Serious analytical inconsistencies challenge the validity of the energy balance theory

Abstract

Energy metabolism theory affirms that body weight stability is achieved as over time the average energy intake equals the average energy expenditure, a state known as energy balance. Here it is demonstrated, however, that weight stability coexists with a persistent energy imbalance. Such unexpected result emerges as a consequence of the answers to three fundamental problems: 1. Is it possible to model body weight fluctuations without the energy balance theory? And if so, what are the benefits over the energy balance strategy? 2. During energy balance, how the oxidized macronutrient distribution that underlies the average energy expenditure is related to the macronutrient distribution of the average energy intake? 3. Is energy balance possible under a low-fat diet that simultaneously satisfies the following conditions? (a) The fat fraction of the absorbed energy intake is always less than the oxidized fat fraction of the energy expenditure. (b) The carbohydrate fraction of the absorbed energy intake is always greater or equal to the oxidized carbohydrate fraction of the energy expenditure. The first of these issues is addressed with the axiomatic method while the rest are managed through analythical arguments. On the whole, this analysis identifies inconsistencies in the principle of energy balance. The axiomatic approach results also in a simple mass balance model that fits experimental data and explains body composition alterations. This model gives rise to a convincing argument that appears to elucidate the advantage of low-carbohydrate diets over isocaloric low-fat diets. It is concluded, according to the aforementioned model, that weight fluctuations are ultimately dependent on the difference between daily food mass intake and daily mass loss (e.g., excretion of macronutrient oxidation products) and not on energy imbalance. In effect, it is shown that assuming otherwise may caused unintended weight gain.

Keywords: Applied mathematics; Biological sciences; Body weight; Energy balance; Health sciences; Low-carbohydrate diet; Low-fat diet; Mass balance; Metabolism; Pathophysiology; Physiology.

Figure 1

A non-zero energy balance can coincide with a null mass change. A. When heat (Q) is supplied, work (W) is done as the expanding gas lifts the mass m through a distance h; energy balance is positive (ΔE > 0) yet the gas mass (m_gas) has not changed since the number of gas molecules is constant during expansion. B. Energy balance may be positive or negative yet the mass change that may occur during energy flux is not required by the First Law of Thermodynamics to mirror the energy balance direction. As illustrated, when a fixed amount of hot water is taken out of a water-heater and simultaneously replaced by the same amount of cold water, energy balance is negative (ΔE < 0) yet the system's mass remains constant.

Figure 2

Axioms of body weight fluctuations describe weight loss dynamics. A. Body weight (BW) remains stable around 100kg (gray trace) when $\overline{EMP}=\left[E{I}_{avg}\left(F/100\right)\right]/{\rho }_F+\left[E{I}_{avg}\left(C/100\right)\right]/{\rho }_C+\left[E{I}_{avg}\left(P/100\right)\right]/{\rho }_P\approx 0.622\text{kg}$ where $E{I}_{avg}=3\text{,}214\text{kcal}=13.454\text{MJ}\left(30\text{\% F,}50\text{\% C,}20\text{\% P}\right)$. If $\overline{EMP}$ is decreased by 15%, 30% and 55% BW stabilizes at the corresponding reduced mean weight (dashed lines). All simulations contain 730 days or iterations. The kth-iteration consisted of the following computations: First, three random numbers $\{x_k,y_k,z_k\}$ were drawn from a normal standard distribution. Second, $x_k$, $y_k$ and $z_k$ were used to compute: $EM{P}_k=\overline{EMP}(1+\frac{CVar}{100}{x}_k);nEM{P}_k=\overline{nEMP}(1+\frac{CVar}{100}{y}_k);F_{loss,k}=\overline{F}{}_{loss}(1+\frac{CVar}{100}{z}_k)$, where $\overline{nEMP}=1\mathrm{kg}$, ${\overline{F}}_{loss}=0.02492$ and CVar (coefficient of variation) = 10%. Finally, body weight was updated according to $w_{k+1}=EP{M}_k+nEP{M}_k+(1-[F_{loss,k}-{\rho }_{O_2}PAL{v}_{O_2}])\cdot w_k$, where ${\rho }_{O_2}=(1/770)\text{kg/L}$, PAL = 1.5 and $v_{O_2}=3.1\text{ml/}\left(\text{kg×min.}\right)\approx 4.49\text{L/}\left(\text{kg×day}\right)$ are fixed. B.Eq. (2) (continuous version, black curve) approximates the weight loss trajectory (gray trace as in A). The red curve is the absolute value of the continuous form of Eq. (3) (|Δw|). As |Δw| approaches zero BW stabilizes. According to Eq. (2) this happens in about 5τ days. Black curve: $w\left(d\right)=\frac{1.28}{0.01622}+\left(100-\frac{1.28}{0.01622}\right){\left(1-0.01622\right)}^d$, d: days; Red curve: $\left|\Delta w\left(d\right)\right|=0.01622\left(100-\frac{1.28}{0.01622}\right){\left(1-0.01622\right)}^d$. C. To simulate process of metabolic adaptation (black trace) the body weight updating formula is change to $w_{k+1}=EP{M}_k+nEP{M}_k+(1-(1-a)[F_{loss,k}-{\rho }_{O_2}PAL{v}_{O_2}])\cdot w_k$ where $a=f(0.622,0.280,\mathrm{1,1})=0.031628$. As shown, the inclusion of this physiological response limits the amount of lost weight (black trace vs. gray trace). The gray trace is the same as in part A where there is no metabolic adaptation. Red curve: $w\left(d\right)=\frac{1.28}{0.0157}+\left(100-\frac{1.28}{0.0157}\right){\left(1-0.0157\right)}^d$. D. The change in fat mass that underlies the weight loss trajectory depicted in part C (black trace) is computed with Eq. (6) generating the shown graph. The initial fat mass (FM) was 35kg and the dashed line represents the average fat mass computed with all FM values after day 300.

Figure 3

The energy proportion from fat, under a clamped caloric intake, determines the amount of ingested mass. The figure exemplifies how the energy proportion from fat impacts the amount of ingested nutrient mass. In the figure the energy densities of F, C and P are as in [9]: ρ_F = 9.4 kcal/g, ρ_C = 4.2 kcal/g, ρ_F = 4.7 kcal/g. A. The graph illustrates that, under clamped energy intake (2,500 kcal = 10.465 MJ), as the fat fraction increases the ingested nutrient mass decreases (black line). B. The effect observed in A is due to the fact that as the fat fraction increases the energy density (black curve) also increases meaning that a same level of energy intake can be achieved with the ingestion of less nutrient mass.

Figure 4

Simulation: low-fat diet vs. low-carbohydrate diet. A. First eight weeks of two simulated 90kg obese individuals under different isocaloric diets: low-fat diet (LFD; 1,300 kcal = 5.44 MJ, F: 30%, C: 55%, P: 15%) vs. low-carbohydrate diet (LCD; 1,300 kcal = 5.44 MJ, F: 70%, C: 15%, P: 15%). The EI of both subjects before the intervention was 2,300 kcal = 9.63 MJ (F: 35%, C: 50%, P: 15%). Although both subjects are expected to experience similar levels of energy imbalance, the LCD resulted in greater weight loss in contrast to the LFD. Plots were computed with Eqs. (8) and (9) by letting k = 0, 7, 14, …, 56. B. The mass balance model predicts a greater decline in fat mass for the LCD in contrast to the LFD. Plots were computed with Eq. (6) using the weight loss sequences depicted in A. C. The figure illustrates the absolute value of the daily weight change ($\left|\Delta w_k\right|$) for both diets. The LCD's daily weight change is greater than that in the LFD. Over time this yields a much faster and greater weight loss as observed in A. Plots were computed with Eqs. (10) and (11) by letting k = 0, 7, 14, …, 56. D. Simulation similar to part A but with reductions in $\overline{nEMP}$ (4% in LFD vs. 6% LCD). Notice that small reductions in $\overline{nEMP}$ enhance the weight loss evoked by reductions in $\overline{EMP}$. This is expected as the $\overline{nEMP}$ is typically the largest component of $\overline{M}$. Such behavior is consistent with experimental data since changes in water intake and minerals result in detectable changes in body weight [26]. FM: fat mass, BW: body weight.

Figure 5

Eqs. (2) and (6) fit experimental data. The figure illustrates fits of the continuous version of Eq. (2) to weight loss data from Brehm et al. [13] in A₁ and Brehm et al. [14] in B₁. Estimates of parameters in Eq. (2) are rounded to four decimal places. The fat mass data from Brehm et al. [13] (A₂) and Brehm et al. [14] (B₂) was fitted with following version of Eq. (6)$F{M}_{k+1}=\beta W\left\{\frac{F{M}_k}{\beta }\exp \left(\frac{w_{k+1}-w_k+F{M}_k}{\beta }\right)\right\}$. During curve-fitting procedure body weights $w_{k+1}$ and $w_k$ were obtained from solid curves in A₁ and B₁. Estimates of β are rounded to two decimal figures. Scatter data in panels A₁ and B₁ were extracted from graphs in the original publications using GetData Graph Digitizer version 2.26.0.20. FM: fat mass, BW: body weight.

Figure 6

Mass balance may occur in the absence of energy balance. Hypothetical macronutrient mass input-output pattern that illustrates that it is possible to achieve weight stability without energy balance. Circles in the diagram represent macronutrient body reserves. The left dashed box contains the input mass while the right box encloses the output mass. Energy densities are ρ_F = 9.4 kcal/g = 39.33 MJ/kg, ρ_C = 4.2 kcal/g = 17.6 MJ/kg, ρ_P = 4.7 kcal/g = 19.7 MJ/kg [9]. Although mass balance is achieved, energy balance is not since some of the absorbed or stored protein may be transformed into glucose (gluconeogenesis, GNG) or lost through EE-independent routs (EEIPL); absorbed or stored fat may also undergo gluconeogenesis; and some of the absorbed glucose may be transformed into fat (de novo lipogenesis, DNL). All these metabolic processes render energy balance not possible. AEI: absorbed EI; EEF: EE fuel; EEIPL: EE-independent protein loss.

Figure 7

The application of the EBT may lead to unintended weight gain. Current dietary guidelines advise subjects with adequate body weight to minimize health risks and avoid weight gain by adopting isocaloric LFDs [43]. Such practice may, however, result in unplanned weight gain. The figure simulates the possible effect of exchanging a high-fat diet (HFD; F: 50%, C: 40%, P: 10%) for an isocaloric LFD (F: 20%, C: 65%, P: 15%). Under the HFD, body weight was stable at ~70kg. After beginning the isocaloric LFD (day 250), body weight increases towards a steady value of ~74.3kg in order to accommodate the increased mass intake (77g) inherent to this diet. The application of the EBT may, therefore, caused unintended consequences as it fails to account the mass balance state and thus it cannot properly predict body weight evolution. Simulation algorithm was similar to that in Figure 2 A. Here $w_0=70\text{kg}$; ${\rho }_{O_2}\cdot PAL\cdot v_{O_2}=0.0087$; $nEM{P}_k=(1+0.05\cdot y_k)$;$F_{loss,k}=0.0278285(1+0.05\cdot z_k)$; and if k < 250 then $EM{P}_k=0.339(1+0.05\cdot x_k)$, otherwise $EM{P}_k=0.416(1+0.05\cdot x_k)$.