REVIEW ARTICLE

Modelling growth and body composition in fish

nutrition: where have we been and where are we going?

Andre¤ Dumas, James France & Dominique BureauDepartment of Animal and Poultry Science, Centre for Nutrition Modelling, University of Guelph, Guelph, Ontario, Canada

Correspondence: A Dumas, Department of Animal and Poultry Science, Centre for Nutrition Modelling, University of Guelph, Guelph,

Ontario N1G 2W1, Canada. E-mail: adumas@uoguelph.ca

Abstract

Mathematical models in ¢sh nutrition have provenindispensable in estimating growth and feed require-ments. Nowadays, reducing the environmental foot-print and improving product quality of ¢sh cultureoperations are of increasing interest. This reviewstarts by examining simple models applied to de-scribe/predict ¢sh growth pro¢les and progresses to-wards more comprehensive concepts based onbioenergetics and nutrient metabolism. Simplegrowth models often lack biological interpretationand overlook fundamental properties of ¢sh (e.g. ec-tothermy, indeterminate growth). In addition, thesemodels disregard possible variations in growth trajec-tory across life stages. Bioenergetic models haveserved to predict not only ¢sh growth but also feedrequirements andwaste outputs from ¢sh culture op-erations. However, bioenergetics is a concept basedon energy-yielding equivalence of chemicals and hassigni¢cant limitations. Nutrient-based models havebeen introduced into the ¢sh nutrition literature overthe last two decades and stand as a more biologicallysound alternative to bioenergetic models. More me-chanistic models are required to expand current un-derstanding about growth targets and nutrientutilization for biomass gain. Finally, existing modelsneed to be adapted further to address e¡ectively con-cerns regarding sustainability, product quality andbody traits.

Keywords: modelling, ¢sh, growth, bodycomposi-tion, nutrition.

Introduction

Aquaculture has become a multinational industryover the last 30 years and is expected to maintain anaverage annual growth rate of 44% over the period2010^2030 (Bruge' re & Ridler 2004). Greater demandfor ¢sh, combined with the reduction in capture ¢sh-eries and more a¡ordable retail prices for several spe-cies, has contributed to foster and sustain theaquaculture industry (NRC 1999; FAO 2006). How-ever, intensi¢cation and potential for development ofthe aquaculture sector have created challenges re-garding pro¢tability, environmental sustainabilityand product quality, most of which are related ulti-mately to nutrition (e.g. Naylor, Goldburg, Primavera,Kautsky, Beveridge, Clay, Folke, Lubchenco, Mooney& Troell 2000; Watanabe 2002). These concernsalong with uncertainties surrounding productioncosts stress, among other things, the need to developaccurate tools to manage productionand predict sce-narios soundly.Here, mathematical modelling ^ de¢ned as the use

of equations to describe or simulate processes in asystem ^ represents an e¡ective approach to takingup the challenges that aquaculture is facing. Mathe-matical models in animal nutrition have proven in-dispensable in estimating growth and feedrequirements that have always represented major¢elds of interest in livestock production (Kellner1911; Murray1914; Brody1945; Blaxter1989; Baldwin1995; Dumas, Dijkstra & France 2008). In aquacul-ture, the quality, safety and health bene¢ts of ¢shproducts are now of increasing interest (Hocquette,

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Richardson, Prache, Me¤ dale, Du¡y & Scollan 2005;Caswell 2006; Moza¡arian & Rimm 2006). Composi-tion of ¢sh with reference to carcass yield, fatty acidcomposition and levels of lipid and contaminants hasrecently received further attention in studies on nu-trition, genetics and health (Rasmussen 2001; Blan-chet, Lucas, Julien, Morin, Gingras & Dewailly 2005;Hamilton, Hites, Schwager, Foran, Knuth & Carpenter2005; Tobin, Kause, M�ntysaari, Martin, Houlihan,Dobly, Kiessling, Rungruangsak-Torrissen, Ritola &Ruohonen 2006).This article begins by summarizing brie£y the bio-

logical properties of ¢sh growth. Thereafter, majorcurrent models applied in ¢sh nutrition are reviewedand challenged. Finally, a global perspective is o¡eredand future directions in modelling are suggested toaddress better the concerns in ¢sh production.

Biological properties of fish growth

Despite its complexity, growth takes place in a highlyorganized scheme in animals. Diverse regulatorystrategies exist in organisms to adjust in£ux of che-micals (amino acids, fatty acids, minerals, etc.) andexcretion of waste products even in a disruptive en-vironment in order to maintain homeostasis (Nelson& Cox 2000). As growth processes do not occur in achaotic manner, they can be generally described andpredicted using conventional mathematics.Growth, body composition and metabolic utiliza-

tion of nutrients or allocation of resources are relatedto each other and change considerably during thelifespan of animals. Growth trajectories of animals ^de¢ned here as the pattern of weight gain achievedthrough time ^ display an almost universally sigmoi-dal shape with an asymptotic body size at adult stage(Fig. 1a). It is well documented that growth rate in-creases during the juvenile stage, i.e. the so-calledself-accelerating phase of growth, and levels o¡whenthe animal approaches the adult stage or induces re-productive growth. This last portion of the growth

curve is also referred to as the self-inhibiting phaseof growth (Brody 1927; Charnov,Turner & Winemil-ler 2001; Lester, Shuter & Abrams 2004). In contrastwith birds and mammals, several species of ¢sh, mol-luscs, crustaceans and amphibians are capable ofgrowing well beyond their size at sexual maturity.These organisms display a much less evident self-in-hibiting phase (Fig. 1b). This phenomenon, also re-ferred to as indeterminate growth, results in adebatable position of asymptotic weight at the adultstage. Indeterminate growth is regulated by environ-ment and genetics (Sebens 1987), which a¡ect thephysiological capacity of an organism to synthesizemuscle ¢bres throughout its life cycle (Biga & Goetz2006). Another peculiarity of ¢sh is their ectother-mic nature. Growth rate of ¢sh is thus highly depen-dent on water temperature. To date, few attemptshave been made to describe ¢sh growthwith an alge-braic expression that accommodates their ectother-mic nature and indeterminate growth.

Current models in fish nutrition

The complexity of interactions in nutrition, the vastamount of information available nowadays and thesubstantial cost of experiments make the use ofmathematical models appealing. Models are helpfultools in that they have the ability to represent com-plex phenomenon (e.g. growth) in a relatively simpleway [e.g. weight gain as a function of protein deposi-tion (PD)]. The following sections review brie£y ex-tant models currently applied in ¢sh nutrition.

Simple growth functions

Growth functions are any models where weight orlength (dependent variable, y) is calculated usingtime, t, as the predictor (independent variable) takingthe form y5 f(t), where f represents some functionalrelationship. Growth functions are usuallyanalyticalsolutions to di¡erential equations that can be ¢tted to

Age (arbitrary units)

Bod

y w

eigh

t(a

rbitr

ary

units

)

(b)(a)

Age (arbitrary units)

Bod

y w

eigh

t(a

rbitr

ary

units

)

Figure 1 Typical growth trajectory of (a) terrestrial animals and (b) ¢sh

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the growth data generally by means of non-linear re-gression analysis (Thornley & France 2007). The sig-moidal or curvilinear shape of the growth trajectoryindicates that linear regression is not suitable to de-scribe growth, unless only small portions of thecurve are considered. For this reason, growth func-tions stand presumably as the best means of estimat-ing animal growth. Because a large number ofgrowth functions had been proposed in the last cen-tury, only those that have been widely applied in ¢shstudies or that have considered the e¡ect of tempera-ture on growth of ectotherms are discussed here. Fora broader description of extant growth functions inanimal science and theories associated with them,the reader is referred to Ricker (1979), Parks (1982),Ratkowski (1990), Seber andWild (2003) and Thorn-ley and France (2007).

von Bertalanffy equation

The equation of von Bertalan¡y (1957) stands as themost studied and applied growth function to predictgrowth of ¢sh and other ectotherms (Ricker 1979;Hernandez-Llamas & Ratkowsky 2004; De Graaf &Prein 2005; Katsanevakis 2006). The equation was¢rst proposed by Pˇtter (1920), a German ¢sh biolo-gist, who conceptualized growth as anabolism pre-vailing over catabolism. The di¡erential and integralforms of his equation, currently referred to as the vonBertalan¡y equation, are

dWdt¼ ZWb � kW

W ¼ Wuf � Wu

f �Wu0

� �e�kt

h i1=u ð1Þ

Parameters : Z > k � 0; 0 < b < 1

t ¼ 0; W ¼W0; t!1; W ¼Wf ¼Zk

� � 11�b

where Z and k are rate parameters for anabolism andcatabolism, respectively, k is a rate constant equal tok(1� b), and u equals 1� b. The allometric exponentfor anabolism b is allowed to vary between 0 and 1.The equation has an asymptote, a £exible point of in-£exion, and adheres to the law of allometry(0obo1).Various rearrangements of the von Berta-lan¡y equation exist in the literature (Ricker 1979;Katsanevakis 2006).The assumption regarding an asymptotic ¢nal size

led to unrealistic values for indeterminate growersand, for this reason, was regarded as a mathematicalartefact rather than a fact of nature (Knight 1968;

Ro¡ 1980). Parker and Larkin (1959) removed thecatabolic part of Eq. (1) in order to relax the con-straint on the ¢nal asymptote and suggested estimat-ing m and b by ¢tting to particular life history groupsand growth stanzas:

dWdt¼ mWb ð2Þ

The assumption that growth is determined by thedi¡erence between anabolism and catabolism hasbeen proven inaccurate because it overlooks the roleof timing of maturation on the shape of the growthcurve (Day & Taylor1997; Lester et al.2004). Evidencesuggests that the change in growth rate of indetermi-nate growers results from the decision to allocatemore resources towards gonad development ratherthan movement towards equilibrium between ana-bolismand catabolism (Day & Taylor1997; Czarnoles-ki & Kozlowski 1998; Charnov et al. 2001). However,the e¡ect of reproduction is not always perceptible inectotherms with indeterminate growth, especially inan environment with £uctuating water tempera-tures (Dumas & France 2008).(Correction added on 9 September 2009, after ¢rstonline publication: In the sentence containing Equa-tion (1),‘u equals1b’was corrected to‘u equals1� b’.)

Thermal-unit growth coefficient (TGC)

The French botanist Re¤ aumur laid the basis of thethermal-unit concept in1735 inanattempt to explainthe time required from sowing to harvesting of cropsby summing the degre¤ -chaleur over that period (Allen1976; Bonhomme 2000).The concept was introducedin ichthyologyat the turn of the 20th century (Be› leh-raŁ dek 1930). Although Norris (1868) noted that thedevelopment rate of trout eggs varies with tempera-ture,Wallich (1901) apparently ¢rst applied the con-cept of the thermal unit to record the developmentof ¢sh eggs.Wallich (1901) de¢ned one thermal unitas 11F above 321F during 1 day, meaning that themean daily water temperature of 361F is equivalentto four thermal units. Krogh (1914) showed that therelationship between developmental rate (usually in% day�1) and temperature ( 1C) exhibited a straightline (slope has degree-dayas denominator) over a cer-tain range of temperature. The time and thermalsummation (degree-day) needed for hatching ¢sheggs can thus be estimated using a simple regressionequation (Krogh1914; Embody1934; Hayes1949).The thermal-unit concept was also applied to esti-

mate growth of hatched ¢sh. Iwama andTautz (1981),

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who did not use the concept explicitly, started fromEq. (2) and related the rate parameter for anabolismto mean daily water temperature averaged over therearing period (T):

dWdt¼ mTWb ð3Þ

where m (40) has units of g1� b( 1C day)�1,T (a con-stant) is water temperature ( 1C) and the allometricexponent b (40) is dimensionless. Integrating Eq. (3)yields

ZW

W0

dWWb ¼ mT

Z t

0

dt

W1�b ¼W1�b0 þ mTð1� bÞt

ð4Þ

whereW0 is the initial (time 0) value ofW. Startingfrom Iwama and Tautz (1981), Cho (1992) explicitlyintroduced the degree-day concept into their modeland proposed, without formal mathematical deriva-tion, a modi¢cation to Eq. (4):

W1=3n ¼W1=3

0 þ c1000

Xni¼1

Ti

where c [g1/3( 1C day)�1)] is TGC and Ti ( 1C) is meandaily temperature.From an inspection of Eq. (1) and Eq. (3), it is evi-

dent that theTGC model is a special case of von Ber-talan¡y’s equationwith

m ¼ 3� TGC� T1000

; b ¼ 2=3; l ¼ 0

The TGC model has since been widely used in theaquaculture literature (e.g. Einen, Holmefjord, —s-g�rd & Talbot 1995; Kaushik 1998;Willoughby 1999;Stead & Laird 2002; Hardy & Barrows 2002). Thissimple model has been adapted recently to the di¡er-ent growth stanzas of rainbow trout (Oncorhynchusmykiss,Walbaum) across life stages (Dumas, France& Bureau 2007). Despite its convenience, the ther-mal-unit approach can entail systematic errors in si-tuations where the temperature moves too far awayfrom the optimum for growth (Krogh 1914; Hayes1949; Ricker1979; Jobling 2003).

Exponential equation or specific growthrate (SGR)

The origin of SGR goes back in 1798 and was devel-oped to address demographic concerns. ReverendThomas Malthus, a mathematician, published an es-say in1798 in which he stated that the human popu-

lation increased according to a geometric progression(Gilbert 1993). His model, known as Malthus’ Law ortheMalthusianModel, corresponds to the exponentialgrowth equation:

W ¼W0emt

whereW is body weight,W0 is body weight at timet50, m is a growth coe⁄cient (in units of per unit oftime) and time t is measured as age.The growth coe⁄cient m is better known as SGR,

which is used ubiquitously in ¢sh studies. The equa-tion for SGR (m) is

m ¼ lnWf � lnW0

tf

whereWf is the ¢nal body weight (g) and tf is the time(days) betweenW0 andWf.The SGR has often been proposed as a growth

model in aquaculture (Willoughby 1999; Alan�r�,Kabri & Paspatis 2001) even though it gives no con-sideration to the e¡ect of body weight and tempera-ture on the growth of ¢sh. Keeping in mind thesedrivers of ¢sh growth, Brett (1974) determined di¡er-ent SGR for various water temperatures and bodyweights and entered the values (observed and extra-polated by eye) into tables that served afterwards topredict ¢sh growth according to prevailing condi-tions (Brett 1974;Willoughby1999). However, the re-lationship between SGR and temperature can bea¡ected by the amplitude of temperature £uctuations(Brett 1979; Xu 1996). In other words, growth ratesobserved at constant temperature (e.g. 15 � 0 1C)might di¡er from those at an average temperature(e.g. 15 � 4 1C), especially when £uctuations occurover a short period of time.The SGR model is based on the incorrect assump-

tion that ¢sh growth is continually exponential. Thishas proven not to be the case and, therefore, growthpredictions have to be re-calculated every time thepredicted growth curve moves too far away from theobserved trajectory (Brett 1979). Unlike Brett (1974),Elliott (1975) plotted the relationships between SGR,body weight and temperature and derived the follow-ing equation to predict the growth of brown troutSalmo trutta (Linne¤ ):

dWdt¼ ðaþ b2TÞW1�b1 ð5Þ

with the integral form (provided T is assumed con-stant):

W ¼ b1ðaþ b2TÞtþWb10

h i1=b1

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where b1, b2 and a are weight exponent (dimension-less), slope [% (day 1C)�1] and intercept (% day�1)of the relationship between SGR (% day�1) and T( 1C) respectively.The Elliott model is often used to investigate ¢sh

growth, especially in the ecology literature (Craig1982; Allen1985; Jensen1990). From an inspection ofEqns (1) and (5), it is evident that the Elliott model isalso a special case of von Bertalan¡y’s equationwith

m ¼ aþ b2T; b ¼ 1� b1; l ¼ 0

Moreover, Eq. (3) of Iwama and Tautz (1981) hasmany similarities to Eq. (5). Therefore, Elliott (1975) in-troduced the e¡ect of temperature into Eq. (2) of Parkerand Larkin (1959) before Iwama andTautz (1981).Equation (5) needs to be solved repeatedly over the

growing period because slope and intercept changewithwater temperature and bodyweight (Fig.2).Thisdrawback limits application of the Elliott (1975) modelbecause predictions can be applicable only to veryshort intervals and preclude comparisonbetween stu-dies, especially under £uctuatingwater temperatures.Elliott, Hurley and Fryer (1995) revised the Elliott

model and included considerations for optimum (Topt)and limiting (Tlim) temperatures for growth (Fig. 2).The resulting equation takes the form

W ¼ Wb0 þ bcðT � TlimÞt100ðTopt � TlimÞ

� �1=bð6Þ

where c is the SGRof a1g ¢sh atTopt,Tlim 5 lower (TL)or upper (TU) temperature at which SGR is 0:Tlim 5TLifT � Topt orTlim 5TU ifT4Topt.Equation (6) is valid as long as the water tempera-

ture does not change. Under £uctuating temperatureconditions, the equation needs to be extendedand body weight at the end of a growingperiod (t1, t2, . . ., tk),Wk, is now predicted using the

following:

Wbk ¼ Wb

0 þbc100

�ðT1 � TlimÞt1Topt � Tlim

þ ðT2 � TlimÞt2Topt � Tlim

þ . . .þ ðTk � TlimÞtkTopt � Tlim

�

ð7Þwhere T1,T2, . . . ,Tk correspond to average tempera-ture ( 1C) for intervals 1,2, . . . ,k, and t1, t2, . . ., tk arein days.The authors reported that Eq. (7) yields signi¢cant

discrepancies when the growing period exceeded 3months (Elliott et al.1995). Furthermore, the assump-tion of a ¢xed growth rate c in Eqns (6) and (7) is con-trary to the biology and growth trajectory of ¢sh.Another exponential ¢sh growth model was pro-

posed more recently by Lupatsch and Kissil (1998):

Y ¼ aXbecT

whereYand X are weight gain (g ¢sh�1day�1) andbody weight (g ¢sh�1), respectively, a and c are con-stants, b is weight exponent (dimensionless) and T iswater temperature ( 1C). This equation is also a spe-cial case of the von Bertalan¡y with Z5 aecT andk50 in Eq. (1):

dWdt¼ a ecTWb

LetdWdt¼ Y;W ¼ X

Therefore,Y ¼ aXbecT

This model has been used successfully to describethe growth trajectory of warmwater ¢sh species suchas gilthead seabream (Lupatsch & Kissil 1998), Eur-opean sea bass (Lupatsch, Kissil & Sklan 2001), whitegrouper (Lupatsch & Kissil 2005) and barramundi(Glencross 2006) within a relatively narrow range oftemperature (�20^27 1C). It assumes an exponentialrelationship between water temperature and growthrate, which can be true only for a certain range of op-timal temperature, and appears in disagreementwith the thermal-unit concept and reaction kineticmodels for ectotherms.The latter showed that growthrate is inhibited at high temperature, and relationshipbetween growth rate and temperature displays anasymmetric bell-shaped curve (Sharpe & DeMichele1977; School¢eld, Sharpe & Magnuson1981).(Correction added on 9 September 2009, after ¢rstonline publication: In the sentence ‘This equation isalso a special case of the von Bertalan¡y withZ5YecT. . .’, the symbolYwas corrected to a.)

0.0

0.5

1.0

1.5

2.0

2.5

5 10 13 15 18Temperature (°C)

Spec

ific

gro

wth

rat

e (%

/d)

3 g BW

5 g BW

50 g BW

300 g BW

TU

Topt

TL

Figure 2 E¡ects of body weight (BW) and temperatureon speci¢c growth rate (SGR). Lower (TL) and upper (TU)temperatures indicate where SGR is zero (adapted fromElliott1975).

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Based on visual appraisal of typical growth curves(e.g. Fig.1), animals do not growgeometrically, i.e. ex-ponentially, across life stages. The exponentialgrowth function is therefore not suitable for accu-rately predicting or describing the growth trajectoryof ¢sh and other animals. Furthermore, this functionyields unavoidably systematic deviations (Fig. 3).Growth data on Arctic charr Salvelinus alpinus(Linne¤ ) obtained from Simmons (1997) are used hereto compare theTGC and SGR models (constant watertemperature:12 1C; duration:112 days). Using the lat-ter equation, growth is underestimated from 11.5 g(W0) to 174.2 g (Wf) whereas body weight increasessteeply from 174.2 to 678 g over a 56-day period,which is unrealistic. This is in agreement with Brett(1974,1979) and Cho (1992) who pointed out that SGRleads to underestimationof growth betweenvalues ofW0 andWk used to compute SGRand to serious over-estimations of weight gain beyondWk.In spite of its limitations, SGR remains widely

accepted by editors and recommended ubiqui-tously in the ¢sh literature likely because of its easeof use (Barton 1996;Willoughby 1999; Alan�r� et al.2001; Stead & Laird 2002). At best, SGR can serve incomparing di¡erent performances, although com-parisons using SGR are valid only if ¢sh have similarW0 andWk and are reared at the same water tem-perature because, as stated earlier, growth rate of¢sh varies with size and temperature. For all thereasons mentioned above, SGR ¢nds very little biolo-gical support and is therefore largely unsuitable as a¢sh growth model and tool to compare short-termgrowth performance.

Simple models of feed conversionto biomass

Goals in animal nutrition are arguably to maximizethe conversion of inputs (e.g. feed, investments) intohigh-quality outputs over a short period of time. Im-proving the conversion of dietary inputs to leanrather than adipose tissue growth is of bene¢t toproducers and consumers. It can also contribute toreduced waste outputs and provide room for man-oeuvre given the volatility of pro¢t margins. As a con-sequence, several studies have turned their attentiontowards feed e⁄ciency, protein utilization and lipiddistribution as a function of ¢sh size, feeding leveland alternative ingredients for example (Aursand,Bleivik, Rainuzzo, J�rgensen & Mohr 1994; Azevedo,Cho, Leeson & Bureau1998; Lupatsch, Kissil, Sklan &Pfe¡er 2001; Cheng, Hardy & Usry 2003). These stu-dies have generated a large amount of information(e.g. on body composition) that still needs to be ex-plored and synthesized.Most of these studies were designed to describe an-

imal responses (e.g. weight gain) within speci¢c ex-perimental conditions. Unfortunately, their ability todescribe awide array of animal responses in varyingsituations is limited because their experimental de-signs prevent representation of the mechanisms inthe internal structure of the organism that are re-sponsible for the observed responses. For this reason,several mathematical modellers have insisted on theneed to move from a requirement-based (input^out-put) to a rate:state approach where the major vari-ables in play can be described and relateddynamically, similar to a metabolic pathway (AFRC1991;Thornley & France 2007; Lo¤ pez 2008).The rate:-state formalism consists of representing the rate ofchange of pools, referred to as state variables, usingdi¡erential equations (Dijkstra, Mills & France2002). Such formalism considers the state of a poolas the result of dynamic exchanges, i.e. in£ux (e.g.protein synthesis) and e¥ux (e.g. protein degrada-tion) of substances. Di¡erential equations are a valu-able tool and have been proven essential in dynamicmodelling in describing the behaviour of a systemconcisely and e⁄ciently (Kleiber1961; France & Keb-reab 2006). The rate:state formalism is discussedfurther in Nutrient-based models.

Bioenergetic models

Animal energetics refers to the quantitative study ofenergy exchanges induced by metabolic processes in

0

100

200

300

400

500

600

700

0 50 100 150 200Time (days)

Bod

y w

eigh

t (g

)

SGR

TGC

Observed

Figure 3 Comparison between observed and predictedbody weight of Arctic charr (Salvelinus alpinus Linne¤ )using the thermal-unit growth coe⁄cient (TGC) and spe-ci¢c growth rate (SGR). Growth data are from Simmons(1997).

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living organisms to stay alive, grow and reproduce(Nelson & Cox 2000). Energy exists in materials ofdietary and body origin and is released in the form ofheat to support work (Blaxter1989).Models constructed on the basis of bioenergetic

principles utilize mathematical equations describingthe heat transactions and adhere generally to a fac-torial scheme, also referred to as an energy budget.The factorial approach follows from the metaboliz-able energy concept (Armsby 1903; HMSO 1975),where energy expenditures or heat productionare al-located to di¡erent metabolic processes according toan order of priority (NRC 1993; Bureau, Kaushik &Cho 2002). Inspired by Ivlev (1939) and Winberg(1956),Warren and Davis (1967, 1968) adhered to thefactorial approach and proposed a simple additiveequation to describe the energy budget of ¢sh:

C ¼ Fþ U þ DBþ R ð8Þ

where C is intake of energy and F and U are energylosses in faeces, and urine and gills respectively (allvariables in units of MJday�1). Variable DB repre-sents growth (energy gain) of the ¢sh and R is energyloss through metabolic processes associated withmaintenance and heat increment of feeding. Eachcomponent of the equation is described using mathe-matical relationships derived mostly using statisticalanalyses.

Equation (8) gained acceptance in ¢sheries andwas adopted by Ricker (1968), Elliott (1976a, b) andKitchell, Stewart andWeininger (1977). A systematicterminology for the description of energy budget andmetabolic processes in animal nutrition was devel-oped later by NRC (1981), and heat losses were cate-gorized as shown in Fig. 4.Fish growth has usually been predicted using two

di¡erent approaches in bioenergetic models. Onewayof forecasting ¢sh growth assumes that energy in-take drives weight gain. This assumption is encoun-tered mostly in ¢sheries and ecology studies becauseavailability of food in natural ecosystems often limits¢sh growth (Elliott1976a, b; Kitchell et al.1977; From& Rasmussen1989). An alternative approach consid-ers genetic or desired growth rate rather than nutri-tion as the factor limiting animal growth (Hubbell1971; Calow 1973; Oldham, Emmans & Kyriazakis1997). Here, intake of energy is a function of the re-quirements of the individual to achieve a givengrowth capability or growth target. This approachwas suggested byWinberg (1956, p.174) and is mostlyused in aquaculture where ¢sh are generally fed tosatiation with nutritionally complete diets (Cho1990; Lupatsch, Kissil & Sklan 2001; Zhou, Xie, Lei,Zhu & Yang 2005). Genetically determined growthcapability of ¢sh is assessed using simple growthfunctions, especially the TGC model and the

Intake Energy

Digestible Energy

Metabolizable Energy

Recovered EnergyGrowth, Fat, Reproduction

Net Energy

MaintenanceBasal metabolismVoluntary activityThermal regulation

Heat IncrementWaste formation and

excretionProduct formation

Digestion andabsorption

Gill excretionsUrine excretions

Faeces excretions

Figure 4 Factorial framework of energy partitioning in typical bioenergetic models intended to evaluate feed require-ments. Each metabolic process results in heat loss that is determined mostly using regression equations. For further in-formationonde¢nitions of terms andmathematical descriptionof metabolic processes, the reader is referred toNRC (1981)and Bureau et al. (2002).

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exponential model of Lupatsch and Kissil (1998) (Cho& Bureau 1998; Lupatsch & Kissil 1998; Lupatsch,Kissil & Sklan 2001; Glencross 2006).Probably the most adapted bioenergetic model for

farmed ¢sh is the FISH-PRFEQ program (Cho & Bureau1998). The model follows a factorial scheme and esti-mates feed requirements and waste outputs from ex-pected growth performance, digestible energy of thediet and body energy deposition.The FISH-PRFEQ modelhas been used or adapted to di¡erent ¢sh species andfor various purposes (Kaushik 1998; Papatryphon,Petit, van der Werf, Kaushik & Claver 2005; Zhouet al. 2005).Bioenergetic models predict energy gain, but they

provide little information on the chemical composi-tion (moisture, protein, lipid and ash) of biomassgain. This characteristic has two signi¢cant draw-backs. Firstly, the bioenergetic models can entail sys-tematic errors because the relationship betweenrecovered energyandweight gain changes across lifestages (Bureau et al. 2002). More energy is containedper unit of biomass gain for a large ¢sh (e.g.10 kJg�1BW) than for a small ¢sh (e.g. 5 kJg�1BW)under typical rearing conditions. Studies haveshown that the composition of biomass gain includesmore lipid and less water in a large ¢sh than in asmall ¢sh (Shul’man 1974; Dumas, de Lange, France& Bureau 2007). Protein and lipid deposition (LD)are two distinct biological processes driven by di¡er-ent factors or determinants that are overlooked inbioenergetic models. Secondly, the recovered energycan serve to determine the energy retention e⁄-ciency, but it is of no utility in assessing the e⁄ciencyof nutrient utilization or rates of depositionunless re-liable equations are developed to describe body com-position across life stages.It has been shown that feed evaluation systems

and animal growth models based on bioenergeticshave major limitations (Birkett & de Lange 2001a; Ba-jer, Whitledge & Hayward 2004; Dijkstra, Kebreab,Mills, Pellikaan, Lo¤ pez, Bannink & France 2007).Feed evaluation systems cannot rely on bioenergeticsexclusively and have to consider dietary proteins andother nutrients, especially with ¢sh that rely heavilyon proteins to meet their metabolic needs. Moreover,digestible proteins, along with dietary amino acids,a¡ect feed e⁄ciencyand nitrogen retention e⁄ciencysigni¢cantly (Azevedo, Leeson, Cho & Bureau 2004a;Encarnac� a� o, de Lange, Rodehutscord, Hoehler, Bu-reau & Bureau 2004; Booth, Allan & Anderson2007). The e¡ect of protein intake, and not only en-ergy, on ¢sh growth performance was soon acknowl-

edged and included in models to estimate feedrequirements, weight gain, and e⁄ciency of energyand protein retention of African cat¢sh (Machiels &Henken1986), tilapia (van Dam & DeVries1995), carp(Schwarz & Kirchgessner 1995), European sea bass(Lupatsch, Kissil & Sklan 2001, Lupatsch et al. 2003),gilthead sea breamandwhite grouper (Lupatsch et al.2003; Lupatsch & Kissil 2005).Although the factorial approach assumes that en-

ergetic costs of metabolic processes are additive, evi-dence suggests that energy is allocated in acompensatory fashion, i.e. according to the meta-bolic scope of the animal at a particular life stage(Wieser 1989; Rombough 1994). This particularitymay explain why the concept of energy requirementfor maintenance remains debatable and is a¡ected bybody composition and other factors such as ambienttemperature and breed (e.g. Close, Mount & Brown1978; ARC 1981; Thompson, Meiske, Goodrich, Rust& Byers 1983; Campbell, Crim, Young & Evans 1994;Knap 2000). For instance, models based on bio-energetic principles assume that growth and feede⁄ciency will be nil when animals are fed a mainte-nance ration (recovered energy50). This assump-tion has been proven inaccurate in ¢sh, as well as inother animals, where positive weight gain was stillobserved even though animals were fed at or below amaintenance ration and the whole-body energy bal-ance was negative (Huisman 1976; Le Dividich, Ver-morel, Noblet, Bouvier & Aumaitre 1980; Meyer-Burgdor¡, Osman & Gˇnther 1989; Lupatsch, Kissil& Sklan 2001; Bureau, Hua & Cho 2006).Bioenergetic models have also been used to esti-

mate feed requirements of ¢sh and waste outputsfrom ¢sh culture operations (Winberg 1956; NRC1993; Cho & Bureau 1998; Lupatsch & Kissil 1998,2005). Assessing waste outputs requires good esti-mates of body composition in order to compute, forexample, nitrogen and phosphorus discharge intothe environment.

Nutrient-based models

Historically, animal nutritionists ¢rst considered nu-trients (i.e. chemicals and macromolecules that pro-vide essential nourishment for maintenance, growthand reproduction) rather than energy to study theconversion of feed to biomass (for a review, see Du-mas et al. 2008). Chemical (water, nitrogen, fat,minerals and carbon) and physical (bone, muscle,adipose tissue, blood, skin, hair and o¡al) composi-

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tions of carcass and chemical composition of feed-stu¡s were estimated for farm animals before the20th century (Wol¡ 1895).Wol¡ (1895) appears to bethe ¢rst to adopt a factorial approach to describe rela-tively and in detail the fate of dietary nitrogen, car-bon and fat with consideration of intake, lossesthrough faeces and urine, and recovery as body fatand body £esh in the carcass.In view of the limitations of bioenergetics, animal

nutritionists and growth modellers have returned tomore nutrient- or biochemical-oriented approaches(e.g. Machiels & Henken 1986; Gerrits, Dijkstra &France 1997; Birkett & de Lange 2001b). These nutri-ent-based models may be de¢ned as mechanistic sys-tems designed to simulate the fate of dietarynutrients, with consideration of utilization of aminoacids, fatty acids and their precursors. Similar tobioenergetics, nutrient-based models serve to predictgrowth, nutrient requirements and waste outputs.

However, these models further explain the proces-sing of nutrients by considering intermediary meta-bolism and are therefore more mechanistic.Bioenergetic models are mostly descriptive, rely on arather simple framework of energy transaction, re-present energy using units of joules or calories andoverlook the stoichiometry of energy-yielding nutri-ents. Nutrient-based models are more explanatory,rely on metabolic pathways of nutrients, representenergy in terms of ATP (e.g. mol ATP per moleculesubstrate), and consider the stoichiometry of chemi-cal reactions.These nutrient-based models have beenshown to be e¡ective for mammals and ¢sh (e.g. Gill,Thornley, Black, Oldham & Beever 1984; van Dam &DeVries1995).Partitioning of nutrients can follow either a factor-

ial or a compartmental scheme. Figures 5 and 6 illus-trate and contrast the factorial and compartmentalapproaches respectively. The former approach is con-sistent with conventional bioenergetic models andadheres to the same assumptions (e.g. energy is allo-cated according to a hierarchy, metabolic processesare additive). The latter was introduced in the 1950sinto animal nutrition by Blaxter, Graham andWain-man (1956)-these authors did not nominate it as com-partmental or mechanistic modelling, though-andconsists of subdividing a given level of organization(e.g. whole animal, tissue, cell) into di¡erent pools(e.g. amino acids in the blood, intracellular glucose)(Thornley & France 2007).Pools are referred to as state variables (i.e. a quan-

tity that de¢nes the size of the pool at a given point intime) and can be in steady state (e.g. blood glucose ina fasting animal) or non-steady state (e.g. muscle pro-tein content ina growinganimal). Flows of substrates(e.g. lysine and other metabolites) between pools andinto and out of the system are represented as termswithin di¡erential equations, which are usually

Intake

Anabolism andCatabolism

Production

Basal

UrinaryExcretion

Faecalexcretion

Figure 5 Example of a factorial framework of nutrientpartitioning (adapted from Blaxter & Mitchell 1948; Bir-kett & de Lange 2001a). Flow of nutrients through eachmetabolic process (intake, faecal and urinary excretion,anabolism and catabolism, basal metabolism and produc-tion) is determined mostly using regression and massbalance equations.

Blood

Fatty acidsAmino acids

Protein inviscera

Lipid inviscera

Protein indressedcarcass

Lipid indressedcarcass

Catabolism

Figure 6 Example of a simple compartmental framework of nutrient partitioning (adapted from Gill et al.1989). Flowofnutrients between each pool (amino acids, fatty acids, protein and lipid in the viscera and dressed carcass) is determinedusing di¡erential and stoichiometric equations.

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based on rules of stoichiometry and saturation ki-netics. Unlike equations based on regressionanalysis,di¡erential equations suit the mathematical descrip-tion of dynamic systems better because they can ex-hibit a wide array of behaviour (May1976; Dijkstra &France1995).The compartmental approach overcomes, to a cer-

tain extent, the lackof £exibility and theoretical basisassociated with the underlying assumptions of thefactorial approach (AFRC 1991; Beever, France & Al-derman 2000), but may require comprehensive data-sets. Compartmental models for animals have beendesigned for a wide range of purposes (e.g. to predictfeed intake, digestion rate, growth) since 1980 (Ma-chiels & Henken1986; Imamidoost & Cant 2005; Bar,Sigholt, Shearer & Krogdahl 2007).In nutrient-based models, growth results from ac-

cretion of chemicals (mostly water, protein, lipid andash), not energy as assumed fundamentally in bioe-nergetic models. Therefore, the accuracy of thesemodels depends on consistent mathematical descrip-tion of the relationships between nutrient depositionand weight gain.

Modelling body composition and rates ofnutrient deposition in farm animals

The reliability of bioenergetic and nutrient-basedmodels depends to a considerable extent onvalid esti-mates of nutrient deposition rates. Moreover, thesearch for optimal nutrient conversion into biomassand maximum pro¢ts, as well as concerns regardingproduct quality (e.g. fatness, fatty acid compositionand bio-accumulation of various constituents) andenvironmental sustainability are strong motives formodelling body composition and nutrient depositionin farm animals.Numerous data exist on body composition of var-

ious ¢sh species, especially rainbow trout, Europeansea bass and white grouper (e.g. Reinitz 1983; Lu-patsch, Kissil & Sklan 2001; Lupatsch & Kissil 2005).From these studies, it can be concluded that whole-body protein contents are comparable between spe-cies and constant across the grow-out phase, andthe contents of moisture, lipid, ash and energy varyin a similar pattern among species as ¢sh size in-creases (cf. Lupatsch, Kissil & Sklan 2001; Bureauet al. 2002).In the past, boundaries or limits to contents of

body water (BH2O), body protein (BP), body lipid(BL) and body ash (BA) in ¢sh have not been deter-

mined from large datasets (cf. Shearer 1994; Jobling2001; Lupatsch, Kissil & Sklan 2001; Bureau et al.2002). Recently, Dumas, de Lange et al. (2007) devel-oped equations to predict body composition in rain-bow trout using data from 66 studies. Theseequations account for the variation in body composi-tion, represent possible benchmarks for future com-parison and provide reliable foundations forassessment of the e¡ects of di¡erent factors on thecomposition of growth in ¢sh.

Estimating body composition andrates of nutrient deposition usingregression analysis

Mathematical description of body composition in an-imal nutrition started 460 years ago. McMeekan(1941) stressed the importance of assessing meatquality in animal productionand addressing require-ments of speci¢c markets. The author recognized thetechnical di⁄culty, high cost and time requirementassociated with chemical analysis and insisted onthe need to develop indices of composition, i.e. math-ematical equations. McMeekan (1941) proposed line-ar regression equations to predict contents of notonly body fat but also muscle and bones in baconpigs. Equations were of the form yi ¼ b0 þ b1xiwhere yi is the ith ¢tted value of the outcome (i.e. ske-leton, muscle or fat) inunits of g, b0 is the intercept, b1is the slope and xi is the ith value of a given predictor(e.g. length of carcass). McMeekan (1941) overlookedthe e¡ect of body weight on carcass composition.Moreover, he did not describe body compositionwithequations of allometric form ðyi ¼ 10b0 � xb1i Þ eventhough, in his days, the concept of allometry wascommonly applied in biology to designate rate ofchange between di¡erent anatomical characteristicsof an organism (for a review, see Gayon 2000).Furthermore, the allometric equation had alreadybeen used in animal production to examine the rateof fat deposition in di¡erent body parts of poultry(Lerner 1939). Almost 30 years later, KotarbinŁ ska(1969) related body protein to fat-free lean mass andbody water to body protein using linear regressionsof allometric form. She also related body ash to bodyprotein assuming an isometric rather than an allo-metric relationship. These isometric and allometricrelationships based on regression analysis still pre-vail in estimating body composition of farmed ani-mals, including ¢sh (Parker & Vanstone1966; Groves1970; ARC 1981; Weatherley & Gill 1983; de Lange,Morel & Birkett 2003; Dumas, de Lange et al. 2007).

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Rates of nutrient deposition have beena topic of in-terest and have found several applications in animalnutrition for the last 40 years (Oslage & Fliegel 1965;Thorbek1969). Assessing rates of nutrient deposition,mostly protein deposition (PD) and lipid deposition(LD) represents a comprehensive way of examininge⁄ciency of utilization of feed components forgrowth, and e¡ects of genetics, nutrition and envir-onment on composition of the growth response anddietary requirements (Black, Davies, Bray, Giles &Chapple 1995; Schinckel & de Lange 1996). Becausegrowth and PD are associated, the amino acid pro¢lein PD can serve in approximating amino acid re-quirements of growing animals (e.g. M˛hn & deLange1998).In ¢sh, description of the nutrient deposition rate

has received little attention to date, despite its simpli-city, relevance and acceptance in modelling growthof livestock species over the past three decades (ARC1981; Black et al.1995; NRC1996,1998).To our knowl-edge, the concept of nutrient deposition rate was in-troduced recently into the ¢sh literature andconsisted of describing rates of PD, LD and ash de-position on a degree-day basis in rainbow troutacross life stages (Dumas, de Lange et al. 2007). Suchquantitative description still needs to be extended toother strains and ¢sh species.

Estimating nutrient deposition usingexplicit partitioning rules

Partitioning rules attempt to represent the utilizationof dietary nutrients or energy for protein relative toLD and have often served as a means to adjust forbody composition, especially in pig nutrition. Parti-tioning rules and their associated partitioning fac-tors ¢gure among the debatable parameters innutrition modelling, and the reader is referred to re-views by de Lange, Morel and Birkett (2008), Sand-berg, Emmans and Kyriazakis (2005a, b) andEmmans and Kyriazakis (1997) for more details. Thepresent section describes the partitioning rules en-countered in ¢sh nutrition models.

Rule1: Body composition regulates dietarynutrient partitioning

The ¢rst rule assumes that growing animals regulatethe breakdown of protein and lipid according to theircurrent and/or target body lipid to body protein ratio(BL:BP). Machiels and Henken (1986) introduced this

rule into ¢sh nutrition. Preferential body lipid to bodyprotein ratio (prefBL:BP), minimum value for this ra-tio (minBL:BP) and mature lipid weight to matureprotein weight (mBL:mBP) are variations of theBL:BP ratio that have been proposed over the last 15years (Whittemore1995; Emmans & Kyriazakis1999).Whittemore (1995) estimated the minBL:BP for

pigs to be 0.5:1. In a study conducted by Reinitz(1983), the BL:BP ratio for juvenile rainbow trout sta-bilized at 0.1:1 during 84 to 140 days of starvation(¢shwere still alive). This value may be considered asthe minBL:BP ratio for that species at that size(o10 g).The BL:BP ratio has been arbitrarily set as a para-

meter in order to avoid unrealistic prediction of bodycomposition (Machiels & Henken 1986). Althoughthese authors did not further justify their concept,the BL:BP ratio ¢nds biological support. Indeed, thecorrection in body composition that occurs duringcompensatory growth lends credence to the concept.For instance, Kyriazakis and Emmans (1992) showedthat farm animals following a period of nutritionallimitation seek to correct their body composition inorder to return to the normal or preferential bodycomposition needed to achieve their growth target.The fraction of energy requirements that is sup-

plied by oxidation of body fat increases non-linearlyas the BL:BP ratio becomes higher (Machiels & Hen-ken 1986). Neither Machiels and Henken (1986) norKyriazakis and Emmans (1992) proposed an equationto describe this non-linear relationship explicitly.Despite some biological support, the concept re-

mains highly empirical and its purpose appears moreto accommodate the prediction of body compositionthan to describe the real metabolic processes (Sand-berg et al. 2005a). Intake of dietary energy and diges-tible amino acids, and not only target BL:BP ratio at agiven body weight, a¡ects the partitioning of energybetween PD and LD in growing animals (Emmans &Kyriazakis1997; Encarnac� a� o et al.2004;Weis, Birkett,Morel & de Lange 2004). Finally, variations in photo-period and temperature are other factors likely to af-fect the BL:BP ratio in ¢sh (e.g. Brown 1957; Jobling2001; Hemre & Sandnes 2008).

Rule 2: Protein intake regulates dietarynutrient partitioning

Starting with Machiels and Henken (1986), van Damand DeVries (1995) moved away from the constrainton BL:BP ratio. They related the proportion of energy

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obtained from oxidation of body fat (or body protein)to protein-feeding level rather than the BL:BP ratio.They did not measure oxidation of body constituentsper se, but rather estimated it using a calibration pro-cedure. Their results indicated a positive relationshipbetween body protein oxidation and both dietaryprotein intake and the protein to gross energy (P:GE)ratio of the diet. These observations are in agreementwith other studies on ¢sh (Halver &Hardy 2002; Aze-vedo et al. 2004a). Their concept, namely AALIRAT(i.e. auxiliary variable determining the proportion ofATP requirement provided by oxidation of body fat),has the advantage of representing the e⁄ciencyof protein use and the protein-sparing e¡ect of extraenergy.The models proposed by Machiels and Henken

(1986) and van Dam and De Vries (1995) estimatedfat deposition poorly, indicating that nutrient parti-tioning is not just a matter of BL:BP ratio or level ofdietary protein intake. These discrepancies result tosome extent from inaccurate assumptions regardingthe energetic costs of metabolic transactions. Bothmodels set a ¢xed value forATP requirement for pro-tein synthesis [0.06 and 0.075mol ATPg�1proteinsynthesized in Machiels and Henken (1986) and vanDam and DeVries (1995) respectively], but this ener-getic cost seems to be rather variable across studies(Jobling1985; Rombough1994).Machiels and Henken (1986), along with van Dam

and De Vries (1995), assumed that no dietary nutri-ents were oxidized to support energy requirementsfor maintenance and growth, an assumption thathas been proven inaccurate in mammals and ¢sh(Lyndsay 1976; Kim, Grimshaw, Kayes & Amundson1992; Stoll, Burrin, Jahoor, Henry, Yu & Reeds 1998;Halver & Hardy 2002). The models of Machiels andHenken (1986) and van Dam and De Vries (1995) re-cognized the existence of amino acid and glucoseblood pools (cf. their £ow diagrams). However, no at-tempt was made to describe these pools mathemati-cally because they were not considered a source ofATP.

Rule 3: Biochemical saturation kineticsregulates dietary nutrient partitioning

The saturation kinetic approach is used in compart-mental modelling and utilizes enzyme-kinetic equa-tions such as the Michaelis^Menton and the Hill (e.g.Gill et al. 1984; Pettigrew, Gill, France & Close 1992;Bar et al. 2007). Flows of biochemical entities (e.g.

amino acids, fatty acids, glucose, volatile fatty acidsand acetyl-CoA) are regulated by their respectiveconcentration in pools and by various constants (e.g.maximum velocity and substrate a⁄nity). The con-straints on nutrient partitioning no longer refer di-rectly to explicit ¢xed ratios (e.g. minBL:BP). Certainbiochemical transactions can exert control overothers and kinetic constants such as maximumvelo-city (Vmax) can serve to achieve a realistic body com-position (Halas, Dijkstra, Babinszky, Verstegen &Gerrits 2004). Nevertheless, this approach can fail torepresent accurately observed empirical relation-ships. For instance, the predicted relationships be-tween energy intake and PD displayed a curvilinearshape, whereas observed data followed a linear pat-tern in evaluating a model designed to partition diet-ary nutrients in growing pigs (Halas et al. 2004).Assumptions are sometimes made in models basedon saturation kinetics that basically mask a lack ofprecise knowledge and may lead to discrepancies.

A global and fresh perspective ondescribing composition of fish growth

Describing body composition and e¡ects of nutrientdeposition on weight gain is crucial in animal nutri-tion modelling (NRC1998; Kyriazakis1999; de Langeet al. 2003). Partitioning of nutrients in models basedon energy and nutrient metabolism needs to rely onequations that re£ect accurately the limits or bound-aries to body nutrient contents in ¢sh (From & Ras-mussen1984; Lupatsch, Kissil & Sklan 2001; Bureauet al. 2002). Assuming the allocation of dietary nutri-ents responds to an organized biological scheme, var-ious deterministic equations can therefore bedeveloped to describe body composition and deposi-tion of body constituents (e.g. protein, lipid, water,ash and, eventually, amino acids and fatty acids).The dependence of bioenergetic growth models on

accurate estimation of body composition has beenstressed in several ¢sh studies (Cui & Wootton 1988;From & Rasmussen 1989; Jobling 1994; Cui & Xie2000). To date, several bioenergetic models have as-sumed a constant energy content per unit of ¢sh bio-mass or energy retention per unit of time (Winberg1956; Solomon & Bra¢eld1972; Elliott 1976a; Kitchellet al. 1977; Brett 1995). However, body lipid has beenshown to vary with plane of nutrition and age or lifestage in ¢sh and unavoidably a¡ects the energy con-tent of the body (Weatherley & Gill 1987; Azevedoet al. 2004a; Azevedo, Leeson, Cho & Bureau 2004b).

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Cui andWootton (1989) observed that energy contentof ¢sh biomass was a major parameter a¡ecting theaccuracy of bioenergetic models, especially at lowand high feeding levels. Quantitative description ofbody composition in growing ¢sh and of feed utiliza-tion thus becomes essential if systematic errors inbioenergetic models are to be avoided.The equations developed in Dumas, de Lange et al.

(2007) addressed, to a certain extent, the need to con-sider the e¡ect of body composition in bioenergeticmodels by indicating that body energy content asprotein can be easily predicted whereas prediction ofbody energy as lipid requires further modelling stu-dies. Moreover, it is easier and probably more accu-rate to predict BW from BP than from recoveredenergy. The results of Dumas, de Lange et al. (2007)provide realistic bounds for di¡erent body constitu-ents that can help setting correct assumptions andaccurate relationships in nutrient-based models.In models based on nutrient partitioning, see-

mingly arbitrary rules founded on untested assump-tions (Cui & Xie 2000) were set to avoid unrealisticestimates of bodycomposition (see,‘‘Estimating nutri-ent deposition using explicit partitioning rules’’).Most nutrient-based models fail to de¢ne quantita-tively what the realistic bounds are for di¡erent bodyconstituents (e.g. BP). This has led to assumptionsthat can be misleading. For instance, van der Meerand van Dam (1998) applied a minimum BL contentof 1% relative to BW, a value that is unlikely to occurin fed ¢sh (restricted or not) weighing43 g (PhillipsJr, Livingston & Dumas1960; Reinitz1983).Body protein in ¢sh remains at a stringent con-

stant fraction of BW across life stages (Groves 1970;Lupatsch et al.2003; Dumas, de Lange et al.2007). Re-gression analysis is therefore an appropriate tool todescribe the relationship between BW and BP.Although BL and BA are highly correlated with BW,large variability across life stages suggests the needfor more comprehensive models. Rates of PD and LDvaried across life stages and higher values of PD andLD were observed in ¢sh with faster growth rates(Dumas, de Lange et al. 2007). Dietary lipid intakepromoted LD, but not PD, in certain studies on rain-bow trout (Rasmussen, Ostenfeld & McLean 2000;Ge¤ lineau, Bolliet, Corraze & Boujard 2002; Chaiyape-chara, Casten, Hardy & Dong 2003). Therefore, geno-type, life stage and life history and feeding regime(diet composition and ration) stand out as explana-tory variables to be included in future mechanisticmodels to predict better the composition of biomassgain.

Relationships between body constituents and BWacross life stages of ¢sh di¡er from that of other farmanimals. In contrast to beef cattle and pigs, BP ishighly and linearly associated with BWeven beyondmarket size (cf. Dumas, de Lange et al. 2007 withBlack, Cambel, Williams, James & Davies 1986 andNRC 1996). Similarly, BL and BH2O contents appearmore linearly related to BW in ¢sh than in other live-stock species (Black et al.1986; NRC1996). Percentageof dressed carcass of ¢sh is, similar to ducks andsmall game birds, relatively constant, whereas it in-creases with BW up to market size in other animalssuch as pigs, broiler chickens and turkeys (Blacket al. 1986; Swatland 1994; Landgraf, Susanbeth,Knap, Looft, Plastow, Kalm & Roehe 2006; Dumas,de Lange et al. 2007). Unlike other livestock species(e.g. NRC 1998), the relationship between daily PDand BW in ¢sh displays a pattern with no negativeslope in older animals (Dumas, de Lange et al. 2007).In contrast to contemporary perceptions (Elliott

1976b; Shearer1994), evidence suggests that BP var-ies isometrically with BW (i.e. assumes the same rateof change between absolute BP content and BW) in¢sh (Dumas, de Lange et al. 2007).

Concluding remarks: towardsmechanistic modelling of fish growth

Variations in growth trajectory, body compositionand rates of nutrient depositionare, to date, better de-scribed than explained, although certain hypotheseshave been suggested. Explanatory studies are there-fore required to further improve our understandingof the underlying mechanisms responsible for thesevariations.Sexual maturation stands out as a mechanism

likely governing growth trajectory and thus rates ofnutrient deposition. Triggering the maturation pro-cess in salmonids involves a reduction or cessationof feed intake, deterioration of £esh quality, develop-ment of large gonads and secondary sexual charac-ters, and greater catabolism of body protein andbody lipid stores to supply nutrients for new tissuesynthesis (Love 1980; Sargent, Tocher & Bell 2002;Roth, Dorenfeld Jenssen,Magne Jonassen, Foss & Ims-land 2007). Ageing entails a decrease in the e⁄ciencyof nutrient utilization in mammals and ¢sh (Brody1945;Weatherley & Gill1987).It is not yet clear as to what the determinants of

sexual maturation in ¢share. It appears that environ-mental conditions at embryonic and larval stages,

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along with genetics, interact to program the growthtrajectoryand the age at sexual maturity throughoutthe ontogeny of rainbow trout. The e¡ect of incuba-tion temperature on growth rate could a¡ect the tim-ing of sexual maturity ultimately. Evidence suggeststhat temperature during egg incubation could be re-sponsible for muscle growth dynamics before the ¢rstspawning in ¢sh (Johnston, Manthri, Alderson,Smart, Campbell, Nickell, Robertson, Paxton & Burt2003; Albokhadaim, Hammond, Ashton, Simbi,Bayol, Farrington & Stickland 2007; Martell & Kie¡er2007). Intermediate temperature during incubationpromotes hyperplasia and thus growth rate atthe juvenile stage (Fauconneau & Paboeuf 2001;Rowlerson & Veggetti 2001; Steinbacher, Haslett,Obermayer, Marschallinger, Bauer, S�nger & Stoiber2007).Body lipid stores are possibly another determinant

of the onset of sexual maturation. Feeding level andenergy intake, which are known to govern the energystores (i.e. body lipid content) in ¢sh (Azevedo et al.1998; Rasmussen & Ostenfeld 2000;Yamamoto, Shi-ma, Furuita & Suzuki 2002), may induce or delay thetiming of sexual maturation because lipid reserveshave been shown to in£uence maturation in ¢sh(Rowe, Thorpe & Shanks 1991; Silverstein, Shearer,Dickho¡ & Plisetskaya 1998). In contrast, other stu-dies concluded that growth rate has a greater role intriggering the maturation process than body lipidstores (Shearer, Parkins, Gadberry, Beckman & Swan-son 2006; Beckman, Gadberry, Parkins, Cooper & Ar-kush 2007). Here, the e¡ect of growth rate andnutrition may have been confounded and new stu-dies are thus required to elucidate the answer to thisquestion.To sumup, current understanding of causality and

relationships between environmental conditionsduring egg incubation, muscle growth dynamics,plane of nutrition, growth rate and timing of sexualmaturation is fragmentary and has not yet been welldescribed quantitatively. These factors a¡ect growthtrajectory, body composition and nutrient depositionin ¢sh and will have to be described better in order todevelop meaningful mathematical models.To conclude, there is a need to synthesize avail-

able information and develop £exible explanatorymodels. Mechanistic models designed to simulatedi¡erent scenarios are crucial to progress towardsoptimization of feed e⁄ciency and growth, reduc-tion of waste outputs, prevention of sexual matura-tion and, therefore, deterioration of growth rateand £esh quality before ¢sh reachmarket size in or-

der to avoid a decline in pro¢tability of ¢sh cultureoperations. Moreover, future models should be de-veloped to accommodate changes in outcomes ofinterest to the private sector. More than 60 yearsago, animal growth stood as the main concern (Ro-bertson 1923; Wright 1926; Brody 1945). Modelswere thus developed or adapted to predict weightgain with respect to time. Nowadays, the composi-tion of weight gain, yield of particular anatomicalparts and food safety represent new topics of inter-est to the animal production industry because ofthe continually evolving eating habits of consu-mers and increasing public awareness of healthi-ness and environmental sustainability (Young,Northcutt, Buhr, Lyon & Ware 2001; Hocquetteet al. 2005; Torstensen, Bell, Roselund, Hendersen,Gra¡, Tocher, Lie & Sargent 2005; Caswell 2006).Here, mathematical modelling serves as a usefultool to meet current and prospective challenges,extract further information and help orient futureresearch programmes in ¢sh nutrition.

References

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