modeling metabolic adaptations and energy regulation in...

NU32CH03-Hall ARI 18 June 2012 11:10

Modeling MetabolicAdaptations and EnergyRegulation in Humans∗

Kevin D. HallLaboratory of Biological Modeling, National Institute of Diabetes and Digestive andKidney Diseases, National Institutes of Health, Bethesda, Maryland 20892;email: [email protected]

Annu. Rev. Nutr. 2012. 32:35–54

First published online as a Review in Advance onApril 23, 2012

The Annual Review of Nutrition is online atnutr.annualreviews.org

This article’s doi:10.1146/annurev-nutr-071811-150705

0199-9885/12/0821-0035$20.00

∗This is a work of the U.S. Government and isnot subject to copyright protection in theUnited States.

Keywords

mathematical model, body weight, body composition, obesity, energybalance, macronutrient balance

Abstract

Mathematical modeling of human energy regulation and body weightchange has recently reached the level of sophistication required foraccurate predictions. Mathematical models are beginning to pro-vide a quantitative framework for integrating experimental data inhumans and thereby help us better understand the dynamic imbalancesof energy and macronutrients that give rise to changes in body weightand composition. This review provides an overview of the various ap-proaches that have been used to model body weight dynamics and en-ergy regulation in humans, highlights several insights that these modelshave provided, and suggests how mathematical models can serve as aguide for future experimental research.

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Contents

INTRODUCTION . . . . . . . . . . . . . . . . . . 36BODY WEIGHT AND BODY

COMPOSITION . . . . . . . . . . . . . . . . . 37ENERGY BALANCE . . . . . . . . . . . . . . . . 38

Conversion Between Mass andMetabolizable Energy . . . . . . . . . . . 38

Energy Partition Models . . . . . . . . . . . 39MACRONUTRIENT BALANCE . . . . 40

Modeling Macronutrient Balance . . . 41ENERGY EXPENDITURE

DYNAMICS . . . . . . . . . . . . . . . . . . . . . . 42Thermic Effect of Feeding . . . . . . . . . 43Resting Metabolic Rate . . . . . . . . . . . . 43Physical Activity Expenditure . . . . . . . 44Adaptive Thermogenesis and

Metabolic Adaptation . . . . . . . . . . . 44INSIGHTS OBTAINED FROM

MATHEMATICAL MODELS . . . . 45Is a Calorie a Calorie? The Effect

of Dietary Macronutrientson Body Composition . . . . . . . . . . . 45

Behavioral Adaptations to EnergyImbalance . . . . . . . . . . . . . . . . . . . . . . 46

Weight Change Variability Due tothe Uncertainty in BaselineEnergy Requirements . . . . . . . . . . . 47

Greater Weight and Body Fat Gainin Obesity for the SameIncrement in Energy Intake . . . . . 48

CONCLUSIONS AND FUTUREDIRECTIONS . . . . . . . . . . . . . . . . . . . . 49

INTRODUCTION

The emergence of the worldwide obesity epi-demic (91) and its public health consequences(101) emphasizes the critical importance of un-derstanding human energy regulation and itseffect on body weight and composition. In-tense public interest in weight control has ledto a proliferation of diet and exercise prod-ucts, with Americans alone spending tens of bil-lions of dollars per year on these products (14).Media advertisements promise miracle weight

loss cures, and every bookshop has an extensiveselection of best-selling diet books. Weight losshas even become a form of entertainment, witha proliferation of reality television programs de-picting dramatic weight loss interventions.

But despite increased public awareness andinvestment in obesity research, a great deal ofmisinformation and confusion remains aboutthe relationship between weight change, nu-trition, and energy metabolism. Unfortunately,these misconceptions are not limited to thegeneral public but are also widespread amongthe professional nutrition and metabolism com-munities. For example, the so-called 3,500-Calorie-per-pound rule has almost ubiqui-tously been misused to imply that cutting500 kcal/d from the daily diet will result in con-stant weight loss rate of about one pound perweek (44). Such prescriptions have been em-braced by official health and nutrition organi-zations around the world (20, 69–71) and havebeen erroneously used for individual weightloss counseling as well as predicting the po-tential impact of policy changes on populationobesity prevalence (25, 32, 88, 89). Althoughit is generally acknowledged that weight losswill gradually slow over time as a result ofthe decreasing energy requirements, the mag-nitude and time course of the slowing weredifficult to determine since this is a dynamicprocess.

To help address these issues, dynamic math-ematical models of human energy regula-tion and weight change have been developed.Although such modeling began in the 1970s(1, 4, 74, 75), only within the past severalyears have mathematical models become suffi-ciently sophisticated to adequately capture themost relevant physiology, guide experimentalresearch, and make accurate weight change pre-dictions (33, 38, 43, 44, 94, 95). This reviewprovides an overview of different approaches tomathematical modeling of body weight changeand energy regulation in humans, highlightsseveral insights that these models have pro-vided, and suggests how mathematical modelscan serve as a guide for future experimentalresearch.

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BODY WEIGHT AND BODYCOMPOSITION

Body weight is an easily measured quantity thathas often been the primary variable of interestin mathematical models of human energy regu-lation (4, 12, 13, 56, 66). However, body weightis less important than body composition whenit comes to health consequences. Thus, manymodels of human energy regulation have fo-cused on predicting body composition changeat some level of detail (1–3, 33, 38, 39, 43, 44,57, 74, 75, 94, 95, 102, 107).

Figure 1a illustrates the body compositionin terms of body fat and fat-free mass in a typ-ical obese man and lean man. Obesity is char-acterized by a greatly expanded body fat massbut also an increased amount of fat-free mass.Figure 1a also illustrates the chemical com-position of the fat-free mass, with water beingits greatest component. The absolute massesof body protein and bone mineral are also in-creased in obesity. Glycogen and cellular solids,such as potassium and nucleic acids, contributea very small fraction of the fat-free mass.

Body fat, protein, and glycogen comprisethe stored energy of the body, and these storesmust be mobilized when the diet is insufficientto meet the body’s energy requirements.Figure 1b illustrates the composition of thebody in terms of its energy content, withfat stored in adipose tissue providing theoverwhelming majority of the available storedenergy, especially in obesity. Despite dietarycarbohydrate providing the majority of thebody’s energy demands on a daily basis, glyco-gen represents a relatively insignificant store ofenergy (∼2,000 kcal). Body protein representsa substantial amount of energy, but in humansit is not a storage pool in the same sense as adi-pose tissue triglyceride. Rather, body proteinsare functionally important and cannot be de-pleted by a significant fraction without seriouscomplications and death. In contrast, fat storesrepresent a considerable energy reserve, andbody fat can be depleted to very low levels with-out substantial functional impairments (30, 61).

Many mathematical models of human en-ergy regulation have represented body fat aswell as fat-free mass components (1–3, 33, 38,

Figure 1(a) The chemical composition of an obese 120 kg man (left) differs from that of a lean 70 kg man (right),primarily as a result of the increased body fat mass. The fat-free mass (FFM) of the body is mostly water,with a significant contribution of protein and bone mineral. Cellular solids and glycogen make up a verysmall part of the chemical composition of the body. (b) Body energy stores are greatly expanded in the obeseman, primarily as a result of the increased body fat mass.

www.annualreviews.org • Modeling Metabolic Adaptations and Energy Regulation in Humans 37

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39, 43, 44, 57, 74, 75, 94, 95, 102, 107). Arecent model also includes rapid body waterchanges as a component of the fat-free mass thatcan be important contributors to early weightchange (44). Only relatively complex compu-tational models have represented the detailedchemical composition of the body (33, 38).

ENERGY BALANCE

Most mathematical models of human energyregulation make the assumption that weightchange is solely determined by an imbalancebetween dietary energy intake and the energyexpended by the body to maintain life and per-form physical work. The theoretical underpin-ning of this energy balance concept is the firstlaw of thermodynamics, and all energy balancemodels can be mathematically described by thefollowing equation:

ddt

(ρBW ) = EI − EE, (1)

where the left side of the equation is the rateof change of body energy, with BW being thebody weight and ρ being an energy density con-verting between units of metabolizable energyand mass. The right side is the energy imbalancebetween the body’s energy intake rate, EI, andenergy expenditure rate, EE. Any of the termsin the above equation can depend on time, t, aswell as other parameters. A conceptual repre-sentation of energy balance models is depictedin Figure 2.

A popular, but erroneous, application of theenergy balance concept involves the assump-tion that the energy expended by the body

EI EEBW

Figure 2Schematic depiction of the energy balance concept,in which changes in body weight, BW, are assumedto result solely from an imbalance between energyintake, EI, and energy expenditure, EE. Translatingan energy imbalance to a change in weight requiresan assumption about the energy density of theweight change.

remains relatively unchanged when energy in-take deviates from an energy-balanced diet byan amount �EI. In that case, the energy balanceEquation 1 gives the following equation forthe rate of weight change, assuming a constantvalue for ρ:

dBWdt

= �EIρ

. (2)

In other words, the rate of weight change is aconstant and depends only on the magnitudeof the diet change, �EI, and the energy den-sity of the weight change, ρ. With the choice ofρ = 3,500 kcal/lb, this erroneous equation en-capsulates the static 3,500-Calorie-per-poundrule that has been ubiquitously misused topredict weight change (44).

The most serious error of Equation 2 is thatthe energy expenditure rate does not stay con-stant but rather dynamically changes. Even theearliest mathematical model of human weightchange recognized that dynamic changes in en-ergy expenditure must be taken into account(4), and it is curious that the persistent erro-neous use of Equation 2 remains so popular tothe present day. The assumption of a constantρ = 3,500 kcal/lb is also an oversimplificationthat will be dealt with forthwith.

Conversion Between Massand Metabolizable Energy

A simple translation between the energy imbal-ance and the rate of weight change, dBW/dt,occurs only if the energy density of the weightchange, ρ, is a constant parameter. However,as described above, the body is composed ofa variety of chemical constituents with widelyvarying energy densities. For example, fat hasan energy density of about 9.4 kcal/g, whereasprotein and carbohydrate have energy densi-ties of about 4.7 kcal/g and 4.2 kcal/g, respec-tively (64). Other major chemical constituentsof the body, water being the most sizable, havemetabolizable energy densities of zero. Thus,translating a given energy imbalance to a rate ofweight change requires additional assumptions

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about the chemical composition of the weightchange.

The historical basis of the common ρ =3,500 kcal/lb assumption can be traced to theidea that body weight changes are primarilydue to loss or gain of adipose tissue, whichis composed of about 87% triglyceride (110).But we know that body water significantlycontributes to overall weight change, especiallyover the first weeks following a reduced-caloriediet (49). Because body water has no metab-olizable energy content, early weight changeshave an energy density substantially lower than3,500 kcal/lb (48). The mathematical modelsof Hall et al. (38, 44) are the only models toaccount for these early body water changes interms of both intracellular and extracellularfluids.

Longer-term changes in body fat are accom-panied by changes in lean tissue mass whose me-tabolizable energy density is significantly lessthan that of body fat (35). To model theselonger-term body composition changes, Forbes(28, 29) hypothesized that the proportion ofweight change resulting from lean versus fattissue is a nonlinear function of body fat. TheForbes hypothesis has since been extended andvalidated implying that ρ is a nonlinear functionof the body composition (34, 43, 93). Neverthe-less, for small changes of weight from an initialbaseline it is valid to approximate the nonlinearForbes curve with a line, and the resulting slopegives a value for ρ. For moderately overweightand obese individuals, the ρ = 3,500 kcal/lb isa reasonable approximation, but this value sig-nificantly overstates ρ for lean individuals (35).

Energy Partition Models

To explicitly model body composition change,the energy balance Equation 1 can be writtenas a pair of equations for the changes in bothlean tissue, L, and body fat, F:

ρLd Ldt

= P (E I − E E)

ρFd Fdt

= (1 − P )(E I − E E). (3)

EI EE

F L

(1-P) P

Δ

Figure 3Schematic depiction of the energy partition concept,where any energy imbalance, �, is partitionedbetween energy stored in body fat, F, or lean tissues,L. The energy partition ratio, P, specifies thefraction of the energy imbalance to/from lean tissue.The energy densities of changes in L and Fsubstantially differ. EE, energy expenditure;EI, energy intake.

Based on assumptions about the chemical com-position of lean and fat changes, the energydensities have been estimated to be aboutρL = 1.8 kcal/g and ρF = 9.4 kcal/g, respec-tively (35). The energy partition ratio, P, rangesbetween 0 and 1 and determines the proportionof an energy imbalance directed to and fromlean versus fat mass. The energy partition con-cept is illustrated in Figure 3.

The seminal energy partition model ofPayne and Dugdale referred to P as the p-ratio,indicating that this factor determines the pro-portion of an energy imbalance accounted forby body protein changes since they ignored thecontribution of glycogen (74, 75). Payne andDugdale hypothesized that P was a constantparameter whose value could vary between peo-ple and could thereby explain variable interindi-vidual weight change for equivalent changes ofdiet.

Applying the Forbes hypothesis to theenergy partition model reveals that P is anonlinear function of F: P = C/(C+F),where C = 10.4 kg × ρL/ρF , as depicted inFigure 4 (11, 34). The initial value of P canbe quite diverse for individuals with very differ-ent fat mass, in agreement with the Payne andDugdale hypothesis. However, the nonlinear-ity of the Forbes body composition curve sug-gests that P is not a fixed parameter because itdepends continuously on the fat mass, whichcan change considerably with large weightchanges.


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0

0.1

0.2

0.3

1007550250

Body fat (kg)

Ene

rgy

part

ition

rat

io (

P)

Figure 4The Forbes hypothesis for the energy partition ratio, P, as a nonlinear functionof the body fat mass.

Weinsier et al. (102) were the first to use thenonlinear Forbes relationship in an energy par-tition model. A recent energy partition modelalso used the Forbes relationship to describebody fat and lean tissue changes but added theeffect of diet on extracellular fluid mass alongwith changes of glycogen and its associatedintracellular water (44). These modificationssubstantially improved the model’s ability tosimulate rapid changes of total body water thatunderlie most of the initial weight loss fol-lowing the induction of low-calorie, and espe-cially low-carbohydrate and low-sodium, diets(49).

More similar to the original Payne andDugdale hypothesis, Kozusko (57) used anonlinear function of the initial body fat todetermine a constant value of P for eachdynamic simulation. Westerterp et al. (107)considered a more complex algorithm for de-termining P based on body fat as well as energyintake. Thomas et al. (93–95) determined Pby generating several polynomial equations tofit cross-sectional body composition data frommen and women of different racial groups.Despite the increased complexity of their pro-posed equations for calculating P, these authorsshowed that the simple Forbes equation pro-vided more accurate predictions of longitudinalbody composition change for all cases except

those in which the body fat approaches zero(93).

MACRONUTRIENT BALANCE

In his early twentieth-century textbook,Food and the Principles of Dietetics, RobertHutchison (50) compared the human bodyto a steam engine, noting that “The buildingmaterial of food corresponds to the metal ofwhich the engine is constructed, the energy-producers to the fuel which is used to heatthe boiler. Where the body differs from theengine is that it is able to use part of thematerial of its construction for fuel also” (50).A more modern engineering analogy mightbe an automobile that can run on an arbitrarymixture of different fuels (37). Such a flex-fuelvehicle would allow the driver to fill the tankwith whatever fuel was cheaper or more readilyavailable, regardless of what mixture is alreadyin the tank. Although designing a flex-fuelvehicle would be a significant engineeringchallenge, imagine the additional complexityif the vehicle could have no fuel tank. Rather,the vehicle itself must be composed of itsfuel and must continually break down andreconstruct its components. Furthermore,despite the daily turnover of its componentsand fluctuations of fuel delivery, the compo-sition of the vehicle must remain relativelystable and maintain similar performancecharacteristics.

Exactly this remarkable engineering featis accomplished by the human body throughits use of the three dietary macronutrients(carbohydrate, fat, and protein) to both fuelmetabolism and provide substrates for bodyconstituents. These macronutrients are ob-tained from the diet, with about 50% of theenergy derived from carbohydrate, 35% fromfat, and 15% from protein (5). However, theseaverage diet proportions can vary widely fromperson to person and also from day to day.Complex physiological mechanisms maintainnormal functioning of the body despite markedfluctuations of diet quantity and composition.

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FI

EE-FatOxFatOx

F

EI-FI

L

Figure 5Schematic depiction of the simple macronutrientbalance concept, where changes in body fat, F, aredetermined by an imbalance between fat intake, FI,and fat oxidation, FatOx, and changes in lean mass,L, are determined by the remaining energyimbalance. EE, energy expenditure; EI, energyintake.

Modeling Macronutrient Balance

Although the molecular, cellular, and phys-iological mechanisms underlying the regula-tion of human macronutrient metabolism areexceedingly complex, the whole-body systemobeys thermodynamic laws that constrain itsdynamics in ways that make the overall systemamenable to mathematical modeling (11). Forexample, macronutrient imbalances betweendietary intake and metabolic utilization under-lie changes of stored fat, glycogen, and proteinand result in changes in the chemical compo-sition of the body. Thus, models of macronu-trient balance calculate dynamic changes in thechemical composition of the body as a resultof imbalances between intake and utilizationof macronutrients (33, 38). The overall bodyweight change is just the sum of the individualchanges in body constituents.

The simplest mathematical model ofmacronutrient balance and its relationship tobody composition change can be expressed as:

ρLdLdt

= (EI − FI) − (EE − FatOx)

ρFdFdt

= FI − FatOx, (4)

where FI is the energy intake from dietary fatand FatOx is the net energy derived from fatoxidation. The simple macronutrient balancemodel is conceptually illustrated in Figure 5.Both energy balance models and energy par-tition models described by Equations 1 and 3

are special cases of the more general macronu-trient balance model (11, 39). For example,when the simple macronutrient balance modelis constrained so that L and F follow along theForbes nonlinear body composition curve, theresulting model is equivalent to the energy par-tition model described by Equation 3 with theenergy partition ratio P shown in Figure 4 (11).Although it is more general than the energy bal-ance and energy partition models, the macronu-trient balance model also obeys the first law ofthermodynamics.

The equivalence of Equations 3 and 4under the constraint of a specified body com-position relationship provides a quantitative ex-planation of how interactions between diet,energy expenditure, and fat oxidation are con-nected to changes of body composition (11, 39).Hall et al. (39) thereby derived an equationthat accurately predicted the observed changesof body composition and metabolic fuel se-lection during both experimental under- andoverfeeding in adult humans when the mea-sured food intake and total energy expenditurewere provided as inputs to the model. A similarapproach was used to calculate metabolic fuelselection during normal human infant growthand provided the first dynamic picture of howmetabolism adapts in concert with changes ofdiet and energy expenditure to give rise to nor-mal tissue deposition over the first two years oflife (51).

Equation 4 represents a simplified versionof a more comprehensive computational modelof human macronutrient balance and its rela-tionship to body composition change (33, 38).The computational model quantitatively tracksthe metabolism of all three dietary macronu-trients and simulates how diet changes result inadaptations of whole-body energy expenditure,metabolic fuel selection, and alterations inthe major whole-body fluxes contributing tomacronutrient balance. The macronutrientbalance model is conceptually depicted inFigure 6 and mathematically represented bythe following equations describing changes inthe body’s energy stores of glycogen (G), fat


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(F ), and protein (P):

ρCd Gdt

= CI − DNL + GNGp

+ GNGf − G3P − CarbOx

ρFd Fdt

= FI + εd DNL − KUexcr

− (1 − εk)KTG − FatOx

ρPd Pdt

= PI − GNGp − ProtOx

, (5)

where ρC , ρF , and ρP are the energy densitiesof carbohydrate, fat, and protein, respectively.The macronutrient intake rates, CI, FI, and PI,refer to the metabolizable energy intake ratesof dietary carbohydrate, fat, and protein, re-spectively. The rates of gluconeogenesis fromamino acids and glycerol are indicated by GNGp

and GNGf , respectively. The efficiencies of denovo lipogenesis, DNL, and ketogenesis, KTG,are represented by the parameters εd and εk, re-spectively. When the ketogenic rate increases,ketones are excreted in the urine at the rateKUexcr . Some flux of carbohydrates is providedfor the production of glycerol 3-phosphate,G3P, which is used in the synthesis of triglyc-eride. The oxidation rates CarbOx, FatOx, andProtOx, sum to the energy expenditure rate,EE, less the small amount heat produced viaflux through ketogenic and de novo lipogenicpathways. Turnover of glycogen, fat, andprotein and the corresponding energy costs arealso included in the model. To account for bodywater shifts with diets that vary in macronu-trient as well as sodium content, the modelalso tracks intracellular and extracellular fluidchanges.

The main model assumptions are thatchanges of the body’s energy stores are givenby the sum of metabolic fluxes entering thepools minus the fluxes exiting the pools. Hence,the computational model obeys the first lawof thermodynamics, and the most recent ver-sion was developed using published human datafrom over 50 experimental studies and was val-idated by comparing model predictions withthe results of several controlled feeding stud-ies not used for model development (38). Todate, this is the only mathematical model of

GNGf

DNLGNGp

FI

CarbOx

PI

FatOx

F

CI

G P

ProtOx

Figure 6Schematic depiction of the macronutrient balancemodel, in which changes in body glycogen, G, fat,F, and protein, P, are determined by the imbalancesbetween various macronutrient fluxes (arrows).Contributions of exchange fluxes, such asgluconeogenesis, GNG, and de novo lipogenesis,DNL, contribute to imbalances between the intakeand oxidation rates of carbohydrate, fat, and protein.CarbOx, carbohydrate oxidation; CI, carbohydrateintake; FatOx, fat oxidation; FI, fat intake;PI, protein intake; ProtOx, protein oxidation.

human metabolism to consider all three di-etary macronutrients and accurately simulatethe metabolic and body composition changes inresponse to diet and physical activity changes ina wide variety of subject groups.

ENERGY EXPENDITUREDYNAMICS

When diet or physical activities are changed,the body’s metabolic fuel selection and the en-ergy expenditure rate dynamically adapt. Allmathematical models designed to accuratelypredict body weight and composition changemust somehow account for these energy ex-penditure dynamics to avoid the same fatal flawof the static 3,500-Calorie-per-pound weightloss rule. Figure 7 depicts the energy expen-diture components for examples of relativelysedentary lean and obese men. The obese manrequires several hundred additional kcal/d tomaintain his increased weight compared to thelean man. Although this general dependenceof weight on energy expenditure may be cap-tured by a simple model of body weight alone,more realistic models of energy expenditure in-clude its multiple physiological components:the thermic effect of feeding, resting metabolicrate, and physical activity.

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Figure 7The total energy expenditure in relatively sedentaryobese and lean men, comprising the thermic effectof feeding, resting metabolic rate, and the physicalactivity.

Thermic Effect of Feeding

The smallest component of the total energyexpenditure rate in humans is the thermiceffect of feeding (also called diet-induced ther-mogenesis or specific dynamic action), definedas the increase of metabolic rate observed forseveral hours following the ingestion of a meal.The thermic effect of feeding is believed torepresent the energy cost of digestion andabsorption as well as the storage and metabolicfate of dietary macronutrients (106). Althoughthe precise mechanisms underlying the thermiceffect of feeding are not fully understood, thereis a clear dietary macronutrient hierarchyin the magnitude of the metabolic rate in-crease after feeding, with protein causing agreater increment than carbohydrate, whichis greater than that of fat. The computationalmodel of macronutrient balance accounts forthis hierarchy (33, 38), but all other energybalance and energy partition models ignorethe macronutrient effect and assume thatthe thermic effect of feeding is given by anoverall proportion of energy intake, typically∼7%–14%.

Resting Metabolic Rate

The resting metabolic rate (RMR) correspondsto the energy expended by the body whennot performing physical work and is typicallythe largest contribution to the total energyexpenditure rate. Contrary to popular belief,obese people generally have a higher absoluteRMR compared to lean people (Figure 7).Readily available clinical measures (e.g., sex,height, weight, and age) have been used alongwith RMR measurements to generate empir-ical prediction equations with RMR being anincreasing function of body weight, commonlya linear or a power law relationship. Althoughseveral mathematical models have used thissimplified approach to modeling RMR (4, 94,95), it has long been recognized that the maincontributor to the RMR is the fat-free massbecause it comprises the metabolically activetissues of the body (15).

Fat-free mass is elevated in obesity alongwith the increased body fat mass, which alsocontributes to increased RMR in obesity. Thelinear relationship between RMR and fat-freemass is identical in obese and lean people (15,108). This means that the elevated RMR inobesity is generally in line with what is expectedfor the body composition of obese people.

Although fat-free mass and, to a lesser ex-tent, fat mass are good predictors of RMR, suchmodels explain only about 70% of interindi-vidual RMR variability, such that for a givenbody composition the RMR standard deviationis about 300 kcal/d (15, 108). Since there is alarge range of specific metabolic rates amongvarious organs that contribute to the fat-freemass (21), some of this residual RMR variabil-ity may be due to differences in organ masses.Magnetic resonance imaging methodologieshave been used to quantify organ sizes and, us-ing assumptions regarding the organ-specificmetabolic rates, RMR prediction equations thatsum the individual metabolic rates of variousorgans explain about 80% of the RMR vari-ability (31, 67, 68). Thus, increasingly detailedknowledge of body composition may improveRMR predictions.


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The Payne and Dugdale energy partitionmodel was the first to discriminate between“fast” and “slow” lean tissue contributionsto RMR (74, 75). More recently, the com-putational model of macronutrient balance(38) incorporated how changes in the sizes ofvarious organs affect RMR, assuming linearrelationships between changes of fat-free massand various organ sizes based on cross-sectionaldata from 110 men and women with bodymass index (BMI) between 18 and 37 kg/m2

(D. Gallagher, personal communication). Ofcourse, longitudinal organ mass changes withweight gain and loss need not follow thecross-sectional relationships; this possibilityrequires experimental investigation.

Another potentially important contributorto RMR dynamics involves flux changesthrough various energy-requiring metabolicpathways. The major macronutrient fluxesof gluconeogenesis, de novo lipogenesis,triglyceride synthesis, and protein turnoverall require energy, and these flux rates can besignificantly influenced by the energy contentof the diet as well as its composition. Forexample, the breakdown and resynthesis ofbody fat requires eight molecules of adenosinetriphosphate (ATP) per molecule of triglyc-eride (22), and the flux through this pathway isstrongly influenced by dietary carbohydrate viainsulin’s inhibition of lipolysis. Similarly, pro-tein synthesis requires four ATP per peptidebond plus one ATP for amino acid transport(6). Such energy-requiring metabolic fluxeshave been incorporated into computationalmodels of macronutrient balance (33, 38) andmay explain the observed energy cost of tissuedeposition that is especially important duringgrowth and weight gain (36).

Physical Activity Expenditure

The physical activities of humans typicallyinvolve locomotion, and the energy costs aredetermined by the duration and intensity ofphysical activity in proportion to the overallbody weight (98). Thus, obese and lean peoplecan have similar daily energy costs for physical

activity despite obese people typically beingless active. With weight loss, it costs less energyto perform most physical activities, and there-fore the physical activity expenditure typicallydecreases unless the quantity or intensity ofphysical activity increases to compensate.

All previous mathematical models of humanenergy expenditure include the body weighteffect on physical activity expenditure. Somemodels further subdivide physical activity ex-penditure into volitional activities (e.g., exer-cise) and low-intensity spontaneous physicalactivity or nonexercise activity thermogenesis(38, 94, 95).

Adaptive Thermogenesis andMetabolic Adaptation

During active weight loss, both RMR and en-ergy expenditure have been observed to de-crease to an extent greater than expected basedon the measured body weight and compositionchange (18, 19, 46, 60, 83). Furthermore, thisimproved energy efficiency appears to persistonce energy balance is established at a lowerbody weight (82), although the magnitude ofthis persistent effect is smaller than during ac-tive weight loss, and its existence has beencontroversial (26, 104). Conversely, overfeed-ing and weight gain can result in highly vari-able increases of energy expenditure that canbe greater than expected based on the observedweight gain (62, 63). Collectively, these phe-nomena have been called adaptive thermogen-esis (65, 83) or metabolic adaptation (58, 80).In the case of overfeeding, the term luxuskon-sumption has been used to describe adaptiveincreases in energy expenditure (79).

Most mathematical models of energy reg-ulation in humans have ignored such adaptivechanges in energy efficiency, although severalhave attempted to account for such effects (33,38, 41, 43, 44, 56, 57, 72, 94, 95, 107). Theenergy partition models of Hall et al. (41, 43,44) incorporated adaptive thermogenesis as anadditive term that was a linear function of theenergy intake change. In these models, thevalue of the adaptive thermogenesis parameter

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was chosen to match changes in overall energyexpenditure measured before and after approx-imately stable weight loss (43). When activeweight loss is followed by subsequent weightstabilization, the change in energy intakerequired for the weight loss phase is greaterthan that required for weight stabilization atthe reduced weight. Thus, modeling adaptivethermogenesis as a function of the energyintake change has the natural consequenceof increasing its magnitude in situations ofactive weight loss, in agreement with obser-vations (104). A similar approach was used byWesterterp et al. (107), who added an energyexpenditure reduction term to account for theeffect of the degree of energy imbalance onmetabolic rate. In contrast, the energy partitionmodels of Thomas et al. (94, 95) used a constantparameter to model the metabolic adaptationof RMR with weight loss and did not distin-guish between active weight loss and weightstabilization. Kozusko (56, 57) considered aset-point model of adaptive thermogenesis asa function of the body weight itself.

Experimental quantification of the adaptivethermogenesis magnitude depends on thedefinition of the expected values for RMRand total energy expenditure. Typically,cross-sectional regression equations are usedto calculate the expected values, using for RMRand total energy expenditure measurementsderived either from baseline body compositiondata in the same subjects (46, 60, 80) or froma separate group of similar subjects (18, 19).But such expected values for RMR and energyexpenditure ignore the possible changes inorgan size distribution as well as changes influxes through energy-requiring metabolicpathways during over- or underfeeding(described above). Whether such consider-ations can explain the observed changes inenergy efficiency is unclear.

Novotny & Rumpler (72) used a mathe-matical model to investigate the impact of adisproportionate reduction of high-metabolic-rate organs during weight loss and foundthat such changes could potentially explainthe observed reduction of RMR, but the

organ sizes were not measured. Conversely,the computational model of macronutri-ent balance that accounts for alterations ofenergy-requiring metabolic fluxes as well asorgan mass changes required an adaptivethermogenesis model variable to explain theobserved average decrease in both RMR andtotal energy expenditure with weight loss (33,38). The adaptive thermogenesis model wasa linear function of the reduction in energyintake below baseline and was used to suppressthe metabolic rate of all organs as well asreduce the energy expended in spontaneousphysical activity. The mechanistic basis of sucha metabolic adaptation is unclear but may berelated to reduced sympathetic drive or bluntedthyroid activity, possibly as a result of decreasedcirculating leptin (58, 81, 83–85, 103).

Interestingly, the adaptive thermogenesiseffect was not required to accurately simulateoverfeeding and weight gain in the compu-tational model of macronutrient balance (33,38). Similarly, the energy partition modelsof Westerterp et al. (107) and Thomas et al.(94, 95) required a metabolic adaptation param-eter to explain changes of energy expenditurewith weight loss but not weight gain. If correct,such an asymmetry in energy regulationsuggests that the body is neutral to overfeedingand weight gain but actively resists weightloss by improving its energy efficiency duringunderfeeding—much to the dismay of over-weight and obese people wishing to lose weight.

INSIGHTS OBTAINED FROMMATHEMATICAL MODELS

Is a Calorie a Calorie? The Effectof Dietary Macronutrientson Body Composition

A topic of great popular interest is the rela-tive effectiveness of weight loss diets varyingin macronutrient composition (92). While fullycomplying with the laws of thermodynamics,macronutrient balance models allow for thepossibility that body weight may depend onthe diet composition because the energy stored


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per unit mass of carbohydrate, fat, and proteinvaries considerably, especially when account-ing for the intracellular water associated withstored glycogen and protein. Furthermore, di-etary carbohydrates have an impact on renalsodium excretion via insulin (16), which re-sults in concomitant changes in extracellularfluid volume. Therefore, when the composi-tion of the diet is altered, transient changesin macronutrient stores and body fluid shiftswill result in an expected body weight changeeven when the energy content of the diet is heldconstant (99).

A more important consideration is whetherbody fat mass depends on the macronutrientcomposition of the diet. Proponents of low-carbohydrate diets for weight loss emphasizethe ability of dietary carbohydrate to influ-ence adipose tissue fat storage via circulatinginsulin (92). Therefore, it is claimed thatbody fat changes depend primarily on dietarycarbohydrate and not the energy content ofthe diet. Macronutrient balance models allowfor this possibility, but energy balance andenergy partition models assume that “a calorieis a calorie” (8) and therefore all equivalentlyreduced energy diets should lead to identicalbody fat loss regardless of their macronutrientcomposition. However, this assumption isnot an obvious or necessary consequence ofthe first law of thermodynamics because themore general macronutrient balance modelsallow for an effect of diet composition whilealso obeying the law of energy conservation.In fact, to achieve an independence of bodyfat on the macronutrient content of the dietrequires a robust physiological control systemto precisely adapt metabolic fuel selection tothe diet composition (37). Nevertheless, mostinpatient studies with adequately controlleddiets have shown little impact of diet composi-tion on body weight and fat mass changes (55,59, 100, 105, 111, 112), but there are notableexceptions (52, 77, 78). Thus, it remains to bedetermined whether the physiology regulatingmetabolic fuel selection can fully adapt todiet composition over extended time periodswithout resulting in body fat mass changes.

Another argument against the concept that“a calorie is a calorie” is that the diet com-position can have a significant impact on theflux pattern through various energy-requiringmetabolic pathways and may thereby affectthe body’s energy expenditure rate (23, 24).Despite the attractive theoretical possibility ofa significant “metabolic advantage” of one dietover another, simulations using a computa-tional model that includes the contributionsof energy-requiring metabolic fluxes (38) sug-gest that the overall effect of diet compositionon energy expenditure appears to be relativelymodest, especially when dietary protein is un-changed. Specifically, the model predicts thatlarge isocaloric exchanges of dietary carbohy-drate and fat will result in energy expenditurechanges of around 100 kcal/d. These model re-sults agree with experimental observations (86),and therefore the assumption that a “calorie is acalorie” may be a reasonable first approximationover relatively short time periods. However,even small differences in energy expenditureand macronutrient balance can theoreticallylead to significant differences of body weightand composition if the diets are maintained overlong periods. A 100 kcal/d difference in energyexpenditure alone could lead to an initial bodyfat imbalance of about 10 g/d. Using currentbody composition methods, it would require asustained period of about 100 days to detectsuch a difference in body fat. Nevertheless, thispossibility requires further investigation.

Behavioral Adaptationsto Energy Imbalance

Energy imbalance and weight change caninfluence behaviors that directly affect energyintake and/or expenditure. For example,outpatient weight loss interventions gearedto reducing energy intake typically result in aperiod of weight loss that plateaus after aboutsix to eight months, followed by slow weightregain (47, 90). The well-known deficienciesin assessing free-living energy intake (97, 109)make it extraordinarily difficult to interpretsuch results, and mathematical models have

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recently been proposed to address this difficultyby estimating changes of free-living energyintake using longitudinal measurements ofbody weight (38, 40, 96). Such models haveshown that the typical weight loss, plateau, andregain trajectory was likely due to short-livedadherence to the prescribed diet that was pro-gressively relaxed over the first year to returnto the preintervention level (38, 44). Thus, anenergy imbalance resulting in transient weightloss leads to an eventual adaptation of behaviorto return to the original lifestyle.

Another example of a behavioral adapta-tion to energy imbalance is the compensatorychanges in energy intake when a physical ac-tivity program is added. In particular, engagingin many weeks of supervised exercise leads to awide range of individual weight changes, withsome people gaining weight and others havinggreater-than-expected weight loss (7, 10, 53,54). These results imply that volitional phys-ical activity can have a wide range of effects onenergy intake and/or other components of totalenergy expenditure.

Under- and overfeeding may also signifi-cantly influence physical activity expenditure,especially nonvolitional spontaneous physicalactivity (62, 63). This effect is very difficultto model because of the highly variable in-terindividual response. Thomas et al. (94, 95)attempted to account for the average change inspontaneous physical activity in the presenceof this variability. However, the mathematicalequation used to model this effect has twoconstraints on its use. First, the model isnot applicable during prolonged substantialunderfeeding because the equation allows forthe possibility of negative values for the spon-taneous physical activity expenditure that isnot physiological. Second, the model requiresthat changes in any other component of totalenergy expenditure are positively correlated tochanges in spontaneous physical activity. Forexample, increased volitional physical activityor exercise in the Thomas model necessarilyresults in an increased spontaneous physicalactivity. Such an effect would not apply topeople where added exercise results in fatigue

and a corresponding reduction in subsequentspontaneous physical activity. Hence, appli-cation of the model of Thomas et al. requiresa priori knowledge of the highly variableresponse characteristics of the individual.

The highly variable behavioral adaptationsto energy imbalance pose a significant challengefor modeling individual weight changes. Ratherthan attempting to directly simulate these be-havior changes, models that start by assum-ing no effect might be used to help estimatethe magnitude of any behavioral adaptations bymeasuring the difference between the measuredand model-predicted weight change. Further-more, models that can quantitatively integratephysiological data collected during the inter-vention may be used to better characterize be-havioral adaptations in the overall context ofenergy and macronutrient balance.

Weight Change VariabilityDue to the Uncertainty in BaselineEnergy Requirements

Calculating the energy imbalance generatedby a given diet requires knowing the energyrequirement to maintain the baseline bodyweight. Unfortunately, even the specializedand expensive doubly labeled water methodcannot measure the initial energy requirementsof a free-living individual with a precisionbetter than ∼5% (87). The uncertainty of thebaseline energy requirements translates to anexpected interindividual variability of weightchange even if adherence to a prescribed dietis perfect (44). This is a fundamental limitationon a model’s ability to precisely predict thebody weight time course of an individual.For example, assuming a ± 150 kcal/d un-certainty in the initial energy expenditurerequirements, the dotted curves in Figure 8aillustrate the minimum expected weight gainvariability during a study intending to over-feed subjects by 500 kcal/d above baselineenergy requirements. In this example, theenergy partition model of Hall et al. (44)was used to simulate multiple runs of thesame 70-kg “virtual study subject,” assuming


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0

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Initial BW = 120 kg

Figure 8(a) Model-predicted body weight gain (solid curve) and its minimum expectedrange of variability (dotted curves) in response to 500 kcal/d of overfeedingassuming perfect adherence. (b) Greater model-predicted weight gain inresponse to 500 kcal/d of overfeeding in an initially obese 120 kg man (dashedcurve) compared with an initially lean 70 kg man (solid curve). BW, body weight.

perfect diet adherence and only differing in theestimate of the baseline energy expenditure.The variability of weight gain in a real over-feeding study would likely be substantiallygreater given that people may have variablechanges in spontaneous physical activity (62,63) and different physiological responses to thediet as well as different levels of diet adherence.

Greater Weight and Body Fat Gainin Obesity for the Same Incrementin Energy Intake

Another source of weight change variabilityresults from physiological differences that arecaptured by mathematical models of human

energy regulation. For example, Figure 8billustrates the predicted average weight changefor a 500 kcal/d increase of daily energy intakein both a 120 kg man and a 70 kg man using anenergy partition model (44). Although the av-erage weight gain over the first year is compara-ble, it takes longer for the 120 kg man to reachhalf of the maximum weight change comparedwith the 70 kg man, and both men require sev-eral years to begin to stabilize at a new weightplateau. Mechanistically, these different aver-age weight change predictions result from thenonlinear relationship between body fat and thefraction of weight change accounted for by in-creased lean tissue. Thus, persons with higherinitial body fat have a greater fraction of theirweight change attributable to increased bodyfat versus lean tissue. Because body fat con-tributes less than lean tissue to RMR and over-all energy expenditure, the person with higherinitial body fat must gain a greater amount ofweight to achieve a new state of energy balance(37, 43, 44).

This finding of increased weight gain inobese versus lean individuals for the same in-crement of energy intake is shared by all math-ematical models that include nonlinear bodycomposition changes as well as the differen-tial effect of lean versus fat tissue on RMR(37, 43, 44, 102). It is also a feature of modelswhere energy expenditure increases sublinearlywith body weight (4, 94, 95) because the sameweight change at a higher initial body weight re-sults in a smaller change of energy expenditure.In other words, a sublinear increase of energyexpenditure with increased body weight indi-rectly reproduces the nonlinear body composi-tion effect on RMR. This results in increasedpredicted weight gain for the same incrementin energy intake for people with greater initialweight.

These mathematical model predictions thatthe same increment of energy intake leads toincreased weight gain in overweight and obesepeople may help explain the observed increasedpositive skewness of the U.S. population’s BMIdistribution over time (27). In other words, ifthe same average increase in energy intake were

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added to all individuals across BMI categories,then mathematical models predict that moreweight would be gained in people with largerBMIs, thereby pushing out the upper tail of theBMI distribution with time. Whether this ef-fect is sufficient to explain the evolving shape ofthe BMI distribution is an intriguing question.

CONCLUSIONS AND FUTUREDIRECTIONS

In the physical sciences, there is a long historyof developing mathematical models that quan-titatively describe past data and predict the re-sults of key new experiments. Such quantitativemodels have been relatively rare in the biomedi-cal sciences, possibly because biological systemsare highly complex, and defining the importantvariables is often difficult. Fortunately, modelsof human energy regulation and macronutri-ent metabolism are constrained by conservationprinciples and knowledge of the main metabolicpathways that contribute to whole-bodyimbalances.

A guiding principle when developing amathematical model is to target its complex-ity to the class of phenomena that the modelis intended to address. Mathematical models ofhuman energy regulation and body weight dy-namics have ranged in complexity from detailedcomputational models that accurately simulatedynamic changes in macronutrient metabolismand body composition to overly simplified staticweight loss models that fail to capture the mostbasic features of body weight dynamics. Mostother models fall between these extremes andhave been used to provide important insightsregarding human energy regulation and bodyweight change.

Despite significant progress, much work re-mains for improving and expanding the existingmathematical models of human energy regu-lation and body weight change. For example,

the anatomical location of body fat, especiallyin visceral adipose tissue, has profound clinicalimportance, and mathematical models haveonly just begun to capture the relationship be-tween body fat changes in various depots duringweight loss and gain (42, 45). Furthermore,even the most detailed computational models ofmacronutrient metabolism implicitly representthe effect of hormones such as insulin, but anexplicit representation of organ systems alongwith concentrations of hormones and metabo-lites would be desirable—especially on shortertime scales so that the response to individualmeals could be simulated (17, 73). Conversely,capturing the dynamics of energy regulationand body composition change during aging aswell as during childhood and adolescent growth(9) will require extending current mathematicalmodels to operate on much longer time scales.

An editorial describing the future ofbiomedical research remarked that “formula-tion of a mathematical model is the ultimatetest of understanding . . . If the model repro-duces the behavior of the system under a rangeof conditions and predicts the consequencesof . . . modifications in any component, one canbe relatively confident about understandingthe system” (76). As highlighted in this review,much progress has been made on integrat-ing past research on human nutrition andmetabolism into self-consistent and predictivemathematical models of energy regulation andbody composition change. Such models canhighlight knowledge gaps, integrate metabolicdata within a broader context of knowledge,and make testable predictions, thereby helpingdesign new experiments. Indeed, the bestmathematical models will never replace exper-imental research but rather will be used to helpdesign the key experiments that, in turn, willhelp improve the mathematical models and ourunderstanding of the human energy regulationsystem.

DISCLOSURE STATEMENT

The author is not aware of any affiliations, memberships, funding, or financial holdings that mightbe perceived as affecting the objectivity of this review.


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ACKNOWLEDGMENTS

The author thanks Caron C. Chow for his critical reading of the manuscript and the suggestedimprovements. The author’s research was supported by the Intramural Research Program of theNIH, National Institute of Diabetes and Digestive and Kidney Diseases.

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Annual Review ofNutrition

Volume 32, 2012Contents

An Unexpected Life in NutritionMalden C. Nesheim � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 1

Endoplasmic Reticulum Stress in Nonalcoholic Fatty Liver DiseaseMichael J. Pagliassotti � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �17

Modeling Metabolic Adaptations and Energy Regulation in HumansKevin D. Hall � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �35

Hypomagnesemia and Inflammation: Clinical and Basic AspectsWilliam B. Weglicki � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �55

Selenoproteins and Cancer PreventionCindy D. Davis, Petra A. Tsuji, and John A. Milner � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �73

The Role of Vitamin D in Pregnancy and Lactation:Insights from Animal Models and Clinical StudiesChristopher S. Kovacs � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �97

Vitamin A Metabolism in Rod and Cone Visual CyclesJohn C. Saari � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 125

Lipoprotein Lipase in the Brain and Nervous SystemHong Wang and Robert H. Eckel � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 147

New Roles of HDL in Inflammation and HematopoiesisXuewei Zhu and John S. Parks � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 161

Nutritional Metabolomics: Progress in Addressing Complexityin Diet and HealthDean P. Jones, Youngja Park, and Thomas R. Ziegler � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 183

Resolvins: Anti-Inflammatory and Proresolving Mediators Derivedfrom Omega-3 Polyunsaturated Fatty AcidsMichael J. Zhang and Matthew Spite � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 203

Visfatin/NAMPT: A Multifaceted Molecule with Diverse Rolesin Physiology and PathophysiologyTuva B. Dahl, Sverre Holm, Pal Aukrust, and Bente Halvorsen � � � � � � � � � � � � � � � � � � � � � � � 229

Gene-Environment Interactions in the Development of Type 2Diabetes: Recent Progress and Continuing ChallengesMarilyn C. Cornelis and Frank B. Hu � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 245

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NU32-FrontMatter ARI 26 June 2012 13:26

Mechanisms of Inflammatory Responses in Obese Adipose TissueShengyi Sun, Yewei Ji, Sander Kersten, and Ling Qi � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 261

Bone Metabolism in Obesity and Weight LossSue A. Shapses and Deeptha Sukumar � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 287

Obesity in Cancer SurvivalNiyati Parekh, Urmila Chandran, and Elisa V. Bandera � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 311

Inflammation in Alcoholic Liver DiseaseH. Joe Wang, Bin Gao, Samir Zakhari, and Laura E. Nagy � � � � � � � � � � � � � � � � � � � � � � � � � � � 343

Lessons Learned from Randomized Clinical Trials of MicronutrientSupplementation for Cancer PreventionSusan T. Mayne, Leah M. Ferrucci, and Brenda Cartmel � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 369

Population-Level Intervention Strategies and Examples for ObesityPrevention in ChildrenJennifer L. Foltz, Ashleigh L. May, Brook Belay, Allison J. Nihiser,

Carrie A. Dooyema, and Heidi M. Blanck � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 391

Type 2 Diabetes in Asians: Prevalence, Risk Factors, and Effectivenessof Behavioral Intervention at Individual and Population LevelsMary Beth Weber, Reena Oza-Frank, Lisa R. Staimez, Mohammed K. Ali,

and K.M. Venkat Narayan � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 417

Indexes

Cumulative Index of Contributing Authors, Volumes 28–32 � � � � � � � � � � � � � � � � � � � � � � � � � � � 441

Cumulative Index of Chapter Titles, Volumes 28–32 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 444

Errata

An online log of corrections to Annual Review of Nutrition articles may be found athttp://nutr.annualreviews.org/errata.shtml

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