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Modeling the relationship between body weight and energy intake: A molecular diffusion-based approach

Biology Direct20127:19

https://doi.org/10.1186/1745-6150-7-19

©  Gong and Gong; licensee BioMed Central Ltd. 2012

Received: 22 March 2012

Accepted: 13 June 2012

Published: 29 June 2012

Abstract

Background

Body weight is at least partly controlled by the choices made by a human in response to external stimuli. Changes in body weight are mainly caused by energy intake. By analyzing the mechanisms involved in food intake, we considered that molecular diffusion plays an important role in body weight changes. We propose a model based on Fick's second law of diffusion to simulate the relationship between energy intake and body weight.

Results

This model was applied to food intake and body weight data recorded in humans; the model showed a good fit to the experimental data. This model was also effective in predicting future body weight.

Conclusions

In conclusion, this model based on molecular diffusion provides a new insight into the body weight mechanisms.

Reviewers

This article was reviewed by Dr. Cabral Balreira (nominated by Dr. Peter Olofsson), Prof. Yang Kuang and Dr. Chao Chen.

Keywords

Molecular diffusionBody weightModelChoice making

Background

Body weight change is a complex behavioral response associated with appetite regulation and energy metabolism [1]. Although changes in body weight involve genetic, metabolic, biochemical, cultural and psychosocial factors, the two main factors that regulate body weight are food intake and energy expenditure [2, 3]. In recent years, mathematical models have become increasingly used in medical research. These models have helped researchers to develop new ways of dealing with animal behaviors. In terms of body weight, behavioral economic models have been developed to address the effects of environmental factors on energy intake and body weight [4]. A series of experimental studies have also been conducted to develop mathematical models to describe the physiological basis of body weight. In fact, these models can quantitatively address the metabolic processes underlying body weight changes and can be used to aid body weight control [5–8]. A mathematical model has also been proposed to address the molecular mechanisms underlying body weight, although the validity of the model has not been verified experimentally [9].

In this paper, we examined the impacts of energy intake and energy expenditure on body weight. Neuropeptides are small protein-like molecules released by neurons to communicate with each other. These neuronal signaling molecules influence specific activities of the brain, including control of food intake [1, 10–12]. Neuropeptides are expressed and released by neurons, and mediate or modulate neuronal communication by acting on cell surface receptors. They have a long half-life, show high affinity for their receptors, and reach their target by diffusion, often over a long distance [13–15]. More specifically, food intake can induce the synthesis of specific neuropeptides that diffuse to activate metabolic processes [10]. Considering the above discussion on the neural regulation of obesity, it seems likely that the molecular mobility (diffusion) of neuropeptides, for example, plays an important role in body weight regulation. In other words, the body converts food stimuli to molecular signaling processes. The molecular mobility of body weight control is at least partly explained by the diffusion of molecules inside or outside of neural cells. Accordingly, changes in body weight are influenced by molecular movements driven by energy intake. Fick’s second law, also known as the diffusion equation, describes non-steady-state diffusion, and is typically used to model molecular mobility [16]. Therefore, we can use the molecular diffusion model to describe body weight behavior, replacing molecular concentration with calorie intake as the driving force in this process. It is known that some biological molecules are synthesized at high concentrations and subsequently affect the concentrations of other molecules by diffusion, until the resulting behavior is established. Therefore, we incorporated the diffusion equation as a model of body weight control and validated this model using experimental data. Because the diffusion equation is nonlinear, the correct parameters can be obtained by global optimization.

In summary, we propose a model in which body weight control is derived from molecular diffusion. We also quantitatively investigate the relationship between energy intake and body weight, by applying Fick’s second law of diffusion in combination with a mathematical algorithm. Validation of the model with experimental data obtained from humans showed that the model dynamically simulates changes in body weight and energy intake very well. This model is suitable for describing the relationship between energy intake and body weight.

Results and discussion

Body weight change: a molecular diffusion based process

Because molecular mobility is accompanied by energy transference, we can describe molecular diffusion with energy diffusion. The human body obeys the law of energy conservation [7], which can be expressed as
d d t ( ρ * V ) = d E d t
(1)

where ρ is the energy density of body mass, V is the body mass, E is the net energy intake, t is the time.

Suppose J is energy flux (amount of energy per unit area per unit time in direction x), p is the energy density of body fat mass. For healthy adults (18-59y), body weight changes largely due to fat mass (FM) [17], so d(ρ*V) is approximately equal to p*dV. We have
d E d t = − d J d x
(2)
and
J = − D p d V d x
(3)
where D is energy diffusion coefficient. Substituting Equation 2 and Equation 3 into Equation 1 leads to the following equation:
d V d t = D d 2 V d x 2
(4)

Equation 4 is actually the form of Fick’s second law of diffusion.

In the initial conditions where t = 0 and x > 0, then V = V 0 . In marginal conditions where t > 0 and x = 0, then V = V s . When t > 0 and x = ∞, V = V 0 . V 0 is the initial body mass, V s is the body mass transformed from energy intake. Therefore, the solution of Equation 4 is:
V ( x , t ) = V s [ 1 − e r f ( x / ( 2 D t ) ) ] + V 0 e r f ( x / ( 2 D t ) )
(5)
where e r f ( c ) = ( 2 / π ) ∫ 0 c exp ( − c 2 ) d c . Because V s = E / p , Equation 5 can be rewritten as the following equation:
V ( t ) = 1 p E [ 1 − e r f ( x / ( 2 D t ) ) ] + V 0 e r f ( x / ( 2 D t ) )
(6)

From the above discussion, we can know the body weight change process is a diffusion process driven by energy intake.

Fitting and the model to experimental data and validation

As described above, changes in body weight can be explained by molecular movement driven by energy intake. Considering that body weight change mimics molecular diffusion, and that diffusive processes are involved in body weight changes at the cellular level, this behavioral activity can be described by Equation 6.

To use the molecular diffusion based model to describe the relationship between energy intake and body weight, because distance x represents body attributes, it is set as a constant in this model. In this way, Equation 6 can be rewritten as:
f ( t ) = b * e r f ( β / t ) + α * l * [ 1 − e r f ( β / t ) ]
(7)
where f(t) = body weight, b = initial body weight, l = energy intake, t = time of feeding, and α and β are constants. If t > 1, this formula can be rewritten as follows:
f ( t ) = f ( t − 1 ) * e r f ( β ) + α * l ( t ) * [ 1 − e r f ( β ) ]
(8)

where l(t) = energy intake, with other parameters identical to those in Equation 7.

Equation 8 can then be applied to simulate experimental data and its validity tested against reference data (in this case human body weight). To best estimate the model parameters, ISCEM algorithm was adopted because this algorithm can not only estimate parameters in complex functions but also conduct global optimization [18]. Energy intake and body weight were recorded for humans in an earlier study [19]. If the experimental data and model-derived data show a good fit, we can conclude that the model is suitable to describe the relationship between energy intake and body weight.

Simulation of body weight change using the developed model

Using the experimental data recorded over 24 weeks (Table 1) and the ISCEM algorithm, the following constants were obtained:
Table 1

Group’s body weight related data (S1-S24) from Minnesota human starvation study

Time (week)

Body weight (kg)

TEE (kcal/day)

Mean energy intake (kcal/day)

Net energy intake (kcal/day)

0

69.39

1934.33

3538.72

1604.39

S1

68.35

1884.53

1658

−226.53

S2

66.8

1835.21

1658

−177.21

S3

65.76

1786.36

1648.88

−137.48

S4

64.29

1737.6

1610.88

−126.72

S5

63.33

1691.55

1645.94

−45.61

S6

62.16

1643.45

1639.16

−4.29

S7

61.11

1595.75

1639.03

43.28

S8

60.31

1548.28

1634.84

86.56

S9

59.56

1500.8

1620.41

119.61

S10

58.71

1453.31

1595.31

142

S11

58.14

1405.63

1578.72

173.09

S12

57.28

1357.94

1525.16

167.22

S13

56.6

1346.8

1515.69

168.89

S14

56.16

1335.67

1492.84

157.17

S15

55.69

1324.54

1459.94

135.4

S16

54.7

1313.39

1430.5

117.11

S17

54.28

1302.28

1488.81

186.53

S18

54.08

1291.1

1486.44

195.34

S19

53.51

1281.22

1519.72

238.5

S20

53.18

1271.34

1515.47

244.13

S21

52.99

1261.46

1538.75

277.29

S22

52.9

1251.58

1554.06

302.48

S23

52.83

1241.7

1581.19

339.49

S24

52.57

1231.83

1641.63

409.8

α = 0.016337, β = 1.7096

Entering these constants yields the following equation:
f ( t ) = f ( t − 1 ) * e r f ( 1.7096 ) + 0.016337 * 7 * l ( t ) * [ 1 − e r f ( 1.7096 ) ]
(9)
Using Equation 9, we can estimate daily body weight from week S1 to week S24. The model-generated body weight data are plotted alongside the actual experimental data in Figure 1. The determination coefficient (R2) for this plot was 0.99666, which indicates that the model-generated data closely match the experimental data. Comparison between the actual experimental body weight and model result of each subject is shown in Appendix A.
Figure 1

Comparison of experimentally recorded and model-generated group’s body weight of humans (weeks S1–S24). Asterisks, experimentally recorded data; circles, model-generated data.

Model validation and body weight prediction

We next sought to validate the model. To achieve this, body weight measured between week S1 and week S12 from Table 1 were entered into the ISCEM algorithm, which yielded the following constants:

α = 0.0170757, β = 1.7029

Entering these constants into Equation 8 yields the following equation:
f ( t ) = f ( t − 1 ) * e r f ( 1.7029 ) + 0.0170757 * 7 * l ( t ) * [ 1 − e r f ( 1.7029 ) ]
(10)
Using Equation 10, it is possible to estimate the daily body weight from week S1 to week S12. The model-generated data are plotted alongside the experimental data in Figure 2. R2 for this model was 0.98499, indicating very close fit between the model and the experimental data.
Figure 2

Comparison of experimentally recorded and model-generated group’s body weight of humans (weeks S1–S12). Asterisks, experimentally recorded data; circles, model-generated data.

Finally, we used this model with the parameters based on the experimental data for weeks S1–S12 to predict body weight change between week S13 and week S24. The body weights predicted for weeks S13–S24 and the corresponding experimental data are plotted in Figure 3. The R2 for this model was 0.94229, indicating the model satisfactorily fits the experimental data. Confidence intervals for predicted body weight of each subject were provided in Appendix B.
Figure 3

Prediction of group’s body weight during weeks S13–S24 based on actual experimental data from weeks S1–S12. The actual experimental data in weeks S13–S24 are also shown. Asterisks, actual experimental data; circles, model-generated data.

Methods

Ethics statement

Because human data were used, approval was obtained from Wuhan University of Technology's Ethics Committee. This research was based on experimental data from literature [19]. As such, no consent statement for participation is required.

The Minnesota starvation study

The study reduced the energy intake of 32 male conscientious objectors (20–33 y old, mean 25.5 y) to decrease body mass comparably to severely undernourished prisoners of war with the aim of testing methods for rehabilitating starved men. The study included a 12-week control phase (weeks C1–C12), 24 weeks of energy restriction (weeks S1–S24), and 20 weeks of recovery (R1–R20). During weeks C1–C12, energy intake was adjusted to bring individuals towards the group norm, based on weight for height, with a mean weight loss of 0.80 kg. Physical activity included 22 miles per week of outdoor walking and additional walking on campus, plus custodial duties. All subjects were also required to walk at 3.5 miles per hr for half an hr per week on a motor-driven treadmill with a 10% grade. The control diet contained about 100 g of protein, 400 g of carbohydrates, and 130 g of fat. Energy intake averaged 3,492 kcal/d (14.62 mJ/d) for the last 3 control weeks, during which group weight declined only 0.3 kg. From then on, subjects were fed at a level that was expected to cause a 24% group average decrease in body mass during the next 24 weeks. Weight loss was induced by reducing food intake to two daily meals with 51 g of protein, 286 g of carbohydrates, and 30 g of fat, with 3 basic menus consisting of cereal, whole-wheat bread, potatoes, turnips, and cabbage, supplemented by scant amounts of meat and dairy products. During the entire starvation period, walking 22 miles a week and custodial work remained mandatory [19].

Total Energy Expenditure (TEE) includes two major parts: Resting Energy Expenditure (REE), the amount of calories needed to maintain basic body systems and body temperature at rest; Activity Energy Expenditure (AEE), the amount of calories used during activity [20]. Net energy intake is the difference between food intake and TEE. Although TEE was not measured in the Minnesota starvation study, TEE can be obtained through calculating REE and AEE [19, 21].

Some useful data are shown in Table 1.

ISCEM algorithm: an improved SCEM-UA algorithm

The shuffled complex evolution metropolis algorithm (SCEM-UA) is a global-searching algorithm developed by Vrugt JA et al. [22]. The SCEM–UA method adopts Markov Chain Monte Carlo theory (MCMC) and uses the Metropolis–Hastings algorithm (MH), replacing the Downhill Simplex method, to obtain a global optimal estimation. Although SCEM-UA can successfully obtain the global optimal solution, its performance depends on correct setting of the minimal and maximal limits. In the current study, we improve the SCEM-UA algorithm so that it can optimize the parameter searching space and obtain the optimal solution. This improved algorithm is termed the ISCEM algorithm.

Suppose ŷ = η(ξ|θ), where ŷ is an N × 1 vector of model predictions, ξ is an N × n matrix of input variables and θ is a vector of n unknown parameters. The SCEM-UA algorithm is given below:
  1. (1)

     To initialize the process, choose the population size s and the number of complexes q. The algorithm tentatively assumes that the number of sequences is identical to the number of complexes.

     
  2. (2)

     Generate s samples from the prior distribution {θ 1 θ 2 ,…,θ s } and compute the posterior density {p(θ(1)|y),p(θ(2)|y),…,p(θ(s)|y)} at each point [22].

     
  3. (3)

     Sort the points in order of decreasing posterior density and store them in an array D[1:s,1:n + 1], where n is the number of parameters, so that the first row of D represents the point with the highest posterior density. The extra column stores the posterior density. Initialize the starting points of the parallel sequences, S1,S2,…,S q , such that S k is D[k,1:n + 1], where k = 1,2,…,q.

     
  4. (4)

     Partition D into q complexes Cl,C2,…,C q , each containing m points, such that the first complex contains every q(j − 1) + 1 ranked point, the second complex contains every q ( j − 1 ) + 2 ranked point of D, and so on, where j = 1,2,…,m.

     
  5. (5)

     Initialize L,T,ARmin, cn. For each C k , call the SEM algorithm [22] and run it L times;

     
  6. (6)

     Unpack all complexes C back into D and rank the points in order of decreasing posterior density.

     
  7. (7)

     Check Gelman and Rubin (GR) convergence statistic. If convergence criteria are satisfied, stop; otherwise, return to step 4.

     
The ISCEM algorithm is given below:
  1. (1)

     Suppose Imin≤θ≤Imax, Imin and Imax are interval vectors of θ. The initial Imax is set to be very large. Run the SCEM-UA algorithm and let the output parameter vector with highest posterior density (p o ) be θ o . Set Imax = θ o .

     
  2. (2)

     Run the SCEM-UA algorithm again, and let the output parameter vector with highest posterior density (p w ) be θ w . If || p o - p w || ≤ ε, where ε > 0, go to step (4); otherwise set θ o = θ w .

     
  3. (3)

     If p o ≤ p w , let Imax = θ w ; otherwise, let Imin = θ w . Let p o  = p w , go to step (2).

     
  4. (4)

     Output θ w .

     

Conclusions

In this paper, we have shown that energy intake and energy expenditure in humans can be simulated using a mathematical algorithm based on molecular diffusion. In the model, only the effects of calorie intake on body weight are considered; other variables that may affect body weight are included as constants. This is because the internal and external environmental factors that may influence body weight can be assumed to be stable when environment is stable. In fact, as shown here, if these factors are kept relatively stable, the prediction of body weight based on energy intake and defined constants matches closely with experimental data.

In this model, only the general relationship between energy intake and body weight was examined. We believe this model will provide new insights into the mechanisms underlying body weight control. In future studies, more information is needed to examine the impact of neuronal signaling mechanisms that control body weight on this model.

Appendix

Appendix A Table 2

Table 2

Comparison of experimental data and Model result of each subject

No.

Weight S1 kg

Weight S2 kg

Weight S3 kg

Weight S4 kg

Weight S5 kg

Weight S6 kg

Weight S7 kg

Weight S8 kg

Weight S9 kg

Weight S10 kg

Weight S11 kg

Weight S12 kg

Weight S13 kg

Weight S14 kg

Weight S15 kg

Weight S16 kg

Weight S17 kg

Weight S18 kg

Weight S19 kg

Weight S20 kg

Weight S21 kg

Weight S22 kg

Weight S23 kg

Weight S24 kg

R2

122(p)

64.6

63

61.6

60.2

59

57.6

56.6

55.5

54.6

53.5

52.8

52.1

51.7

51.5

51

50

49.4

49

48.5

48.2

47.7

47.8

47.2

47.4

 

122(m)

63.974

62.658

61.434

60.249

59.226

58.294

57.461

56.718

56.046

55.424

54.867

54.309

53.763

53.204

52.615

52.002

51.523

51.067

50.696

50.34

50.049

49.808

49.636

49.593

0.891

123(p)

63.8

62.6

61.8

60.6

60.1

59.1

58.2

57.6

57.3

56.8

56.5

55.8

55.2

54.9

54.8

53.9

53.4

53

52

51.9

51.5

52.1

52.2

52.1

 

123(m)

63.285

61.98

60.767

59.591

58.579

57.657

56.834

56.101

55.438

54.826

54.279

53.73

53.192

52.642

52.062

51.458

50.988

50.54

50.177

49.829

49.546

49.313

49.149

49.113

0.685

119(p)

65.5

64.1

63

61.5

60.7

59.4

58.4

57.6

56.9

55.9

55.4

54.5

53.9

53.4

53.2

52.2

51.4

51

50.8

50.5

50.3

50.7

49.8

49.1

 

119(m)

64.86

63.531

62.293

61.094

60.058

59.113

58.267

57.511

56.827

56.193

55.624

55.054

54.496

53.926

53.325

52.702

52.212

51.745

51.363

50.997

50.696

50.444

50.263

50.21

0.988

120(p)

69.6

68.2

67.1

65.6

64.4

63.2

61.9

61.1

60.5

59.7

58.7

57.7

56.8

56

55.5

54.7

54

53.4

53

52.3

52.1

51.3

51.2

51.6

 

120(m)

69.29

67.891

66.585

65.319

64.218

63.207

62.297

61.479

60.732

60.037

59.409

58.78

58.163

57.536

56.879

56.2

55.655

55.135

54.7

54.282

53.929

53.627

53.396

53.294

0.947

129(p)

64.7

63.3

62.4

61

60.3

59.6

58.6

58.1

57.7

57

57

56.1

55.8

55.5

54.8

53.8

53.4

53.4

53.3

53.2

53

52.8

52.8

52.2

 

129(m)

64.171

62.852

61.625

60.436

59.411

58.476

57.64

56.894

56.219

55.595

55.036

54.475

53.926

53.364

52.773

52.158

51.676

51.218

50.844

50.486

50.193

49.949

49.776

49.73

0.731

130(p)

64.8

63.4

63

61.5

60.7

60.1

59

58.5

58

58.1

57.8

57.5

56.9

56.6

56.6

55.4

55.7

55.7

55.7

55.6

54.2

53.9

54.5

53.6

 

130(m)

64.565

63.24

62.007

60.812

59.781

58.84

57.998

57.247

56.566

55.937

55.372

54.806

54.252

53.685

53.089

52.469

51.982

51.519

51.141

50.778

50.48

50.232

50.054

50.004

0.104

126(p)

82.6

81.2

79.8

78.1

77.1

75.4

74.1

72.8

71.8

70.6

69.8

69.1

68.1

67.4

67.6

66

65.7

65.4

65.3

63

62

62.6

61.6

60.6

 

126(m)

81.89

80.294

78.795

77.338

76.049

74.853

73.762

72.764

71.842

70.973

70.174

69.377

68.595

67.804

66.987

66.15

65.45

64.777

64.191

63.625

63.126

62.681

62.308

62.067

0.988

127(p)

63.1

61.2

60.2

58.3

57.4

56.1

55.5

54.6

54.2

53.2

52.9

52.5

51.8

51.3

51.2

50.8

50.4

49.6

49.2

48.7

49.2

48.7

49.2

49.3

 

127(m)

62.793

61.496

60.29

59.122

58.117

57.202

56.386

55.66

55.004

54.399

53.858

53.316

52.785

52.241

51.667

51.069

50.605

50.164

49.806

49.464

49.187

48.959

48.801

48.771

0.971

22(p)

64.2

62.8

61.4

60.2

59.2

58.1

57.2

56.8

56.2

55.4

55

53.8

53.4

53

52.4

51.2

51.2

51.3

50.6

50.5

50

49.4

49.9

49.4

 

22(m)

63.679

62.368

61.148

59.967

58.949

58.021

57.192

56.453

55.785

55.168

54.615

54.061

53.518

52.963

52.378

51.769

51.294

50.841

50.473

50.121

49.834

49.596

49.427

49.387

0.995

23(p)

68.3

66.6

65.4

64

62.8

61.6

60.4

59.6

58.5

57.6

56.9

55.8

55

54.6

53.9

53.4

52.8

52.7

52.2

51.8

51.5

51.4

51.4

51.4

 

23(m)

67.715

66.341

65.059

63.817

62.739

61.751

60.864

60.068

59.344

58.67

58.063

57.455

56.86

56.252

55.616

54.956

54.431

53.93

53.514

53.114

52.78

52.496

52.282

52.197

0.954

19(p)

69.6

68.3

67.6

65.9

64.6

63.7

62.6

61.7

60.8

59.4

58.6

57.5

56.8

56.1

55.7

54.3

54

51.5

51.4

51.4

52.5

52.4

52.2

50.4

 

19(m)

69.093

67.697

66.395

65.131

64.033

63.025

62.118

61.302

60.559

59.867

59.241

58.614

58

57.375

56.721

56.044

55.502

54.984

54.552

54.136

53.786

53.486

53.257

53.157

0.933

20(p)

63.7

62.5

61.5

60.1

59

58

57.1

56.4

55.5

54.9

54.7

53.9

53

52.8

52.1

50.8

49.8

49.4

49.2

48.4

47.8

48.1

48.2

48

 

20(m)

63.285

61.98

60.767

59.591

58.579

57.657

56.834

56.101

55.438

54.826

54.279

53.73

53.192

52.642

52.062

51.458

50.988

50.54

50.177

49.829

49.546

49.313

49.149

49.113

0.974

29(p)

69.7

68.1

67.3

66.5

65.5

65.4

64.5

63

60.3

57.4

55.1

54.5

55.2

55.5

54.1

53.3

54.2

54

52.7

52.3

53

54.2

53.8

53.5

 

29(m)

69.585

68.182

66.871

65.601

64.495

63.48

62.566

61.743

60.993

60.294

59.661

59.028

58.408

57.777

57.116

56.433

55.885

55.361

54.922

54.501

54.145

53.839

53.605

53.5

0.863

30(p)

67.1

65.6

64.6

63.1

62.3

61

60.4

59.3

59

58.2

58.6

57.6

57

56.2

56

54.8

54.5

54.2

53.6

53.8

53.9

53.1

53.2

52.4

 

30(m)

66.632

65.275

64.01

62.784

61.722

60.75

59.879

59.098

58.389

57.731

57.138

56.545

55.963

55.37

54.747

54.101

53.589

53.101

52.698

52.311

51.989

51.718

51.516

51.444

0.944

26(p)

70.3

69.3

67.7

65.8

65

63.5

62.3

61.7

60.6

59.7

59.3

58.1

57.7

58

56.7

56.5

56

57.2

55.8

55.4

55.4

54.7

53.2

53.1

 

26(m)

69.782

68.376

67.062

65.789

64.68

63.662

62.745

61.92

61.166

60.465

59.829

59.194

58.571

57.937

57.274

56.589

56.038

55.512

55.071

54.647

54.289

53.981

53.744

53.637

0.98

27(p)

74.5

73.2

72

69.6

68.4

67

65.5

64.4

63.1

61.9

61.5

60.8

60.3

60

58.8

58.1

57.6

57.4

56.8

56.1

55.8

55.3

55.6

55.7

 

27(m)

73.621

72.155

70.782

69.451

68.285

67.21

66.238

65.358

64.551

63.797

63.109

62.423

61.749

61.066

60.354

59.62

59.022

58.45

57.963

57.493

57.091

56.739

56.46

56.31

0.958

4(p)

60.9

59.6

58.6

57.2

56

54.9

53.8

53.1

52.3

51.5

51

50.4

50

49.4

48.6

47.9

48.3

48.3

47.5

47.3

47.1

47.1

47.3

47.4

 

4(m)

60.627

59.364

58.191

57.056

56.084

55.2

54.415

53.72

53.095

52.519

52.008

51.494

50.992

50.476

49.93

49.359

48.921

48.506

48.175

47.858

47.606

47.403

47.269

47.263

0.971

5(p)

79.6

77.9

76.4

74.8

73.4

72.2

70.6

69.6

68.2

66.9

65.5

64.6

63.7

62.8

62.3

60.7

60

59.6

58.8

58.6

58.1

57.8

57.2

57.1

 

5(m)

79.134

77.581

76.124

74.709

73.461

72.306

71.254

70.296

69.411

68.581

67.819

67.059

66.313

65.558

64.776

63.974

63.308

62.668

62.115

61.581

61.115

60.7

60.359

60.148

0.893

1(p)

75.5

73.8

72.1

70.1

68.8

67.3

66

65.3

64.8

64.3

63.8

64.9

64.4

62.6

60.6

59.2

58.8

57.7

57.3

56.6

56.5

56.6

58.2

57

 

1(m)

75.59

74.093

72.69

71.329

70.133

69.03

68.029

67.122

66.287

65.505

64.791

64.078

63.379

62.67

61.933

61.175

60.553

59.956

59.446

58.953

58.528

58.154

57.852

57.681

0.932

2(p)

72.1

70.1

69.1

67.9

67

65.7

64.6

63.5

62.6

61.5

61

60.1

58.8

58.2

58.2

57.1

56.5

56.2

54.6

55.1

55.2

57.2

57.9

55.9

 

2(m)

71.948

70.507

69.161

67.854

66.713

65.664

64.715

63.859

63.076

62.344

61.68

61.015

60.364

59.702

59.011

58.299

57.722

57.169

56.702

56.253

55.869

55.537

55.276

55.145

0.956

11(p)

65.7

63.9

62.6

61.6

60

59.1

58.2

57.6

56.7

56

55.3

54.4

54.1

53.5

53.2

52.5

51.8

51.9

50.8

50.3

50

49.5

49.9

49.6

 

11(m)

64.958

63.627

62.388

61.188

60.151

59.204

58.356

57.599

56.914

56.278

55.709

55.137

54.578

54.006

53.404

52.78

52.288

51.821

51.437

51.07

50.768

50.515

50.333

50.278

0.989

12(p)

79.7

77.5

75.8

74

73.2

71.5

70.4

70

69.5

68.1

68.1

68.7

67.6

67.8

67.7

66.2

65.2

65.1

64.9

63.8

62.9

61.8

61.6

63.2

 

12(m)

79.33

77.775

76.315

74.897

73.646

72.488

71.433

70.472

69.585

68.752

67.987

67.224

66.476

65.719

64.934

64.129

63.461

62.818

62.263

61.727

61.258

60.842

60.498

60.285

0.896

8(p)

63.8

62.8

62.7

61.6

60.7

59.2

58.6

58.2

57.2

57.1

56.2

53.9

50.7

50.2

50.3

50.5

49.9

48.8

47.8

47.7

48.8

48.9

48.3

47.5

 

8(m)

63.187

61.883

60.671

59.497

58.487

57.566

56.744

56.012

55.351

54.74

54.195

53.647

53.111

52.562

51.983

51.38

50.911

50.465

50.103

49.756

49.474

49.242

49.079

49.045

0.905

9(p)

71.5

69.6

69.1

68

67.2

66.2

64.6

64.1

63.6

64.1

63

60.4

59.6

59.9

59.6

58

57.3

57.6

57.1

57.5

57.4

57.2

56.6

58.1

 

9(m)

71.357

69.926

68.588

67.291

66.159

65.118

64.178

63.33

62.555

61.832

61.175

60.518

59.875

59.221

58.537

57.832

57.262

56.717

56.257

55.815

55.438

55.113

54.858

54.733

0.917

104(p)

66.7

64.7

63.9

63

62.5

61.1

60.2

59.3

58.9

58

57.6

56.6

55.8

54.9

53.8

52.9

52.6

52.3

51.8

51.4

51.1

51.4

51.4

51.6

 

104(m)

66.632

65.275

64.01

62.784

61.722

60.75

59.879

59.098

58.389

57.731

57.138

56.545

55.963

55.37

54.747

54.101

53.589

53.101

52.698

52.311

51.989

51.718

51.516

51.444

0.985

105(p)

67.4

66

65.4

63.7

63

61.7

61.3

59.9

59.3

58.1

57.5

56.5

55.7

55

54.7

53.5

52.7

52.6

51.4

50.8

51.5

51.4

51.8

51.8

 

105(m)

67.223

65.856

64.582

63.347

62.276

61.296

60.416

59.627

58.91

58.243

57.643

57.041

56.452

55.851

55.221

54.568

54.048

53.553

53.143

52.749

52.42

52.142

51.934

51.855

0.974

101(p)

63.7

62.4

61.6

60.2

59.2

58.2

57.2

56.5

55.8

55

54.5

53.6

53.1

53.3

53.1

51.9

51.6

51.8

51.7

51.4

49.3

48.4

48.4

49.7

 

101(m)

63.088

61.786

60.576

59.404

58.394

57.475

56.654

55.924

55.264

54.655

54.111

53.564

53.029

52.482

51.904

51.303

50.835

50.389

50.029

49.683

49.403

49.171

49.01

48.976

0.962

102(p)

67

65.5

64.6

63.4

62.2

61

59.9

59

58

57.5

57.2

56.3

55.5

54.5

55.2

53.7

53.4

53.6

53

52.8

53

51.9

51.8

51.9

 

102(m)

66.435

65.081

63.819

62.596

61.537

60.568

59.7

58.922

58.215

57.56

56.97

56.379

55.8

55.209

54.589

53.946

53.436

52.951

52.55

52.165

51.846

51.576

51.377

51.306

0.987

111(p)

62.5

60.6

59.4

58.1

57.2

56

54.9

54.3

53.9

53

52.9

52.1

51.8

51.3

50.9

50.4

50.1

50.1

49.8

49.5

49.4

49.1

49

49.1

 

111(m)

61.612

60.333

59.145

57.995

57.008

56.11

55.311

54.602

53.963

53.373

52.849

52.322

51.807

51.278

50.719

50.137

49.687

49.26

48.916

48.588

48.325

48.11

47.965

47.948

0.976

112(p)

60.6

58.9

58

56.3

55.9

54.8

53.4

52.5

51.9

51.4

50.8

50.5

50.4

50.7

50.2

48.9

49

50.3

50.6

50.9

50.9

50.4

49

49

 

112(m)

60.332

59.073

57.905

56.774

55.806

54.927

54.147

53.456

52.834

52.263

51.756

51.246

50.747

50.236

49.693

49.126

48.692

48.28

47.952

47.639

47.391

47.191

47.06

47.057

0.776

108(p)

66

64.6

63.5

62

61.4

60.6

59.8

59.8

60.5

60

59.1

57.4

56.5

55.8

55.7

55.5

55

55.1

54.4

54.6

55.1

56.6

57.1

54.1

 

108(m)

65.451

64.112

62.865

61.657

60.613

59.658

58.804

58.04

57.347

56.705

56.129

55.551

54.985

54.407

53.799

53.168

52.671

52.197

51.808

51.435

51.127

50.869

50.681

50.621

0.324

109(p)

78.3

76.3

75

73.4

72.5

70.9

69.6

68.6

67.8

66.9

66.3

65.4

64.8

64.5

63.7

62.2

61.4

61.5

60.8

60.2

59.6

58.9

59.2

59.5

 

109(m)

77.657

76.128

74.693

73.3

72.074

70.941

69.91

68.973

68.11

67.3

66.558

65.817

65.091

64.355

63.592

62.808

62.16

61.538

61.003

60.486

60.037

59.639

59.314

59.12

0.996

Note: 1) No. is subject number, 122(p) is actual experimental body weight value of subject 122, 122(m) is model result of subject 122, etc.

2) R 2 is determination coefficient.

3)The experimental data of Subject 130 is not fulfilled. It shows this subject is a special case.

Appendix B Table 3

Table 3

Confidence interval of estimation (Confidence level is 95%)

Week

Weight(kg)

Weight(kg)

Weight(kg)

Weight(kg)

Weight(kg)

Weight(kg)

Weight(kg)

Weight(kg)

Weight(kg)

Weight(kg)

Weight(kg)

Weight(kg)

S13

S14

S15

S16

S17

S18

S19

S20

S21

S22

S23

S24

Subject No.

            

122(p)

51.7

51.5

51

50

49.4

49

48.5

48.2

47.7

47.8

47.2

47.4

122(model)

51.588 ± 2.592

51.063 ± 2.443

50.504 ± 2.332

49.919 ± 2.241

49.476 ± 2.145

49.057 ± 2.059

48.728 ± 1.983

48.414 ± 1.919

48.17 ± 1.86

47.977 ± 1.817

47.858 ± 1.767

47.876 ± 1.745

123(p)

55.2

54.9

54.8

53.9

53.4

53

52

51.9

51.5

52.1

52.2

52.1

123(model)

55.229 ± 4.239

54.645 ± 3.992

54.029 ± 3.788

53.387 ± 3.637

52.889 ± 3.493

52.415 ± 3.364

52.032 ± 3.255

51.666 ± 3.143

51.369 ± 3.044

51.125 ± 2.952

50.956 ± 2.903

50.924 ± 2.881

119(p)

53.9

53.4

53.2

52.2

51.4

51

50.8

50.5

50.3

50.7

49.8

49.1

119(model)

53.95 ± 1.32

53.386 ± 1.244

52.79 ± 1.179

52.168 ± 1.149

51.69 ± 1.1

51.235 ± 1.068

50.871 ± 1.036

50.523 ± 1.001

50.245 ± 0.969

50.019 ± 0.94

49.868 ± 0.966

49.853 ± 0.941

120(p)

56.8

56

55.5

54.7

54

53.4

53

52.3

52.1

51.3

51.2

51.6

120(model)

57.099 ± 1.013

56.485 ± 0.975

55.839 ± 0.973

55.168 ± 0.949

54.641 ± 0.947

54.14 ± 0.975

53.729 ± 1.017

53.335 ± 1.051

53.012 ± 1.139

52.741 ± 1.188

52.547 ± 1.335

52.489 ± 1.436

129(p)

55.8

55.5

54.8

53.8

53.4

53.4

53.3

53.2

53

52.8

52.8

52.2

129(model)

55.524 ± 3.382

54.936 ± 3.191

54.314 ± 3.045

53.668 ± 2.915

53.165 ± 2.79

52.687 ± 2.681

52.3 ± 2.61

51.929 ± 2.572

51.628 ± 2.567

51.38 ± 2.575

51.207 ± 2.587

51.171 ± 2.62

130(p)

56.9

56.6

56.6

55.4

55.7

55.7

55.7

55.6

54.2

53.9

54.5

53.6

130(model)

56.902 ± 4.128

56.291 ± 3.888

55.648 ± 3.691

54.981 ± 3.561

54.457 ± 3.416

53.958 ± 3.349

53.55 ± 3.356

53.16 ± 3.422

52.839 ± 3.525

52.571 ± 3.48

52.379 ± 3.437

52.325 ± 3.48

126(p)

68.1

67.4

67.6

66

65.7

65.4

65.3

63

62

62.6

61.6

60.6

126(model)

68.316 ± 1.859

67.522 ± 1.757

66.699 ± 1.667

65.855 ± 1.677

65.156 ± 1.607

64.486 ± 1.571

63.91 ± 1.589

63.353 ± 1.691

62.869 ± 1.646

62.441 ± 1.649

62.09 ± 1.604

61.88 ± 1.576

127(p)

51.8

51.3

51.2

50.8

50.4

49.6

49.2

48.7

49.2

48.7

49.2

49.3

127(model)

51.982 ± 1.518

51.45 ± 1.435

50.885 ± 1.364

50.294 ± 1.312

49.845 ± 1.289

49.42 ± 1.274

49.085 ± 1.231

48.766 ± 1.19

48.515 ± 1.152

48.317 ± 1.165

48.193 ± 1.145

48.206 ± 1.205

22(p)

53.4

53

52.4

51.2

51.2

51.3

50.6

50.5

50

49.4

49.9

49.4

22(model)

53.261 ± 1.182

52.709 ± 1.117

52.123 ± 1.074

51.512 ± 1.037

51.044 ± 1.008

50.6 ± 0.971

50.246 ± 1.006

49.908 ± 0.988

49.639 ± 1.0

49.423 ± 0.985

49.282 ± 0.957

49.277 ± 0.973

23(p)

55

54.6

53.9

53.4

52.8

52.7

52.2

51.8

51.5

51.4

51.4

51.4

23(model)

55.229 ± 1.538

54.645 ± 1.457

54.029 ± 1.381

53.387 ± 1.318

52.889 ± 1.261

52.415 ± 1.211

52.032 ± 1.176

51.666 ± 1.139

51.369 ± 1.104

51.125 ± 1.073

50.956 ± 1.05

50.924 ± 1.041

19(p)

56.8

56.1

55.7

54.3

54

51.5

51.4

51.4

52.5

52.4

52.2

50.4

19(model)

56.902 ± 1.893

56.291 ± 1.784

55.648 ± 1.696

54.981 ± 1.615

54.457 ± 1.594

53.958 ± 1.551

53.55 ± 1.982

53.16 ± 2.207

52.839 ± 2.307

52.571 ± 2.243

52.379 ± 2.18

52.325 ± 2.123

20(p)

53

52.8

52.1

50.8

49.8

49.4

49.2

48.4

47.8

48.1

48.2

48

20(model)

53.36 ± 1.413

52.805 ± 1.353

52.218 ± 1.282

51.606 ± 1.223

51.136 ± 1.258

50.691 ± 1.414

50.335 ± 1.524

49.996 ± 1.582

49.726 ± 1.723

49.508 ± 1.909

49.365 ± 1.968

49.359 ± 1.986

29(p)

55.2

55.5

54.1

53.3

54.2

54

52.7

52.3

53

54.2

53.8

53.5

29(model)

53.95 ± 5.381

53.386 ± 5.136

52.79 ± 5.047

52.168 ± 4.87

51.69 ± 4.705

51.235 ± 4.722

50.871 ± 4.779

50.523 ± 4.709

50.245 ± 4.641

50.019 ± 4.69

49.868 ± 4.956

49.853 ± 5.142

30(p)

57

56.2

56

54.8

54.5

54.2

53.6

53.8

53.9

53.1

53.2

52.4

30(model)

57 ± 2.118

56.388 ± 1.995

55.743 ± 1.895

55.074 ± 1.811

54.549 ± 1.74

54.049 ± 1.67

53.639 ± 1.611

53.247 ± 1.556

52.925 ± 1.53

52.656 ± 1.556

52.463 ± 1.525

52.407 ± 1.522

26(p)

57.7

58

56.7

56.5

56

57.2

55.8

55.4

55.4

54.7

53.2

53.1

26(model)

57.492 ± 1.308

56.872 ± 1.24

56.22 ± 1.373

55.543 ± 1.339

55.01 ± 1.394

54.503 ± 1.444

54.086 ± 1.995

53.687 ± 2.116

53.358 ± 2.217

53.082 ± 2.362

52.881 ± 2.416

52.819 ± 2.355

27(p)

60.3

60

58.8

58.1

57.6

57.4

56.8

56.1

55.8

55.3

55.6

55.7

27(model)

60.149 ± 2.482

59.486 ± 2.34

58.792 ± 2.241

58.074 ± 2.135

57.501 ± 2.043

56.953 ± 1.961

56.497 ± 1.904

56.059 ± 1.845

55.692 ± 1.786

55.379 ± 1.732

55.142 ± 1.683

55.043 ± 1.651

4(p)

50

49.4

48.6

47.9

48.3

48.3

47.5

47.3

47.1

47.1

47.3

47.4

4(model)

49.916 ± 1.246

49.417 ± 1.175

48.884 ± 1.114

48.325 ± 1.074

47.908 ± 1.056

47.514 ± 1.037

47.209 ± 1.082

46.92 ± 1.055

46.7 ± 1.038

46.531 ± 1.025

46.435 ± 1.03

46.476 ± 1.077

5(p)

63.7

62.8

62.3

60.7

60

59.6

58.8

58.6

58.1

57.8

57.2

57.1

5(model)

63.888 ± 2.476

63.165 ± 2.336

62.412 ± 2.226

61.636 ± 2.121

61.006 ± 2.099

60.402 ± 2.09

59.891 ± 2.057

59.399 ± 2.063

58.978 ± 2.036

58.612 ± 2.018

58.323 ± 1.997

58.173 ± 2.009

1(p)

64.4

62.6

60.6

59.2

58.8

57.7

57.3

56.6

56.5

56.6

58.2

57

1(model)

64.183 ± 2.634

63.456 ± 2.485

62.698 ± 2.416

61.917 ± 2.622

61.282 ± 2.953

60.674 ± 3.146

60.159 ± 3.416

59.662 ± 3.608

59.237 ± 3.807

58.867 ± 3.918

58.574 ± 3.95

58.42 ± 3.848

2(p)

58.8

58.2

58.2

57.1

56.5

56.2

54.6

55.1

55.2

57.2

57.9

55.9

2(model)

59.46 ± 0.854

58.808 ± 0.916

58.125 ± 0.948

57.418 ± 0.904

56.855 ± 0.884

56.318 ± 0.871

55.872 ± 0.841

55.444 ± 1.041

55.087 ± 1.022

54.783 ± 0.992

54.556 ± 1.483

54.466 ± 2.094

11(p)

54.1

53.5

53.2

52.5

51.8

51.9

50.8

50.3

50

49.5

49.9

49.6

11(model)

53.852 ± 0.897

53.29 ± 0.861

52.695 ± 0.827

52.075 ± 0.844

51.597 ± 0.843

51.144 ± 0.817

50.782 ± 0.883

50.435 ± 0.853

50.158 ± 0.828

49.934 ± 0.806

49.784 ± 0.809

49.771 ± 0.789

12(p)

67.6

67.8

67.7

66.2

65.2

65.1

64.9

63.8

62.9

61.8

61.6

63.2

12(model)

67.922 ± 1.654

67.135 ± 1.573

66.318 ± 1.548

65.48 ± 1.691

64.787 ± 1.67

64.123 ± 1.619

63.552 ± 1.643

63.001 ± 1.73

62.523 ± 1.72

62.1 ± 1.678

61.755 ± 1.636

61.551 ± 1.593

8(p)

50.7

50.2

50.3

50.5

49.9

48.8

47.8

47.7

48.8

48.9

48.3

47.5

8(model)

53.36 ± 4.876

52.805 ± 4.92

52.218 ± 4.943

51.606 ± 4.846

51.136 ± 4.68

50.691 ± 4.543

50.335 ± 4.49

49.996 ± 4.525

49.726 ± 4.524

49.508 ± 4.409

49.365 ± 4.292

49.359 ± 4.205

9(p)

59.6

59.9

59.6

58

57.3

57.6

57.1

57.5

57.4

57.2

56.6

58.1

9(model)

59.755 ± 3.082

59.099 ± 2.905

58.411 ± 2.799

57.699 ± 2.759

57.132 ± 2.646

56.59 ± 2.541

56.14 ± 2.506

55.708 ± 2.469

55.346 ± 2.548

55.039 ± 2.661

54.807 ± 2.774

54.714 ± 2.82

104(p)

55.8

54.9

53.8

52.9

52.6

52.3

51.8

51.4

51.1

51.4

51.4

51.6

104(model)

56.016 ± 1.381

55.42 ± 1.308

54.791 ± 1.282

54.137 ± 1.358

53.627 ± 1.48

53.141 ± 1.529

52.746 ± 1.539

52.369 ± 1.563

52.061 ± 1.587

51.806 ± 1.606

51.626 ± 1.572

51.583 ± 1.533

105(p)

55.7

55

54.7

53.5

52.7

52.6

51.4

50.8

51.5

51.4

51.8

51.8

105(model)

55.918 ± 1.544

55.323 ± 1.462

54.695 ± 1.401

54.043 ± 1.334

53.534 ± 1.314

53.05 ± 1.342

52.657 ± 1.314

52.281 ± 1.423

51.974 ± 1.56

51.721 ± 1.53

51.542 ± 1.493

51.501 ± 1.458

101(p)

53.1

53.3

53.1

51.9

51.6

51.8

51.7

51.4

49.3

48.4

48.4

49.7

101(model)

53.064 ± 2.0

52.515 ± 1.884

51.933 ± 1.853

51.325 ± 1.898

50.859 ± 1.846

50.418 ± 1.818

50.067 ± 1.898

49.732 ± 2.015

49.467 ± 2.117

49.253 ± 2.054

49.114 ± 2.035

49.112 ± 2.007

102(p)

55.5

54.5

55.2

53.7

53.4

53.6

53

52.8

53

51.9

51.8

51.9

102(model)

55.721 ± 1.479

55.129 ± 1.401

54.505 ± 1.385

53.856 ± 1.383

53.35 ± 1.327

52.869 ± 1.274

52.478 ± 1.287

52.105 ± 1.271

51.801 ± 1.277

51.55 ± 1.365

51.374 ± 1.336

51.336 ± 1.315

111(p)

51.8

51.3

50.9

50.4

50.1

50.1

49.8

49.5

49.4

49.1

49

49.1

111(model)

51.588 ± 0.896

51.063 ± 0.855

50.504 ± 0.824

49.919 ± 0.82

49.476 ± 0.832

49.057 ± 0.869

48.728 ± 1.003

48.414 ± 1.113

48.17 ± 1.204

47.977 ± 1.308

47.858 ± 1.375

47.876 ± 1.435

112(p)

50.4

50.7

50.2

48.9

49

50.3

50.6

50.9

50.9

50.4

49

49

112(model)

50.014 ± 1.162

49.514 ± 1.124

48.979 ± 1.3

48.419 ± 1.438

48 ± 1.404

47.605 ± 1.455

47.299 ± 2.001

47.008 ± 2.567

46.786 ± 3.144

46.616 ± 3.632

46.519 ± 3.945

46.558 ± 4.001

108(p)

56.5

55.8

55.7

55.5

55

55.1

54.4

54.6

55.1

56.6

57.1

54.1

108(model)

56.804 ± 4.93

56.194 ± 4.648

55.553 ± 4.413

54.887 ± 4.204

54.364 ± 4.038

53.867 ± 3.891

53.461 ± 3.805

53.072 ± 3.705

52.752 ± 3.664

52.486 ± 3.728

52.295 ± 4.098

52.242 ± 4.545

109(p)

64.8

64.5

63.7

62.2

61.4

61.5

60.8

60.2

59.6

58.9

59.2

59.5

109(model)

64.675 ± 0.839

63.94 ± 0.795

63.174 ± 0.832

62.386 ± 0.852

61.744 ± 0.823

61.128 ± 0.812

60.605 ± 0.807

60.102 ± 0.785

59.669 ± 0.761

59.293 ± 0.739

58.993 ± 0.741

58.832 ± 0.727

Note: 1)122(p) is actual experimental body weight value of subject 122, 122(model) is model body weight confidence interval of subject 122, etc.

2) In all 384 confidence intervals, 26 actual body weight values are outside the confidence interval, but 9 values from these 26 values are within the area of statistical handling error. So the unsatisfied rate of estimation is from 4.4% to 6.77%. It shows our model estimation is acceptable. 3)α is 5%. The confidence interval: estimated body weight ± t α / 2 ( n − 2 ) * S E , where SE is standard error, degree of freedom is n-2.

Reviewer’s report

Reviewer 1 (Dr. E. Cabral Balreira)

The authors propose an interesting model between Energy intake and body weight based on Fick’s second law of diffusion. This is an important are of research and the present article provides an interesting contribution to this research area and can provide new insight into the body weight mechanisms. The work justifying the use of Fick’s second law of diffusion went to a substantial justification and it is well motivated and explained. The referee agrees with the authors that there should be an investigation of the model.

The referee recommends that a major revision of the paper is required in order to be published in the Biology Direct.

The main issues are outlined as follows.

· In the results section, the authors did not fully disclosed their hypotheses that the energy density of body mass is independent of time. They needed this fact to arrive at their model equation (4). This is not supported by the previous discussion and it is big hypotheses that needs more explanation.

Author reply: Generally, for adult men (20-33y) in the Minnesota human starvation study, the change in body weight is largely due to fat mass (FM), but not fat-free mass(FFM). As we can see from Kyle et al., fat-free mass does not change much at middle age (from 18-34y to 35-59y), especially when compared with fat mass which changes significantly during the same period [17]. Considering that the energy density of FM is much higher than that of FFM, the energy change is largely decided by change in FM. Thus, the change in energy intake, d(ρ*V), is approximately the change in energy in fat mass, which can be represented as p*d(V), in which p is the energy density of fat mass. The energy density of fat mass is supposed to be a constant, so we think the formula d(ρ*V) = p*dV is valid and the possible error here won’t affect our conclusion significantly.

· In the section titled simulation of body weight change using the developed model, it is unclear how the authors obtained the experimental data.

Author reply: Total Energy Expenditure (TEE) includes two major parts: Resting Energy Expenditure (REE), the amount of calories needed to maintain basic body systems and body temperature at rest; Activity Energy Expenditure (AEE), the amount of calories used during activity [20]. Net energy intake is the difference between food intake and TEE. Although TEE was not measured in the Minnesota starvation study, TEE can be obtained through calculating REE and AEE [19,21].

REE is calculated from Basal oxygen (cc/min) and kcalorie equivalent per cc/min. The daily energy expenditure at rest converts cc of oxygen/min into liters of oxygen/day, multiplied by the kcalorie equivalent of oxygen. The caloric equivalent of each cc of oxygen consumed in the resting state is calculated on the basis of Thorne Martin Carpenter’s 1921 table [19]. The group’s REE of 994.2 kcal/day at S24 equals group oxygen consumption of 139.1 cc/min multiplied by 1.44 (1440 min divided by 1000) and the groups’ caloric equivalent of oxygen of 4.964 kcal/cc.

We here give an example to show how AEE is calculated. 22 miles per week of outdoor walking means 3.14 miles walking per day. A man’s normal walking speed is 3 miles per hour or so. When a 54 kg man walks with speed of 3 mph, the energy expenditure is 3.6 kcal/min. At S24, the group’s body weight is 52.57 kg. The group’s energy expenditure is (52.57/54)*(3.14/3)*3.6*60 = 220.1(kcal/day) [21]. When a 54 kg subject walks at 3.5 mph for half hour per week on a treadmill, his energy expenditure is 4.2 kcal/min. The group’s energy expenditure is (52.57/54)*4.2*30/7 = 17.52 (kcal/day) [21]. These two parts of walking energy expenditure added, we can know AEE is 237.62 kcal/day.

So at S24, TEE is 1231.83 kcal/day, net energy intake is 409.8 kcal/day.

From their work, in page 8 below equation (9), the authors state that they simply generated data from their own model and use that same data to validate the model. Such approach is circular and does not support the model validation. It simply shows that the ISCEM algorithm is working properly.

The authors must validate their model using the actual experimental data which they display in Table 1. Using the data from the Minnesota human starvation study, the authors need to estimate the parameters of their model, plot the actual results against the model predictions and report the R2 value.

Author reply: In fact, we actually estimated the model parameters using the experimental data from Table1. We actually used the experimental data from Table1to validate the model. We also plotted the actual experimental results against the model predictions and reported the R 2 value.

· Finally, the authors need to better explain how the ISCEM algorithm works and how is the SCEM-UA algorithm optimizing the parameters in their nonlinear problem.

Author reply: Corrected.

Reviewer 2 (Prof. Yang Kuang)

This paper address an interesting but potentially controversial modeling problem that due to the quality or simplicity of the data, may be modeled by other simple or simpler models. There seems to be no real difficulties in fitting the data sets used in the three Figures. For example, using the first few weeks' data, we can find a energy and mass conversion rate for each subject and then use their weekly Total Energy Expenditure (TEE) to predict their weekly weight. Maybe the authors can comment on why such a simple and intuitive approach was not explored?

Author reply: We proposed a molecular diffusion based model to uncover the relationship between energy intake and body weight. We used the data from the Minnesota human starvation study to verify the validity of our molecular diffusion based model. Because the relationship between body weight and energy intake is not linear, to predict body weight simply using the energy and mass conversion rate is not feasible, even if from a pure data fitting purpose.

Reviewer 3 (Dr. Chao Chen)

The authors propose a mathematical model in which body weight at time t is a function of linear combination of an error function, erf(#/#t) (a monotonic increasing function), and its complement 1-erf(#/#t)(a monotonic decreasing function), derived from the hypothesis of molecular diffusion following Fick’s second law. The model is found to have a good fit to a set of data taken from the Minnesota human starvation study. However, only data from the second phase of the study during the 24 weeks starvation period are used for model fitting; excluding data of the control and recovery phases from the same study.

Author reply: In order to make clear how the body weight is affected by energy intake, we chose the data of starvation period from the Minnesota human starvation study.

The authors claim: “This model provides valuable insights into the neural basis of behavioral decisions and their resulting effects”. It is difficult to see, on the basis of the presentation, any mechanistic connection as claimed. This article is just a data fitting exercise because similar models that are linear combination of two monotonic functions of opposing trends can also adequately fit the data.

Author reply: This sentence, “This model provides valuable insights into the neural basis of behavioral decisions and their resulting effects”, is deleted.

We considered that molecular diffusion (of, for example, neuropeptides) plays an important role in body weight changes. Because molecular diffusion is accompanied by energy transference, we then describe the molecular diffusion based process with energy diffusion.

Our purpose is not to do data fitting exercise, but to use the data from the Minnesota human starvation study to verify the validity of our molecular diffusion based model.

Furthermore, this data fitting exercise leaves a lot to be desired: e.g., only the mean body weight over time were analyzed, as presented in Figures 1–3; no body weight changes from individual’s baseline was analyzed; and no statistical analysis, such as confidence intervals, for predicted body weight changes were provided.

Author reply: Please see Appendix A and Appendix B.

Editorial issues:

Pages 7–8. Something must be wrong: it is unlikely that parameters are estimated to be identical when different data sets from S1-S24 and S1–S12 are used.

Author reply: Corrected.

First line on top of p9: “are” should be deleted.

Author reply: Corrected.

Declarations

Acknowledgment

This research is supported by the National Natural Science Foundation of China (No. 31070944).

Authors’ Affiliations

(1)
College of Logistics Engineering, Wuhan University of Technology, Wuhan, China
(2)
School of Medicine, Zhejiang University, Hangzhou, China

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Copyright

© Gong and Gong; licensee BioMed Central Ltd. 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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