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Modeling weight-loss maintenance to help prevent body weight

regain1–3

Kevin D Hall and Peter N Jordan

ABSTRACT

Background:Lifestyleinterventioncansuccessfullyinduceweight

lossinobesepersons,atleasttemporarily.However,therecurrently

is no way to quantitatively estimate the changes of diet or physical

activity required to prevent weight regain. Such a tool would be

helpful for goal-setting, because obese patients and their physicians

could assess at the outset of an intervention whether long-term ad-

herence to the calculated lifestyle change is realistic.

Objective: We aimed to calculate the expected change of steady-

state body weight arising from a given change in dietary energy

intakeand,conversely,tocalculatethemodificationofenergyintake

required to maintain a particular body-weight change.

Design: We developed a mathematical model using data from 8

longitudinal weight-loss studies representing 157 subjects with ini-

tialbodyweightsrangingfrom68to160kgandstableweightlosses

between 7 and 54 kg.

Results:Modelcalculationscloselymatchedthechangedata(R2?

0.83, ?2? 2.1, P ? 0.01 for weight changes; R2? 0.91, ?2? 0.87,

P ? 0.0004 for energy intake changes). Our model performed sig-

nificantly better than the previous models for which ?2values were

10-foldthoseofourmodel.Themodelalsoaccuratelypredictedthe

proportion of weight change resulting from the loss of body fat (R2

? 0.90).

Conclusions: Our model provides realistic calculations of body-

weightchangeandofthedietarymodificationsrequiredforweight-

loss maintenance. Because the model was implemented by using

standard spreadsheet software, it can be widely used by physicians

and weight-management professionals.

88:1495–503.

Am J Clin Nutr 2008;

INTRODUCTION

The increasing prevalence of obesity poses a serious health

concern (1). Whereas lifestyle interventions can result in signif-

icantweightlossinobesepatients,weightregainisverycommon

(2,3).Therefore,ratherthanfocusingexclusivelyonweightloss,

several investigators have emphasized the importance of main-

tainingbodyweightatalowerlevelandpreventingweightregain

(4–7). Data from the National Weight Control Registry have

beenusedtogleanusefulinsightsregardingthestrategiesusedby

persons who have successfully maintained significant weight

change over extended periods (6, 7). However, there currently

are no available quantitative tools to estimate, at the outset of

obesity treatment, the lifestyle changes required to maintain a

specific weight-loss goal. In other words, the question remains:

If a patient wishes to change his or her body weight by a certain

amount (?BW), how would his or her diet or physical activity

have to change to maintain the goal weight? A quantitative an-

swer to this question would be helpful for goal-setting, because

both patient and physician could assess whether long-term ad-

herence to the calculated lifestyle change is a realistic proposi-

tion. Such a calculation is not currently possible.

In this report, we propose a mathematical model for calculat-

ingthechangesindietaryintakeandphysicalactivityrequiredto

maintain a given body-weight change and to prevent weight

regain. To facilitate use of the model by physicians and weight-

managementprofessionals,themodelwasimplementedbyusing

standard spreadsheet software that can be downloaded (see

Spreadsheet files under “Supplemental data” in the current on-

lineissue;thespreadsheetfilesandanonlineversionofthemodel

are available at http://www2.niddk.nih.gov/NIDDKLabs/LBM/

lbmHall.htm). We developed our model by using longitudinal

weight-change data from studies that measured energy expendi-

ture (EE) and body composition during periods of weight stabil-

ity both before and after weight loss (8–15). We compared the

performance of our model with that of previous mathematical

models(16–18)regardingthecapacitytomatchthesteady-state

weight-change data, and we found that our model was superior

andcouldproviderealisticestimatesofboththemagnitudeofthe

weight change and the changes in dietary energy intake (EI)

required to prevent weight regain.

MATERIALS AND METHODS

Proposed model of steady-state body-weight change

Wedevelopedasimplemodelofthesteady-stateEErateofthe

bodyasafunctionofbodycomposition,EI,andphysicalactivity:

EE ? K ? ?EI ? ?FFMFFM ? ?FMFM ? ??FFM ? FM?

(1)

where K is a constant, ?FFM? 22 kcal ? kg?1? d?1, and ?FM?

3.2 kcal ? kg?1? d?1are the regression coefficients for resting

metabolic rate versus fat-free mass (FFM) and fat mass (FM),

1From the Laboratory of Biological Modeling, National Institute of Dia-

betes and Digestive and Kidney Diseases, National Institutes of Health,

Bethesda, MD.

2Supported by the Intramural Research Program of the National Institute

ofDiabetesandDigestiveandKidneyDiseases,NationalInstitutesofHealth.

3Reprints not available. Address correspondence to KD Hall, NIDDK/

NIH, 12A South Drive, Room 4007, Bethesda, MD 20892-5621. E-mail:

kevinh@niddk.nih.gov.

Received April 28, 2008.Accepted for publication August 28, 2008.

doi: 10.3945/ajcn.2008.26333.

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respectively(19).Weassumedthattheenergycostofmostphys-

icalactivitiesisproportionaltobodyweightandisthusspecified

by the parameter ?. The parameter ? accounts for the thermic

effectoffeedingaswellasanyadaptationsoftheEEratebeyond

that predicted by body-composition change alone (20). Because

there is significant debate as to whether such adaptations occur

withweightloss(20,21),thenumericalvalueoftheparameter?

was determined by the best fit to the weight-change data (de-

scribed in the final paragraph of the Materials and Methods

section).

When the body weight is maintained at a constant reduced

level, the EE rate equals the EI rate. Therefore, if the dietary

intake changes by ?EI, and the physical activity changes by ??,

the following equation is satisfied at the new steady state:

?EI ? ??EI ? ??FFM? ?init? ????FFM

? ??FM? ?init? ????FM ? BWinit??

(2)

whereBWinitistheinitialbodyweight,and?FMand?FFMare

the changes in body fat and FFM, respectively. (Note that the

constant, K, no longer appears in equation 2, so we need not

specifyitsvalue.)Usingthefactthat?BW??FFM??FM,the

following equation holds for the change in FFM:

?FFM ? ??1 ? ???EI ? BWinit??

? ??FM? ?init? ????BW?/??FFM? ?FM?

(3)

Our previously published modification of the classic Forbes

equation of body-composition change (22) predicts that the

change in FFM is given by the following nonlinear function:

?FFM ? ?BW ? FMinit? C

? W?

FMinit

C

exp?

FMinit? ?BW

C

??

(4)

where FMinitis the initial body FM, C ? 10.4 kg is the constant

providing the best fit to the empirical Forbes body-composition

curve (23), and W is the Lambert W function (24). Equations 3

and4canbesolvedfortheexpectedchangeinsteady-statebody

weight, as shown in the following equation:

?BW ??1 ? ???EI ? BWinit?? ? ??FFM? ?FFM?FMinit

??FM? ?init? ???

?

C??FFM? ?FM?

??FM? ?init? ???

? W?

exp?

??FM? ?init? ???FMinit

C??FFM? ?init? ???

C??FFM? ?init? ????

exp?

??FM? ?init? ???FMinit

?1 ? ???EI ? BWinit??

C??FFM? ?init? ????

(5)

Equation5allowsustocalculatethechangeinsteady-statebody

weight, given information about the initial body weight and FM

and about the changes in dietary EI and physical activity. We

recognize that most readers cannot easily calculate expected

weightchangeresultsbyusingequation5becauseoftheappear-

ance of the nonelementary Lambert W function. To address this

issue,wehaveprovidedstandardspreadsheetfilesthatwillallow

readers to examine the predicted changes in steady-state body

weightasafunctionofthemodelparameters(seeSpreadsheetfiles

under “Supplemental data” in the current online issue; the spread-

sheetfilesandanonlineversionofthemodelareavailableathttp://

www2.niddk.nih.gov/NIDDKLabs/LBM/lbmHall.htm).

Previous models of steady-state body-weight change

Previous mathematical models have been proposed to calcu-

late ?BW resulting from a specified change of diet or physical

activity(16–18).Thefirstmodel,byChristiansenetal(16),was

given by the following equation:

k??

where k ? 13.5 kcal ? kg?1? d?1for men and k ? 11.6

kcal ? kg?1? d?1for women, and PAL ? the physical activity

level,definedastheratiooftotalEEratetotherestingmetabolicrate.

The second model, by Kozusko (18), expressed the predicted

body-weight change according to the following equation:

?BW ????1 ? ??2? 4??1 ? ?EI/EIinit?

?BW ?1

EI

PAL?

(6)

? ? ? 1?BWinit/2?

(7)

where ? was given by the following equation:

2tanh?

? ?1

3

2?

1

2?FMinit

BWinit???

FMinit

BWinit?1 ?FMinit

BWinit???

?1

2

(8)

In the third model under consideration, Heymsfield et al (17)

used the Institute of Medicine of the National Academies of

Science (IOM-NAS) equations to estimate the expected weight

change for a given dietary intervention, but those investigators

did not allow for changes in physical activity. We accounted for

physicalactivitychangesbysolvingtheIOM-NASequationsfor

the expected body-weight change as follows:

?BW ? ?EI/?K1PAfinal? ? ?BWinit? K2h/K1??PA/PAfinal ?9)

where h is height in meters, K1? 14.2 kcal ? kg?1? d?1and K2

?

503kcal ? m?1? d?1

for

kcal ? kg?1? d?1and K2? 660.7 kcal ? m?1? d?1for women

(25). PA stands for a dimensionless physical activity parameter

related to PAL, but it takes on discrete values within a range of

PAL values (17, 25).

men,and

K1

?

10.9

Proposed model of diet and physical activity changes

required to prevent weight regain

Our main goal was to calculate the changes in dietary intake

and physical activity required to maintain a specified body-

weight change. The following rearrangement of equation 5 ex-

presses the EI change required for a specified body weight

change and physical activity change:

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?EI ? BWinit??/?1 ? ?? ? C2??FFM? ?FM?FMinit/?1 ? ??

? C2?BW??FFM? ?init? ???/?1 ? ??

? W?FMinitC exp ??1 ? FMinit? ? ?BW]?

? C??FFM? ?FM?/?1 ? ??

(10)

We have provided a spreadsheet file for this calculation (see

Spreadsheet files under “Supplemental data” in the current on-

lineissue;thespreadsheetfilesandanonlineversionofthemodel

are available at http://www2.niddk.nih.gov/NIDDKLabs/LBM/

lbmHall.htm).WecomparedthepredictedEIchangefromequa-

tion10withthechangeinthesteady-statetotalEEratemeasured

in the longitudinal weight-loss studies described below.

Comparison of model results with longitudinal weight-

change data

We compared the various model calculations for ?BW with

longitudinalweight-changedatathatwereachievedoverperiods

ofweightlossrangingfrom1to14mo.Thewhole-bodyEErates

wereaccuratelymeasuredduringperiodsofweightstabilityboth

before and after weight loss either by using the doubly labeled

water method (8, 9, 12–15) or by titrating the food intake in an

in-patient setting to achieve weight stability for ?2 wk (10, 11).

Because the measurements were made during periods of weight

stability,weassumedthattheEErateequaledthedietaryEIrate.

Therefore,themeasuredchangesofEEafterweightlossgavean

accurate estimate of the dietary EI changes required to maintain

the measured body-weight change and prevent regain.

To provide all of the parameter values used in the above cal-

culations, we required measurements of the initial body FM and

thePAL.Wefound8longitudinalweight-lossstudies,represent-

ing 157 patients, that satisfied these criteria, in which a wide

range of weight losses were induced either by bariatric surgery

(9, 13, 15) or by diet restriction (8, 10–12, 14). The parameters

for each study are shown in Table 1.

Weevaluatedeachmodelincomparisontotheweight-change

data by using the following measures. First, we calculated the

Pearson correlation coefficient (r) between the model calcula-

tions and the data according to the following equation:

r ? ?N ? 1?? 1?

i ? 1

N

?yi? ?y???mi? ?m??

(11)

where yiwas the model calculation corresponding to the mea-

suredvaluemiforeachgroupandthebrackets(?and?)denote

the mean value. Second, we calculated the chi-square value,

which is a weighted measure of the distance between the model

calculations and the data according to the following equation:

?2??

i ? 1

N

?yi? mi?2/?i

2

(12)

where ?iwas the SD of the measurement. The chi-square value

was our primary measure of model fit to the data. Using the

incomplete gamma function, we computed the probability that

the chi-square for a correct model should be less than the chi-

square calculated for the model (26). Finally, we calculated the

coefficientofdeterminationaccordingtothefollowingequation:

TABLE 1

Data and parameters used for model comparisons in Figures 1 and 21

Amatruda

et al (8)

Das et al (9)

Keys (10)

Leibel et al (11)

Martin et

al (12)

van Gemert

et al (13)

Weinsier

et al (14)

Westerterp

et al (15)

Subjects (n)

18

30

12

11

9

10

12

8

32

5

BWinit(kg)

83.7 ? 8.52

139.5 ? 35.3

67.5 ? 5.1

70.5 ? 11.7

132.1 ? 26.9

124.8 ? 29.6

82.3 ? 6.6

130.1 ? 17.5

78.8 ? 6.5

158.6 ? 33.7

Finit(kg)

34.3 ? 3.3

71.6 ? 23

9 ? 4.7

17.5 ? 12.6

68 ? 19.8

64.4 ? 24.8

26.1 ? 3.1

68.3 ? 11.7

30.7 ? 4.8

74 ? 29

?init(kcal ? kg?1? d?1)

11.2

6.9

24.5

9.0

5.5

6.7

12.7

7.0

6.3

6.1

?BW (kg)

?22 ? 10

?53.4 ? 22.2

?15.8 ? 6.1

?6.8 ? 15.4

?17.8 ? 34.4

?29.2 ? 37.1

?12 ? 2.1

?46.3 ? 21.4

?12.9 ? 8.6

?53.9 ? 17.8

?EI (kcal/d)

?231 ? 668

?862 ? 598

?2018 ? 252

?428 ? 664

?551 ? 853

?886 ? 891

?417 ? 687

?861 ? 609

?114 ? 522

?908 ? 477

?? (kcal ? kg?1? d?1)

3.5

1.0

?16.5

?1.7

?1.0

?2.1

?1.3

1.2

2.0

2.4

1BWinit,initialbodyweight;FMinit,initialfat;?init,initialbodycompositionandinitialphysicalactivityparameter;?,change;EE,totalenergyexpenditure;RMR,restingmetabolicrate;?FFMand?FM,regression

coefficients for resting metabolic rate versus fat-free mass (FFM) and fat mass (FM), respectively; C, the constant providing the best fit to the empirical Forbes body-composition curve. Physical activity was

calculated by using ? ? (0.9 ? EE – RMR)/BW, and all models used the following parameters: ?FFM? 22 kcal ? kg?1? d?1, ?FM? 3.2 kcal ? kg?1? d?1, C ? 10.4 kg, and ? ? 0.24

2x ? ? SD (all such values).

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R2? 1 ??

i

(yi? mi)2??

i

(mi? ?m?)2

(13)

Because the models are generally nonlinear, the coefficient of

determination (R2) is not identical to the correlation coefficient

squared.

The model parameter ? was determined by using a weighted

least-squares optimization procedure to find the best fit value of

? that corresponded to the weight-change data listed in Table 1

(26). To calculate the uncertainty of our estimate of ?, we per-

formed a Monte-Carlo analysis in which we re-fit ? with the use

of5000setsofsynthesizeddatawiththesamestatisticsasthereal

data. In other words, each synthesized data-point was randomly

selected from a normal distribution with a mean and SD corre-

sponding to a real data-point. The uncertainty of the model pa-

rameter ? was then calculated as the SD of the 5000 best-fit ?

parameters from the Monte-Carlo simulations. All model calcu-

lations and statistical comparisons were performed with the use

of MATLAB software (version R2008a; The MathWorks Inc,

Natick, MA).

RESULTS

Comparison of results from the proposed model with

longitudinal weight-change data

A comparison of results from our model with the measured

changesofsteady-statebodyweightgiventhemeasuredchanges

of EI and physical activity are plotted in Figure 1. For our

proposed model, the best-fit value for the parameter ? was 0.24

? 0.13. Statistical evaluation of our model showed a Pearson

correlation coefficient of 0.983 between our model’s body-

weight calculations and the measured values. The coefficient of

determination was 0.83, which indicated that our model de-

scribed 83% of the data variability. The chi-square value was

2.08 and, according to an evaluation of the incomplete gamma

function, the probability was ?0.01 that the chi-square for a

correct model should be less than the chi-square calculated for

our model (26).

Our model results are compared (Figure 1B) with the mea-

sured changes in EI rate required to maintain the average mea-

suredweightlossobservedineachstudy(R2?0.91).Inthiscase,

the chi-square value was 0.87, and the incomplete gamma func-

tion gave a probability of 3.1 ? 10?4. Therefore, our model

provided an excellent match to the data over a wide range of

changes in body weight (Figure 1A) and dietary intake (Figure

1B), which suggests that our model successfully captured the

salient physiologic changes.

Thecomparisonofthepredictedandthemeasuredfatfraction

oftheweightlossisplottedinFigure2;and(R2?0.90)indicates

that our model accurately described the observed body-

compositionchanges.Therewasawiderangeofobservedbody-

compositionchanges,whichwereduetothedifferentinitialbody

compositionsandtherangeofweightlossesinthevariousstudies

(22).Forexample,theinitiallyleansubjectsfromtheMinnesota

starvation experiment of Keys et al (10) mostly lost FFM, and

only34%oftheirweightlosswasaccountedforbyalossofbody

fat.Incontrast,bodyfatlossaccountedfor84%oftheweightloss

in the in the obese subjects studied by Leibel et al (11) who

reduced their weight by 10%.

Greater steady-state weight change is predicted for

persons with higher initial body fat

The fact that our model accounts for variations in body-

composition change leads to an interesting result when compar-

ing the calculated steady-state weight changes in people with

different initial body weights. The predicted steady-state body-

weight changes as a function of the diet change for 2 example

subjects are illustrated in Figure 3: the solid line represents the

first subject, who corresponds to an average participant from a

study by Martin et al (12), and the dashed line represents the

secondobesesubject,whocorrespondstoanaverageparticipant

-70

-60

-50

-40

-30

-20

-10

0

-70-60 -50-40-30-20-100

Predicted ∆BW (kg)

Measured ∆BW (kg)

Amatruda

Das

Keys

Leibel

Martin

van Gemert

Weinsier

Westerterp

Identity

-2500

-2000

-1500

-1000

-500

0

-2500-2000-1500-1000-5000

Predicted ∆EI (kcal/d)

Measured ∆

(kcal/d)

EI

Amatruda

Das

Keys

Leibel

Martin

van Gemert

Weinsier

Westerterp

Identity

A

B

FIGURE 1. A: Predicted versus measured changes in steady-state body

weight (?BW) resulting from the measured changes in energy intake and

physical activity (R2? 0.83). B: Predicted versus measured changes in

dietary energy intake (?EI) required to achieve the observed changes in

steady-state body weight (R2? 0.91). The dotted lines are lines of identity,

and error bars are SEs. BW, body weight; ?, change.

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inastudybyDasetal(9).Fortheobesesubjectstartingatabody

weightof139.5kg,a400-kcal/dreductionindietaryintake(with

no change of physical activity) resulted in a predicted body-

weight loss of 23.4 kg, whereas the overweight subject whose

initialweightwas82.3kglostonly13.7kgforthesamereduction

indietaryEI.Thedottedcurvesoneithersideofeachlineindicate

therangeofpredictedweightchangeswhenthemodelparameter

? was swept by ? 1 SD.

Practical calculation of dietary changes required to

prevent weight regain

Our proposed model calculations for sedentary 40-y-old men

and women are shown in Table 2 and Table 3, respectively,

under the assumption that these persons’ physical activity level

does not change with weight loss (ie, ?? ? 0). Separate calcu-

lations were required for men and women because women have

ahigheramountofbodyfatthandomenofsimilarweight.Thus,

whereas sex was not an explicit variable in our equations, spec-

ification of the initial body composition at a given body weight

differedbetweenmenandwomen.Theinitialbodycomposition

and the initial physical activity parameter (?init) were estimated

byusingtheregressionequationsofJacksonetal(27)andMifflin

et al (28), respectively, with an assumed initial PAL of 1.4.

Whereas these calculations provided a useful estimate of the

expected steady-state weight change as a function of EI changes

forsedentarypeople,wehavealsoprovidedspreadsheetfilesfor

calculationsofchangesinphysicalactivityalongwithchangesin

EI (for these files, see Spreadsheet files under “Supplemental

data” in the current online issue; the spreadsheet files and an

online version of the model are available at http://www2.niddk.

nih.gov/NIDDKLabs/LBM/lbmHall.htm). These spreadsheet files

can also be used to investigate the sensitivity of the model calcula-

tionstochangesinparameters,muchaswereillustratedinFigure3,

where we examined a range of values for the parameter ?.

Comparison with previous steady-state weight-change

models

The residuals between the various model calculations and the

measured body-weight changes for each group of subjects are

compared in Figure 4. Our proposed model performed better

thantheothermodelsbecausetheresidualbody-weightchanges

wereclosertozeroandlessspreadoutthanwerethoseintheother

models.WhereasthemodelbyChristiansenetal(16)resultedin

a respectable correlation coefficient of 0.855 and an R2of 0.64,

it had a high chi-square value of 20.9 and a resulting high prob-

ability of 0.98 that the chi-square for a correct model would be

lessthanthechi-squarecalculatedfortheirmodel.Themodelby

Kozusko (18) resulted in a correlation coefficient of 0.722 but

had a poor R2(ie, 0.067) and a high chi-square value of 19.1,

which gave a probability of 0.98 that the chi-square for a correct

modelwouldbelessthanthechi-squarecalculatedforhismodel.

The predictions based on the IOM-NAS equations (25) resulted

inacorrelationcoefficientof0.60,butaverypoorR2(ie,?1.79),

which indicated that the mean of the data provided a better de-

scriptionofthebodyweightchangethanthepredictionsfromthe

IOM-NAS equations. Correspondingly, the chi-square value of

184gaveaprobabilityof1thatthechi-squareforacorrectmodel

would be less than the chi-square calculated for the IOM-NAS

equations.

DISCUSSION

Current methods estimate body weight loss on the basis of

simplified rules such as “3500 kcal ? 1 pound” (5, 29). Because

suchrulesdonotaccountforchangesinEEwithweightloss,they

donotallowforstabilizationofbodyweightatanewsteadystate

despite continued adherence to a lifestyle intervention. Unfortu-

nately,thiscrudeapproximationcurrentlyistheonlywidespread

0

0.2

0.4

0.6

0.8

1

00.2 0.40.60.81

Predicted ∆FM/ ∆BW

Measured ∆

∆

FM/

BW

Amatruda

Das

Keys

Leibel

Martin

van Gemert

Weinsier

Westerterp

Identity

FIGURE 2. Predicted versus measured changes (?) in the fat fraction

(FM) of body weight (BW) loss (?FM/?BW) (R2? 0.90). The dotted lines

are lines of identity, and error bars are SEs.

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

-800-700-600-500-400 -300-200-1000

∆ ( EI kcal/d)

∆BW (kg)

FIGURE3.Predicteddiet-inducedweightlosses(?BW)asafunctionof

dietary intake changes (?EI) for 2 subjects with very different initial body

weightscorrespondingtoaveragedatafromMartinetal(12)(—,initialBW

? 82.3 kg) and Das et al (9) (- - -, initial BW ? 139.5 kg). ?, change; BW,

bodyweight.Thedottedlinesindicatethesensitivityofthemodelpredictions

when the parameter ? was changed by ? 1 SD.

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clinical tool available for predicting weight change, and any

improvement represents a significant step forward.

Weaddressedthisproblembyusingtheavailablelongitudinal

weight-loss data to develop a mathematical model that accounts

for reduced energy requirements at the lower steady-state body

weight. Mathematical modeling of human body-weight change

hasbeenattemptedmanytimesandinvolvessolvingdifferential

equations that model the rate of body-weight change (16, 18,

30–43). However, these mathematical models typically require

specialized software to numerically approximate the equation

solutions via computer simulation. In contrast, we solved our

nonlinear model equations at the new steady state and provided

standard spreadsheet files so that readers can easily perform the

calculations for themselves (see Spreadsheet files under “Sup-

plemental data” in the current online issue; the spreadsheet files

and an online version of the model are available at http://www2.

niddk.nih.gov/NIDDKLabs/LBM/lbmHall.htm).

Previousmathematicalmodelsofhumanbody-weightchange

have been used to explicitly calculate the steady-state body-

weightchangeforprescribedchangesindietorphysicalactivity

(16–18). In the present study, we showed that, whereas 2 of the

previous steady-state models provided a reasonably good corre-

lation with the data (16, 18), our model performed significantly

better with a chi-square value one-tenth that of the previous

models.

The predictions based on the IOM-NAS EE equations were

surprisingly poor, and they indicated that these cross-sectional

equations should not be used to predict weight change. This

conclusion is in contrast to the recent study by Heymsfield et al

(17)thatreportedgoodagreementbetweentheIOM-NASequa-

tionsandthemeasuredEErateofformerlyobesesubjects.How-

ever,mostoftheanalysisperformedinthatstudyinvolvedcross-

sectional comparisons. Only 3 longitudinal data sets were used

by Heymsfield et al, and their analysis pooled these data and did

not report information about variability of the results or model

statistics. Our analysis included these same 3 longitudinal data

sets, but the data were not pooled, and we included 5 additional

longitudinal weight-loss studies. Therefore, we believe that the

existing longitudinal weight-loss data do not support the use of

the cross-sectional IOM-NAS equations to predict weight

change.

Our model was also used to calculate the dietary EI changes

requiredtomaintainagivenamountofweightlossandtoprevent

weight regain. This important question has been previously ad-

dressed by Heymsfield et al (44) in another study using the

IOM-NAS cross-sectional EE equations. However, given the

poor performance of these equations in predicting steady-state

weight change, we do not recommend their use for predicting

changes of EI. Furthermore, the assumed linear dependence of

EE on body weight implies that a specified decrement in dietary

intake leads to the same weight change regardless of the initial

body weight or body composition. This result ignores the facts

that body-composition changes are likely to be nonlinear func-

tionsoftheinitialfatmassandthemagnitudeofweightloss(22,

23)andthatFFMcontributestoEEtoagreaterdegreethandoes

body fat (19, 45). Our proposed model incorporates nonlinear

changes in body composition with weight loss, and Figure 2

TABLE 2

Predicted body weight (BW) changes in 40-y-old sedentary men (PAL ? 1.4) of average height (1.77 m) assuming no change of physical activity (?? ? 0)1

BWinit(kg)

Change in energy intake (kcal/d)

?100

?200

?300

?400

?500

?600

?700

?800

?900

?1000

?1100

?1200

?1300

?1400

?1500

kg

80

85

90

95

?4.7

?5.0

?5.3

?5.5

?5.8

?6.0

?6.2

?6.4

?6.6

?6.8

?7.0

?7.1

?7.3

?7.5

?7.6

?7.8

?7.9

?8.0

?8.2

?8.3

?8.4

?8.5

?8.6

?8.7

?9.2

?9.8

?10.3

?10.8

?11.3

?11.7

?12.2

?12.6

?13.0

?13.4

?13.7

?14.1

?14.4

?14.7

?15.1

?15.3

?15.6

?15.9

?16.2

?16.4

?16.6

?16.9

?17.1

?17.3

?13.5

?14.3

?15.1

?15.8

?16.5

?17.2

?17.9

?18.5

?19.1

?19.7

?20.3

?20.8

?21.3

?21.8

?22.3

?22.7

?23.2

?23.6

?24.0

?24.4

?24.7

?25.1

?25.4

?25.7

?17.6

?18.6

?19.6

?20.6

?21.5

?22.4

?23.3

?24.2

?25.0

?25.8

?26.6

?27.3

?28.0

?28.7

?29.3

?29.9

?30.5

?31.1

?31.6

?32.2

?32.7

?33.1

?33.6

?34.0

?21.4

?22.7

?23.9

?25.1

?26.3

?27.4

?28.5

?29.6

?30.7

?31.7

?32.6

?33.6

?34.4

?35.3

?36.1

?36.9

?37.7

?38.4

?39.1

?39.8

?40.4

?41.0

?41.6

?42.2

?26.5

?28.0

?29.4

?30.8

?32.1

?33.5

?34.7

?36.0

?37.2

?38.4

?39.5

?40.6

?41.7

?42.7

?43.6

?44.6

?45.5

?46.3

?47.2

?47.9

?48.7

?49.4

?50.1

?31.8

?33.4

?35.0

?36.6

?38.1

?39.6

?41.1

?42.5

?43.9

?45.2

?46.5

?47.8

?49.0

?50.1

?51.2

?52.3

?53.3

?54.3

?55.3

?56.2

?57.0

?57.9

100

105

110

115

120

125

130

135

140

145

150

155

160

165

170

175

180

185

190

195

?39.0

?40.8

?42.5

?44.2

?45.9

?47.5

?49.1

?50.6

?52.1

?53.6

?55.0

?56.3

?57.6

?58.9

?60.1

?61.2

?62.3

?63.4

?64.4

?65.4

?44.8

?46.7

?48.5

?50.4

?52.2

?54.0

?55.7

?57.4

?59.1

?60.7

?62.2

?63.7

?65.1

?66.5

?67.9

?69.2

?70.4

?71.6

?72.8

?50.6

?52.6

?54.7

?56.6

?58.6

?60.5

?62.4

?64.3

?66.1

?67.8

?69.5

?71.1

?72.7

?74.2

?75.7

?77.1

?78.5

?79.8

?56.5

?58.7

?60.8

?63.0

?65.1

?67.1

?69.2

?71.2

?73.1

?75.0

?76.8

?78.6

?80.3

?82.0

?83.6

?85.1

?86.6

?64.8

?67.1

?69.4

?71.6

?73.8

?76.0

?78.1

?80.1

?82.2

?84.1

?86.0

?87.9

?89.7

?91.4

?93.1

?71.0

?73.4

?75.8

?78.1

?80.5

?82.8

?85.0

?87.2

?89.4

?91.5

?93.5

?95.5

?97.4

?99.3

?79.7

?82.2

?84.7

?87.2

?89.6

?92.0

?94.3

?96.6

?98.8

?101.0

?103.1

?105.2

?86.1

?88.7

?91.3

?93.9

?96.4

?98.9

?101.4

?103.8

?106.2

?108.5

?110.7

1Blank cells indicate that the prescribed intake changes would result in a BMI (in kg/m2) ? 18.5. PAL, physical activity level; ?, physical activity

parameter; BWinit, initial BW.

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showsthattheresultingmodelpredictionsforbody-composition

change match the data quite well.

Further illustrating the importance of this effect, Figure 3

showed that different initial body compositions can lead to dif-

ferentdegreesofweightchangeforthesamereductionindietary

intake. In our model, higher initial body fat leads to greater

predicted weight change for an equal decrement in dietary EI.

Thiscanbeshownmathematicallybyconsideringthesimplified

case in which the fraction of weight lost as body fat is specified

by a parameter (?). Equation 3 can then be rearranged to obtain

the following equation for the expected weight change:

?BW ?

?1 ? ???EI ? BWinit??

?FM? ?init? ?? ? ??FFM? ?FM??

(14)

Assumingthatthereisnochangeinphysicalactivity(ie,???0),

thepredictedsteady-stateweightchangeperunitreductionofEI

varieswiththeparameter?accordingtothefollowingequation:

???

?

?BW

?EI??

??FFM? ?FM??1 ? ??

??FFM? ?init? ??FFM? ?FM???2? 0(15)

Thisquantityisgreaterthanzerofor?FFM??FM,sotheexpected

weightlossisenhancedasthefractionofweightlostasbodyfat,

?,increases.BecausetheForbesbody-compositiontheorystates

that ? is an increasing function of the initial body fat, persons

withhigherinitialbodyfatwilleventuallylosemorebodyweight

for a specified decrement in dietary EI.

At first glance, this result appears to be contradictory to our

previous study showing that the required energy deficit per unit

of weight loss is higher in obese than in lean persons (29). The

physiologic explanation derives from the fact that persons with

higherinitialbodyfatloseasmallerproportionoftheirmetabol-

icallyexpensiveFFMandaretherebybetterabletopreservetheir

total EE rate during weight loss. In contrast, an initially lean

TABLE 3

Predicted long-term body weight (BW) changes in 40-y-old sedentary women (PAL ? 1.4) of average height (1.63 m) assuming no change in physical

activity (?? ? 0)1

BWinit(kg)

Change in energy intake (kcal/d)

?100

?200

?300

?400

?500

?600

?700

?800

?900

?1000

?1100

?1200

?1300

?1400

?1500

kg

80

85

90

95

?5.9

?6.2

?6.4

?6.7

?6.9

?7.1

?7.3

?7.5

?7.7

?7.8

?8.0

?8.2

?8.3

?8.4

?8.6

?8.7

?8.8

?8.9

?9.0

?9.2

?9.3

?9.3

?9.4

?9.5

?11.5

?12.1

?12.6

?13.1

?13.5

?14.0

?14.4

?14.8

?15.2

?15.5

?15.8

?16.2

?16.5

?16.7

?17.0

?17.3

?17.5

?17.8

?18.0

?18.2

?18.4

?18.6

?18.8

?19.0

?16.9

?17.7

?18.5

?19.2

?19.9

?20.6

?21.3

?21.9

?22.4

?23.0

?23.5

?24.0

?24.4

?24.9

?25.3

?25.7

?26.1

?26.4

?26.8

?27.1

?27.4

?27.7

?28.0

?28.3

?22.0

?23.1

?24.1

?25.1

?26.1

?27.0

?27.9

?28.7

?29.5

?30.2

?30.9

?31.6

?32.2

?32.8

?33.4

?34.0

?34.5

?35.0

?35.4

?35.9

?36.3

?36.7

?37.1

?37.5

?26.9

?28.2

?29.5

?30.8

?32.0

?33.1

?34.2

?35.3

?36.3

?37.2

?38.1

?39.0

?39.8

?40.6

?41.3

?42.0

?42.7

?43.3

?43.9

?44.5

?45.1

?45.6

?46.1

?46.6

?33.0

?34.6

?36.1

?37.5

?38.9

?40.3

?41.6

?42.8

?44.0

?45.1

?46.1

?47.1

?48.1

?49.0

?49.9

?50.7

?51.5

?52.3

?53.0

?53.7

?54.3

?55.0

?55.5

?39.4

?41.1

?42.8

?44.5

?46.1

?47.6

?49.0

?50.4

?51.8

?53.0

?54.2

?55.4

?56.5

?57.6

?58.6

?59.5

?60.4

?61.3

?62.1

?62.9

?63.6

?64.4

100

105

110

115

120

125

130

135

140

145

150

155

160

165

170

175

180

185

190

195

?47.8

?49.7

?51.5

?53.3

?55.0

?56.6

?58.2

?59.6

?61.1

?62.4

?63.7

?65.0

?66.1

?67.3

?68.3

?69.4

?70.3

?71.3

?72.2

?73.0

?54.6

?56.7

?58.7

?60.6

?62.5

?64.3

?66.0

?67.6

?69.2

?70.7

?72.1

?73.5

?74.8

?76.1

?77.2

?78.4

?79.5

?80.5

?81.5

?63.8

?65.9

?68.0

?70.0

?72.0

?73.8

?75.6

?77.4

?79.0

?80.6

?82.1

?83.5

?84.9

?86.2

?87.4

?88.6

?89.8

?73.3

?75.5

?77.7

?79.8

?81.8

?83.7

?85.6

?87.4

?89.1

?90.7

?92.3

?93.8

?95.2

?96.5

?97.8

?83.0

?85.3

?87.6

?89.8

?91.8

?93.8

?95.8

?97.6

?99.4

?101.1

?102.7

?104.2

?105.7

?90.6

?93.1

?95.5

?97.8

?100.0

?102.1

?104.2

?106.2

?108.1

?109.9

?111.6

?113.3

?103.4

?105.8

?108.2

?110.5

?112.6

?114.8

?116.8

?118.7

?120.6

?113.9

?116.4

?118.8

?121.1

?123.4

?125.5

?127.6

1Blank cells indicate that the prescribed intake changes would result in a BMI (in kg/m2) ? 18.5. PAL, physical activity level; ?, physical activity

parameter; BWinit, initial BW.

-80

-60

-40

-20

0

20

40

Predicted – Measured ∆BW (kg)

Proposed Model

NAS-IOM

Christiansen

Kozusko

equations

FIGURE 4. The difference between the predicted and measured weight

losses(?BW)areplottedforeachmathematicalmodel.?,change;BW,body

weight. Each data-point compares the model prediction with the mean value

for each group of subjects.

MODELING WEIGHT-LOSS MAINTENANCE

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individual will lose a greater amount of FFM, which concomi-

tantly decreases the EE rate to a greater degree. Although a

greater cumulative energy deficit is required per unit of body

weight lost by persons with higher initial body fat (29), this

energy deficit will be more readily achieved by persons with

higherinitialbodyfat,becausethemetabolicallyexpensiveFFM

isspared,whereasthemetabolicallymoreinertbutenergy-dense

FM is lost.

Some studies suggested that, after the steady-state body

weighthasbeenreached,noadaptationofEEoccursbeyondthat

predicted by body composition change alone (8, 9, 14, 46–52).

OthersmaintainedthatchangesinEEafterweightlosscannotbe

accounted for by body composition changes (11–13, 15, 20,

53–57). Many of these previous studies evaluated resting meta-

bolic rate in the overnight fasted state by using indirect calorim-

etry, which accounts for only a fraction of the total daily EE, or

24-h EE in a metabolic chamber, which does not represent the

free-living situation.

Incontrast,thepresentstudyusedtheavailabledoublylabeled

water data (8, 9, 12–15)—the gold standard for measuring free-

living total EE (58)—to model changes of total free-living EE

after a new steady-state body weight was achieved. The best fit

value of the unknown parameter ? (ie, 0.24 ? 0.13) was clearly

greater than zero and also ?0.1, which is the typical value used

torepresentthethermiceffectoffeeding(11).Thisindicatesthat

somedegreeofadaptationoftotalEEbeyondthatexpectedfrom

body weight change alone was required to explain the experi-

mental observations. We do not propose a physiologic mecha-

nism for such an adaptation. But when using our simplified

model of EE rate, with body FM and FFM as the only indepen-

dent variables, an additional term related to the EI change was

required to adequately represent the data.

Wedonotwanttoleavethereaderwiththeimpressionthatour

model can make precise calculations of weight change or the

requiredlifestylechangestopreventweightregainbyindividual

subjects. The variability in the weight loss data would suggest

thatprecisecalculationswouldbeverydifficultwiththeuseofa

general model whose parameters are not adjusted for individual

subjects. Prospective evaluation of our model using individual

subjects, with the possibility of developing personalized models

of weight change, will be the subject of future research.

Nevertheless, given the current inability to make any quanti-

tative estimates regarding the expected level of body-weight

stabilization or the required lifestyle changes to prevent weight

regain, our model represents a significant step forward that we

believe is useful for setting goals before an obesity intervention.

Given a weight-change goal, ?BW, equation 10 allows for test-

ing of various lifestyle intervention scenarios by specifying the

prescribedphysicalactivitychange(??)andthencalculatingthe

dietary EI change (?EI) that would be required to maintain

weightlossandpreventweightregain.Thephysicianandpatient

can then evaluate whether long-term adherence to such an inter-

vention is a realistic possibility. For example, consider a 150-kg

woman who wants to lose 50% of her body weight and prevent

regain. Our model (Table 3) suggests that this would require a

permanent reduction of ?1000 kcal/d from the diet to prevent

weight regain, a goal that may not be realistic in the long term.

However, by increasing her physical activity, she will be able to

cutbacklessonherdiet.Asmentionedearlier,wehaveprovided

spreadsheetfilestofacilitatesuchacalculation(seeSpreadsheet

files under “Supplemental data” in the current online issue; the

spreadsheetfilesandanonlineversionofthemodelareavailable

at http://www2.niddk.nih.gov/NIDDKLabs/LBM/lbmHall.htm).

Itisimportanttoemphasizethatthepresentstudyhasnotdealt

withthetime-courseofweightloss.Ifthecalculatedintervention

formaintainingadesiredweightchangewasimplementedatthe

onset of obesity treatment, it would likely take several years for

the body weight to reach the steady state (59). Therefore, it may

be beneficial to partition an obesity intervention into a weight-

loss phase followed by a weight-maintenance phase. The model

proposedherewouldbemostusefulforhelpingdefinethedietary

and physical activity changes required for the weight mainte-

nance phase—a problem for which no clinical tool is currently

available.

We thank SR Smith for suggesting the creation and inclusion of Tables 2

and 3.

The authors’ responsibilities were as follows—both authors contributed

to the design of the study, the analysis of the results, and the writing of the

manuscript. Neither author had a personal or financial conflict of interest.

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MODELING WEIGHT-LOSS MAINTENANCE

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Erratum

Hall KD, Jordan PN. Modeling weight-lossmaintenance to help preventbodyweight regain. Am J Clin Nutr 2008;88:1495–503.

On page 1497, Equation 10 contains an error in the exponential term. The correct equation appears below.

DEI ¼ BWinitDd= 1 ? b

þ C2DBW cFFMþ dinitþ Dd

? W FMinitC exp C 1 þ FMinit

3C cFFM? cFM

ð Þ þ C2cFFM? cFM

ðÞFMinit= 1 ? b

ðÞ

ðÞ= 1 ? b

Þ þ DBW

ðÞ

ð½?

fg

ðÞ= 1 ? b

ðÞð10Þ

doi: 10.3945/ajcn.2009.27585.

Erratum

Landberg R, A˚man P, Friberg LE, Vessby B, Adlercreutz H, Kamal-Eldin A. Dose response of whole-grain biomarkers:

alkylresorcinols in human plasma and their metabolites in urine in relation to intake. Am J Clin Nutr 2009,89:290–6.

The first paragraph of Results on page 292 should read: ‘‘All subjects except for one completed the study, and one was excluded

because of noncompliancewith the advised intake (tick-off list) and unsatisfactory urine collections [.100% within-subject CV

in creatinine excretion (26)]. Blood sampling was completed by 16 subjects (samples from 15 subjects were included in the

statistical analysis), and 90% of urine collections were reported as complete.’’

In Figure 1 on page 294, ‘‘(n ¼ 16 3 3)’’ should be replaced with ‘‘(n ¼ 15 3 3).’’

Figure 4 on page 295 is incorrect and should be replaced with the figure below.

doi: 10.3945/ajcn.2009.27667.

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LETTERS TO THE EDITOR