ARTICLE ORIGINAL

Mathematical models applied to prenatal growthof the ovine pineal gland1

° A. FRANCO, ° E. REDONDO, ° A.J. MASOT, °° Y. LIGNEREUX and ° S. REGODÓN

° Department of Anatomy and Histology, Faculty of Veterinary Medicine, University of Extremadura, 10071 Cáceres, Spain, Fax : 00-34-927-257110.°° Laboratoire d’Anatomie, École Nationale Vétérinaire, 23, chemin des capelles, F-31076 Toulouse Cedex 3

SUMMARY

In order to obtain mathematical models applicable to prenatal growth ofthe ovine pineal gland, 420 embryos and fetuses at varying stages of deve-lopment were used for morphometric analysis. For each of the followingvariables studied, results were expressed as mean ± standard error : bodyweight, head weight, brain weight, pineal gland weight and pineal glandvolume. The possible influence of sex on variables was determined by ana-lysis of variance. The set of variables analysed was fitted to mathematicalgrowth models. The results obtained indicate that during prenatal develop-ment all the variables examined fit a curvilinear-multiplicative growthmodel; additionally, the increase in these weights exceeds increases in length, particularly in the case of the pineal gland, suggesting considerableprenatal development of the gland, especially at the perinatal stage, wherethere is a marked increase in both weight and volume. It’s also discussed thepossible influence of morphologic changes of the pineal gland in its phy-siological maturation during prenatal development and it is established aclear correspondence between the different phases of the growing curves ofthe pineal gland whit the ontogenic-proliferative and hypertrophic-differen-tiative phases, previously described by the authors in their description of theovine pineal gland development. Finally, the data presented are offered tothe scientific community to be used on future researches (i.e. in order toobtain the value of the volume or weight of pineal gland using the curve andthe formulae displayed in a specific phase of development).

KEY-WORDS : growth - mathematical models - ontoge-nesis - pineal gland - sheep.

RÉSUMÉ

Modèles mathématiques appliqués au développement prénatal de laglande pinéale ovine. Par A. FRANCO, E. REDONDO, A.J. MASOT, Y.LIGNEREUX et S. REGODÓN.

Dans le dessein d’obtenir des modèles mathématiques appliqués à lacroissance de la glande pinéale ovine pendant la vie prénatale, nous avonsutilisé 420 embryons et foetus à différents stades de développement, afin deréaliser une analyse morphologique. Pour chacune des variables utiliséespendant l’expérimentation (poids : corporels, céphaliques, encéphaliques,glandulaires et volumes épiphysaires), nous avons décrit leur moyenne ±écart-type. L’éventuelle influence du facteur sexe sur les différentesvariables a été déterminée par une analyse de variance. Grâce à l’ensembledes variables analysées nous pouvons les adapter aux modèles mathéma-tiques de croissance. Les résultats obtenus nous indiquent que, pendant lavie prénatale, toutes les variables analysées se rapportent au modèle decroissance curviligne multiplicative. Cela implique donc un grand dévelop-pement de la glande pinéale durant la vie intrautérine et surtout en périodepérinatale où elle montre un notable développement aussi bien en poidsqu’en volume.

Il est de même discuté une possible influence des variations morpholo-giques de la glande pinéale dans sa maturation physiologique pendant sondéveloppement parmi les différentes phases des courbes de croissance de laglande pinéale avec les phases ontogéniques prolifératives et hypertro-phiques différencielles décrites, au préalable, par les auteurs dans leursétudes sur le développement de la glande pinéale. Finalement, les donnéesprésentées sont communiquées à la communauté scientifique afin d’être uti-lisées dans de futures recherches (par exemple : pour obtenir des valeurs devolume et de poids de la glande pinéale à partir de la courbe et de la formulede croissance dans une phase particulière du développement).

MOTS-CLéS : croissance - modèles mathématiques -ontogenèse - glande pinéale - ovin.

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1. This works was supported by El Fondo Social Europeo y la Consejería de Educación y Juventud de la Junta de Extremadura (PR197D007).

IntroductionGrowth and development are time-dependent biological

processes (W=f[t]), and can thus be expressed by means of amathematical model which, according to FRANCE andTHORNLEY [6], is simply an equation representing thebehavior of a system. A dynamic model is used here, sincethe time factor accounts for the evolution of events occurringin the course of prenatal development. The concept of allo-metry put forward by HUXLEY [14] is not without interest,in that it compares growth of a part with growth of the wholeof the subject.

The purpose of the present experiment was to approach theanatomy of ovine pineal gland growth (weight and volume)within the context of the developing anatomical background(head and brain weights) with a view to obtaining — byapplication of mathematical models — formulae applicableto head growth in general, and the growth and developmentof the pineal gland in particular. Few published studies haveaddressed the ontogenesis of the pineal gland [11, 19, 20, 21],and none apply mathematical growth models to embryodevelopment, with the exception of some studies of stomachdevelopment [7, 8, 9, 10].

Material and methodsA total of 420 Merino embryos and fetuses (210 males and

210 females) from the Cáceres municipal abattoir were usedfor this experiment. Study material was obtained frommothers routinely slaughtered on the abattoir line ; specimenswere extracted immediately on opening of the genital tract.Once outside the uterus, embryos and fetuses were measured(C-R, crown rump length) using a slide calliper in order toobtain age-size estimates following the criterion laid downby EVANS and SACK [5]. The following measurementswere obtained :

— Total embryo weight.

— Head weight following separation of the head from thebody by incision behind the occipital condyles.

— Brain weight following extraction ; skull opened follo-wing the method proposed by GAZQUEZ et al. [12].

— Pineal gland weight following identification by carefuldissection. This variable could only be obtained in embryosand fetuses at a stage of development allowing gross visuali-sation of the pineal gland. All gland weights were measuredusing an electronic precision balance (d=0.1mg).

Once weighed, the gland was fixed by immersion in 10 %formol in phosphate buffer and processed by usual paraffin-embedding methods to obtain a complete set of transversesections 5 µm thick, which were stained with hematoxylin-eosin. Samples for morphometric analysis were viewedthrough a Nikon Optiphot microscope equipped with a video-camera. The image was reflected onto the screen of a imageanalyzer, using a VIDS-IV program. Gland volume wasobtained as the sum of the areas of each of the serial sections,applying Simpson’s numerical integration method [15].

For each of the following variables studied, results wereexpressed as mean ± standard error : body weight, headweight, brain weight, gland weight and pineal gland volume.The possible influence of sex on variables was determined byanalysis of covariance. The set of variables analysed was fit-ted to mathematical growth models [2 ]. The model used ineach case depended on the shape of the real growth curve,taking initially the model which «a priori» provided the bestfit and estimating goodness of fit by the correlation coeffi-cient r2. The models initially tested were the linear model y =a + bX, the multiplicative model y = aXb and the exponentialmodel y = EXP(a + bX). In all cases, the best data fit wasobtained with the multiplicative model (r2 > 0.95).

Embryo body length, expressed in cm, always served asindependent variable ; dependent variables were head weight(g), brain weight (g), pineal gland weight (mg) and pinealgland volume (mm3).

Coefficients of allometry b (relative growth rate of part ofan organism in relation to another part of the whole body),in HUXLEY’s terms [14 ], were calculated using a multipli-cative correlation equation y = aXb, in which the «whole»(independent variable) is represented by body length andthe "part" (dependent variable) by body weight, headweight, brain weight, gland weight and gland volume. Bodyweight was then taken as the "whole" compared to the"parts" represented by head weight, brain weight, and glandweight and volume ; this process was continued for headweight, brain weight, and gland weight and finally glandvolume.

All statistical analyses [1] were performed on a personalcomputer using the statistical package Statgraphics version5.0 (Statistical Graphics Corporation, Maryland, USA,1991).

Results

RELATIONSHIP BETWEEN HEAD WEIGHT AND BODYLENGTH (Table I, Fig. 1)

The growth curve for head weight consists of several sec-tions. The initial phase (1.6-9.5 cm C-R, crown rump length,27-56 days of gestation) suggests that there is not a great cor-relation betwen body length and head weight since headweight scarcely varies. In this phase, no sex-related diffe-rences were apparent. In the second phase, from 9.6 cm C-Rto 27.5 cm C-R (56-101 days), sex-related differences weresmall but significant (p < 0.05). In this phase, head weight isclosely related with the body length because both increase ata similar rate. Finally, in the third phase, from 27.6 cm C-Ronwards, there was a marked initial increase in head weight,in a short stage of body length (34-37 cm C-R). Thenceforth,and to the end of gestation, head weight continued toincrease, although at a slower rate ; significant sex-relateddifferences were more marked (p < 0.001), with a mean dif-ference of roughly 35 g between sexes.

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TABLE. I. — Data for head weight, expressed in grams (g). The asterisks denote statistical significant diffe-rences between male and female values (p < 0.05).

FIG. 1. — Growth and modeling of head weight as a function of body length. Growth equa-tion and analysis of variance for the proposed model. Real and estimated values (SS : Sums of squares, df : degrees of freedom, MS : Mean squares, Fs : ratio of the meansquares).

RELATIONSHIP BETWEEN BRAIN WEIGHT ANDBODY LENGTH (Table II, Fig. 2)

The analysis of the curve for brain weight reveals thatduring the early stages of development (1.6-11.5 cm C-R, 27-61 days) brain weight barely changed with age and fewsex-related differences were observed. From 61 to 110 days(11.6 cm - 31.5 cm C-R) brain weight recorded showedmoderate growth, matching the increase in body length ; weobserved sex-related differences, with higher growth inmales. Thenceforth, and up until birth, there was a more mar-ked increase in brain weight in both males and females, appa-rent in the steep slope of growth curve.

RELATIONSHIP BETWEEN PINEAL GLAND WEIGHTAND BODY LENGTH (Table III, Fig. 3)

The growth curve for pineal gland weight vs. body lengthwas totally curvilinear, making it difficult to identify phasesof growth. Nevertheless, three phases can be defined. In theinitial phase, up until 92 days (24 cm C-R), there was littlechange in pineal gland weight compared to the increase inbody length and a greater increase in pineal gland weight wasobserved at around 80 days (20 cm C-R) ; statistically signi-ficant differences were observed between sexes. The secondphase, from 92 days (24 cm C-R), was marked by greaterincrease of pineal gland weight in males than in females, des-pite an equal increase in body length. Increases in pinealgland weight in this phase matched with the increases in bodylength because the pineal gland grows at a similar rate ofbody length. Finally, the third phase from 116 to 150 days(34.5 cm to 41cm C-R) was marked by a considerableincrease in pineal gland weight compared to a relativelysmall increase in body length.

RELATIONSHIP BETWEEN PINEAL GLAND VOLUMEAND BODY LENGTH (Table IV, Fig. 4)

Pineal gland volume values were broadly similar in malesand females ; however, at certain stages of developmentvolume tended to be greater in males, differences sometimesbecoming significant (p < 0.05). With regard to the slope ofthe growth curve, during the initial phase, up until 92 days ofgestation (24 cm C-R), there was little change in pineal glandvolume compared to the increase in body length. From 92 days (24 cm C-R), increase in pineal gland volume mat-ched those of body length ; growth was moderate until 34.5 cm (116 days). Thenceforth, and until birth, curve dis-play a greater increase in pineal volume than in body length ;this increase became more marked over the last 20 days ofprenatal development, during which volume values almostdoubled.

COEFFICIENTS OF ALLOMETRY (Table V)

Analysis of the allometry coefficients correlating or com-paring the development of each variable studied with theoverall increase growth in specimen length, and with each ofthe other variables, revealed the following :

— During prenatal development, all variables increased ata greater rate than body length ; this was especially true ofpineal gland weight and volume, which yielded very highallometry coefficients (3.80 and 3.45, respectively), sugges-ting considerable development of the pineal gland duringgestation.

— Where the "whole" was represented by body weight, thepineal gland again showed the fastest rate of growth anddevelopment, whilst head and brain showed negative allome-try with respect to body weight.

— Comparison of pineal gland weight and volume withhead weight also revealed greater growth of the pineal gland.Brain weight growth-rates remained slower.

— Pineal gland weight and volume increased faster thanbrain weight.

— Finally, pineal gland volume increased more slowlythan pineal gland weight, although development was almostisometric (b = 1), as might be expected.

CORRELATIONS (Table VI)

All correlation values were positive and statistically signi-ficant. There was a particularly strong correlation betweenpineal gland weight and body weight, and between headweight and brain weight. Pineal gland weight also correlatedwell with the other variables. Poorer correlations were recor-ded between pineal gland volume and body length, betweenbody length and body weight, and between pineal glandvolume and head weight : all these showed lower coefficientsclose to 0.90.

DiscussionGiven the paucity of research addressing the prenatal deve-

lopment of the pineal gland in sheep, reference is madebelow to planimetric studies of postnatal development, inother animal species.

Values for pineal gland weight obtained in the presentstudy for the later stages of sheep development are compa-rable to those reported by LEGAIT and OBOUSSIER [17],in rabbits (12.4 mg) and adult cats (10.4 mg). Nevertheless,pineal gland weights in ovine fetuses differed considerablyfrom those obtained in small adult ruminants : 128 mg, [22] ;182 mg, [3]. During early stages of development, pinealgland volume, like pineal gland weight, was similar to those obtained by VOLLRATH [23] in adult guinea-pigs(0.436 mm3).

There was no great variation in pineal gland weight andvolume between age-matched individuals, although suchvariation is reported by other authors for both pineal glandweight [22, 3] and pineal gland volume [16].

Significant differences between sexes were found for allvariables studied, in terms of both absolute and relativevalues (i.e. with respect to body weight). A similar finding isreported only by VOLLRATH [23], who found that the adult

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TABLE. II. — Data for brain weight, expressed in grams (g). The asterisks denote statistical significant diffe-rences between male and female values (p < 0.05).

FIG. 2. — Growth and modeling of brain weight as a function of body length. Growth equa-tion and analysis of variance for the proposed model (SS : Sums of squares, df : degreesof freedom, MS : Mean squares, Fs : ratio of the mean squares).

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FIG. 3. — Growth and modeling of pineal gland weight as a function of body length.Growth equation and analysis of variance for the proposed model (SS : Sums of squares,df : degrees of freedom, MS : Mean squares, Fs : ratio of the mean squares).

TABLE. III. — Data for pineal gland weight, expressed in milligrams (mg). The asterisks denote statisticalsignificant differences between male and female values (p < 0.05).

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TABLE IV. — Morphometric data for pineal gland volume, expressed in cubic millimeters (mm3). The asterisksdenote statistical significant differences between male and female values (p < 0.05).

FIG. 4. — Growth and modeling of pineal gland volume as a function of body length.Growth equation and analysis of variance for the proposed model (SS : Sums of squares,df : degrees of freedom, MS : Mean squares, Fs : ratio of the mean squares).

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TABLE V. — Coefficients of allometry correlating variables (parts) with body length(whole), and variables with each other (BW : body weight ; C-R : body length ;HW : head weight ; BrW : brain weight ; GW : pineal gland weight ; GV : pinealgland volume).

TABLE VI. — Correlation coefficients between the variables studied (BW : body weight ;C-R : body length ; HW : head weight ; BrW : brain weight ; GW : pineal glandweight ; GV : pineal gland volume ; n : number of animals).

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guinea-pig pineal gland was of slightly greater volume thanits male counterpart.

With regard to correlation of pineal gland weight andvolume with the other variables, the high degree of correla-tion recorded here contrasts with the results reported byROUX [22] and BLIN and MAURIN [3], who failed to esta-blish any correlation between body weight and pineal glandweight in large and small adult ruminants. The latter authors,however, report a good correlation between body and pinealgland weight in the adult horse. The results obtained herealso agree with those of LEGAIT et al. [16], who report foradults of several species a greater correlation between pinealgland volume and body weight than between pineal glandvolume and brain weight.

With regard to the proposed growth model, this study sug-gests a single exponential growth phase for all variables exa-mined, compared to the polynomial model proposed byFRANCO et al. [7, 8, 9, 10] for the increase in wall thicknessof all compartments of the developing sheep stomach. Pinealgland weight and volume display identical phases of deve-lopment in their growing curves :

The initial phase, up until 92 days of gestation (24 cm C-R), is characterised by a slow growing of pineal glandvolume and weight refered to body length. In this phase,according our previous results [21] the progressive growth ofthe pineal gland outline is accompanied by a gradual reduc-tion of the recess lumen and by an interruption of its commu-nication of a fine layer of mesenchyma connective tissue.The pinealoblasts are the only cellular type detected in thefirst phases of development and with low numerical density[21]. This initial phase coincides with the ontogenic-prolife-rative phase described by REDONDO et al. [19], whichbegins at about 30 days of prenatal life and commences withdifferentiation of the rudimentary pineal and includes inva-sion of the pineal parenchyma by ependymary cells and theproliferation pineal cells themselves.

From 92 days of gestation (24 cm C-R) onwards, pinealgrowth becomes dependent on body length, and is moderateuntil the final third of the development process. In this phase,according our previous results [19] at around 98 days of ges-tation, the oclusion of the pineal recess is finalised as a resultof the folding and fusion of its walls. The moderated growthof the pineal gland in this phase may be due to the increase ofpinealoblast number [21] and to the arise of a second cellulargroup : the intersticial cells at 98 days of gestation [11]. Thissecond phase matches with the end of the ontogenic-prolife-rative described in structural development of the ovine pinealgland [21].

From at around 116 days of gestation (34.5 cm C-R) anduntil birth, curves show greater growth of the pineal glandwith respect to body length, an increase which becomes moremarked over the last 20 days of gestation, particularly in thecase of pineal gland volume. This strong growing of pinealgland matches in the time with the increase in pinealoblastsvolume [21], differentiation of pinealoblasts in electron-lucent and electron-dense at 118 days of gestation [19] andincrease of the numerical density of the interstitial cells [21].

So that, this latter phase may corresponds in time to thehypertrophic-differentiative phase proposed by REDONDOet al. [19]. This includes the increase in volume of the pinea-loblasts and occurs at about 118 days of gestation and corres-ponds to the morphology reorganization phase described byCALVO and BOYA [4]. We do not agree with those ofQUAY [18], who places this second phase of hypertrophyand cell differentiation in varying stages of postnatal deve-lopment because ours appear to suggest that growth conti-nues at a similar rate after birth, the same is deduced from themodel put forward by GOEDLOED [13], who distinguishestwo phases in the postnatal evolution of pineal gland weightin rats : an initial exponential stage followed by a phase ofmoderate growth.

AcknowledgementsThe authors are grateful to the Municipal Abattoir at

Cáceres for their assistance in collecting material and toGermán FERNÁNDEZ of the Faculty of VeterinaryMedicine, Cáceres, for technical assistance.

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