Introduction

Natural skin tension was discovered in 1834 by Guillaume Dupuytren on the occasion of an attempted suicide by a young man1. He noticed that driving a round punch into the skin did not produce a round wound but a linear incision. Dupuytren then confirmed this finding on cadavers and showed that the orientation of the resulting oval incision was different on different body areas. Joseph Malgaigne confirmed these results2, and in 1861, Karl Langer published the results of his studies performed on the cadavers of whom he perforated the skin with a 2 mm round punch3,4. By connecting the longitudinal axes of the ellipses resulting from the deformed wounds, Langer mapped the tension lines all over the body, known as Langer lines. In 1897, Emil Kocher1 suggested that surgical incisions should follow these lines.

Skin tension is present in all directions of the skin but mainly in favored directions. Other researchers have been interested in the study of skin tension, and the mapping of tension lines in relation to the direction of surgical incisions and the main direction of skin tension. In 1935, Jérôme P. Webster5 suggested that surgeons should place incisions in natural wrinkle lines. In 1951, Cornelius Kraissl6 extended the use of wrinkle lines to the whole body in vivo using photography and drawing. These lines are mainly perpendicular to the muscles. He noted that collagen is generally oriented parallel to the wrinkles and that in a scar placed perpendicular to the underlying muscle, the collagen will be deposited in the same direction as that usual in the wrinkles, which allows for optimal healing of the incision. Relaxed Skin Tension Lines (RSTLs) define tension lines in living subjects rather than in cadavers7. These lines follow the furrows when the skin is relaxed and correspond to the main direction of skin tension according to Borges5,8. Relaxation is obtained by joint mobilization, muscle contraction or pinching9. Of these three methods, Borges9 used the pinching method, which is the most reliable according to him. The grooves formed during joint mobilization or muscle contraction can be unreliable, depending on the type of joint mobilization or the type of muscle contraction. Pinch-formed grooves are best created by applying the pinch at right angles to the RSTLs. When the pinch is exerted perpendicular to the RSTLs, the furrows formed become fewer, taller and longer and can be formed more easily. If the pinch is applied obliquely, the furrows form an S-shaped pattern. RSTLs exert constant traction in a certain direction and are only temporarily modified by muscular action or external stress10. Although this method is often used by surgeons to establish the orientation of skin tension, the pinch technique can only provide a rough orientation of the line due to the limited angular resolution that can be achieved by manual manipulation. Other authors (Rubin11, Cox12, Straith13, Bulacio14 and Sarifakioglu15) have also described skin lines. Most of them are similar to Langer lines, wrinkles or RSTLs. A comparison of different representations of facial skin tension lines was presented by Alhamdi5 (Fig. 4b).

Natural skin tension changes continuously throughout life, from birth to adulthood (Fig. 4a). Zahouani et al.16,17 were interested in studying the evolution of the state of skin tension in newborns from the analysis of the skin microrelief. The change in skin tension is characterized by the size of the area of skin topology plateaus. The area of these plateaus changes depending on skin tension. This development begins in infants 4—5 weeks old and peaks at 4—5 years of age. The maturation of skin tension continues into adulthood.

Techniques for characterizing the skin’s properties, presented in the literature, are limited to the evaluation and characterization of the elastic and/or viscoelastic properties of the skin18,19,20,21,22,23,24,25. These studies have explored skin tension through experimental protocols and numerical simulation models26,27,28,29,30,31. They were interested in studying the anisotropic behavior of human skin by performing ex vivo tensile tests and by characterizing the anisotropic properties of human skin in vitro at the molecular level. These types of tests (ex vivo and in vitro) help to understand behavior, but have major drawbacks that must be taken into account when interpreting the results, as the sampling method and experimental protocol, for example. Excised skin does not have the same properties as skin in vivo3. Nowadays, it is possible to study skin tension by in vivo tests thanks to the progress of biomedical imaging techniques32,33,34,35,36,37,38,39,40. The results of all these studies showed a difference in the amplitude and distribution of skin tension forces with age as a function of direction and highlighted directions characterized by higher resistances than others, likely corresponding to the directions of Langer. All these studies characterized the natural skin tension. However, ex vivo and in vitro tests appear to be unsuitable for studying natural skin tension and the causal links between its physiology in vivo and its mechanical, topographical properties, its age and the area of the body. In vivo studies use experimental tests with direct contact with the skin which disturbs its initial natural tension state and give only qualitative characterization of skin tension.

In the current study we propose a quantitative characterization of skin tension in vivo in seven body areas using a new model. This model consists in calculating the tension indices from skin topology images. A feasibility study has also been carried out to apply the model to other image types such as LC-OCT (Line-field Confocal Optical Coherence Tomography) and confocal microscopy images. These tension indices, developed at the LTDS (Ecole Centrale de Lyon, France), provide a quantification to characterize the skin’s tension forces in vivo without disturbing the skin's initial state of tension. This method is described and detailed in materials and methods section. The experimental results are presented in results section and discussed and analyzed in discussion section.

Results

Skin tension indices calculated from the skin topology images

This new approach to analyze the state of skin tension was applied to characterize tension in seven body areas: the forearm, the thigh, the cheek, the upper chest, the belly, the anterior face of the arm and the posterior face of the arm.

Figure 1 shows the skin tension indices calculated from the skin topology images of the seven body areas by age group (young and elderly).

Fig. 1
figure 1

Skin tension indices calculated from the skin topology images of the seven body areas (the forearm, the thigh, the cheek, the upper chest, the belly, and the anterior and posterior faces of the arm) for the young group (n = 21) and the elderly group (n = 21). (* for 0.01 < p-value < 0.05, ** for 0.001 < p-value < 0.01, *** for p-value < 0.0001, and – for p-value > 0.05).

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For the forearm, the ICx index tends to increase with age (p-value = 0.094) (ICx, young = 0.24 ± 0.07, ICx, elderly = 0.34 ± 0.20). The ICy index increases significantly with age (ICy, young = 0.07 ± 0.17, ICy, elderly = 0.23 ± 0.22).

For the thigh, the ICx and ICy indices increase significantly with age (ICx, young = 0.11 ± 0.12, ICx, elderly = 0.20 ± 0.15; ICy, young = 0.08 ± 0.05, ICy, elderly = 0.21 ± 0.11).

For the cheek, the results show that tension indices ICx and ICy increase significantly with age along the two body axes x and y (ICx, young = 0.057 ± 0.04, ICx, elderly = 0.19 ± 0.1; ICy, young = 0.044 ± 0.05, ICy, elderly = 0.18 ± 0.1).

For the upper chest, the results show that the ICx index increases significantly with age (ICx, young = 0.226 ± 0.3, ICx, elderly = 0.624 ± 0.5), and the ICy index tends to increase with age, but the difference is not significant (p-value = 0.095).

For the belly, the results show that the ICx and ICy indices increase significantly with age. The measured ICx indices are 0.107 ± 0.1 for the young group and 0.236 ± 0.1 for the elderly group. The ICy indices measured are 0.065 ± 0.05 for the young group and 0.310 ± 0.2 for the elderly group.

For the anterior and posterior faces of the arm, the ICx and ICy indices increase significantly with age (for the anterior face: ICx, young = 0.054 ± 0.03, ICx, elderly = 0.127 ± 0.1; ICy, young = 0.125 ± 0.1, ICy, elderly = 0.266 ± 0.1, and for the posterior face: ICx, young = 0.363 ± 0.2, ICx, elderly = 0.982 ± 0.4; ICy, young = 0.347 ± 0.1, ICy, elderly = 0.950 ± 0.5).

Using the calculated tension indices (ICx and ICy), it was possible to map the state of skin tension (Fig. 2). The axes of the ellipses correspond to the indices ICx and ICy along the x and y axes. The isovals correspond to the variation of IC (Figs. 1, 2). This graphical representation gives graphic information on the skin tension state and on the state of the tension force equilibrium between the family of skin lines from 0° to 90° and the family of lines from 90° to 180°. The radius of these ellipses gives an idea of the state of skin tension within the two families of lines: the smaller the radius, the greater the tension. The radius also gives information on the state of equilibrium of the tension forces between the two families of lines: a perfect balance between the two families of lines results in identical and symmetrical parts of the graph.

Fig. 2
figure 2

Graphical representation of the state of skin tension as a function of age group on the forearm, the thigh, the cheek, the upper chest, the belly, the anterior face of the arm and the posterior face of the arm. The axes of the ellipses correspond to the indices ICx and ICy along the x and y axes. The isovals correspond to the variation of IC.

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For \({IC}_{x}\approx \) \({IC}_{y}\), the state of cohesion can be represented by a circle of diameter ICx and ICy (Fig. 2).

For \({IC}_{x} \ne {IC}_{y}\), the state of cohesion can be represented by an ellipse of axes ICx and ICy (Fig. 2).

Figure 2 shows that, whatever the body area, the imbalance of the tension forces between the two families of lines increases with age. This imbalance results in an increase in the radius of the curves. This figure also shows that the skin has greater tension in certain body areas which are for the young group: the cheek, the belly, the thigh and the anterior face of the arm. This result is most remarkable on the graphs, in which the radius of the curves for the thigh, for example, is smaller than the forearm regardless of the age group. It should be noted that with age, the relaxation of the skin tension is greater for the posterior face of the arm.

Skin tension indices calculated from confocal microscopy and LC-OCT images

The skin tension analysis model is applied to characterize skin tension based on confocal microscopy (for the cheek) and LC-OCT images (for the forearm and the thigh). For the thigh, tension indices are calculated from LC-OCT images and skin replicas at rest and during the folding test.

Figure 3 shows illustrations and comparisons of tension indices calculated on confocal microscopy or LC-OCT images and skin replicas of young and elderly volunteers for the skin of the cheek, forearm and thigh at rest and during the folding test.

Fig. 3
figure 3

Illustrations and comparisons of tension indices calculated on confocal microscopy and LC-OCT images of young and elderly volunteers for the skin of the cheek, forearm and thigh at rest and during folding test35,39. Black arrows represent the direction of folding.

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For the cheek, the results show that tension indices ICx and ICy, calculated on confocal microscopy images, increase with age along the two body axes x and y (ICx, young = 0.10, ICx, elderly = 0.22; ICy, young = 0.15, ICy, elderly = 0.26). Tension indices on skin replicas images also increase with age (ICx, young = 0.009, ICx, elderly = 0.20; ICy, young = 0.001, ICy, elderly = 0.41).

For the forearm, the ICx and ICy indices increase with age (ICx, young = 0.11, ICx, elderly = 0.27, ICy, young = 0.006, ICy, elderly = 0.23). Tension indices on skin replicas images also increase with age in the y-axis direction (ICx, young = 0.28, ICx, elderly = 0.20; ICy, young = 0.03, ICy, elderly = 0.22).

For the thigh skin at rest, the ICx and ICy indices increase with age for both the LC-OCT and skin replica images (ICx, young, LC-OCT = 0.10, ICx, young, Replica = 0.21, ICx, elderly, LC-OCT = 0.27, ICx, elderly, Replica = 0.38; ICy, young, LC-OCT = 0.12, ICy, young, Replica = 0.22, ICy, elderly, LC-OCT = 0.26, ICy, elderly, Replica = 0.53).

For the thigh during the folding test, the ICx and ICy indices also increase with age for both the LC-OCT and skin replica images (ICx, young, LC-OCT = 0.008, ICx, young, Replica = 0.09, ICx, elderly, LC-OCT = 0.21, ICx, elderly, Replica = 0.73; ICy, young, LC-OCT = 0.08, ICy, young, Replica = 0.05, ICy, elderly, LC-OCT = 0.34, ICy, elderly, Replica = 0.12). We observe that for the skin of the young volunteer, tension indices decrease after folding for both the LC-OCT and skin replica images. For the skin of the elderly volunteer, for the LC-OCT images, the tension indices tend to increase along the y-axis after skin folding. For the skin replica images, the indices tend to increase along the x-axis and decrease along the y-axis after folding.

Discussion

It was shown in the previous studies of skin microrelief and fiber organization34,35,39 that skin tension is mainly due to the tension applied by collagen and elastin fibers. This fiber tension is reflected in the skin's topology in the form of tension lines. These lines are divided into two families: the family of tension lines parallel to the main direction of skin tension (or Langer's direction) between 0° and 90°, and the family of lines perpendicular to Langer's lines between 90° and 180°.

This study presents a new approach to characterize natural human skin tension in vivo. This method is based on the analysis of the skin microrelief and on the analysis of the state of the tension forces, particularly in the family of lines parallel to Langer lines (between 0° and 90°) and in the family of lines perpendicular to Langer lines (between 90° and 180°). This method was also tested on images of skin fibers taken by confocal microscopy and LC-OCT and for skin at rest and during a folding test presented in previous studies35,39.

The results showed, on the one hand, that skin tension is not the same in the two families of skin lines regardless of body areas and age group. For the young group, the skin has a higher tension, and the tension forces are relatively balanced between the two families of lines. With age, there is a loss of tension38,39, the skin relaxes, and therefore the imbalance of the tension forces increases.

For the forearm and the thigh, the imbalance of forces increases with age along both body axes and particularly along the y axis for the forearm (Fig. 2). Based on the results of27, this imbalance in the tension forces appears to be linked to the dermal fiber network. The imbalance would cause a reorientation of the dermal fibers because, although decreasing, the amplitudes of the tension forces are not the same in both directions. This reorientation will take place in a preferred direction which has been shown to be the main direction of skin tension or direction of Langer lines. Regarding the skin topology of the elderly group, a reorientation of the skin lines occurs towards the main direction of the skin tension or the direction of the Langer lines. The results of the indices calculated from the LC-OCT images of skin at rest are consistent with the results obtained from the skin topology analysis. For the thigh skin during the folding test, it is difficult to explain the result obtained on the basis of two examples. An extended study is necessary and will be carried out to better understand how skin tension varies under mechanical stress such as folding.

For the cheek, the indices quantify the imbalance of skin tension forces. The indices for the young group and for the elderly group are plotted as circles (Fig. 2), meaning that the tension forces are not in equilibrium for the young group or for the elderly group. The radius of the circle is greater for the elderly group compared with that of the young group. The imbalance of the tension forces increases with age along the two body axes. The indices also underline that the radii increase with quite similar orders of magnitude in both the x-direction and the y-direction for the elderly group. This result is not straightforward when analyzing the skin topology images recorded for the elderly group. The relaxed images of the elderly group show furrows, plateaus and lines oriented in the preferred direction of the Langer lines27 and a very anisotropic topology. Here the calculated indices characterize this anisotropic behavior through the non-zero radii equivalent to an imbalance in the tension forces. The indices also underline that this anisotropy is due to possible different responses of the skin tissues in the x-direction and in the y-direction. But whatever the tension forces that develop in the volume of the skin tissues, their imbalance is almost the same in both directions. The results of the indices calculated from the confocal microscopy images are consistent with the results obtained from the skin topology analysis. An extended study is necessary and will be carried out to better compare results obtained on different images for all the subjects of this study.

For the upper chest, the indices are plotted as ellipses with the main axis parallel to the y-axis. The amplitudes of the main and minor axes of the ellipses are far bigger for the elderly group compared with those of the young group. Again, for the upper chest, the skin tissues exhibit a more relax state for the elderly group. Here the indices make it possible to quantify this more relaxed state. It is linked to the imbalance in the tension forces which appears to be equal to 0.27 for the young group and to 0.65 for the elderly group. Moreover, it can be noted that the imbalance in the skin tension forces appears to be greater along the x axis of the body than along the y axis (Fig. 2). Mechanically speaking, the physiology of the skin tissue of the upper chest seems to have a greater influence on its tension state than the gravity forces applied to the skin tissue material of this body zone.

For the belly, the indices are plotted as ellipses with the main axis parallel to the x-direction for the young group and to the y-direction for the elderly group. The amplitudes of the main and minor axis of the ellipses are bigger for the elderly group compared with those of the young group. The classical result of a more relaxed state for the skin tissues of the elderly group is also obtained here. The indices link this relaxed state to an imbalance of the skin tensile forces at the belly which seems to be greater along the y axis of the body with age. Mechanically speaking, the physiology of the skin tissues of the belly of the young group appears has a bigger influence on its tension state than the gravity forces applied on the material of the skin tissues of this body area. Here, we return to the observation made previously for the chest. Contrary for the skin tissues of the elderly group, the influence of the physiology and the gravity forces combine and lead to a far greater influence in the tension forces than that which could have been reached by each contribution alone.

For the anterior face of the arm, the indices are plotted as circles for the young group and the elderly group. An increasing radius can be seen between both age groups. This imbalance of the skin tension forces between the two families of lines increases with age, although the non-significant differences in the amplitudes of the indices tend to signify tension forces for which the imbalance is not so different from one direction to the other. This would mean that with age, there is a loss of skin tension, resulting in an imbalance of the tension forces between the two families of skin lines. Thus, this loss results in an equivalent state of the skin tissues of the anterior face of the arm in both directions.

These results show the specific behavior of the skin in each area of the body. In all areas of the body, skin tension relaxes with age. The degree to which the skin relaxes varies from area to area. This may be due to a number of factors, such as physiology, which differ from one area to another. For example, the thickness of the skin layers is different34, the skin on the cheek is thinner than the skin on the stomach. Skin tension can be influenced by the accumulation of fat in certain areas of the body, such as the abdomen and back of the arms, pregnancy and breastfeeding, the effects of gravity and exposure to the sun41.

The discussion is based on the statistical analysis presented in the Results part of this paper. It should be kept in mind that when the indices are plotted as circles, it means that the differences between the index in the x-direction and that in the y-direction are non-significant. Despite their smallness, differences exist which means that there is not a perfect balance of the tension forces between these two families. Skin tension is higher in the family of tension lines (parallel to Langer lines), but lower in the family of lines perpendicular to the tension lines (perpendicular to Langer lines). This imbalance of forces between the two families is at the origin of the anisotropic behavior of skin which is accentuated with age.

The tension indices provide information on the state of skin tension forces in each body area and how it changes with age. They also allow classifying body areas according to their state of tension. This ranking in descending order (from the zone with the highest tension to the zone with the lowest tension) is as follows: 1- the cheek, 2- the anterior face of the arm, 3- the thigh, 4- the belly, 5- the upper chest, 6- the forearm and 7- the posterior face of the arm.

Testing the model presented in this study on in-depth skin images taken by confocal microscopy or LC-OCT, shows the potential of this model and its ability to study and quantify skin tension at different scales. The first results of this feasibility study show differences between the indices calculated for the different image types. This difference could be related to the difference in the scale of the images and the difference in the depth of their acquisition. The feasibility study should extend to calculate the tension indices on the LC-OCT and confocal microscopy images of all the volunteers in order to be able to compare and study correlations between skin tension indices derived from the replicas images and those derived from LC-OCT or confocal microscopy images.

This study presented a new model developed to characterize natural human skin tension in vivo in seven body areas: forearm, thigh, cheek, upper chest, belly, the anterior and posterior face of the arm. This model uses the analysis of the skin microrelief by analyzing the balance of tension forces between the family of lines parallel to Langer lines and the family of lines perpendicular to Langer lines, and by studying the age effect on skin tension. The results showed that skin tension is higher in the family of lines parallel to Langer lines, but lower in the family of lines perpendicular to Langer lines. The results also showed that, whatever the body area, skin tension relaxes as a function of age. This relaxation is greater in the family of lines perpendicular to the Langer lines. The degree of relaxation of skin tension depends on the body area. It is greater in the posterior face of the arm and in the upper chest.

The results of this study show that the model can be used on deep skin images to study fiber tension. The cutaneous tension indices will allow morpho-mechanical changes in the fibrillar microstructure of connective tissue to be studied and modelled. Analysis of a large number of subjects suffering from specific pathologies, such as scleroderma, osteogenesis imperfecta, cutis laxa, Marfan's disease, etc., can be detected preventively by studying the topography of their tension lines in comparison with a population of subjects (of the same age and gender) with healthy skin. This model for calculating skin tension could also be used to assess the effectiveness of a medical treatment or cosmetic product designed to improve skin tension. This opens up significant potential for using this model to study the tension of different organs and biomaterials.

Materials and methods

Volunteers

The tests were performed on seven body areas: the forearm, thigh, cheek, belly, upper chest, and arm (front and back faces) of 42 Caucasian French women volunteers divided into two groups: a young group20,21,22,23,24,25,26,27,28,29,30 years-old (21 volunteers) and an elderly group [45–65] years-old (21 volunteers).

Informed consent was obtained from all volunteers in advance of recruitment, after a full explanation of the nature of the study and receipt of both written and oral procedural information. The research adhered to the principles of the Declaration of Helsinki and was approved by the Clarins Laboratories Ethics Committee (R-17-E-012/03, R-17-E-012/04).

The volunteers were non-smokers, in good health and had healthy skin without scars or tattoos. The volunteers had a Body Mass Index (BMI) between 18.5 and 27 kg/m2, a cellulite index less than 2 for the thigh (a visual assessment according to the photonumerical scale42 was carried out) and a phototype between I and III.

The volunteers did not apply any cosmetic products on their bodies on the test day. The tests were carried out after an acclimatization period of at least 10 min in an air-conditioned room (T = 21 ± 2 °C and H = 50 ± 10%). For the measurements on the cheek, the anterior face of the arm, the belly and the upper chest, the volunteer laid on her back, her feet stretched out, not crossed, and with her right arm resting on an armrest. For the measurements on the posterior surface of the arm, the volunteer laid on her belly with both hands parallel to the body and the palm facing downwards. The position of the volunteers was chosen so as to minimize tension due to the structure underlying the skin (muscles and bones), in order to observe mainly the natural skin tension. The volunteer was asked not to move for the duration of the measurements to ensure their homogeneity and to prevent noise in the data recorded as much as possible.

The measurement zones with a surface area of 9 cm2 were delimited using a mask. These zones were (Fig. 4c): the right forearm (12 cm above the wrist), the left thigh (12 cm above the knee), the right cheek (4 cm from the ear and 2 cm from the nose), the anterior and posterior faces of the arm (in the mid-distance elbow-shoulder), the belly (4 cm to the right of the navel), the upper chest (5 cm from the bottom of the neck).

Fig. 4
figure 4

(a) Evolution of skin tension throughout life, from birth to adulthood, tension directions are more defined and mature for adults 3,16. (b) Comparison of different skin tension line representations5. (c) Test zones and body axes.

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Skin replicas

Skin replicas were taken using silicone SILFLO (MONADERM, Monaco)38. The replicas were made on relaxed skin. The three-dimensional skin topology was then reconstructed using the chromatic confocal microscopy system (from AltiMet—AltiSurf 500®, France). The 3D images obtained were studied with a program developed by Zahouani43 using TopoSurf®. This approach allows the identification of the anisotropy of the line network at different scales of depth and with different orientations38. For each plane at a certain depth of the skin surface, are determined three parameters of the point belonging to the line of tension: the density of depth z, the width of the line and the rose of directions between 0 and 180°. It then makes it possible to extract the density of skin lines in different directions from 0° to 180° with a step of 20°35.

Confocal microscopy and LC-OCT imaging

Confocal laser scanning microscopy was used to image the cheek. This technique uses a near-infrared (830 nm) laser beam. The laser penetrates the skin and is reflected by the skin tissue. Only reflected light from the focal region is detected. Variations in the refractive index of tissue microstructures provide images with contrast. The confocal microscope used in this study was the Vivascope 1500 (MAVIG). Measurements were taken on the right cheek. Three image stacks were taken, with the following parameters: acquisition from the surface down to 118.87 μm in depth, with a step of 3.05 μm between each image.

LC-OCT (Line-field Confocal Optical Coherence Tomography) images were taken on the forearm and thigh for resting and folded skin. This technique and the detailed study have been presented already in35.

Only a few images were used in this study to illustrate the efficiency of the tension index calculation model (4 youth volunteers and 4 elderly volunteers).

Model of skin tension and definition of tension indices

In vivo, the skin is subjected to tension forces in different directions with different intensities. The trace of these tension forces is reflected on the skin surface by its particular topology. This relief has lines oriented in different directions. From the skin topology image, it is possible to extract the network of these skin lines oriented in different directions.

In this study, we developed a model based on the fundamental hypothesis of the natural pre-tension exerted by the collagen and elastin fibers of the connective tissue34,35. All the tissues and organs in our body are subjected to the natural supporting pre-tension provided by the connective tissue. Previous studies35,39 of skin microrelief and fiber organization enabled us to divide skin lines into two families: the family of tension lines parallel to the main direction of skin tension (or Langer's direction) between 0° and 90°, and the family of lines perpendicular to Langer's lines between 90° and 180° (Fig. 5).

Fig. 5
figure 5

(a) Extraction of skin lines in different directions from 0° to 180°. The depth of the skin lines is illustrated by the range of colors from blue to red, secondary lines (colors of green, yellow, and red correspond to a variation of depths between -17 microns and 50 microns). The family of skin tension lines in the main color blue are in a scale between -17 and -84 microns43. (b) Decomposition of skin lines by the Fourier transform into two-line families: the family of tension lines parallel to the main direction of skin tension (or Langer's direction) between 0° and 90°, and the family of lines perpendicular to Langer's lines between 90° and 180°.

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To analyze the state of equilibrium of this natural tension and quantify it in the form of tension indices (or cohesion anisotropy) along the two axes of the image (x and y), we consider that the skin tissue is subjected to a biaxial state of anisotropic tension along the two axes x and y. In the literature there are several mathematical models that simulate the balance of tension through the tensegrity model (tension integrity)44,45,46,47. In our model we have chosen to study this balance along the two orthogonal directions x and y of the image by projecting the static balance equation onto the two axes.

For the study of this balance, we do not know the forces undergone by the tissue. However, based on a previous work27, we assume that the image of the tissue is made up of patterns having an orientation, a spatial frequency and a wavelength which represent the effect of pre-tension (Fig. 5). The orientation and wavelength of each pattern represent the important parameters for studying the equilibrium state. The wavelengths of the image in the Fourier spectrum are determined as follows:

Let \(f\left( {x,y} \right)\) be the function representing the image of Fig. 5. The two-dimensional Fourier transform (Cooley and Tukey) of the image \(f\left( {x,y} \right)\) is calculated with the expression (1). It is denoted by FT for direct Fourier transform. The FT of the image \(f\left( {x,y} \right)\) gives a spatial frequency spectrum in mm-1. It should be remembered that the inverse Fourier transform exists: FT-1. In Fourier space, the spatial frequencies are noted: \(\nu_{x,i} { }and{ }\nu_{y,j}\), the discrete Fourier transform of the image has the expression:

$$ F\left( {\nu_{x,i} ,\nu_{y,j} } \right) = { }\mathop \sum \limits_{x = 0}^{N - 1} \mathop \sum \limits_{y = 0}^{M - 1} f\left( {x,y} \right){\text{ exp}}\left[ { - j2\pi \left( {x\frac{{\nu_{x,i} }}{N} + y\frac{{\nu_{y,j} }}{M}} \right)} \right] $$
(1)

with the complex j defined as \({j}^{2}=-1\)

The spatial frequencies whose unit is the inverse of a length (mm-1) are defined with the expressions (2):

$$ \nu_{x,i,k} \left( {m^{ - 1} } \right) = { }\frac{k}{Ndx}{ }\;{\text{and}}\;_{y,j,l} \left( {m^{ - 1} } \right) = { }\frac{l}{Mdy} $$
(2)

where \(k and l\) are indices of frequencies and dx and dy are respectively increments in the x and y directions. High frequencies are identified in the Fourier spectrum for \(k=\frac{N}{2}\text{ and }l= \frac{M}{2}\). The expressions (2) rewritten for the high frequencies indices lead to the expressions (3):

$$ \nu_{{x,i, N/2{ }}} \left( {m^{ - 1} } \right) = { }\frac{1}{2dx} ,\quad \nu_{{y,j, M/2{ }}} \left( {m^{ - 1} } \right) = { }\frac{1}{2dy} $$
(3)

Low spatial frequencies are identified in the spectrum for \(k = l = 1\). The expressions (2) rewritten for the low frequencies indices lead to the expressions (4):

$$ \nu_{{x,i,1{ }}} \left( {m^{ - 1} } \right) = { }\frac{1}{Ndx} ,\quad \nu_{{y,j,1{ }}} \left( {m^{ - 1} } \right) = { }\frac{1}{Mdy} $$
(4)

The high and low spatial frequencies given by expressions (3) and (4) are respectively linked to the wavelengths of the image \(f\left( {x,y} \right)\) giving access to the spectrum representation of the image \(f\left( {x,y} \right)\) in the Fourier space. The following expressions (5) give the spectrum of the image in wavelengths for short wavelengths:

$$ \lambda_{x} \left( m \right) = { }\frac{1}{{\nu_{{x,i,N/2{ }}} }}\;{\text{and}}\;\lambda_{y} { }\left( m \right) = { }\frac{1}{{\nu_{{y,j, M/2{ }}} }} $$
(5)

while those below (6) give the long wavelengths:

$$ \lambda_{x} \left( m \right) = { }\frac{1}{{\nu_{{x,i,1{ }}} }}\;{\text{and}}\;\lambda_{y} { }\left( m \right) = { }\frac{1}{{\nu_{{y,j,1{ }}} }} $$
(6)

For any point in the Fourier spectrum, the spatial frequency and its direction are defined as (7) and (8):

$$ \nu \left( {m^{ - 1} } \right) = \sqrt {\nu_{x,i}^{2} + \nu_{y,j}^{2} } $$
(7)
$$ \theta \left(^\circ \right) = tan^{ - 1} \left( {\frac{{\nu_{y,j} }}{{\nu_{x,i} }}} \right) $$
(8)

Based on these relations, the model proceeds in four steps:

i) Separation of image orientations into two sub-images \(f_{1} \left( {x,y} \right)_{{\left( {0^\circ - 90^\circ } \right)}} \) and \(f_{2} \left( {x,y} \right)_{{\left( {90^\circ - 180^\circ } \right)}}\).

From the Fourier transform in polar coordinates of the image f(x,y), we determine two sub-images \(f_{1} { }and{ }f_{2}\) (Fig. 5) with respect to the orientations of the patterns θ which are from 0° to 90° for \(f_{1}\) and from 90° to 180° for \(f_{2}\). The expression for \(f_{{1\left( {0^\circ - 90^\circ } \right)}} { }\) is given by relation (9) and that for \(f_{{2\left( {90^\circ - 180^\circ } \right)}}\) by relation (10):

$$ f_{1} \left( {x,y} \right)_{{\left( {0^\circ - 90^\circ } \right)}} = \left[ {FT^{ - 1} \left( {F\left( {\nu ,\theta } \right)} \right)} \right]_{0^\circ - 90^\circ } ;\;{\text{for}}\;0^{ \circ } \le \;\theta \le 90^\circ $$
(9)
$$ f_{2} \left( {x,y} \right)_{{\left( {90^\circ - 180^\circ } \right)}} = \left[ {FT^{ - 1} \left( {F\left( {\nu ,\theta } \right)} \right)} \right]_{90^\circ - 180^\circ } {;}\;{\text{for}}\;90^{ \circ } < \;\theta \le 180^\circ $$
(10)

As shown in Fig. 5, decomposing the image of the tension lines into two sub-images makes it possible to study the anisometric distribution of residual tension forces on the anisometric distribution of the wavelengths of the skin image on the x and y axes.

The projection of the balance equation onto the x and y axes can be written as (11) and (12):

$$ \mathop \sum \limits_{i}^{N} F_{{i\left( {0 \to 90^\circ } \right)}}^{x} + \mathop \sum \limits_{j}^{N} F_{{j\left( {90^\circ \to 180^\circ } \right)}}^{x} = 0 = K \left( {\mathop \sum \limits_{i}^{N} \overline{{\lambda_{x, i,0^\circ - 90^\circ } }} - \mathop \sum \limits_{j}^{N} \overline{{\lambda_{x, j,90^\circ - 180^\circ } }} } \right) $$
(11)
$$ \mathop \sum \limits_{i}^{M} F_{{i\left( {0 \to 90^\circ } \right)}}^{Y} + \mathop \sum \limits_{j}^{M} F_{{j\left( {90^\circ \to 180^\circ } \right)}}^{y} = 0 = K \left( {\mathop \sum \limits_{i}^{M} \overline{{\lambda_{y, i,0^\circ - 90^\circ } }} - \mathop \sum \limits_{j}^{M} \overline{{\lambda_{y, j,90^\circ - 180^\circ } }} } \right) $$
(12)

ii) Calculation of the wavelength’s gravity center: Centroid

Under the effect of the cohesion forces, the centroids of the spatial frequencies of the decomposed images are the signatures of the cohesion forces from 0° to 90° and from 90° to 180°. The spectral centroid represents the frequency of the spectral center of gravity of the image. Its value corresponds to a spatial frequency (mm-1) on either side of which the energy of the spectrum is equally distributed. The spectral centroid of the image along the x-axis for 0° to 90° and for 90° to 180° are given by (13) and (14) respectively:

$$ \nu_{x,i, 0^\circ - 90^\circ } \left( {m^{ - 1} } \right) = { }\frac{{\mathop \sum \nolimits_{i = 1}^{N - 1} \mathop \sum \nolimits_{j = 1}^{M - 1} { }\nu_{x,i,j, 0^\circ - 90^\circ } \left( {FT(f_{1} \left( {x,y} \right)_{0^\circ - 90^\circ } )} \right)}}{{\mathop \sum \nolimits_{i = 1}^{N - 1} \mathop \sum \nolimits_{j = 1}^{M - 1} { }\left( {FT(f\left( {x,y} \right)_{0^\circ - 90^\circ } } \right))}} $$
(13)
$$ \nu_{x,i, 90^\circ - 180^\circ } \left( {m^{ - 1} } \right) = { }\frac{{\mathop \sum \nolimits_{i = 1}^{N - 1} \mathop \sum \nolimits_{j = 1}^{M - 1} { }\nu_{x,i,j, 90^\circ - 180^\circ } \left( {FT(f_{1} \left( {x,y} \right)_{90^\circ - 180^\circ } )} \right)}}{{\mathop \sum \nolimits_{i = 1}^{N - 1} \mathop \sum \nolimits_{j = 1}^{M - 1} { }\left( {FT(f\left( {x,y} \right)_{90^\circ - 180^\circ } )} \right)}} $$
(14)

The spectral centroids in the y-direction for 0° to 90° and for 90° to 180° are calculated with relations (15) and (16):

$$ \nu_{y,j, 0^\circ - 90^\circ } \left( {m^{ - 1} } \right) = \frac{{\mathop \sum \nolimits_{i = 1}^{N - 1} \mathop \sum \nolimits_{j = 1}^{M - 1} { }\nu_{y,i,j, 0^\circ - 90^\circ } \left( {FT(f_{2} \left( {x,y} \right)_{0^\circ - 90^\circ } } \right))}}{{\mathop \sum \nolimits_{i = 1}^{N - 1} \mathop \sum \nolimits_{j = 1}^{M - 1} { }\left( {FT(f\left( {x,y} \right)_{0^\circ - 90^\circ } } \right))}} $$
(15)
$$ \nu_{y,j, 90^\circ - 180^\circ } \left( {m^{ - 1} } \right) = { }\frac{{\mathop \sum \nolimits_{i = 1}^{N - 1} \mathop \sum \nolimits_{j = 1}^{M - 1} { }\nu_{y,i,j, 90^\circ - 180^\circ } \left( {FT(f_{2} \left( {x,y} \right)_{90^\circ - 180^\circ } } \right))}}{{\mathop \sum \nolimits_{i = 1}^{N - 1} \mathop \sum \nolimits_{j = 1}^{M - 1} { }\left( {FT(f\left( {x,y} \right)_{90^\circ - 180^\circ } } \right))}} $$
(16)

From the spectral centroids, the centroids of wavelengths are determined as (17) and (18):

$$ \lambda_{x, 0^\circ - 90^\circ } \left( m \right) = { }\frac{1}{{\nu_{x,i, 0^\circ - 90^\circ } }}\;{\text{and}}\;\lambda_{x, 90^\circ - 180^\circ } \left( m \right) = { }\frac{1}{{\nu_{x,i, 90^\circ - 180^\circ } }} $$
(17)
$$ \lambda_{y, 0^\circ - 90^\circ } \left( m \right) = { }\frac{1}{{\nu_{y,j, 0^\circ - 90^\circ } }}\;{\text{and}}\;\lambda_{y, 90^\circ - 180^\circ } \left( m \right) = { }\frac{1}{{\nu_{y,j, 90^\circ - 180^\circ } }} $$
(18)

iii) Analysis of wavelengths in both the x and y axes of the image

The quantification of wavelength centroids is fundamental in determining the indices of cohesion. We consider that the centroids of the wavelengths \(\lambda_{x, 0^\circ - 90^\circ }\), \(\lambda_{x, 90^\circ - 180^\circ }\) along the x axis are the consequences of the cohesion forces which we do not know.

\(\lambda_{x, 0^\circ - 90^\circ }\) is due to the cohesive force \(F_{x, 0^\circ - 90^\circ }\) and \(\lambda_{x, 90^\circ - 180^\circ }\) is due to the cohesive force \(F_{ - x, 90^\circ - 180^\circ }\). Similarly, along the y axis: \(\lambda_{y, 0^\circ - 90^\circ }\) is due to the cohesive force \(F_{y, 0^\circ - 90^\circ }\) and \(\lambda_{y, 90^\circ - 180^\circ }\) is due to the cohesive force \(F_{ - y, 90^\circ - 180^\circ }\).

The analysis mainly focuses on the centroid wavelengths of the images \(f_{1} \left( {x,y} \right)_{{\left( {0^\circ - 90^\circ } \right)}} { }\) and \(f_{2} \left( {x,y} \right)_{{\left( {90^\circ - 180^\circ } \right)}}\) which were defined by relations (9) and (10). We define the deviation from equilibrium by analyzing the differences in wavelengths given by relations (19) and (20):

$$ \overline{{\Delta \lambda_{x} \left( {\sum F_{x} } \right)}} = { }\left| {\overline{{\lambda_{x, 0^\circ - 90^\circ } - \lambda_{x, 90^\circ - 180^\circ } }} } \right|{ } $$
(19)
$$ \overline{{\Delta \lambda_{y} \left( {\sum F_{y} } \right)}} = \left| {\overline{{\lambda_{y, 0^\circ - 90^\circ } - { }\lambda_{y, 90^\circ - 180^\circ } }} } \right| $$
(20)

Based on relations (19) and (20), a completely balanced state in cohesion forces corresponds to the equality of the wavelengths in the directions x and y:

$$ \Delta \lambda_{x} = \Delta \lambda_{y} $$
(21)

This equality (21) in terms of wavelengths is the result of a balance between the mutual cohesive forces of the sub-images \(f_{1} \left( {x,y} \right)_{{\left( {0^\circ - 90^\circ } \right)}} { }\) and \(f_{2} \left( {x,y} \right)_{{\left( {90^\circ - 180^\circ } \right)}}\) and leads to relations (22):

$$ F_{x, 0^\circ - 90^\circ } = { }F_{ - x, 90^\circ - 180^\circ } \;{\text{and}}\;F_{y, 0^\circ - 90^\circ } = F_{ - y, 90^\circ - 180^\circ } $$
(22)

In the case where we have an equality close to zero on the two axes x and y, we are in a situation of a very cohesive tissue:

$$ \Delta \lambda_{x} = \Delta \lambda_{y} \approx 0{ } $$

When the values of \(\Delta \lambda_{x} {\text{ and }}\Delta \lambda_{y}\) are different from zero, this situation indicates a state of cohesion far from equilibrium, which implies anisotropy of the cohesion forces:

$$ F_{x, 0^\circ - 90^\circ } \ne F_{ - x, 90^\circ - 180^\circ } \;{\text{and}}\;F_{y, 0^\circ - 90^\circ } \ne F_{ - y, 90^\circ - 180^\circ } $$

iv) Definition of cohesion indices and study of anisotropy

The indices of cohesion \(IC_{x}\) in the x-direction and \(IC_{y}\) in the y-direction are defined as (23):

$$ IC_{x} = \frac{{\Delta \lambda_{x} }}{{\lambda_{x,min} }},\quad IC_{y} = \frac{{\Delta \lambda_{y} }}{{\lambda_{y,min} }}{ } $$
(23)

where \(\lambda_{x,min} { }\) is the minimum between the two wavelengths \(\lambda_{x,i, 0^\circ - 90^\circ }\) and \(\lambda_{x,j, 90^\circ - 180^\circ }\), and \(\lambda_{y,min}\) is the minimum between the two wavelengths \(\lambda_{y,i, 0^\circ - 90^\circ }\) and \(\lambda_{y,j, 90^\circ - 180^\circ }\).

Now, relation (21) leads to identifying an isotropic state of cohesion when the two indices ICx and ICy are equal:

$$ IC_{x} = IC_{y} $$

The anisotropy of cohesion is identified when the ICx and ICy indices are different:

$$ IC_{x} { } \ne { }IC_{y} $$

Therefore, the entire cohesion index of the element studied is defined as (24):

$$ IC = \sqrt {IC_{x}^{2} + IC_{y}^{2} } $$
(24)

The IC index is very useful for quantifying the intensity of the deviation from the cohesion balance, and it can be analyzed in a wide range of values between 0 and 100%. In different biological situations it allows carrying out a status study on cohesion. In our study, this IC index proved interesting for statistically studying the effect of mechanical stimulation on the balance of cohesion forces.

Statistical analysis

A statistical analysis was performed using the XLStat software (version 2019.4.2, Addinsoft, France). A descriptive analysis was performed upstream to identify data trends and anomalies, and to provide essential information such as: the minimum, maximum, mean, median, and standard deviation of a sample of data. The Shapiro–Wilk normality test was then performed by age groups and body areas. It led to the conclusion that the data in this study did not follow a normal distribution. These data are presented as mean ± standard deviation (Fig. 1). Therefore, the Mann–Whitney non-parametric test was performed with a significance level of 5% to assess the significance of the differences observed between the two age groups (young and elderly).