# Proposal of a Mathematical Model to Monitor Body Mass over Time in Subjects on a Diet

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

_{V}> 0

_{i}

_{v}dt; ∫ dV(t)/V(t) = −k

_{v}∫ dt; lnV(t) = −k

_{v}t + cost.;

_{i}; lnV(t) = lnM

_{i}− k

_{v}t;

_{i}= −k

_{v}t; ln[V(t)/M

_{i}] = −k

_{v}t; V(t)/M

_{i}= e

^{− k}

_{v}

^{t};

_{old}(t) = Mi e

^{−k}

_{v}

^{t}

_{n}> 0

_{f}

_{n}N(t) + Q(t)

_{n}n = q/s; n = q/s (s + k

_{n});

_{n}) = A/s + B/(s + k

_{n}); A = q/k

_{n}; B = −q/k

_{n}

_{n}) (1 − e

^{−knt})

_{f},

_{new}(t) = M

_{f}(1 − e

^{−k}

_{n}

^{t})

- with: k
_{vn}, k_{nv}, k_{ve}, k_{ve}> 0 - with:
**k**= k_{e}_{ve}= k_{ne};**k**= k_{v}_{vn}+ k_{ve};**k**= k_{n}_{nv}+ k_{ne} - where: k
_{v}− k_{vn}= k_{n}− k_{nv}and k_{v}+ k_{nv}= k_{n}+ k_{vn}=**K**

_{v}V(t) + k

_{nv}N(t)

_{n}N(t) + k

_{vn}V(t)

_{v}v − k

_{nv}n = V

_{x}

_{n}n − k

_{vn}v = N

_{x}

_{v}) v − k

_{nv}n = V

_{x}

_{vn}v + (s + k

_{n}) n = N

_{x}

v | n | Known |

(s + k_{v}) | −k_{nv} | V_{x} |

−k_{vn} | (s + k_{n}) | N_{x} |

_{t}= (s + k

_{v}) (s + k

_{n}) − k

_{nv}k

_{vn}= (s + α) (s + β); (the roots of the equation in s

^{2}are α, β)

_{v}= V

_{x}(s + k

_{n}) + N

_{x}k

_{nv}

_{n}= N

_{x}(s + k

_{v}) + V

_{x}k

_{vn}

_{x}(s + k

_{n})/(s + α)(s + β) + N

_{x}k

_{nv}/(s + α)(s + β)

_{x}(s + k

_{v})/(s + α)(s + β) + V

_{x}k

_{vn}/(s + α)(s + β)

_{x}(s + k

_{n})/(s + α)(s + β) = A/(s + α) + B/(s + β)

_{x}k

_{nv/}(s + α)(s + β) = C/(s + α) + F/(s + β)

_{x}(s + k

_{n}) = A (s + β) + B (s + α) = s(A + B) + A β + B α

_{x}, and V

_{x}k

_{n}= A β + B α

_{x}− B

_{x}k

_{n}= V

_{x}β − B β + B α = V

_{x}β − B (β − α)

_{x}(k

_{n}− β) = −B (β − α)

_{x}(k

_{n}− β)/(β − α)

_{x}− B = V

_{x}+ V

_{x}(k

_{n}− β)/(β − α)

_{x}(k

_{n}− α)/(β − α)

_{x}k

_{nv}= C (s + β) + F (s + α) = s (C + F) + C β + F α

_{x}k

_{nv}= C β + F α

_{x}k

_{nv}= −F β + F α = −F (β − α);

_{x}k

_{nv}/(β − α)

_{x}k

_{nv}/(β − α)

_{x}(s + k

_{n})/(s + α)(s + β) + N

_{x}k

_{nv}/(s + α)(s + β)

_{x}(s + k

_{v})/(s + α)(s + β) + V

_{x}k

_{vn}/(s + α)(s + β)

**V(t) =**V

_{x}(k

_{n}− α)e

^{−}

^{αt}/(β − α) − V

_{x}(k

_{n}− β)e

^{−}

^{βt}/(β − α) + N

_{x}k

_{nv}e

^{−}

^{αt}/(β − α) − N

_{x}k

_{nv}e

^{−}

^{βt}/(β − α)

**N(t) =**N

_{x}(k

_{v}− α)e

^{−}

^{αt}/(β − α) − N

_{x}(k

_{v}− β)e

^{−}

^{βt}/(β − α) + V

_{x}k

_{vn}e

^{−}

^{αt}/(β − α) − V

_{x}k

_{vn}e

^{−}

^{βt}/(β − α)

_{x}(k

_{n}− α + k

_{vn})e

^{−}

^{αt}/(β − α) + N

_{x}(k

_{v}− α + k

_{nv})e

^{−}

^{αt}/(β − α)+

_{x}(k

_{n}− β + k

_{vn})e

^{−}

^{βt}/(β − α) − N

_{x}(k

_{v}− β + k

_{nv})e

^{−}

^{βt}/(β − α)

_{x}(k

_{n}− α + k

_{vn}) + N

_{x}(k

_{v}− α + k

_{nv})] e

^{−}

^{αt}/(β − α)+

_{x}(k

_{n}− β + k

_{vn}) + N

_{x}(k

_{v}− β + k

_{nv}) ] e

^{−}

^{βt}/(β − α)

_{e}= k

_{v}− k

_{vn}= k

_{n}− k

_{vn}; k

_{v}+ k

_{vn}= k

_{n}+ k

_{vn}= K

_{x}(K − α) + N

_{x}(K − α)] e

^{−}

^{αt}/(β − α) − [V

_{x}(K − β) + N

_{x}(K − β)] e

^{−}

^{βt}/(β − α)

_{redistributive}(t) = (V

_{x}+ N

_{x})(K − α) e

^{−}

^{αt}/(β − α) − (V

_{x}+ N

_{x})(K − β) e

^{−}

^{βt}/(β − α)

_{redistributive}(t) = (M

_{i}+ M

_{f}) (e

^{−αt}− e

^{−βt})

_{redistributive}(0) = 0

_{redistributive}(t→∞) = 0

_{c}(t) = M

_{i}e

^{−kv}

^{t}+ M

_{f}(1 − e

^{−kn}

^{t}) + (M

_{i}+ M

_{f})(e

^{−}

^{α}

^{t}− e

^{−}

^{β}

^{t})

_{c}(t) = M

_{i}e

^{−}

^{kmi}

^{t}+ M

_{f}(1 − e

^{−}

^{kmf}

^{t}) + (M

_{i}+ M

_{f})(e

^{−}

^{α}

^{t}− e

^{−}

^{β}

^{t}

**)**

_{c}is the body mass at time t, M

_{i}the body mass at the beginning of the observations, M

_{f}the body mass at the end, respectively, and α and β are the constants controlling the distributive process. To facilitate the calculations necessary to obtain the numerical solution of the model on the experimental data set, specially compiled software (Dies4) was used. Body mass values were measured monthly by means of an electronic balance (sensitivity of 0.1 kg). The metabolic rate constants were calculated with the following formula:

_{m}[h

^{−1}] = −(1/24) ln [(M

_{c}− M

_{a})/M

_{c}]

_{c}(kg) is the body mass and M

_{a}(kg) the daily food mass.

_{C}and M

_{a}, the metabolic constants of the body mass can be calculated at any time. A balanced diet was assigned for each of the patients with the aim of making the patients tend toward normal with respect to their BMI interval. Body mass was measured initially and then subsequently at each control. The initial food mass was reported by the patients through the food diary and subsequently replaced by the dietary mass of the assigned diet. Patients were required to follow the diet scrupulously and to maintain the same lifestyle at least as long as necessary to carry out the checks. For each of the patients, the metabolic constants were calculated during the control visits using the same formula with which the initial values were calculated. Therefore, the constants varied because the body mass, not the food mass, varied over time. The metabolic constants are stable when body mass stabilizes. Considering body mass as a result of the three catabolic, anabolic, and redistributive phases for each patient, a temporal trend was constructed by inserting the initial, intermediate, and final constants into the formal solution of the model. The expected body masses were calculated, whereas the observed values were measured experimentally during the control visits. At every time on the curve, each patient was characterized by a couple of expected–observed values. A database was constructed by randomly inserting the following parameters for each patient: sex, age, initial body mass, body mass index (BMI), initial k

_{m}, final body mass, final k

_{m}, number of control visits, deviance, and variance. Figure 1 shows an example of fitting a patient’s body mass.

_{observed}− Y

_{expected})

^{2}. The variance is Var = $\frac{\mathrm{Dev}}{\mathrm{DF}}$, where Dev = deviance and DF = degrees of freedom. Table S1 shows all values obtained for the studied subjects and Table S2 reflects all values with statistical analysis.

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**An example of fitting a patient’s body mass. The expected body mass (blue line and points) is the sum of three contributions from the catabolic (dark line), anabolic (green line), and redistributive phases (yellow line).

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**MDPI and ACS Style**

Soriano, J.M.; Sgambetterra, G.; Boselli, P.M.
Proposal of a Mathematical Model to Monitor Body Mass over Time in Subjects on a Diet. *Nutrients* **2022**, *14*, 3575.
https://doi.org/10.3390/nu14173575

**AMA Style**

Soriano JM, Sgambetterra G, Boselli PM.
Proposal of a Mathematical Model to Monitor Body Mass over Time in Subjects on a Diet. *Nutrients*. 2022; 14(17):3575.
https://doi.org/10.3390/nu14173575

**Chicago/Turabian Style**

Soriano, Jose M., Giovanna Sgambetterra, and Pietro Marco Boselli.
2022. "Proposal of a Mathematical Model to Monitor Body Mass over Time in Subjects on a Diet" *Nutrients* 14, no. 17: 3575.
https://doi.org/10.3390/nu14173575