A total of ten healthy males (25–37 years) participated in the study. Exclusion criteria were acute abdominal or low back pain, limited range of motion and trunk injuries, nervous system disorders, or skin diseases. All participants were fasting at least 2 h before the start of the study and wore loose pants without a restrictive waistband. Anthropometric characteristics of the participants were taken as shown in Table 1. This included body weight and skinfold thicknesses (Clauser et al., 1988; Norton, 2018). The latter was measured three times at each of seven positions (Norton, 2018) with a calibrated Harpenden Skinfold Caliper. For consistency, all data was recorded by a single examiner. The participants’ body mass index (BMI) was within the normal range (19.5–24.6 kg/m²). The Ethics Committee of the Medical Faculty of the Ruhr-University Bochum approved the study (23-7868 08/10/23) and participants provided written informed consent.

A custom-built mechatronic tissue indenter with test rig (Figure 1) was used to gather the force-displacement curves and the surface displacements simultaneously. The indenter tip was moved towards the participant through a ring [in analogy to Carter et al. (2001)] mounted on the indenter housing. The ring was not continuous, but had cutouts laterally, allowing for visual measurement of the skin deformation (Figure 1C). To define the zero position for each test and to ensure a perpendicular measurement orientation, this contact ring rested lightly on the skin so that the tissue was just visibly compressed. The plastic indenter tip had the shape of a hemisphere with radius r t = 10  mm . The feed rate was 5 ± 1 mm/s and realized by an electronic micro linear drive. The feed force was measured with a 222.4 N (50 lbf) load cell, mounted to the side of the indenter facing away from the participant, which was subjected to the force applied by the linear drive. Maximum travel and maximum feed force were adjustable. A test rig made of torsion-resistant 40 × 40 mm aluminum profiles was used to mount the mechatronic indenter above the participant (Figure 1A). Both height and alignment of the indenter were adjustable on the test rig for each participant and measurement position. To avoid injury, the feed force was electronically and mechanically limited based on the estimated pressure between skin and indenter tip. The limits were taken from findings for the design of workplaces with collaborating robots (Muttray et al., 2014; Melia et al., 2015) and correspond to the point at which the increasing perception of pressure from an indenter tip turns into a noticeable pain. The lower pain limit for abdominal muscles was 35 N/cm². Because no pressure limits were known for paraspinal tissue, we used a software-controlled force limit of 110 N for all positions. A hardware force limitation was additionally implemented in case the electronic limitation failed. The dead weight of the free-standing test rig was 10% above the software-controlled force limit, so that it lifted off the ground when limits were exceeded. A wired remote control was used to allow participants to start and interrupt a measurement by themselves.

To calibrate our force-displacement test setup with tissue-like material (Wells and Liang, 2011), we used cylinders made of ballistic gelatin (GELITA BALLISTIC 3 gelatin, 255–265 g Bloom). The cylinders, 100 mm in diameter and 80 mm in height, were produced according to the preparation procedure of GELITA: Comprised of heating distilled water and gelatin to a maximum of 55°C ± 5°C, curing the mixture in molds for 24 h, demolding the specimens, and conditioning for 60 h within a moisture-sealing barrier. Conditioning temperatures for our cylinders with 10 wt% gelatin solution were 4°C according to FBI recommendations (Fackler and Malinowski, 1988; Maiden et al., 2015) and 15°C for increased compliance. Since gelatin is very sensitive to shear stress (Wells and Liang, 2011; Valliere et al., 2018), we covered the entire top of the gelatin cylinder with a rigid acrylic plate during calibration. Ultrasonic gel was applied in between. A cylinder with 10 mm radius was used as indenter tip. We compressed the differently tempered cylinders three times each at 5 mm/s up to 200 N using our experimental test setup (Figure 1A) and a materials testing machine (Zwick Z010 with GTM GmbH series K, 10 kN, 2 mV/V). Maximum deformation was 24.6 mm at 15°C. Comparing the measurement accuracy, maximum root mean square errors (RMSE) were 0.389 and 0.433 N, and percentage deviations were below 2.4% and 3.1%, for maximum compression forces of 60 and 120 N.

To optically measure the 3D surface deformations, two 8×8 Multi-Zone ToF sensors (VL53L5CX, STMicroelectronics) were used. These were mounted sagittal symmetrically to the sides of the indenter tip on the housing, each at a 12° angle to the observation plane (Figure 1C). The sensor distances perpendicular to the reference plane (zero position) were 35.1 mm. Each trapezoidal field of view, starting at the tip of the indenter, had a diagonal of 63° and a minimum length of 39.1 mm laterally to the reference plane. This resulted in a minimum observable area of approximately 16.74 cm². When the tissue is deformed starting from its initial state, the measurable area increases. For calibration of the ToF sensors and their data processing, we used 3D printed PLA (polylactic acid) profiles in matt white after ensuring that the optically measured distances did not differ from those of the skin [all participants had a Fitzpatrick skin type (Fitzpatrick, 1988) of I-III]. The seven profiles (including horizontal planar, bevelled at 12° and thus parallel to the sensor plane, convex or concave converging towards the sensor tip) represented continuous surface deformations from 0 to 45 mm (Figure 1B). The processing of the 15 Hz raw sensor data included a transformation of the absolute distance values for the 64 measuring zones into the 3D displacement of the trapezoidally measured surface, a calculation of the means over a sliding window of length 5 across the neighboring elements, and a linear time interpolation to 0.1 s. The comparisons to the fully known geometries resulted in absolute measurement deviations of ±1.6 mm. Maximum deviations occurred in the peripheral measurement zones. Mean RMSE and standard deviation (SD) over all profiles was 0.805 ± 0.45 mm.

During the indenter measurements, bipolar sEMG activity of the three main trunk muscles was recorded using 42 × 24 mm Ag/AgCl disposable surface electrodes with hydrogel (Kendall H93SG). Following skin preparation, three pairs of electrodes were placed on the right side of the body on the anterior abdominal (E1), the lateral abdominal wall (E2), and the paraspinal musculature (E3) (Criswell, 2011) with a center-to-center distance of 24 mm. To not interfere with the optical measurements of the skin deformations, the placements of E1, E2, and E3 were cranial to the subcostal plane. E1 was centered on the rectus abdominis and E3 was centered on the muscle belly of the erector spinae. E2 was located at the level of the most caudal palpable costa spuria in the transition between the hypochondric and right lumbar region (Figure 2). The reference electrode was placed caudally to the lateral abdominal wall muscles in the region of the anterior superior iliac spine. For sEMG signal acquisition, we used three bipolar preamplifiers of type ToMEMG V1.2 and a Tower of Measurement (DeMeTec GmbH, Langgöns, Germany). sEMG signals were sampled at 1024 Hz, filtered, quantified, visualized, and recorded using custom-built software. To mitigate the influence of electrocardiographic and power line artifacts, a band-pass filter between 45 and 500 Hz and a notch filter at 50 Hz were applied to the raw signals. For quantization, root mean square of the filtered signal was calculated on a sliding window of 200 samples (195.3 ms).

To create a frame of reference for normalization, participants performed maximum voluntary contractions (MVC) in three positions and received the following instructions:

1) MVC in supine position: Crunch with legs bent 90° and abdominal muscles actively tensed after inhaling. Arms at sides of torso, shoulders and head not touching the floor. Gaze is centered on the test bench.

For the sEMG amplitude normalization, the MVCs from the same positions as in the measurements were used in each case. The baseline sEMG activity was recorded when the participants were lying fully relaxed in all three positions before the start of the measurements.

Six measurement regions were deduced from the muscular structures and anatomical characteristics of the abdomen (Rohen et al., 2015; Netter, 2017; Schünke et al., 2018; Tayebi et al., 2021) as visualized in Figure 2 and explained in more detail in Table 2. The regions were identified by palpation (Van Sint Jan, 2007) and marked with a water-soluble pen. To eliminate individual and uncontrollable influence of the trunk muscles to stabilize the spine (Hodges et al., 2015), the participants were lying in three positions: supine, lateral, and prone. Four measurements were taken at each region with fully relaxed (R) and controlled activated (A) musculature, leading to a total of 48 measurements per participant. The regions are numbered from 1 to 6 and measurements were carried out in the same order (for instance: indentation with fully relaxed muscles in measurement region 1 is named R1). To maximize comparability, participants were given the following instructions:

• All measurements: Exhale before starting a measurement and hold the breath while the indenter tip moves out.

By comparing the monitored sEMG amplitudes to that of other participants, the postures were evaluated before and during each measurement and adjusted by interacting with the participant if necessary. To reduce tilting of the pelvis and chest in coronal plane, a folded towel was placed under the waist.

After the posture instructions, ultrasound gel was applied to the measurement region to minimize friction between skin and indenter tip. The test rig was set up so that the indenter was perpendicular to the body (Figure 1D) and the contact ring was in light contact with the skin. This was checked before each single measurement. To familiarize participants with the measurement procedure, test measurements were performed at region 1 with R and A at least once. If the patient breathed during an indentation, significant changes in muscle activity were observed, or other disturbances were detected, the respective measurement was repeated.

Data was processed and analyzed using MATLAB R2022b (MathWorks, Inc., Natick, MA). Three measurements were manually selected for each participant at each measurement region. Data recorded before a detectable contact force and after maximum stroke was not considered for further data processing. For the force-displacement curves, the raw data was cleaned of recording-related outliers. Polynomial curve fits f δ of degree n = 1 , 2 , … , 5 were used for the uniform and continuous description of the non-linear curves for each participant (Figure 3A) and as a mean for each measurement region (Figure 3B). The polynomials (1) were selected according to two criteria: the least possible degree with minimum RMSE, and validity in the range 0 ≤ δ ≤ δ max , based on the available cleaned measurement data. f δ = f 1 δ n + f 2 δ n − 1 + … + f n δ + f n + 1 (1)

The coefficients f n are in descending powers, and the length of f is n + 1 . The mean force-displacement curves for each measurement region with R or A were based on all the respective raw data sets of the participants. To quantify entire test ranges, these combined data sets were also used to create conforming 2D boundaries, the upper and lower limits of which were each described by a curve fit (Figure 4). Goodness of fits (Sum of Squares Due to Error, R-Squares, and RMSEs) are provided with all coefficients f n in Supplementary Material S1.

For the relative skin or surface displacements, we used the measuring points from the fifth row from the anterior of both ToF sensors (Figure 5A), which were synchronized in time with the force-displacement data. These two sets of eight ToF measuring points were located to the left and right of the indenter tip respectively and were used to deduce continuous surface displacement curves: In relation to the reference plane, the vertical components, as visualized in Figure 5B, served as the relative surface displacements. Offsets to the indenter tip were eliminated on the surface side. The first measurable contact force as a result of an incipient deformation was taken as reference. Assuming that measuring points close to the indenter tip are distorted, raw data in the range − 0.9   r t ≤ x ≤ 0.9   r t were discarded. The direct connection of the remaining most central measuring points from the left and right side resulted in two intersection points with the indenter tip profile. From these, together with the measuring points and three additional base points on the reference plane, curve fits of third degree were created separately for each side. The base points on the reference plane were introduced because the field of view of the ToF sensors did not always capture the entire displacement. Consequently, the start of the deformation on the reference plane could be unknown. To approximate it, the positions x B of the base points located within an 8 mm interval were shifted in 5 mm steps from the outermost ToF measuring point to x B ≤ r C = r t + 100  mm and a fit was calculated in each case. This is subject to the assumption that the surface displacements are continuous and can be described by a third degree polynomial. The selection of a participant-specific fit v x per side and indentation depth δ I is done by minimizing the cost function ϴ x B in (2). ϴ x B = R M S E v x + Q , (2) Q = 0.75 x x m a x r C  if  v x ≤ r C ∩ reference plane ≠ ∅ 10  if  v x ≤ r C ∩ reference plane = ∅ R M S E v x is the standard error of the regression of v x and x x m a x is the intersection point of v x with the reference plane for a given x B . If no intersection existed, the penalty factor was applied instead.

For mean surface displacement per measurement region u x (Figure 5D), a fit of third degree was calculated over all v x (Figure 5C) of both sides combined (mirrored left side) for one δ I each. Each of these complete experimental surface deformations is defined for the range 0  mm ≤ x ≤ r C and includes the profile of the indenter tip in the negative z-direction for x < x min and the reference plane for x > x max . With n = 4 , u x is analogous to (1). As part of the objective function analysis (Section 2.9), the transitions to the indenter tip and the reference plane were automatically smoothed in an x-axis section of up to 10 mm for a continuous displacement curve.

The experimental force-displacement results were statistically analyzed for T1) differences between the six measurement regions with similar muscle activities (R or A), and T2) differences between muscle activities for the six measurement regions. A p-value of <0.05 was considered statistically significant. Statistical tests were conducted for δ = 7 , 14 , and  20 mm where available. As a result of these multiple comparisons, Bonferroni correction factors were used. All data sets for all subjects were tested for normal distribution using the one-sample Kolmogorov-Smirnov test. Because the test was rejected for a number of data sets, we used the nonparametric Kruskal–Wallis test for T1), considering each measurement region independent from one another. Post hoc comparisons were made with Tukey’s honestly significant difference procedure (multiple comparison test) for comparisons between regions. Assuming that the respective measurements from R and A are paired, we conducted a non-parametric Friedman test with a Bonferroni factor of 2 ( α = 0.025 ) for T2).

For the inverse FEA of an axisymmetric indentation test, we utilized the framework indentify (https://github.com/SolavLab/indentify, v1.0.1) developed and made available by Oddes and Solav (2023) and adapted it to our needs. For pre- and postprocessing of the simulations, running in the FE solver FEBio v4.3 (Maas et al., 2012), we used MATLAB R2022b with the open-source toolbox GIBBON (Moerman, 2018). The indentation model comprises an FE cylinder with radius and height r C and Ogden material model (cf. Section 2.8). No differentiations were made between the different layered abdominal tissue components. Skin, fat, muscles, and all subsequent structures were lumped and modeled as a homogeneous material. The indenter tip was modelled rigid with the same geometry as in our experiments and was displaced vertically downwards to the respective indentation depths δ I . In the tangential section, the 90° circular sector of the indenter tip including a 20 mm cylinder section (Figure 6B) had 60 triangular shell elements. Four equally spaced δ I were simulated for each region (Table 2) to match with the experimental data. Maximum displacement was δ max = δ I = 4 = 25 mm for R2-R5. Friction between indenter tip and cylinder surface was omitted with a friction coefficient of zero using FEBio’s sliding-elastic contact formulation. To reduce simulation time, only a cylinder sector of 2° (Figure 6A) was solved with the meshing characteristics and boundary conditions described in detail by Oddes and Solav (2023).

An FE mesh convergence analysis was conducted to check the numerical simulation accuracy for different mesh densities. The mesh density bias was 1.1 towards the center and 0.8 towards the top of the FE cylinder. No larger mesh density bias could be set to ensure model stability until δ max and mesh refinement factors N < 7. A fine mesh with a mesh refinement factor of N ∼ = 6 served as reference for data comparison at δ = 20 mm. Target was a convergence below 1% of the relative errors for the indentation force (3) and the surface displacement (4). E F N = F N − F N ∼ F N ∼ ∙ 100 % (3) E u N = ∑ i = 1 N n N u i N − u i N ∼ u i N ∼ ∙ 100 % (4) F N is the maximum indentation force and the vector u i N denotes the final displacement of the i t h node on the upper outer edge of the FE cylinder segment (cf. Figure 6B) using a mesh refinement factor N. N n N is the number of the nodes used. To take into account varying material behaviors, three material sets covering the entire parameter space were analyzed. As a result of the mesh convergence analysis (compare with Section 3.3), we used a mesh refinement factor of N = 3 and quadratic elements (hex20 and penta15) in the objective function analysis (Section 2.9). This resulted in 30 elements in radial, 15 elements in axial, and one element in tangential direction.

Consistent with previous mechanical descriptions of human soft tissue (Holzapfel, 2000; Erdemir et al., 2006; Flynn et al., 2011; Halloran and Erdemir, 2011; Maas et al., 2012; Moerman et al., 2017; Calvo-Gallego et al., 2018; Wang et al., 2019; Marinopoulos et al., 2020; Alawneh et al., 2022; Lohr et al., 2022), we used a hyperelastic, nearly-incompressible Ogden material model (Simo and Taylor, 1991) to describe the nonlinear force-displacement and surface deformation behaviors gathered in this study. With a set of material parameters p = p 1 , p 2 = c , m and the bulk-like modulus κ , the uncoupled first-order Ogden strain energy density function Ψ integrated in FEBio is defined in (5), Ψ λ i = c m 2   ∑ i = 1 3 λ ∼ i m − 1 + κ 2 ln   J 2 (5) where λ i is the i t h deviatoric principal stretch and J represents the volume ratio. For a nearly isochoric deformation, κ = c 2 · 10 3 was selected (Moerman et al., 2017; Oddes and Solav, 2023).

Based on the objective function analysis with synthetic reference values of Oddes and Solav (2023), we defined the combined objective function F f u p ; δ as the modulated sum of the overall normalized errors between the indentation forces F f p ; δ and the associated overall normalized relative errors between the surface displacements F u p ; δ in Eq. (6). F f u p ; δ = η   F f p ; δ + 1 − η   F u p ; δ ,   η   ϵ   0 , 1 (6)

The modulation factor η enables a convex combination of both residual errors. This means that η = 0 or η = 1 corresponds to a sole evaluation of the superficial displacements or the indentation reaction forces. The residual errors between the indentation forces at an indentation depth δ are defined in (7). F f p ; δ = f p ; δ − f δ 2 f δ 2 (7) f δ is the experimental mean and f p ; δ is the simulated indentation reaction force in negative z-direction, respectively. F u p ; δ quantifies the combined surface displacement deviations in (8). F u p ; δ = ∑ i = 1 N ¯ n w n , i · u i p ; δ − u i δ 2 u i δ 2 (8) u i p ; δ and u i δ are the simulation and the mean reference surface displacements during an indentation depth δ of the i ^th-node in negative z-direction. No displacements in directions other than the z-direction could be tracked using our ToF measurement setup. After interpolating the mean reference displacements (Table 5) to all nodes of the simulation model, only the nodes N ¯ n that fulfill x min ≤ x n ≤ x T r i m were considered. x n is the node position in x-direction and x min represents the outer edge of the indenter tip. x T r i m in turn results from the reference displacements δ T r i m = u x T r i m , which must at least be given so that a reasonable evaluation of F u p ; δ is possible. Due to the processing of the measurement data (Section 2.5), u x ≥ x max = 0 applies to all mean displacements. For the calculation of F u p ; δ , this leads to an underrepresentation of displacement errors near the indenter tip due to lim   x → x max u x = 0 , despite an inversely proportional weighting to their initial radial coordinates w n . w n = 1 r 1 , 1 r 2 , … , 1 r N ¯ n T 1 r 1 , 1 r 2 , … , 1 r N ¯ n T (9)

The nodes visualized in Figure 6B (vertex and edge nodes in the range x min ≤ r N n ≤ r C ) on the outer edge of the cylinder section were represented by their coordinates r 1 , r 2 , … , r N n . The number of all surface nodes used in each case was N ¯ n ≤ N n ∊ N .

To analyze the objective functions combined over all indentation depths, δ I (10) was specified for the resulting parameter space, where p is an element of the discrete parameter space P = C × M . F f u t o t p = 1 4 ∑ I = 1 4 F f u p ; δ I ,  for I = 1 , 2 , 3 , 4 (10)

Consequently, the minimum of F f u t o t p constitutes the resulting material parameter p r e s for the FE model, which approximately results in the smallest error to the experimental data (best fit) over all indentation depths and the selected η . The 2D grid of P is based on C and M , which comprise the evenly spaced Ogden material parameters c and m: C = 1 : 1 : 60  kPa , and M = 4 : 1 : 125 .

We evaluated a trim factor δ T r i m in the range 0.1 − 1.2  mm best suitable for evaluating the objective function for each measurement region (Table 2) using six parameters: 1) The SD of distances between F f u t o t p and all F f u p ; δ I , 2) the proportion of F f u t o t p > 1.5 in P , 3) the circularity measurement for F f u t o t p ≤ 1.5 in P , 4) the relative change of mean F f p ; δ to F u p ; δ over all δ I , and 5) and 6) the change in the resulting relative material parameters c and m. Their results were analyzed for the entire parameter space and are visualized as an example in Supplementary Material S2 over the range of δ T r i m . To determine the shape parameters 1–3, the parameter space was nondimensionalized using the respective p max = c max , m max . For parameter 4, the minimum difference to 0.5 (equal weighting), assuming the lowest possible trim factor, was aimed for. Convergence was the aim for all other parameters. For 1, 2, and 3 with <2% and for 5, and 6 <1%. The rounded mean value of all six individual target trim factors resulted in the δ T r i m per region used in the objective function analysis.

The processed force-displacement curves for the measured regions with distinction between R (fully relaxed musculature) and A (controlled activated musculature) are visualized in Figure 4 and the mean curve fits f δ are listed in Table 4. As usual for mechanically loaded layered biological soft tissue (Zhang et al., 1997; Huang and Zheng, 2015), non-linear courses with a toe region with very small stiffnesses at low deformation, followed by tissue stiffening with increasing indentation depths, can be observed for the relaxed curves. Transitions between the two sections were on average 12.1 ± 2.5 mm for R and 4.3 ± 1.6 mm for A. The measurements for A were less compliant, and the curves were almost linear after the toe region. The activation of the muscles under the skin and subcutaneous tissue after uniaxial compression therefore resulted in a stiffer overall system that behaved less like usual soft tissue. In comparison, participant 6 (P06) had the lowest body fat percentage at 10.3% (cf. Table 1) and a pronounced lateral abdominal musculature due to his athletic background. The measured force at A4 was therefore more than twice as high as the average. For A1, A4, and A5, P06 forms the upper range limits (cf. Supplementary Material S3) but shows no deviations from the average when his muscles were relaxed. It could also be seen that the lack of subcutaneous fat <13.6% led to an approximate linearization of the overall anterolateral force-displacement curves in P01 and P06. P02 set the upper limit for A2. He had pronounced trunk muscles to prevent back pain and an average body fat percentage of 17.2%. After a toe region of less than 1.3 mm, the force-displacement curves of P02 were almost linear. The force ranges increased with greater displacements and were 14.6 ± 11.57 N at δ max * .

Overall, measurement regions 2, 3, 4, and 6 showed large linear sections with muscles activated. In the center of the linea alba and laterally (regions 1 and 5) the curve curses remained predominantly non-linear. Mean gradients of the linear sections for regions 1 to 6 with activated musculature could be approximated with 1.42, 1.39, 1.43, 1.08, 1.36, and 5.22 N/mm. For better comparability, we subdivided the curves with relaxed muscles at 0.45 · δ max and specified a stiffness value for each segment. From region 1 to 6, these were 0.34, 0.29, 0.24, 0.26, 0.25, and 0.72 N/mm for the first segment. For the second segment, linearly approximated stiffnesses were 0.56, 0.56, 0.49, 0.41, 0.62, and 2.31 N/mm. The subjects with a body fat >21% represented the lower anterolateral limits of the ranges. For relaxed musculature, no participant represented a systematic upward or downward outlier. The lowest variance between the subject data was observed for R3 and the highest for A5. The resulting RMSEs for the mean curve fits f were 1.18 ± 0.765 N/mm and 4.97 ± 2.996 N/mm for all regions with R and A. Consequently, interindividual differences have increased due to the activation of the trunk muscles.

The Kolmogorov-Smirnov tests (T1) were statistically significant ( p < 0.001 ) for the indentation depths considered across all participants. This means that neither the six measurement regions for relaxed nor for active musculature originate from the same distribution. Analysis of variances of the various measurement regions also showed that the results for R6 differed significantly ( p < corrected  α ) from R3, R4, and R5 as well as R3 from R1. Limited to δ = 7 mm, this also applied for R6 to R2. For an activated musculature, only region 6 did not differ significantly from region 2 (cf. Figure 3B). Highly significant differences were also found in the Friedman test (T2): Depending on a relaxed or activated musculature, the group means of measurement regions 1 to 6 differ from each other ( p < 0.001 ).

The experimental mean surface displacements for all measurement regions at δ I are listed in Table 5 and are visualized for region 3 in Figure 5D. All other regions and indentation depths are visualized in Supplementary Material S4. Each displacement curve was calculated using the individual curves of all participants from both body sides. As can be seen in Figure 5, the left and right sides of the measurements differ. The mean deformation curve therefore represents an approximation for the respective region. The weakest symmetry is found in regions 5 and 6 (see colored indication of the sides). Fits of third degree were suitable for describing the measured displacement in all cases. In individual cases, lower or higher degrees led to abrupt transitions, no intersection with the reference plane, or oscillations. A qualitative comparison of relaxed and activated musculature showed that curve shapes for R had greater slope, x min were smaller (closer to the center of the indenter tip), and x max were therefore also relatively smaller. In addition, the displacement curves for regions 3, 4, and 5 were concave and for 1, 2, and 6 they were more linear.

We ran all simulations on a PC with Windows 11 Pro, Intel Core i7-10700K 3.80 GHz CPU, and 32 GB RAM. The results for the FE mesh convergence analysis are shown in Figure 6C. Given a short calculation time of ≤60 s, the errors for E F N and E u N are convergent and <1% with N = 3 . Lower N, however, provided a barely relevant time advantage, but increased the error for the surface displacement almost threefold, especially with more compliant material.

Four parameter spaces P with a total of 3321, 2565, 2565, and 2115 parameter sets for δ max * = 13, 16, 20, and 25 mm were calculated respectively, depending on the properties of the experimental reference data. Sets with δ max * = 25 mm took an average of 69.1 ± 23.84 s each. Figure 7 shows a representative example for the numerical interim results of the objective function analysis. The resulting shapes of F f u p ; δ I are visualised as contour plots for each specific indentation depth step δ I . The centers highlighted with markers define the minima of these contour plots and thus the respective parameter set p I , which represent optimal solutions for δ I and η . The optimization results for η = 0.5 (center column) together with the intermediate results (Figure 7) are compiled in Figure 8 for each measurement region investigated. The resulting material parameter sets p r e s at the minimum of F f u t o t p [see Eq. (10)] are indicated as well. All results are listed in Table 6.

From the closed shapes of F f u t o t p , a unique p r e s could be identified for all regions with the optimization conditions used. The smallest absolute error occurred for R6 with F f u t o t = 0.12 and the largest for R1 with F f u t o t = 0.43 . Partial errors of F f p ; δ I and F u p ; δ I were accumulated in F f u t o t over all δ I . As an example, for region 3, Figure 9 compares the FE results of the indentation FE model with the experimental data used. Deviations between the surface displacements are thus recognizable. However, their quantitative comparison for F u was conducted solely in the surface evaluation range indicated, between the intersection of the indenter tip profile and δ t r i m (see Section 2.9). In general, a more concave surface deformation was observed for the material model used compared to some of the experimental data. The resulting material parameters for the different abdominal regions can be characterized as follows: With relaxed musculature c = 6.6 ± 1.14 kPa, m = 18 ± 7.04 anterior-lateral, and c = 25 kPa, m = 19 posterior. With activated musculature c = 17.8 ± 1.79 kPa, m = 21.4 ± 2.97 anterior-lateral, and c = 100 kPa, m = 23 posterior. The absolute deviations between f * and f s i m were maximum at δ max * with 3.4 ± 2.36 N.