily applicable to large numbers of eggs or (b) they are too simplified and fail to give precise measurements essential for work on surface area dependent effects. Were we present a different experimental approach for estimation of egg surface area and compare it with other approaches.
METHODS
Unincu~t~ eggs of house sparrow (fusser ~om~sr~~~s), rock dove (Co&&r liria). domestic fowl--Sussex stock-(Gallrts [~~~e.sf~~us)and goose (Anser rrnser) as well as an ostrich (~t~~r~~~~ ~it#~~~~.~) egg were used. Their externai measures are presented in Table 1. Egg volumes were determined by differential weighing of the eggs suspended in a wire basket immersed in water. Water density was corrected for temperature using a thermistor couple. The eggs were dissected and the shell carefully peeled otTof the internal membranes. Shell weight was then measured, as well as its apparent density (determined by differential weighing in water of a large piece of shell in a controlled temperature bath). She11thickness was determined by measuriu~ with a micrometric screw in at least five different places. The volume of the ostrich egg was measured by estimating the volume of dry fine sand displaced by the egg. Three measurements were done for each egg and the mean was selected. Derermintrtion
ofegg sucfuce ctren
(1) ,4~~~ffsr~~~r to 61sphere.From the sole value of egg volume (EV) the hypothetical radius of a sphere with the same volume was determ~ued from: EV
=
4rri 3
as egg surface area (ES) is ES = 4nrZ: then ES =
3EV. f
(2) Adjustntcnr to (I prolatr spheroid. From the values of (I and h (semi-major and semi-minor axes, respectively) measured BS4 of maximal height and radius of equatorial girdle, the value of ES can be calculated: ES = 2nh’ + 2n ‘2 arcsin E: e where l
‘
M. GONZALEZ
102 (3) E.stimution
of meun shell surfuce ureu from
This method considers the egg shell volume from its weight SW and density SD:
she// o&es.
ESV calculated
ESV = g; the egg mean surface ES can be deduced from ESV because the mean shell thickness ST is known; then the mean surface area is calculated from these ESV = ES.ST:
SW ES = ___ SD.ST
(4) Estimution tf meun she// surfuce ureu from shell rulues corrected by thickness. This is a further refinement of method (3) that takes into account the differences between internal and external surface areas of the shell (IES and EES). In method (3) the mean IES + EES 2 egg shell surface area is obtained. This value corresponds to an ideal sphere (if for the sake of simplicity the values are corrected to those of a sphere) of radius r. Thus, from the above calculated data: ES = TDs:
the EES corresponds
ES = 4d:
to a sphere with radius
R where
R=r+y:
thus,
(5) Estimation of she// surfuce ureu by inteyrution puruboloids ofrecolution. Each egg was photographed
of two
so as to obtain sagittal projections. The figures were projected and drawn over cross section paper. The main axes were also traced along the maximal length and perpendicular to the equatorial circle. The shorter axis was used to separate the two parabolae (upper and lower halves of the egg) with a common longer axis. From the drawings, sets of points were used to fit parabolae by the method of least squares yielding a parabolic correlation coefficient >0.99. From the formulas and the length of the semiaxes a computer program integrated the successive tronchoconic segments into which the curves were divided. (6) Direct integrcttiou oj’ egg surfice uretr Jvm strgitttrl c’ross section projec~tions. This laborious method was used as a means to contrast the other methods presented. It was based on the same drawings as method (5). and the drawings were carefully transferred to millimetre paper. A series of parallel lines spaced exactly 0.5, 1 or 2mm (depending on the size of the egg) were used to determine at each level the width of the egg. A simple computing program was used to determine the tronchoconic surfaces of the segments obtained and, thus, the egg surface area was estimated. The method had a repeatability of more than 99.4yzC, on measurements done on projections (checked with goose eggs).
RESULTS
In Table
I the
mean
AND
of the same egg
DISCUSSION
values of experimental data of egg surface area are shown. The different approaches used to determine the egg surface areas are presented in Table 2 in which the correlation between the different fittings can be compared. used for the calculation
ef ul.
103
Avian egg surface area Table 2. Comparison of different methods used for the determination Species domesticus Columhu liuia Gtrllus domesticus Anser unser Struthio ctrmelus
Passer
of egg surface area
N*
Method (1)
Method (2)
Method (3)
Method (4)
Method (5)
Method (6)
4 5 5
9.18 + 0.18 31.05 + 2.02 71.14 + 1.73 131.7 + 0.8 591.3
11.17 + 0.28 37.33 + 2.83 75.17 + 2.87 141.6 f 0.5 664.8
11.93 * 0.47 28.62 + 2.04 70.26 + 1.24 145.5 + 0.7 589.0
12.07 f 0.47 28.99 + 2.06 71.33 + 1.26 148.0 + 0.7 606.3
12.17 f 0.52 28.79 + 2.11 72.51 k 3.23 144.0 * 1.2 613.4
12.15 + 0.52 28.33 + 1.91 69.92 k 2.05 145.6 k 1.3 613.4
5
1
All values are the mean f SEM surface areas expressed in cm’ from the same eggs indicated in Table 1 The number of the methods correspond to their description in the text.
The direct integration method (6) was selected as more representative of the real egg surface area because of the degree of precision obtained in the measurements as well as in the simple integrative process, which had a small residual error. However, it must be taken into account that the method used has some built-in bias towards slightly smaller values than the real egg, because of the conversion of small segments of arc into straight lines for calculation of trapezoid circular segments areas. The mean variations for each of the methods studied (data calculated from the means) expressed as a percentage ( f SEM) values of the deviation from the assumed -5.3 + 5.1 real (direct integration) data were: (method l), 4.9 + 7.2 (method 2), -0.9 + 0.9 (method 3), 0.8 + 0.7 (method 4) and 1.9 & 1.7 (method 5). It is directly apparent that methods (1) and (2) were poor fits, the first biased towards smaller surface areas than the actual values and the second with higher estimations; in both cases the degree of error as observed by the gross deviation (high SEM values) was extremely marked. The method presented here showed some bias towards higher values (method 4) or the reverse (method 3). However, they showed better fit to the calculated results of method (6) than the other methods presented, including that of the two parabolas, where despite a very good fit with both poles of the egg, the curves met at the egg equator creating a ridge that significantly altered the results. The best approximation to the direct integration results was, in the average, that of method (4), as it gave some positive bias (a situation that can be better tolerated in terms of real values than the reverse because of the estimated negative bias of the control method) with the lowest average deviation from the control method values. It also gave the lowest dispersion of data of the methods contrasted. This method (4) has the advantage that is self-correcting and does not require complex measurements and mathematical treatment of the data. The form of the egg is a factor that deeply influences the surface area measurement. depending on the method used, as most systems try to fit the complex form of the egg into the constraints of a given mathematical figure (Besch et al., 1968; Paganelli et al., 1974; Hoyt, 1976). The precision of the method used depends directly on the precision of the measurement done, being es-
pecially critical in two points: the estimation of shell thickness and the determination of shell apparent density. However, when these factors are adequately controlled and the repeatability is adequate, the method poses no problem, despite different egg shapes and varying curvatures (Preston, 1953: Hoyt, 1976), allowing for routine determinations of egg surface area in most studies. The main problem it presents is that it requires the destruction of the egg, a situation that is not always feasible; it also assumes a uniform shell thickness despite changes in curvature. Acknowledgement-Thanks are given to Dr. Ralph Mansfield for his help with some mathematical criteria and for reading and critical revision of the text. REFERENCES
BESCHE. L., SLUKAS. J. & SMITHA. H. (1968) Determination of surface area using profile recordings. Poultr) Sci. 47, 82-85. BRODYS. & PROCTERR. C. (1932) XXII. Relation between basal metabolism and body weight in laboratory animals: published data. Missouri ugric. Co//. exp. Stn. Rex Bull. 166, 83-101. BRODYS.. FUNK E. M. & KEMPSTER H. L. (1932) XIX. Relation between basal metabolism and body weight in the growing domestic fowl. Missouri trqric. Co/l. exp. Stn. Res. Bull. 166, 67-70. DIACKS. L. (1930) The determination of the surface area of the white rat. J. Nutr. 3, 289-296. HEMMINGSEN A. M. (1960) Energy metabolism as related to body size and respiratory surfaces, and its evolution (Part 11). Reporfs of the Steno Memorid Hospitul und Nordisk Insu/inlahorutorium, 9, l-l 10. HONG C. C., EDIGERR. D., RAETZ R. & DJURICKOVKS. (1977) Measurement of guinea pig body surface area. Ltrh. Animctl Sci. 27, 474476. HOYT D. F. (1976) The effect
of shape on the surfacevolume relationships of birds aggs. Condor 78. 343-349. PAGANELLIC. V., OLSZOWAA. & AR A. (1974) The avian eggs: surface area, volume and density. Condor 76. 319-325.
PRESTONF. W. (1953) The shapes of birds’ eggs.
Auk
70,
16&182.
THOMPSOND’A.W. (1942) On Growth und Form, pp. 934944. Cambridge University Press. WANGENSTEEN 0. D. & RAHN H. (197c-1971) Respiratory gas exchange by the avian embryo. Resp. Pkysiol. 11, 3145.