A general overview of the frontal impact simulation is shown in Fig. 3. The events occurring over time are described in detail in Appendix.
In the following, the effect of different volumetric muscle tissue stiffnesses is shown in comparison with literature regarding resulting stress–strain curves during impact, tissue motion, computation time and injury prediction. To achieve different muscle tissue stiffnesses, the ordinate value (stress value) of the engineering stress–strain curve was scaled, which is defined within the hyperelastic material model MAT_SIMPLIFIED_RUBBER/FOAM used to model muscle and soft tissues in THUMS. Different values of this scaling factor of the ordinate value (SFO) were selected to achieve different stiffness states, as further described in “Methods” of this work.
Each of the following subsections will be discussed in the respective subsection in “Discussion”.
Volumetric muscle element stiffness
As shown in Fig. 4, the slope of the effective stress–strain curve for elements of the buttock and thigh region increases with increasing SFO values. The element stiffness is directly defined by the slope of this curve. The slope of the linear fits, calculated for the timespan of highest loading during impact, approximates the Young’s modulus for each curve. The slope increases with the SFO value, but not proportionally. Arrows indicate points of maximum effective stress and strain. Literature data [25] incorporated in the model showed the highest slope in comparison [25].
In Fig. 5, Young’s moduli of different human and animal muscles from literature data are compared to those calculated from numerical simulation (Fig. 4) for different material stiffness parameters. The data are subdivided into three regions of entirely independent data. From left to right, data on ‘relaxed’, ‘partly contracted’ and ‘contracted’ muscle samples are shown. ‘Relaxed’ includes experimental data from ex vivo experiments from, e.g., isolated bovine and porcine muscle, as well as in vivo human muscle measurements without voluntary contraction. ‘Partly contracted’ includes muscle stiffness states at various levels (e.g., 20%, 30% voluntary contraction, lifting of 7.5 kg weight, etc.) from human volunteers. Data classified as ‘contracted’ includes experimental data from human volunteers, during which the muscle was contracted to the maximum voluntary level or where heavy weights were lifted (15 kg). Further details can be found in Table 1 of Appendix. Experiments during which tetanic excitation was reached in rabbit tibialis anterior muscle by nerve excitation was also assigned to the ‘contracted’ samples [25]. An averaged Young’s modulus of different strain rates in the ‘contracted’ state is shown, as well as separate values of Young’s moduli for different strain rates at the ‘relaxed’, passive muscle state.
Results from SFO0.5 and SFO1 were both classified as ‘Relaxed’. SFO2 as ‘Partly contracted’ and SFO10 as ‘Contracted’. As the Young’s modulus of SFO10 was much higher than in the literature data, respective values are listed separately. The same accounts for literature values of [25], who performed their experiments at higher strain rates than the other sources, resulting in much higher Young’s moduli. Data to the presented injury risk curve [25] were incorporated in MAT_SIMPLIFIED_FOAM via a Table ID and are shown as the stress–strain curve ‘Myers 1998’, cf. Fig. 4. Linear fits, approximating the Young’s modulus, are in range of the literature data (Fig. 5) obtained during tensile tests [25].
Tissue motion
If the material stiffness parameter (SFO value) is increased, the tissue motion caused by the frontal impact is reduced. This can be observed in Fig. 6 in the distal region of the thigh, close to the knee joint. Only optical evaluation on tissue motion based on Figs. 6 and 7 was performed.
Computation time and model stability
Varying the stress–strain load curve (LC) from the default value (SFO = 1) hardly had an impact on the computation time (Figs. 8, 9). The largest difference with approximately 8% to the default SFO1 case was found for the SFO0.5 case with a predefined time-step size of 0.001 ms achieved by selective mass scaling. Generally, the current server utilization by other users and processes as well as hyperthreading seemed to have a much higher impact on the computation time than the change in material model parameters. This was found for the four different SFO values and ‘Myers’, which were calculated multiple times at various states of server utilization. Depending on the state of server utilization, computation times varied by several hours. Differences in computation time were smallest for the isolated CPU. Some of the simulations with undefined time-step size were error-terminated after a minimum computation time of 60 ms (SFO0.5, Isolated CPU). All computations passed cycle 100,000 and 400,000, which were therefore used for computation time comparison.
Results from the simplified cuboid model showed that any changes in SFO value or stress–strain curve including different strain rates (Myers) had no significant impact on the computation time. SFO0.5 again showed the highest decrease in computation time of 3.1% compared to SFO1, while computation time of SFO2 decreased by 0.2% and of SFO10 by − 1.1%. Myers had the highest increase of 2.0%. The final number of cycles calculated was constant for all material parameter variations, for both defined and undefined timestep size. For all cuboid model simulations normal termination occurred.
Regarding model stability, all simulations reached the final calculation cycle of explicit time integration (160 ms simulation time) for the mass-scaled solution (dt2ms), except the SFO0.5 simulation, which was error-terminated after 125 ms due to negative volume errors in the knee region. Therefore, reducing the muscle stiffness resulted in model instabilities.
Injury prediction
Results regarding injury prediction will be presented in two subsections: injuries of bone tissues and injuries of muscle and soft tissues. Resultant forces (contact forces), first principal strain and effective plastic strains were used as injury criteria for injury prediction.
Bone tissues
Force-dependent injury prediction
The seat bottom was exclusively in contact with the THUMS pelvis and proximal region of the thigh. The knee bolster was exclusively in contact with the knee. Resultant forces were obtained from contact forces as described in subsection “Injury prediction”.
Contact forces between the THUMS skin and the vehicle parts and the resulting injury probability are shown in Fig. 10. For the buttock, the peak resultant force increases with increasing SFO value, but not by the same factor. The highest peak force between buttock and seat was found after 75 ms for all stiffness values except the Myers stiffness value. For the latter, the peak resultant force is reached after 65 ms. Increasing muscle stiffness accordingly leads to an increasing probability of hip fracture or dislocation according to the presented injury risk curve [52].
The muscle and soft tissue stiffness changes in the hip and thigh region do not have an impact on the probability of knee injuries (Fig. 10c, d, all except purple). For SFO0.5, SFO1, SFO2 and SFO10 the stiffness of knee surrounding muscle and soft tissues were unchanged from the default settings. For the Myers stiffness case, higher tissue stiffness was defined for all volumetric muscle and soft tissue elements modeled with MAT_SIMPLIFIED_FOAM, including the tissues surrounding the knee joint. Increasing the stiffness of knee surrounding tissues reduced the probability of knee injuries (AIS2+) from 9–11 to 2%.
Strain-dependent injury prediction
The threshold of 3% ultimate strain was hardly exceeded by any cortical or spongy bone elements when analyzed regarding effective plastic strains and first principal strains. Therefore, a lower limit of 1.5% was selected to allow a better comparison of muscle stiffness effects on the predicted injury risk of the bones. The volume of shell and solid elements that failed the threshold of 1.5% effective plastic strain or first principal strain was added up, resulting in a ‘failed element volume’. Depending on the strain type and bone tissue, different impact of muscle stiffness on injury risk prediction can be found.
Data on effective plastic strains of cortical bones (Fig. 11) show that the predicted injury risk based on failed element volume is increasing with increasing muscle stiffness, as well as the number of locations, where the threshold of 1.5% strain is exceeded. In contrast, the Myers stiffness case which is stiffer than SFO10 (Fig. 4) has a slightly lower injury risk based on failed element volume (Fig. 11). Body regions that exceeded the threshold were the right hipbone 1.5 mm thickness (all simulations), 3rd lumbar vertebrae posterior (all except SFO0.5), left hipbone 1.5 mm thickness (SFO2, SFO10) and the 1st lumbar vertebrae posterior (Myers).
Data on first principal strains of cortical bones (Fig. 12) show an increase in predicted injury risk based on failed element volume in the following order: SFO2, Myers, SFO0.5, SFO1 and SFO10. Affected body regions of THUMS were the left femur upper null shell (all except Myers) and the right femur upper null shell (SFO1, Myers).
Data on first principal strains of spongy bones (Fig. 13) show an increase in predicted injury risk based on failed element volume in the following order: SFO1, SFO2, SFO0.5, Myers and SFO10. SFO10 had by far the highest failed element volume. Affected body regions were the right hipbone (all), left hipbone (all except Myers), left patella (all except Myers).
Therefore, no clear tendency between muscle stiffness changes and predicted injury risk of bones of the lower part of the body based on first principal strain evaluation (Figs. 12, 13) was found in this study.
Muscle and soft tissues
For muscle and soft tissues, effective strains were used for injury prediction, as well as the Cumulative Strain Damage Measure (CSDM) for different body regions and tissues based on effective strains (“Injury prediction”).
Effective strain color plots in the dorsal view are shown in Figs. 14 and 15. They were obtained after 75 ms, where for most regions, maximum loading was found. Highest strains in muscle and soft tissue were found in the periphery of the anus and in the dorsoproximal region of the thigh. In the buttock, higher strains were found for the soft tissue (outer layer) compared to the muscle tissue (inner layer). In contrast, high strains were distributed over a larger area for the thigh muscle tissue than for the soft tissue. Peak effective strain values were continuously decreasing with increasing muscle stiffness for all analyzed tissue types and body regions, which is described in detail below. The CSDM value (Figs. 16, 17) also decreases with increasing muscle stiffness for all analyzed tissue types and body regions. Therefore, it becomes evident that increasing muscle stiffness reduces the predicted muscle and soft tissue injury risk in frontal impact scenarios based on strain-dependent data.
Peaks in effective strain of muscle tissue (Fig. 14) for each stiffness case are: SFO0.5 = 103.9% (t = 70 ms); SFO1 = 95.3% (t = 70 ms); SFO2 = 80.4% (t = 65 ms); SFO10 = 61.2% (t = 65 ms); Myers = 30.2% (t = 75 ms). Data on 1st principal strain showed a similar distribution of peak strains, but lower maximum values. Peaks in 1st principal strain are: SFO0.5 = 82.1% (t = 60 ms); SFO1 = 80.5% (t = 60 ms); SFO2 = 78.6% (t = 60 ms); SFO10 = 64.4% (t = 65 ms); Myers = 20.5% (t = 85 ms).
Peaks in effective strain of soft tissue (Fig. 15) for each stiffness case are: SFO0.5 = 81.4% (t = 80 ms); SFO1 = 80.6% (t = 80 ms); SFO2 = 77.6% (t = 80 ms); SFO10 = 56% (t = 65 ms); Myers = 21.5% (t = 75 ms). Data on 1st principal strain showed a similar distribution of peak strains, but lower maximum values. Lower strain peaks are found for the periphery of the anus. Peaks in 1st principal strain are: SFO0.5 = 73:3% (t = 75 ms); SFO1 = 67% (t = 70 ms); SFO2 = 57.9% (t = 70 ms); SFO10 = 45% (t = 65 ms); Myers = 20.2% (t = 85 ms).