# Construction of fitted behavior laws from experimental data for muscle aging simulation

Inès Douania, Jeremy Laforet, Sofiane Boudaoud

► To cite this version:

Inès Douania, Jeremy Laforet, Sofiane Boudaoud. Construction of fitted behavior laws from experimental data for muscle aging simulation. Multidisciplinary Biomechanics Journal, 2025, 49th congress of the Société de Biomécanique, 10.46298/mbj.14581. hal-04738074v2

HAL Id: hal-04738074

https://hal.science/hal-04738074v2

Submitted on 31 Jan 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Construction of fitted behavior laws from experimental data for muscle aging simulation

Jeremy Laforet*, Inès Douania, Sofiane Boudaoud

a Université de technologie de Compiègne, CNRS, UMR 7338 Biomechanics and Bioengineering, Centre de recherche Royallieu, 60203 Compiegne cedex, France

* Corresponding author: jeremy.laforet@utc.fr

Received date: 07/04/2024

Accepted date: 28/06/2024

Publication date: 31/01/2025

**Keywords:** aging, muscle model, parameter fitting

© 2025 The Authors
Licence CC-BY 4.0
Published by Société de Biomécanique

## Introduction

Developing realistic numerical model of physiological systems generated recently a rising interest from the medical community. These models should help the clinician to refine his diagnosis related, for example, to muscle aging pathologies as sarcopenia. In order to simulate muscle aging, it is mandatory to be able to set model parameters to their correct values at any age (Douania et al. 2019). However, literature data being sparse we propose to set relationship between age and parameter values. We focused on the Biceps brachii (BB) for our multiscale and multiphysics models (Carriou et al. 2016) as it is a fusiform muscle, used in many daily tasks and practical for the recording of HD-sEMG signals. Hence this study will focus of this particular muscle. Our approach in this work is to use regression analysis to interpolate the parameter value between the ones available in the literature and be able to simulate realistic configurations at any age, but also to simulated aging of a given virtual subject. The abstract is organized as follows: a literature report of the available parameter values for each subject category (by age and sex), data sources with high standard deviation or inadequate measurement protocol were excluded, modeling the relationship between aging and parameter values using classical regression models.

The Cross-Sectional Area (CSA) of BB muscle are rarely reported in the literature. Usually the BB, Brachialis and Triceps are reported together globally. Terms like Anatomical Cross-Sectional Area (ACSA)

and Physiological Cross-Sectional Area (PCSA) are also frequently used in these studies. ACSA design the muscle CSA measured in the plane perpendicular to its tendons (the longitudinal axis), at the widest point along the muscle. PCSA is the muscle CSA measured in the plane perpendicular to the muscle fibers. It is obtained by dividing the muscle volume by its true fiber length. For the BB muscle (fusiform muscle, the PCSA do not vary which is not the case for pinnate muscles.

## 2. Methods

The Table 1 shows CSA values of BB muscle with the elbow positionned at 90° of flexion. We should mention that many techniques for muscles CSA measurement are reported in the literature. The most used are the computed tomography (CT), the MRI and Ultrasound Extended-Field-Of-View (UEFOV). No significant difference are noted between results using these techniques, only the time cost is a major advantage for the UEFOV method. We assumed the young group to be bellow the age of 30 and the elder one above 60 in accordance with most studies we reviewed. Studies or subjects groups with intense sports practice were excluded for this work.

### 2.2 Growth models

Four growth model, linear, exponential, logistic and Gompertz model, were fitted from experimental data and the plausibility fo the obtained relationship were assessed.
<table><caption>**Table 1:** Comparison between the CSA of the biceps brachii muscle reported in the literature (YM: Young Men; EM: Elder Men).</caption><thead><tr><th>Subject</th><th>Age (yr)</th><th>CSA (cm²)</th><th>Number of subject</th><th>Modality</th><th>Reference</th></tr></thead><tbody><tr><td rowspan="4">YM</td><td>23.3±3</td><td>12.7±2.3</td><td>20</td><td>MRI</td><td>Klein et al. 2002</td></tr><tr><td>23.3±1.3</td><td>12.4±1.2</td><td>8</td><td>CT</td><td>Miller et al. 1993</td></tr><tr><td>28±0.1</td><td>15.3±2.5</td><td>8</td><td>CT</td><td>Klitgaard et al. 1990</td></tr><tr><td>12~25</td><td>11.8±2.7</td><td>8</td><td>MRI</td><td>McCall et al. 1996</td></tr><tr><td rowspan="2">EM</td><td>81±13</td><td>9.1±1.7</td><td>23</td><td>MRI</td><td>Klein et al. 2002</td></tr><tr><td>68±0.5</td><td>12.5±3.3</td><td>8</td><td>CT</td><td>Klitgaard et al. 1990</td></tr></tbody></table>

Linear growth: the relationship with age is evolution with aging can be modeled with a linear equation:

$$p(x) = a \cdot x + p_0 \quad (1)$$

Where $p(x)$ is the muscle parameter at age $x$, $a$ is the growth rate, $p_0$ the initial parameter value.

Exponential growth: the parameter growth rate evolves with age, in proportion to the parameter values, getting larger or smaller. The general form of this model is: $p(x) = p_0 \cdot e^{a \cdot x}$ (2) Where $p(x)$ is the muscle parameter at age $x$, $a$ is the growth factor, $p_0$ the initial parameter value.

Logistic growth: In logistic growth, a parameter growth rate increases (or decreases) as it approaches an imposed extremum value, known as the carrying capacity. The general form of this model:

$$p(x) = \frac{k \cdot p_0}{p_0 + (k - p_0) \cdot e^{-a \cdot x}} \quad (3)$$

Where $p(x)$ is the muscle parameter at age $x$, $a$ is the growth rate, $k$ is the carrying capacity, $p_0$ the initial parameter value.

Gompertz-Weibull growth: It is also a sigmoid function. In contrast to the simple logistic function in which both asymptotes are approached by the curve symmetrically, here the asymptotes have different growth rates. The general form of this model is:

$$p(x) = c \cdot e^{-b \cdot x} \quad (4)$$

Where $p(x)$ is the muscle parameter at age $x$, $a$ is the growth factor, $b$ is the halfway point, and $c$ is an asymptote, since $l_{x→∞} \cdot c \cdot e^{-b \cdot x} = c$.

The equations 1, 2, 3, and 4 were all fitted from the selected experimental data. The model equation parameters are

estimated using the Levenberg-Marquardt algorithm (non-linear least squares optimization function).

### 3. Results and discussion

To obtain the estimated relationship between the muscle CSA and age, we first eliminate experimental data with high standard deviation. Then we have applied a regression using growth equations of section 2.2. Four curves of CSA decline are estimated (Figure 1). Given the high variance of the data, the simple quality of fit isn't enough to discriminate between the possible models. We use our priori knowledge of muscle aging to guide our choice. Muscle force starts to decline around the age of 40 and can be associated with a reduction of muscle mass. The decline is constant, only its speed is modulated. Taking into account these informations we selected the Gompertz-Weibull. This model shows a plateau for younger male and then a progressive diminution of CSA with age. The other models present behavior that are unrealistic when put against our knowledge.

We will use the Gompertz-Weibull equation to obtain the muscle CSA at each age for future models. However it can only hint at possible values for a generic virtual subject.

<figure><img src="image_2.png" /><figcaption>**Figure 1.** CSA fitted functions for male subjects.</figcaption></figure>

Similar work has been performed for other morphological parameters of the muscle and for aging female subjects as well.
Finally, such behavior laws are mandatory for personal-
izing a future neuromuscular multiscale and multiphysics
model to subject age. This model aims to mimic muscle
aging with a high physiological realism.

Funding

This work was carried out and funded in the framework
of the Labex MS2T. We thank equally the Region
Hauts-de-France for co-funding this work.

Conflict of Interest Statement

The authors declare no conflict of interest.

References

Douania, Ines et al. (2019). “Assessment and sensitiv-
ity analysis of a motor units recruitment model
during isometric contractions of the Biceps
Brachii”. In: *2019 Fifth International Conference on
Advances in Biomedical Engineering* (ICABME), 1–4.
doi: 10.1109/icabme47164.2019.8940248

Carriou, Vincent et al. (July 2016). “Fast generation model of high density surface EMG signals in a cylindrical conductor volume”. en. In: *Computers in Biology and Medicine* 74, pp. 54–68. issn: 0010-4825. doi: 10.1016/j.compbiomed.2016.04.019

Klein, C. S. et al. (July 2002). “Muscle size, strength, and bone geometry in the upper limbs of young and old men”. eng. In: *The Journals of Gerontology*. Series A, Biological Sciences and Medical Sciences 57.7, pp. M455–459. doi: 10.1093/gerona/57.7.m455

Klitgaard, H. et al. (1990). “Function, morphology and protein expression of ageing skeletal muscle: a cross-sectional study of elderly men with different training backgrounds”. en. In: *Acta Physiologica Scandinavica* 140.1, pp. 41–54. doi: 10.1111/j.1748-1716.1990.tb08974.x

McCall, G. et al. (1996). “Muscle fiber hypertrophy, hyperplasia, and capillary density in college men after resistance training.” In: *Journal of applied physiology*. doi: 10.1152/jappl.1996.81.5.2004

Miller, A. E. J. et al. (Mar. 1993). “Gender differences in strength and muscle fiber characteristics”. en. In: *European Journal of Applied Physiology and Occupational Physiology* 66.3, 254–262. doi: 10.1007/BF00235103

Mitchell, W. K. et al. (2012) "Sarcopenia, Dynapenia, and the Impact of Advancing Age on Human

Skeletal Muscle Size and Strength; a Quantitative
Review.” *Frontiers in Physiology*, 3:260, 2012.