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Erratum

Hall KD, Jordan PN. Modeling weight-loss maintenance to help prevent body weight 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 BW initDd= 1 À bð Þ þ C 2 cF F M À cF M ð ÞF M init= 1 À bð Þþ C 2DBW cF F M þ dinit þ Ddð Þ= 1 À bð Þ

À W F M initC exp C 1 þ F M initð Þ þ DBW ½ �f g

3C cF F M À cF M ð Þ= 1 À bð Þ ð10 Þ

doi: 10.3945/ajcn.2009.27585.

Erratum

Landberg R, Aman 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 excludedbecause of noncompliance with the advised intake (tick-off list) and unsatisfactory urine collections [.100% within-subject CVin creatinine excretion (26)]. Blood sampling was completed by 16 subjects (samples from 15 subjects were included in thestatistical 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.

1478 LETTERS TO THE EDITOR

a t G u a d al a j ar a onA pr i l 1 3 ,2 0 1 0

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Modeling weight-loss maintenance to help prevent body weightregain1–3

Kevin D Hall and Peter N Jordan

ABSTRACT

Background: Lifestyle interventioncan successfullyinduce weight

loss in obese persons, at least temporarily. However, there currently

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,to calculate the modificationof energy intake

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-

tial body weights ranging from 68 to 160 kg and stable weight losses

between 7 and 54 kg.

Results: Model calculations closely matched the change data ( 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 2 values were

10-fold those of our model. The model also accurately predicted the

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

0.90).

Conclusions: Our model provides realistic calculations of body-

weight change and of the dietary modifications required for weight-

loss maintenance. Because the model was implemented by using

standard spreadsheet software, it can be widely used by physicians

and weight-management professionals. Am J Clin Nutr 2008;

88:1495–503.

INTRODUCTION

The increasing prevalence of obesity poses a serious health

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

icant weightlossin obese patients,weight regainis very common(2, 3). Therefore, rather thanfocusingexclusivelyon weight loss,

several investigators have emphasized the importance of main-

taining bodyweight at a lower level and preventing weight regain

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

beenused to glean useful insightsregarding the strategies usedby

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-

ingthe changes in dietary intakeand physical activity required to

maintain a given body-weight change and to prevent weightregain. To facilitate use of the model by physicians and weight-

management professionals, the model was implementedby using

standard spreadsheet software that can be downloaded (see

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

lineissue;the spreadsheetfiles and an online version of the model

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) regarding the capacity to match the steady-state

weight-change data, and we found that our model was superiorandcould provide realistic estimates of both themagnitude of the

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

We developeda simple modelof thesteady-state EE rate of the

bodyas a functionof bodycomposition, EI,and physicalactivity:

EE K EI FFMFFM FMFM FFM FM (1)

where K is a constant, FFM 22 kcal kg1 d1, and FM

3.2 kcal

kg1

d1

are the regression coefficients for restingmetabolic rate versus fat-free mass (FFM) and fat mass (FM),

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

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

Bethesda, MD.2 Supported by the Intramural Research Program of the National Institute

of Diabetes and Digestive and KidneyDiseases,NationalInstitutesof Health.3 Reprints not available. Address correspondence to KD Hall, NIDDK/

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

[email protected] April 28, 2008. Accepted for publication August 28, 2008.

doi: 10.3945/ajcn.2008.26333.

1495 Am J Clin Nutr 2008;88:1495–503. Printed in USA. © 2008 American Society for Nutrition


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http://www.ajcn.org/cgi/content/full/88/6/1495/DC1Supplemental Material can be found at:















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respectively(19).We assumed that theenergy cost of most phys-

ical activities is proportionalto body weightand is thus specified

by the parameter . The parameter accounts for the thermic

effectof feeding as well as anyadaptations of theEE rate beyond

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

there is significant debate as to whether such adaptations occur

with weight loss (20, 21), thenumerical value of theparameter

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

scribed in the final paragraph of the Materials and Methodssection).

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)

where BWinit is the initial body weight, andFM and FFM are

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

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

specify itsvalue.)Usingthe fact that BWFFMFM,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 F M init

C expFMinit BW

C (4)

where FMinit is 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

and 4 can be solved for the expected change in steady-state body

weight, as shown in the following equation:

BW

1 EI BWinit FFM FFMFMinit

FM init

C FFM FM

FM

init

W FM init FMinit

C FFM init

exp FM init FMinit

C FFM init

exp1 EI BWinit

C FFM init (5)

Equation 5 allows us to calculatethe changein steady-state body

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

weightchange results by using equation 5 because of the appear-

ance of the nonelementary Lambert W function. To address this

issue, we haveprovidedstandard spreadsheet files thatwill allow

readers to examine the predicted changes in steady-state body

weight as a function of the model parameters (see Spreadsheet files

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

sheet files and an online version of the model are available at http://

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). The first model, by Christiansen et al (16), was

given by the following equation:

BW

1

k EI

PAL (6 )

where k 13.5 kcal kg1 d1 for men and k 11.6

kcal kg1 d1 for 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 41 EI/EIinit

1BWinit /2 (7 )

where was given by the following equation:

1

2tanh3

21

2

FMinit

BWinitFMinit

BWinit1

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

physical activity changes by solving the IOM-NAS equations for

the expected body-weight change as follows:

BW EI/ K 1PAfinal BWinit K 2h / K 1PA/PAfinal 9)

where h is height in meters, K 1 14.2 kcal kg1 d1 and K

2

503 kcal m1 d1 for men, and K 1 10.9kcal kg1 d1 and K

2 660.7 kcal m1 d1 for 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).

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:

1496 HALL AND JORDAN


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EI BWinit / 1 C 2 FFM FMFMinit / 1

C 2BW 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;the spreadsheetfiles and an online version of the model

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

lbmHall.htm). We compared the predicted EI change from equa-

tion 10 with thechange in thesteady-state total EE rate measured

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

longitudinal weight-change data that were achieved over periods

of weightloss ranging from 1 to 14 mo.The whole-body EE rates

were accurately measured during periods of weight stability bothbefore 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 for2 wk (10, 11).

Because the measurements were made during periods of weight

stability, we assumed that the EE rate equaled the dietary EI rate.

Therefore, the measured changes of EE after weight loss gave an

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

the PAL.We found 8 longitudinalweight-loss studies,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.

We evaluatedeach modelin comparison to theweight-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 1i 1

N

yi ymi m (11)

where yi

was the model calculation corresponding to the mea-

sured value mifor each group and the brackets (� 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 mi2 / i

2 (12)

where i

was 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

coefficient of determination according to the followingequation: T A B L E

1

D a t a a n d p a r a m e t e r s u s e d f o r m o d e l c o m p a r

i s o n s i n F i g u r e s 1 a n d 2 1

A m a t r u d a

e t a l ( 8 )

D a s e t a l ( 9 )

K e y s ( 1 0 )

L e i b e l e t a l ( 1 1 )

M a r t i n e t

a l ( 1 2 )

v a n G e m e r t

e t a l ( 1 3 )

W e i n s i e r

e t a l ( 1 4 )

W e s t e r t e r p

e t a l ( 1 5 )

S u b j e c t s ( n )

1 8

3 0

1 2

1 1

9

1 0

1 2

8

3 2

5

B W

i n i t

( k g )

8 3 . 7

8 . 5

2

1 3 9 . 5

3 5 . 3

6 7 . 5

5 . 1

7 0 . 5

1 1 . 7

1 3 2 . 1

2 6 . 9

1 2 4 . 8

2 9 . 6

8 2 . 3

6 . 6

1 3 0 . 1

1 7 . 5

7 8 . 8

6 . 5

1 5 8 . 6

3 3 . 7

F i n i t

( k g )

3 4 . 3

3 . 3

7 1 . 6

2 3

9

4 . 7

1 7 . 5

1 2 . 6

6 8

1 9 . 8

6 4 . 4

2 4 . 8

2 6 . 1

3 . 1

6 8 . 3

1 1 . 7

3 0 . 7

4 . 8

7 4

2 9

i n i t ( k c a l

k g 1

d 1 )

1 1 . 2

6 . 9

2 4 . 5

9 . 0

5 . 5

6 . 7

1 2 . 7

7 . 0

6 . 3

6 . 1

B W ( k g )

2 2

1 0

5 3 . 4

2 2 . 2

1 5 . 8

6 . 1

6 . 8

1 5 . 4

1 7 . 8

3 4 . 4

2 9 . 2

3 7 . 1

1 2

2 . 1

4 6 . 3

2 1 . 4

1 2 . 9

8 . 6

5 3 . 9

1 7 . 8

E I ( k c a l / d )

2 3 1

6 6 8

8 6 2

5 9 8

2 0 1 8

2 5 2

4 2 8

6 6 4

5 5 1

8 5 3

8 8 6

8 9 1

4 1 7

6 8 7

8 6 1

6 0 9

1 1

4

5 2 2

9 0 8

4 7 7

( k c a l

k g 1

d 1 )

3 . 5

1 . 0

1 6 . 5

1 . 7

1 . 0

2 . 1

1 . 3

1 . 2

2 . 0

2 . 4

1

B W

i n i t , i n i t i a l b o d y w e i g h t ; F M

i n i t , i n i t i a l f a t ; i n i t , i n i t i a l b o d y c o m p o s i t i o n a n d i n i t i a l p h y s i c a l a c t i v i t y p a r a m e t e r ; , c h a n g e ; E E , t o t a

l e n e r g y e x p e n d i t u r e ; R M R , r e s t i n g m e t a b o l i c r a t e ; F F M

a n d F M , r e g r e s s i o n

c o e f f i c i e n t s f o r r e s t i n g m e t a b o l i c r a t e v e r s u s f a t - f r e e m a s s ( F F M ) a n d f a t m a s s ( F M ) , r e s p e c t i v e l y ; C , t h e c o n s t a n t p r o v i d i n g t h e b e s t f i t t o t h e e m p i r i c a l F o r b e s b o d y - c o m p o s i t i o n c

u r v e . P h y s i c a l a c t i v i t y w a s

c a l c u l a t e d b y u s i n g

( 0 . 9

E E – R M R ) / B W , a n d a l l m o d e l s u s e d t h e f o l l o w i n g p a r a m

e t e r s : F F M

2 2 k c a l

k g 1

d 1 , F M

3 . 2

k c a l

k g 1

d 1 ,

C

1 0 . 4 k g , a n d

0 . 2 4

2

x

S D ( a l l s u c h v a l u e s ) .

MODELING WEIGHT-LOSS MAINTENANCE 1497


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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

of5000 sets ofsynthesizeddatawiththe same statisticsas thereal

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

changes of steady-state body weight given the measured changes

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-

sured weightlossobserved ineach study ( R20.91).In this case,

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

tion gave a probability of 3.1 104. 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.

The comparison of the predicted and themeasured fatfraction

oftheweightlossisplottedin Figure2;and( R20.90) indicates

that our model accurately described the observed body-

composition changes. There wasa wide range of observed body-

compositionchanges,which weredue to the different initialbody

compositionsand therange of weight losses in thevariousstudies

(22). For example, the initially lean subjects from the Minnesota

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

only34% oftheir weightloss was accounted for bya lossof body

fat. In contrast, body fatlossaccounted for84%of theweight loss

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

second obese subject, who corresponds to an average participant

-70

-60

-50

-40

-30

-20

-10

0

-70 -60 -50 -40 -30 -20 -10 0

Predicted ∆BW (kg)

M e a s u r e d ∆ B W

( k g )

Amatruda

Das

Keys

Leibel

Martin

van Gemert

Weinsier

WesterterpIdentity

-2500

-2000

-1500

-1000

-500

0

-2500 -2000 -1500 -1000 -500 0

Predicted ∆EI (kcal/d)

M

e a s u r e d ∆

( k c a l / d )

E I

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 andphysical activity ( R2

0.83). B: Predicted versus measured changes indietary 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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in a study by Das et al(9).For the obese subject starting ata body

weightof 139.5 kg, a 400-kcal/d reductionin dietary intake(with

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

weight loss of 23.4 kg, whereas the overweight subject whose

initial weightwas 82.3 kg lost only 13.7 kgfor thesamereduction

indietaryEI. Thedotted curveson eithersideof each lineindicatetherange of predictedweight changes when themodel parameter

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

a higheramount of body fat than do men of similar weight. Thus,

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

ification of the initial body composition at a given body weight

differed between men and women. The initial body composition

and the initial physical activity parameter (init) were estimated

by using theregression equations of Jacksonet al (27)and Mifflin

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

forsedentary people, we have also provided spreadsheet files for

calculationsof changes in physicalactivity along withchangesin

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-

tions to changes in parameters, much as were illustrated in Figure 3,

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 betterthan the other models because the residual body-weight changes

wereclosertozeroandlessspreadoutthanwerethoseintheother

models. Whereas the model by Christiansen et al (16) resulted in

a respectable correlation coefficient of 0.855 and an R2 of 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

less than the chi-square calculated for their model. The model by

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

model would be less than thechi-squarecalculatedfor hismodel.

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

in a correlationcoefficient of 0.60, buta very poor R2

(ie,1.79),which indicated that the mean of the data provided a better de-

scriptionof thebody weightchange than thepredictionsfrom the

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

184gave a probability of 1 that thechi-squarefor a correct model

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

such rulesdo notaccountforchangesin EEwith weightloss, they

do notallow forstabilizationof body weightat a newsteady state

despite continued adherence to a lifestyle intervention. Unfortu-

nately, thiscrude approximation currently is the only widespread

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Predicted ∆FM/∆BW

M e a s u r e d ∆

∆

F M /

B W

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 -100 0

∆ (EI kcal/d)

∆ B W

( k g )

FIGURE 3. Predicted diet-induced weight losses (BW)as a function of dietary intake changes (EI) for 2 subjects with very different initial body

weights corresponding to average data from Martin et al (12) (—,initial BW

82.3 kg) and Das et al (9) (- - -, initial BW 139.5 kg). , change; BW,bodyweight. The dottedlines indicate the sensitivityof the modelpredictions

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.

We addressed this problem by using the available longitudinalweight-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

has been attempted many times and involves solving differential

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).

Previous mathematical models of human body-weight change

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

weight change for prescribed changes in diet or physical activity

(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) that reported good agreement between the IOM-NAS equa-

tions and themeasured EE rate of formerly obese subjects. How-

ever, most of theanalysis performedin that study involved cross-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

required to maintain a given amount of weightlossand to prevent

weight regain. This important question has been previously ad-

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

tions of the initial fat mass and the magnitude of weight loss (22,

23) and that FFM contributes to EE to a greater degree than does

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 4.7 9.2 13.5 17.6 21.4

85 5.0 9.8 14.3 18.6 22.7 26.5

90 5.3 10.3 15.1 19.6 23.9 28.0 31.895 5.5 10.8 15.8 20.6 25.1 29.4 33.4

100 5.8 11.3 16.5 21.5 26.3 30.8 35.0 39.0

105 6.0 11.7 17.2 22.4 27.4 32.1 36.6 40.8 44.8

110 6.2 12.2 17.9 23.3 28.5 33.5 38.1 42.5 46.7 50.6

115 6.4 12.6 18.5 24.2 29.6 34.7 39.6 44.2 48.5 52.6 56.5

120 6.6 13.0 19.1 25.0 30.7 36.0 41.1 45.9 50.4 54.7 58.7

125 6.8 13.4 19.7 25.8 31.7 37.2 42.5 47.5 52.2 56.6 60.8 64.8

130 7.0 13.7 20.3 26.6 32.6 38.4 43.9 49.1 54.0 58.6 63.0 67.1 71.0

135 7.1 14.1 20.8 27.3 33.6 39.5 45.2 50.6 55.7 60.5 65.1 69.4 73.4

140 7.3 14.4 21.3 28.0 34.4 40.6 46.5 52.1 57.4 62.4 67.1 71.6 75.8 79.7

145 7.5 14.7 21.8 28.7 35.3 41.7 47.8 53.6 59.1 64.3 69.2 73.8 78.1 82.2 86.1

150 7.6 15.1 22.3 29.3 36.1 42.7 49.0 55.0 60.7 66.1 71.2 76.0 80.5 84.7 88.7

155 7.8 15.3 22.7 29.9 36.9 43.6 50.1 56.3 62.2 67.8 73.1 78.1 82.8 87.2 91.3

160 7.9 15.6 23.2 30.5 37.7 44.6 51.2 57.6 63.7 69.5 75.0 80.1 85.0 89.6 93.9

165 8.0 15.9 23.6 31.1 38.4 45.5 52.3 58.9 65.1 71.1 76.8 82.2 87.2 92.0 96.4

170 8.2 16.2 24.0 31.6 39.1 46.3 53.3 60.1 66.5 72.7 78.6 84.1 89.4 94.3 98.9

175 8.3 16.4 24.4 32.2 39.8 47.2 54.3 61.2 67.9 74.2 80.3 86.0 91.5 96.6 101.4

180 8.4 16.6 24.7 32.7 40.4 47.9 55.3 62.3 69.2 75.7 82.0 87.9 93.5 98.8 103.8

185 8.5 16.9 25.1 33.1 41.0 48.7 56.2 63.4 70.4 77.1 83.6 89.7 95.5 101.0 106.2

190 8.6 17.1 25.4 33.6 41.6 49.4 57.0 64.4 71.6 78.5 85.1 91.4 97.4 103.1 108.5

195 8.7 17.3 25.7 34.0 42.2 50.1 57.9 65.4 72.8 79.8 86.6 93.1 99.3 105.2 110.7

1 Blank cells indicate that the prescribed intake changes would result in a BMI (in kg/m 2) � 18.5. PAL, physical activity level; , physical activity

parameter; BWinit

, initial BW.



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showsthat the resulting model predictions for body-composition

change match the data quite well.

Further illustrating the importance of this effect, Figure 3showed that different initial body compositions can lead to dif-

ferentdegrees of weightchange forthe same reduction in dietary

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

predicted weight change for an equal decrement in dietary EI.

This can be shown mathematically by considering the simplified

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

by a parameter (). Equation 3 can then be rearranged to obtainthe following equation for the expected weight change:

BW

1 EI BWinit

FM init FFM FM(14)

Assumingthat there is no change in physicalactivity (ie,0),

the predicted steady-state weight change per unit reduction of EI

varies with the parameter according to the following equation:

BW

EI FFM FM1

FFM init FFM FM2 0 (15)

This quantity isgreaterthan zero for FFM� FM, so theexpected

weight loss is enhanced as the fraction of weight lost as body fat,

, increases. Because the Forbes body-compositiontheory states

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

withhigher initial bodyfat willeventually losemore bodyweight

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

higher initial body fat lose a smaller proportion of their metabol-

ically expensiveFFM and are thereby better ableto preservetheir

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 5.9 11.5 16.9 22.0 26.9

85 6.2 12.1 17.7 23.1 28.2 33.090 6.4 12.6 18.5 24.1 29.5 34.6 39.4

95 6.7 13.1 19.2 25.1 30.8 36.1 41.1

100 6.9 13.5 19.9 26.1 32.0 37.5 42.8 47.8

105 7.1 14.0 20.6 27.0 33.1 38.9 44.5 49.7 54.6

110 7.3 14.4 21.3 27.9 34.2 40.3 46.1 51.5 56.7

115 7.5 14.8 21.9 28.7 35.3 41.6 47.6 53.3 58.7 63.8

120 7.7 15.2 22.4 29.5 36.3 42.8 49.0 55.0 60.6 65.9

125 7.8 15.5 23.0 30.2 37.2 44.0 50.4 56.6 62.5 68.0 73.3

130 8.0 15.8 23.5 30.9 38.1 45.1 51.8 58.2 64.3 70.0 75.5

135 8.2 16.2 24.0 31.6 39.0 46.1 53.0 59.6 66.0 72.0 77.7 83.0

140 8.3 16.5 24.4 32.2 39.8 47.1 54.2 61.1 67.6 73.8 79.8 85.3 90.6

145 8.4 16.7 24.9 32.8 40.6 48.1 55.4 62.4 69.2 75.6 81.8 87.6 93.1

150 8.6 17.0 25.3 33.4 41.3 49.0 56.5 63.7 70.7 77.4 83.7 89.8 95.5

155 8.7 17.3 25.7 34.0 42.0 49.9 57.6 65.0 72.1 79.0 85.6 91.8 97.8 103.4

160 8.8 17.5 26.1 34.5 42.7 50.7 58.6 66.1 73.5 80.6 87.4 93.8 100.0 105.8

165 8.9 17.8 26.4 35.0 43.3 51.5 59.5 67.3 74.8 82.1 89.1 95.8 102.1 108.2 113.9

170 9.0 18.0 26.8 35.4 43.9 52.3 60.4 68.3 76.1 83.5 90.7 97.6 104.2 110.5 116.4

175 9.2 18.2 27.1 35.9 44.5 53.0 61.3 69.4 77.2 84.9 92.3 99.4 106.2 112.6 118.8

180 9.3 18.4 27.4 36.3 45.1 53.7 62.1 70.3 78.4 86.2 93.8 101.1 108.1 114.8 121.1

185 9.3 18.6 27.7 36.7 45.6 54.3 62.9 71.3 79.5 87.4 95.2 102.7 109.9 116.8 123.4

190 9.4 18.8 28.0 37.1 46.1 55.0 63.6 72.2 80.5 88.6 96.5 104.2 111.6 118.7 125.5

195 9.5 19.0 28.3 37.5 46.6 55.5 64.4 73.0 81.5 89.8 97.8 105.7 113.3 120.6 127.6

1 Blank cells indicate that the prescribed intake changes would result in a BMI (in kg/m 2) � 18.5. PAL, physical activity level; , physical activity

parameter; BWinit

, initial BW.

-80

-60

-40

-20

0

20

40

P r e d i c t e d – M e a s u r e d

∆ B W

( k g )

Proposed Model

NAS-IOM equations

Christiansen

Kozusko

FIGURE 4. The difference between the predicted and measured weightlosses(BW)are plotted foreachmathematical model., change; BW,bodyweight. Each data-point compares the model prediction with the mean value

for each group of subjects.



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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

higher initial body fat,because the metabolically expensive FFM

is spared, whereas the metabolically moreinert but energy-dense

FM is lost.

Some studies suggested that, after the steady-state bodyweighthas been reached, no adaptation of EE occursbeyond that

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

Othersmaintained that changes in EE after weightloss cannotbe

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.

In contrast, the present study used the available doubly labeled

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

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

to representthe thermic effectof feeding (11). This indicates that

some degreeof adaptation of total EE beyondthat expected from

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.

We do notwant to leave thereader with theimpressionthatour

model can make precise calculations of weight change or the

required lifestyle changes to prevent weight regain by individualsubjects. The variability in the weight loss data would suggest

that precise calculations would be very difficult with the use of a

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

prescribed physical activity change () andthen calculating the

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

weightloss andprevent weightregain. Thephysician andpatient

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

cutback less on herdiet. As mentioned earlier, we have provided

spreadsheet files to facilitate such a calculation (see Spreadsheet

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

spreadsheet files andan onlineversion of the model areavailable

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

It is importantto emphasize that thepresent study hasnot dealt

with the time-course of weight loss. If the calculated intervention

for maintaining a desired weight change was implemented at the

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 modelproposedhere would be mostusefulfor helping define the dietary

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

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