## Abstract

This study presents a model describing lipid and protein depletion of an individual facing total starvation. The model distinguishes two compartments of body mass: a metabolic compartment and a structural compartment. It is considered that the lipids and the proteins of the metabolic compartment ensure the totality of physiological functions. The main assumptions of the model lie in the definitions of lipid mass and protein mass of the metabolic compartment, which are related to total lipid mass and total body mass, respectively. Under these assumptions, for a given individual, the ratio of lipid and protein utilization rates is proportional to the adiposity. The model accounts for the protein sparing observed at high adiposity levels and enables us to discuss the individual's survival in relation to the levels of lipid and protein depletion. The time course of changes in lipid and protein depletion rates can be calculated by introducing the energy expenditure of the individual. In simulations, it was assumed that specific energy expenditure was constant during starvation and that mortality occurred at a critical level of protein depletion. The most characteristic results derived from these simulations concern the kinetics of protein depletion, which depend markedly on initial adiposity. Accordingly, in obese subjects, the rate of protein losses remains fairly constant during fasting, whereas it increases from the onset of the fast in lean subjects, in agreement with experimental observations. In the model, protein and lipid depletion rates are both proportional to energy expenditure, which needs to be confirmed from complementary data.

- body composition
- protein sparing
- energy expenditure

during total starvation, the energy required to maintain physiological functions is derived from the oxidation of body lipids and proteins, because glycogen stores are exhausted at the onset of fasting (3). Lipids, however, represent the main energy source. They have a high energy density and can be almost entirely depleted to cover maintenance requirements (45, 49). By contrast, proteins, which have a much lower energy density, can be only partly depleted due to their vital enzymatic, mechanical, and structural roles. Accordingly, their contribution to energy demand is rather modest (17, 28, 33, 37). The result is that, conversely to lipid catabolism, protein catabolism is not clearly related to energy production and remains essentially unexplained (27). It could be attributed to gluconeogenesis, which converts proteins into glucose to meet the glucose requirements of the brain after the exhaustion of glycogen stores (2, 4). However, this hypothesis is unlikely, given the large variability in the rate of protein depletion among individuals (14, 16, 36). To better understand the physiological basis for protein catabolism and its consequences for the survival of individuals, many studies have intended to quantify protein losses during starvation. They have succeeded in pointing out characteristic trends that were observed in humans as well as in animals, fasting spontaneously or submitted experimentally to food deprivation.

From changes in protein metabolism, three phases can be distinguished during a fast (see Ref. 41 for review; also Refs. 20, 32, 34, 43). The first phase (phase I) corresponds to a phase of transition between the fed state and starvation, during which the individual finishes utilizing diet-derived energy. This transient phase is relatively short, between several hours and several days, depending on the individual. It is characterized by a rapid decrease in daily protein losses, usually measured from nitrogen excretion. Throughout the following phase (phase II), daily protein losses remain approximately constant. The duration of this phase depends on the initial lipid mass and was shown to last for several months in obese individuals (8, 43). At the end of phase II, protein utilization increases, which is reflected by a rise in nitrogen excretion (20, 34). This feature, characterizing the entrance to the terminal phase of the fast (phase III) is always very brief, since the important daily protein losses lead shortly to the death of the individual. It must be noted that this last phase is not systematically observed (22, 41). It seems to have never been observed in obese subjects (33).

One of the most fundamental results, pointed out by Voit (48) at the beginning of the last century and later confirmed in many studies, is that, for the same body mass, the level of nitrogen excretion during phase II depends on the initial adiposity of the individual. The higher the initial adiposity is, the lower the nitrogen excretion will be (9, 15, 18, 20, 25, 35). This relationship between protein losses and adiposity could be a determining factor for survival, because it enables the individual to spare its proteins in the case of a prolonged fast. The mechanism of this protein sparing, however, has not yet been entirely elucidated.

The first modeling approach describing the response of an individual to changes in energy balance was proposed by Payne and Dugdale (39, 40). The model postulates the existence of an internal control of lipid and protein utilization processes. The basic assumption is that the ratio of the rates of lipid and protein mobilization during starvation, which is supposed to be equal to the ratio of the rates of lipid and protein deposition during refeeding, is an individual characteristic. Its value is constant within the same individual but varies between individuals. In a recent study, Dulloo and Jacquet (14) analyzed the interindividual variability of this ratio, which could be explained from the sizes of both lipid and protein reserves. However, the basic assumption of the model implies that the lipid and protein depletion rates vary similarly during starvation, which is not compatible with the marked rise in protein catabolism measured at the end of fasting in lean subjects (9, 20).

The present study suggests an alternative approach, which does not assume a constant ratio in the rates of lipid and protein utilization. The proposed model distinguishes two compartments of the body mass, a metabolic compartment and a structural compartment. It is considered that the lipids and the proteins of the metabolic compartment ensure the totality of physiological functions of the individual. The particularity of the model lies on an attempt of quantification of lipid and protein masses of the metabolic compartment, which are related to total lipid mass and total body mass, respectively. Under the assumptions of the model, for a given individual, the ratio of the lipid and protein utilization rates is proportional to the adiposity of the individual. Therefore, this ratio depends on body composition and varies during fasting. The model has been developed with the main objective of describing quantitatively the effect of initial adiposity and energy expenditure on lipid and protein depletion. In the course of the study, the predictions of the model are discussed in relation to some of the most characteristic data previously reported.

## DESCRIPTION OF THE MODEL

The model is developed at the whole body level, and no distinction is made between the various lipid and protein sources. Phase I of fasting is excluded from this study, since during this initial period the individual finishes utilizing its food stores. During total starvation, the body mass is considered as having two compartments, a metabolic compartment and a structural compartment. The metabolic compartment represents the proteins and the lipids, which, at a given time of fasting, are involved in the biochemical processes ensuring the whole physiological homeostasis. The structural compartment is constituted by the other components of the body mass. It includes in particular the proteins and the lipids that, in the course of the fast, will be transferred into the metabolic compartment to supply the substrates required for the maintenance of the physiological functions.

The model is based on the following three assumptions.

*1*) The lipid mass of the metabolic compartment, L_{m}, is proportional to the total lipid mass, L, so that (1) where α is a constant for a given individual.

*2*) The proteins of the metabolic compartment govern the totality of physiological functions of the individual. It is considered that their mass, P_{m}, is proportional to the total body mass, M, so that (2) The β coefficient is also assumed to be constant for a given individual.

*3*) The energy required to maintain physiological functions is provided by the oxidation of the lipids and proteins of the metabolic compartment. The proportion of substrates oxidized is considered to reflect the composition of the metabolic compartment. This can be expressed by the relationship (3) where dL_{m} and dP_{m} represent the lipid and protein losses of the metabolic compartment. Excluding any nitrogen recycling during fasting, which up to now has been observed only in bears during hibernation (38), the oxidized substrates are eliminated from the organism. Therefore, the lipid and protein losses of the total body mass, dL and dP respectively, are equal to the lipid and protein losses of the metabolic compartment, so that (4)

Combining relationships *Eqs. 1*–*4*, one obtains the basic equation of the model (5) where l = L/M is the adiposity, and γ = β/α is a constant characteristic of the individual. According to this relationship, the relative parts of lipids and proteins used during fasting vary with the adiposity of the individual.

In the following sections, it is first demonstrated that *Eq. 5* may account for characteristic data collected in fasting birds or mammals, indicating that the proportion of energy derived from protein at the beginning of starvation is mainly dependent on initial adiposity. This analysis leads to an estimate of the γ coefficient, which is the only characteristic parameter of the model. On the basis of *Eq. 5*, it is then shown that, for a given individual, the changes in lipid and protein masses during starvation can be related to the change in adiposity. A relationship between the initial and final adiposity and the level of protein depletion is deduced, which could explain the differences in body composition observed between individuals at the end of starvation. In the last section, a kinetic extension of the model is derived by introducing the energy expenditure of the individual. In this extension, it will be assumed that specific energy expenditure is constant during fasting. This assumption enables one to calculate the time course of change in adiposity and, subsequently, to express the changes in lipid and protein masses as a function of time.

## APPLICATIONS

### Protein Sparing in Relation to Initial Adiposity

One of the most characteristic results emphasized in pioneer studies is the relationship between the rate of protein depletion and the adiposity of the individual. For a given body mass, the daily protein loss during phase II is inversely related to the initial adiposity. Therefore, a high level of lipid stores enables the individual to spare its protein during starvation.

To compare the effect of initial adiposity on protein losses between individuals or species, the effectiveness of protein sparing is generally expressed in terms of the P_{R} ratio defined as the contribution of protein to energy production (13), so that (6) where c_{L} and c_{P} are the energy equivalents of lipids and proteins, 39.3 and 17.8 kJ/g, respectively (45). It was found that the experimental values of this ratio varied only slightly during phase II (26, 33). The data in Fig. 1, collected by Jenni and Jenni-Eiermann (27), present the P_{R} values determined at the beginning of phase II or during phase II as a function of the initial adiposity of individuals. Cherel and Groscolas (7) proposed a similar compilation, which exhibits the same experimental trends. Although these data are related to various animal species and concern active as well as inactive fasting individuals, it appears that the P_{R} values in phase II depend markedly on the initial adiposity: they decrease when the initial adiposity increases. This result can be discussed using the model.

Taking into account *Eq*. *5*, the P_{R} ratio (*Eq. 6*) can be written as (7)

This equation relates P_{R} to the adiposity for a given γ value. Due to the short duration of phase I, the adiposity at the beginning of phase II does not differ significantly from the initial adiposity l_{i}. Therefore, the contribution of protein to energy production during phase II, P_{R2}, which is expected to be similar the P_{R} value at the beginning of phase II, can be estimated using the relationship (8)

According to *Eq 8*, P_{R2} is mainly dependent on the initial adiposity of the individual and on the value of the γ coefficient. The curves in Fig. 1 illustrate *Eq. 8* for three values of γ: 2.5, 4.5, and 6.5 × 10^{−2}. It can be observed that the upper and lower curves envelop most of the data and that the middle curve describes the general trend of experimental variation. It should be pointed out that the large dispersion of data could reflect the inaccuracy in the determinations of protein and lipid losses and/or initial adiposity. It could also be attributed to interindividual or interspecies variability in the γ coefficient, which cannot be assessed from this small number of data. On the basis of this analysis, the γ value used in the rest of this study will be set at the intermediate value of 4.5 × 10^{−2}. In the model, the reduction in the P_{R} values at high levels of initial adiposity is due to the higher dilution of proteins by lipids in the metabolic compartment.

### Lipid and Protein Masses as a Function of Adiposity

Neglecting glycogen stores, which represent usually <1% of fuel stores and are depleted at the onset of fasting, (3, 23, 30, 45), the total body mass during starvation can be described by (9) where W, P, L, and R are the masses of water, proteins, lipids, and minerals, respectively.

The mass of minerals, generally measured by the mass of ashes after complete combustion, represents ∼4% of body mass in fed individuals. Previous studies have shown that the mass of minerals varies only slightly during fasting (6, 9, 47). On the basis of these results, it will be assumed that the mass of minerals is constant during a fast and that it can be estimated by R = r_{i}M_{i}, where M_{i} is the initial body mass and r_{i} is the proportion of minerals in initial body mass. Because phase I is excluded from this study, in the following developments the initial values of the variables will correspond to the values of these variables at the beginning of phase II.

Because water is relatively insoluble in lipids, body water is essentially associated with proteins. The hydration coefficient of proteins, defined by w = W/P, ranges generally between 2.5 and 3.5. It varies between individuals and species (19). For a given individual, it also varies between organs. Thus muscle proteins, preferentially mobilized during fasting (21, 23, 46), have higher water content than proteins constitutive of the skeleton. Therefore, it can be expected that the hydration coefficient will decrease in the course of fasting. Such a decrease, however, is low in king penguins (6) and is not significant in rats or petrels (9, 24). Accordingly, in this approach the hydration coefficient will be considered constant during starvation.

Under these two assumptions, R and w constant during fasting, *Eq. 9* can be written as (10) with h = 1 + w. At the onset of the fast, the protein mass,P_{i}, calculated from *Eq. 10*, is given by (11)

During fasting, the changes in the masses of the different components can be related by differentiating *Eq. 10*, which gives (12)

For a given change in total body mass, dM, the changes in lipid and protein mass, dL and dP, cannot be calculated from *Eq. 12* alone; it is necessary to introduce a complementary relationship between these components. In the model, such a relationship is given by *Eq. 5*. Taking this equation into account, the change in lipid mass during fasting can be related to the change in adiposity by the relationship (see appendix a) (13a) where C(l_{i}) = [(a − l_{i})/l_{i}]^{δ} and F(l)= [l/(a − l)]^{δ}. The coefficients a = 1 − hγ and δ = 1/a are constants for a given individual. Total body mass and protein mass are then simply calculated as (13b) and (13c)

Throughout this study, the parameter values will be taken as r_{i} = 0.04, h = 4, and γ = 0.045. One deduces that a = 0.82 and δ = 1.22.

According to *Eq. 13, a*–*c*, the masses of the different components are proportional to the initial body mass. The result is that, for a given individual, the change in the M/M_{i}, L/M_{i}, and P/M_{i} ratios during fasting are only dependent on the change in adiposity. Figure 2, *A* and *B* presents the variations of these ratios for two values of initial adiposity, 10 and 45%. They are independent of the energy expenditure of the individual.

The validity of the preceding relationships was examined using data obtained by Robin et al. (44) and presented in Fig. 3, *A* and *C*. They illustrate the changes in protein and lipid mass as a function of total body mass in emperor penguins during a 4-mo fast. The mean body mass of individuals was ∼38 kg at the beginning of the fast and 18 kg at the end of the fast. Their initial adiposity was close to 30%. Experimental data clearly show two phases. In the first phase, for body mass values between 38 and 25 kg, lipid and protein masses decreased linearly with total body mass. In the second phase, mass loss increased for proteins, and it decreased for lipids. Figure 3, *B* and *D* present the theoretical variations calculated from *Eq. 13, a*–*c*. There is good agreement with the experimental trends. It can be noted that the phase of linear decrease was also observed in other species, in particular in the great-winged petrel (24) and in the European hedgehog (5). However, the second phase did not appear in these studies, for the fast was stopped before the entrance in the terminal phase.

### Survival of Individuals in Relation to Levels of Lipid and Protein Depletion

When mortality occurs at the end of a fast, lipid depletion generally exceeds 80%, whereas protein losses represent <50% of initial protein mass. Because lipids constitute the main energy store, a good estimate of the duration of survival can be obtained from initial lipid mass, converted into energy units, and daily energy expenditure measured at the onset of starvation (45).

However, fundamentally, the relationship between the duration of survival and lipid exhaustion is not clearly established. Indeed, in some experiments death occurred before all of the lipid stores had been fully used. Thus, in obese rats having an initial adiposity of 45%, adiposity was still 25% when death occurred, whereas lean rats with an initial adiposity of 10% could deplete their lipid stores almost entirely and survive a 2-wk fast (9). These results support the assumption that mortality is not necessarily associated with lipid exhaustion but results from a critical level of protein depletion leading to an irreversible alteration of physiological functions (17, 33, 37). This assumption can be discussed in the framework of the model.

Let P_{i} and P_{f} designate the initial and final protein masses. The corresponding level of protein depletion, d, is defined by (14)

For a given individual, the change in protein mass during fasting is related to the change in adiposity (Fig. 2, *A* and *B*). The result is that the adiposity at the end of the fast can be calculated from the initial adiposity of the individual and the level of protein depletion. This calculation is addressed in appendix b. It is illustrated in Fig. 4, which presents the final adiposity as a function of the initial adiposity for three levels of protein depletion. It can be observed that a level of protein depletion ranging between 30 and 35% accounts for the results reported in both lean and obese rats, since the final adiposity calculated using *Eq. B3* is ∼2% in lean rats and 22–26% in obese rats. Therefore, the difference in the residual adiposity measured between lean and obese rats at the end of the fast could be explained by the existence of a critical level of protein depletion. This assumption will be retained in the rest of this study. However, it needs to be discussed more accurately from complementary data.

### Kinetics of Lipid and Protein Depletion

A kinetic extension of the model can be achieved by introducing the energy expenditure of the individual. Because during starvation the energy needs are met by the oxidation of lipids and proteins, the energy expenditure E is related to the lipid and protein depletion rates by the relationship (15)

Therefore, by combining *Eqs. 5* and *15* the lipid and protein depletion rates can be expressed as (16a) and (16b) They vary during starvation with the adiposity and the energy expenditure of the individual.

Energy expenditure is not constant during fasting. In particular, it varies with the change in the body mass of the individual. For this reason, it is usually expressed in terms of specific energy expenditure ε, defined as ε = E/M. Some studies have characterized the change in the ε value during fasting. However, the results are not all concordant; they may differ according to the experimental situations (10). Thus, in king penguins, specific energy expenditure was found to remain approximately unchanged in the course of the fast (10, 31), whereas it increased slightly in emperor penguins (12) or decreased slightly in domestic geese (34). In this study, it will be considered that specific energy expenditure is constant during fasting, so that (17) where E_{i} is the initial energy expenditure and ε_{i} is the initial specific energy expenditure. Under this assumption, *Eq. 16, a* and *b* can be written as (18a) and (18b)

It follows from *Eqs. 13* and *18* that lipid and protein depletion rates are proportional to ε_{i} and M_{i} and depend on the initial adiposity of the individual and on its adiposity during fasting. Therefore, their time dependence is only governed by the time dependence of adiposity, which can be derived from *Eqs. A3* and *18a* (see appendix c) and is given by the relationship (19) where a = 1 − hγ, C_{1} = c_{L} + c_{P}γ/a and C_{2} = c_{P}γ/a. It is illustrated in Fig. 5 for two values of initial adiposity and specific energy expenditure. In these simulations, the final level of protein depletion is 40%. According to *Eq. 19*, the effect of a change in the ε_{i} value on the time dependence of adiposity can be taken into account by a change in the time scale, the ratio of the time scales being inversely proportional to the ratio of the specific energy expenditures.

#### Calculation of kinetics of lipid and protein depletion.

For a set of initial values M_{i}, l_{i}, and ε_{i}, lipid and protein depletion rates are first calculated as a function of adiposity using *Eqs. 13* and *18*. They are then expressed as a function of time using *Eq. 19*.

In most fasting experiments, rough experimental data concern daily body mass loss and daily nitrogen loss, which can be measured continuously during starvation with noninvasive methods. These data can be simply derived from the preceding calculations, as total body mass loss is related to lipid and protein mass losses by *Eq. 12*, and nitrogen mass loss can be deduced from protein mass loss by considering that 6.25 g of protein contain 1 g of nitrogen (*45*).

#### Effect of initial adiposity on kinetics of lipid and nitrogen depletion.

Kinetics of daily nitrogen excretion and daily body mass loss measured in fasting lean and obese rats (9) were compared with the simulations of the model. This comparison is presented in Fig. 6. The theoretical curves were calculated using the initial values M_{i} = 250 g and l_{i} = 0.1 for lean rats, and M_{i} = 500 g and l_{i} = 0.45 for obese rats. The specific energy expenditure, ε_{i}, which was not measured experimentally, was set in the simulations to account approximately for the level of nitrogen excretion in phase II (0.30 and 0.27 kJ·g^{−1}·day^{−1} for lean and obese rats, respectively). In Fig. 6, *B* and *D*, the final level of protein depletion was 30% for the continuous lines and 40% for the dashed lines.

At the beginning of the fast, experimental curves (Fig. 6, *A* and *C*) show a phase of rapid decrease in daily nitrogen excretion and daily body mass loss. As mentioned in the presentation of the model, this phase, which corresponds to the transition between fed state and starvation (phase I) is not taken into account in this analysis. If one excludes this initial phase, it can be observed that the curves calculated using the same parameter values (Fig. 6, *B* and *D*) describe satisfactorily the experimental data for both lean and obese rats. In particular, the level of nitrogen excretion (Fig. 6*B*) remains approximately constant in obese rats, whereas it increases from the onset of fasting in lean rats.

The effect of initial adiposity on the kinetics of lipid and nitrogen depletion is presented more specifically in Fig. 7. The different curves correspond to four levels of initial adiposity, 10, 20, 30, and 45%. They were calculated for the same initial values of body mass and specific energy expenditure (M_{i} = 1 kg and ε_{i} = 0.3 kJ·g^{−1}·day^{−1}). In this analysis, the initial M_{i} and ε_{i} values may be chosen arbitrarily, since a modification in the M_{i} value can be accounted for by a change in the scale of the *y*-axis (*Eqs. 13* and *18*) and a modification in the ε_{i} value by a change in the scale of both axes (*Eqs. 18* and *19*). As expected, for the same protein depletion, the duration of fasting increases with the initial adiposity of individuals. In Fig. 7*A*, it can be noted that, at the beginning of starvation, daily lipid losses depend only slightly on initial adiposity. For the four levels of initial adiposity, the rate of lipid depletion decreases almost linearly with time, but the decrease is much more pronounced at the low l_{i} values. However, the most characteristic results are related to daily nitrogen losses (Fig. 7*B*). In accord with the protein-sparing process discussed in the first section, daily nitrogen losses at the beginning of fasting decrease when initial adiposity increases. For levels of initial adiposity >30%, the rate of nitrogen depletion remains approximately unchanged during fasting, but it increases progressively with time at the lower levels of initial adiposity. Therefore, conversely to what was suggested by Goodman et al. (20), the final rise in nitrogen excretion observed in lean subjects does not necessarily reflect a change in the regulatory controls of protein metabolism. In the model, it could result from the increase in the protein to lipid ratio of the metabolic component, due to the exhaustion of lipid stores.

#### Effect of energy expenditure on kinetics of lipid and protein depletion.

According to *Eq. 16, a* and *b*, lipid and protein depletion rates are proportional to energy expenditure. The proportionality between lipid depletion rate and energy expenditure is expected, at least qualitatively, since lipids are the main energy source during the major part of the fast. In contrast, due to the low contribution of proteins to energy production, the rate of protein depletion is not necessarily related to energy expenditure, and the validity of *Eq. 16b* needs to be assessed.

To date, only one study (5) analyzed the effect of a reduction in energy expenditure on lipid and protein depletion rates. In that study, lipid and protein losses were measured in two groups of hedgehogs, one group being maintained at the constant temperature of 20°C (shallow hypothermal fast) and the second one being transferred at 5°C (deep hypothermal fast). The energy expenditure was ∼2.4 times lower in shallow- than in deep-hypothermal fasting hedgehogs. This study concluded that there was a similar dependence of lipid and protein losses with energy expenditure, which agrees with *Eqs. 16*.

In the case of an increase in energy demand, the data from the literature are less accurate due to the complexity of experimental approaches. Studies have been carried out in birds fasting in the cold or during migration, which are generally considered sustained periods of starvation. Their metabolic requirements were two or more times those of inactive birds fasting at thermoneutrality (1, 29, 42). The few data reported on migrating birds suggested proportionality between lipid and protein losses and energy expenditure (see Ref. 27), which also agrees with *Eqs. 16*. However, in the case of cold-stressed birds, the results are contradictory, as Klaassen and Biebach (29) reported a similar increase in lipid and protein catabolism, whereas Thouzeau et al. (47) concluded that there was an increase in lipid catabolism without significant change in protein catabolism. Complementary data are therefore required to specify this effect.

## CONCLUSION

The model presented here attempts to describe the changes in the rates of lipid and protein depletion during starvation. The basic equation of the model stipulates that during fasting, the ratio of lipid and protein depletion rates is proportional to the adiposity of the individual. The coefficient of proportionality, *γ*, is the only characteristic parameter of the model. The model accounts for the change in the contribution of proteins to energy production with the initial adiposity of the individual. It enables discussing the survival of individuals in relation to the level of lipid and protein depletion. Thus it was shown that in lean and in obese subjects, mortality could be associated to a critical level of protein depletion. These above results are independent of the energy expenditure of the individual.

The kinetics of lipid and protein losses can also be calculated from the model by introducing energy expenditure. They are found to be mainly dependent on initial adiposity and energy expenditure. In the simulations, it was considered that specific energy expenditure was constant during fasting, and that mortality occurred for a critical level of protein depletion. From the results relative to daily protein losses, it was shown that the observation of phases II and III of fasting described in previous studies depends essentially on the initial adiposity of the individual. For levels of adiposity higher than 30%, the rate of protein depletion remains approximately constant over all the duration of the fast. Only phase II is observed. By contrast, for levels of adiposity less than 20% the rate of protein depletion increases from the onset of fasting, reflecting a rapid entrance in phase III. In the model, the rates of lipid and protein depletion are both proportional to energy expenditure, which cannot be confirmed from the data presently available. Complementary experiments associating measurements of body composition, nitrogen excretion and energy expenditure will be required to specify the validity of the model, which could constitute a simple framework to compare quantitatively the response to fasting between individuals and species.

## APPENDIX A

#### Change in lipid mass as a function of the change in adiposity.

Using the basic equation of the model (*Eq. 5*), the change in total body mass during fasting (*Eq. 12*) can be written as (A1)

On the other hand, differentiation of the *l* = L/M ratio leads to the relationship (A2)

Combining *Eqs. A1* and *A2*, one obtains the differential equation (A3) which relates the change in lipid mass to the change in adiposity. Integration of this equation gives (see Ref. 11) (A4) where a = 1 − hγ and δ = 1/a are constants for a given individual. C is the constant of integration, which can be determined by introducing the initial condition L = L_{i} for l = l_{i}

## APPENDIX B

#### Relationship between change in adiposity and level of protein depletion.

The protein mass at the end of the fast, P_{f}, can be calculated from the initial protein mass and the level of protein depletion. Combining *Eqs. 11* and *14* yields (B1)

Using *Eq. 13, a*–*c*, P_{f} can also be expressed as (B2) where l_{f} is the final adiposity. Identification of *Eqs. B1* and *B2* leads to the equation (B3) which relates initial adiposity, final adiposity, and the level of protein depletion.

## APPENDIX C

#### Time course of change in adiposity during fasting.

On the assumption that specific energy expenditure is constant during fasting, the change with time in adiposity can be calculated by combining *Eqs. A3* and *18a*, which leads to the differential equation (C1)

On the right side of this equation, the minus sign corresponds to the fact that *Eq. 18a* describes the kinetics of lipid losses. This preceding equation can be integrated analytically (see Ref. 11). Introducing the initial condition l = l_{i} for *t* = 0, one obtains the time dependence of adiposity given in *Eq. 19*.

## Acknowledgments

I thank Dr. Yvon Le Maho for helpful comments. My thanks are also due to Alexandre Zahariev for technical assistance in the manuscript preparation.

## Footnotes

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- Copyright © 2004 by American Physiological Society