Estimating Apparent Maximum Muscle Stress of Trunk Extensor Muscles in Older Adults using Subject-Specific Musculoskeletal Models

Abstract

Maximum muscle stress (MMS) is a critical parameter in musculoskeletal modeling, defining the maximum force that a muscle of given size can produce. However, a wide range of MMS values have been reported in literature, and few studies have estimated MMS in trunk muscles. Due to widespread use of musculoskeletal models in studies of the spine and trunk, there is a need to determine reasonable magnitude and range of trunk MMS. We measured trunk extension strength in 49 participants over 65 years of age, surveyed participants about low back pain, and acquired quantitative computed tomography (QCT) scans of their lumbar spines. Trunk muscle morphology was assessed from QCT scans and used to create a subject-specific musculoskeletal model for each participant. Model-predicted extension strength was computed using a trunk muscle MMS of 100 N/cm². The MMS of each subject-specific model was then adjusted until the measured strength matched the model-predicted strength (± 20 N). We found that measured trunk extension strength was significantly higher in men. With the initial constant MMS value, the musculoskeletal model generally over-predicted trunk extension strength. By adjusting MMS on a subject-specific basis, we found apparent MMS values ranging from 40–130 N/cm², with an average of 75.5 N/cm² for both men and women. Subjects with low back pain had lower apparent MMS than subjects with no back pain. This work incorporates a unique approach to estimate subject-specific trunk MMS values via musculoskeletal modeling and provides a useful insight into MMS variation.

INTRODUCTION

Trunk extension strength is an important aspect of health status and overall quality of life in older adults, as mobility and balance in older adults are associated with trunk extension strength.^1–3 Furthermore, a longitudinal study reported that postmenopausal women who performed back-strengthening exercises had a reduced incidence of vertebral fractures.⁴ While aging is associated with declines in both muscular strength and lean mass, the strength declines more rapidly than the concurrent loss of muscle mass.⁵ This suggests that aging is associated with decreasing muscle strength per unit size as well as declines in muscle mass.

In musculoskeletal models, muscle strength is typically directly proportional to muscle size, and characterized by specifying a value for maximum muscle stress (MMS). MMS, also referred to as specific tension, specific strength or stress ratio, relates the maximum force a muscle can produce to physiological cross-sectional area:

$$\mathit{MMS}=\frac{F_{\mathit{Max}}}{\mathit{PCSA}}$$ (1)

where F_Max is the maximum isometric muscle strength, and PCSA is the muscle physiological cross-sectional area. A wide range of MMS values have been reported in literature, from 25 N/cm² up to 134 N/cm²,^6,7 representing measurements in different muscle groups, age groups, and sexes. Indeed, while musculoskeletal models typically use a single constant MMS value, MMS is not necessarily constant between muscle groups, as the MMS of elbow flexors is greater than that of elbow extensors,⁸ and the MMS of ankle dorsiflexors is more than twice that of ankle plantar flexors.⁹ Optimal fiber length and pennation angle, if used to calculate PCSA from measured anatomical cross-sectional area, may also affect MMS.¹⁰ Several cross-sectional studies have reported that MMS declines with increasing age,^11–14 while other studies report no change in MMS with aging.^15–17 Similarly, most studies report no difference in MMS with respect to sex,^18,19 although some suggest that men tend to have larger MMS than women.^15,20 Few studies have examined other factors that may be related to MMS. For example, back pain and muscle attenuation (a measure of density from QCT scans) may influence strength in general or MMS in particular. Low back pain is associated with lower muscle attenuation²¹ and reduced muscle strength,^22–24 with most studies finding no difference in trunk muscle area,^21,23 although one study found a reduction in multifidus, but not other trunk muscles.²⁵ Since strength is reduced without a corresponding reduction in trunk muscle area, this suggests that low back pain could be associated with a reduction in MMS. Muscle attenuation is lower in older adults than young adults,²⁶ and muscle attenuation may be associated with physical function and muscle strength.²⁷ These observations indicate that MMS is likely not a singular value as frequently modeled, and as such there is a need for evaluation of MMS variation across different muscle groups and populations.

As musculoskeletal models become more widely used in studies of the trunk and spine in older adults^28–32, reasonable estimates of MMS are necessary for models to be functionally realistic. However, measurements of MMS in spinal muscles are limited, as are measurements of MMS in elderly populations. For trunk and spine muscles, the MMS of 46 N/cm² reported by Bogduk et al.³³ has been used in a variety of modeling studies.^33–35 Another study used an EMG driven model to predict MMS by basing muscle activation on the EMG signal from the muscle group, but trunk muscle anatomy was still taken from previous studies.^36,37 Other spine modeling studies appear to choose an MMS value without direct reference to literature estimates.^38,39 Thus, current approaches to specifying spinal muscle MMS are clearly lacking, and an examination of MMS in a sample of older adults would provide a useful insight into MMS magnitude and variation in the spine.

The primary aim of this study was to determine subject-specific apparent trunk muscle MMS in a group of older adults using experimental trunk extension strength, QCT-based muscle morphology, and musculoskeletal modeling. Our second aim was to examine whether subject-specific estimates of MMS vary with age, sex, muscle attenuation and back pain. We hypothesized that MMS would be higher in men than women, lower in individuals with back pain, positively correlated with muscle attenuation and negatively correlated with age.

METHODS

Study Participants

This study was a cross-sectional analysis conducted in a subset of subjects from the Boston Rehabilitative Impairment Study of the Elderly (Boston RISE) at Spaulding Rehabilitation Hospital in Boston, MA. The Boston RISE cohort consists of 430 community dwelling patients aged 65 years or older at risk for decline in mobility.⁴⁰ A group of 49 participants (mean age = 78, 69% female) from the Boston RISE cohort volunteered for the current study,⁴¹ and had a mean age and population similar to that of the entire Boston RISE cohort (mean age = 77 years, 68% female). This study was approved by the Institutional Review Board of Spaulding Rehabilitation Hospital, and all participants provided written informed consent before participation.

Measured Trunk Extension Strength

During a study visit within the 3 months prior to their QCT scan, quasi-static maximum trunk extension strength was measured on a customized trunk extension machine (Keiser Pneumatic Lower Back; Keiser Corporation, Fresno, CA), as previously described.¹ Participants were instructed to push back slowly and steadily through a small range of motion (15 degrees total range of motion centered on a neutral spine position), pause at the end of the motion, and release. Each participant practiced repetitions on the machine at low levels of resistance (30 lbs, 133N) before starting their actual strength measurement. Subjects started with a trunk extension resistance of 50 lbs (222 N), and resistance was incrementally increased with each successful repetition until the subject was either unable to perform the task or verbally acknowledged that their maximum was reached. The mean number of repetitions to reach a subject’s trunk extension strength resistance was 6.2 ± 1.8. Within the Boston RISE study, intrarater reliability for trunk extension strength measurements was excellent (r=0.96). The strength outcome variable was reported as the one-repetition maximum trunk extension force that could be resisted (N).

Low Back Pain Survey

During a study visit within the 3 months prior to their QCT scan, participants were asked if they had experienced chronic low back pain over the last 4 weeks and to rate their average low back pain from 1–10, with 10 being the most severe or excruciating pain you could imagine. In the current analysis, a pain score of less than 5 was deemed as ‘low pain,’ while a score of 5 or greater was deemed ‘moderate to severe pain’.

QCT Imaging and Analysis

Participants visited an imaging center to have a lumbar spine QCT scan. A 16-slice multidetector scanner (LightSpeed Pro 16; General Electric, Milwaukee, WI) was used to acquire volumetric QCT scans of the trunk at vertebral levels L3-L5. Scans had a nominal slice thickness of 1.25mm and in-plane pixel size of 0.98 × 0.98 mm.

Muscle cross-sectional area (CSA, cm²), position relative to vertebral body centers (cm) and attenuation (Hounsfield units, HU) were determined in the mid-plane slice of each measured vertebral level. Using previously described methods,⁴² we measured the following muscles: rectus abdominis, external obliques, internal obliques, latissimus dorsi, psoas major, quadratus lumborum, erector spinae and multifidus. To identify the CSA, the outer boundaries of each muscle were contoured using an imaging processing program (Analyze; Biomedical Imaging Resource, Mayo Clinic, Rochester, MN).⁴³ The position of several major trunk muscle groups at the mid-plane of vertebral bodies L3 to L5 were calculated using the center of the CSA contours. Combined paraspinal muscle attenuation values were adjusted at each vertebral level by the water-equivalent chamber from a five-chamber phantom (Model 3 QCT Calibration Phantom, Mindways Software, Inc., Austin, TX) scanned with each subject. The voxels within muscle contour boundaries were analyzed on each side and averaged to get a muscle group CSA and attenuation. Voxels outside the range of −50 to 150 HU were excluded since this tissue is outside the range of normal muscle. Overall interreader and intrareader intraclass correlation coefficients for muscle CSA and attenuation (more than 70% greater than 0.90) are good with this method.²⁶

Subject-Specific Musculoskeletal Models of the Spine

A subject-specific musculoskeletal model was created for each subject based on our previously developed musculoskeletal model of the thoracolumbar spine⁴² in OpenSim, an open-source musculoskeletal modeling software.⁴⁴ Vertebral compressive loading and erector spinae muscle tension predicted by the generic model were previously validated and found to be highly correlated with measured data.⁴² To create models that are anatomically similar to the Boston RISE participants, we used subject-specific height, weight and QCT-based measurements of trunk muscle CSA and position. In biomechanical modeling, muscle moment arms and CSA have a profound effect on predicted muscle forces.⁴⁵ A more detailed description of musculoskeletal model scaling can be found in Bruno et al.⁴² Briefly, the CSA and moment arms of the trunk muscles in the musculoskeletal model were adjusted to match the each individual’s muscle values that were measured from the QCT scans. Pennation angles for the trunk muscle were used as reported in previous model development and anatomy literature³⁴ to derive PCSA from CSA measurements. Trunk muscle MMS in this generic model was 100 N/cm².⁴²

Simulation of Trunk Extension Strength Testing

Model trunk extension strength was determined by simulating the trunk extension strength tests performed by the subjects. The positioning of the OpenSim model was set to reflect the seated posture during the measurement, with a posterior pelvic tilt of 35° and a flattened lumbar lordosis of 21° (Fig. 1).⁴⁶

Figure 1.

Schematic diagram showing body posture during trunk extension and variables used to find the location at which vertebral extension force was applied to the model, with an origin at the hip joint center (HJC). The black box shows the location of the seat in relation to the subject. The white arrow denotes the location of applied extension force.

The height of the seatback extension bar that subjects pushed against during testing was measured for each individual with respect to the top of the seat (L_{bar_to_seat}). Sagittal scout views from the QCT scans were used to measure the locations of the hip joint center and ischial tuberosity of the pelvis, providing the distance between them (L_{HJC_IT}). Furthermore, the soft tissue thickness overlying the ischial tuberosity was measured (T_ST), but as the QCT scans were taken with the subjects in a supine position, the tissue thickness over the ischial tuberosity will be reduced in a seated position to approximately 19% of the tissue thickness when supine (0.19 T_ST).⁴⁷

With this information, the force height from the hip joint center was calculated as (Fig. 1):

$$L_{\mathit{bar}\_\mathit{HJC}}=L_{\mathit{bar}\_\mathit{to}\_\mathit{seat}}-L_{\mathit{HJC}\_\mathit{IT}}-0.19T_{\mathit{ST}}$$ (2)

The applied force was exerted on a specific thoracic vertebra in each model. The vertical height from hip joint center (L_{bar_HJC}) was compared to the vertebral center of mass heights in the subject-specific model, and the external force was applied to the thoracic vertebra whose center of mass height was closest to the height of the bar.

Static Optimization

The individual musculoskeletal models were solved using an inverse dynamics based static optimization with a cost function that minimized the sum of cubed muscle activation.⁴⁸ To determine the model-predicted strength, we increased the externally applied force, beginning at 200N, in 10N increments. When the applied external force exceeded the strength of the model, the optimization consistently failed to find a solution because the muscles were too weak to balance the forces acting on it. Thus, the model-predicted strength was taken to be the highest external force for which the static optimization successfully solved.

Subject-Specific MMS Estimations

When measured strength was different than model-predicted strength, the MMS of that subject’s model was iteratively changed until the model-predicted strength matched the measured strength within 20 N. The same MMS value was used for all muscle groups.

Statistical Methods

The effects of muscle attenuation, sex, subject age and back pain on measured extension strength, predicted extension strength, and estimated MMS were examined individually in separate bivariate linear regression models, and were combined within in separate multivariable linear regression models describing each dependent variable. Measured and predicted trunk extension strengths of men and women were compared using a paired 2-tailed student’s t-test. Calculated MMS values were compared between men and women using a 2-tailed student’s t-test. Statistical analyses were performed in Stata (StataCorp LP, College Station, TX) with significance set at α = 0.05.

RESULTS

A total of 49 participants (34 women and 15 men, 67–96 years old) took part in this study, with subject characteristics shown in Table 1. On average, the men were older and taller than the women. Abdominal QCT scans were obtained in all 49 participants, but one subject lacked a phantom in their QCT scan, resulting in usable muscle CSA and moment arms, but not attenuation values.

Table 1.

Mean (standard deviation) characteristics of study participants

Characteristic	All (n=49)	Women (n=34)	Men (n=15)
Age (yrs)	78.24 (7.2)	76.82 (6.2)	81.47 (8.5)
Weight (kg)	75.87 (16.1)	74.63 (18.2)	78.67 (9.6)
Height (m)	1.62 (0.1)	1.59 (0.1)	1.69 (0.1)
Muscle Attenuation (HU)	26.56 (11.5)	25.53 (12.2)	28.90 (9.5)
Pain Score:
No or Low Pain (Score<5)	84% (n=41)	82% (n=28)	87% (n=13)
Moderate to Severe Pain (Score≥5)	16% (n=8)	18% (n=6)	13% (n=2)

The vertebral levels at which extension force was applied in the musculoskeletal models was specific to each subject. Among women, 59% had the extension force applied at T5, with some higher (24% between T1 and T4) and some lower (18% at T6 or T7). The level of extension force application was slightly lower in the spine for men, with 13% at T5, 47% at T6, and 40% at T7.

Measured trunk extension strength was 30% higher in men than in women, and model-predicted trunk extension strength with a constant MMS value of 100 N/cm² was similarly 30% higher in men than women (Table 2). However, with a constant MMS value of 100 N/cm², the model significantly over-predicted trunk extension strength by about 38% (p<0.001) (Table 2, Fig. 2). The slope of the regression line between predicted and measured trunk extension strength was 0.83 (95% CI 0.51 – 1.16). Estimated MMS values were not different for men and women (p=0.99), both having an average of 75.5 N/cm². MMS was not associated with age (p=0.74) or muscle attenuation (p=0.06) in single regressions, however muscle attenuation was negatively associated with MMS in a multiple regression (p=0.04).

Table 2.

Mean (SD) of measured strength and musculoskeletal modeling results

	All	Women	Men	P-value
Measured Trunk Extension Strength (N)	542.0 (148.5)	496.2 (121.9)	645.9 (154.5)	<0.001
Predicted Trunk Extension Strength (N) *From subject-specific model, constant MMS =100	747.6 (206.5)	684.7 (175.5)	890.0 (205.5)	<0.001
Predicted Trunk Extension Strength (N) *From subject-specific model, subject-specific MMS	539.8 (151.8)	495.3 (125.2)	640.7 (162.2)	<0.001
Subject-Specific MMS (N/cm2)	75.5 (20.4)	75.5 (20.2)	75.5 (21.7)	0.999

Figure 2.

Measured versus model-predicted trunk extension strength with a constant MMS of 100 N/cm2. Subjects with moderate-severe low back pain are depicted by triangles, while those with low or no back pain are depicted by circles. Shaded markers indicate male, while open markers indicate female. The black line of y=x represents the ideal situation where model predictions match the measured trunk extension strength.

Subjects reporting moderate or severe back pain had a significantly lower apparent MMS than subjects with little or no back pain, with values of 60.9±10.8 N/cm² and 78.3±20.7 N/cm² respectively (p<0.05) (Fig. 3). There was no significant difference in measured or predicted trunk extension strength based on back pain.

Figure 3.

Average MMS prediction from subject-specific models in subjects with and without back pain. Black column represents adults with no or low pain scores; grey column represents adults with moderate to severe pain scores. Error bars are standard deviation. * p<0.05 vs no/low back pain group

DISCUSSION

This study estimated apparent trunk extensor muscle MMS in 49 mobility-limited older men and women using a unique combination of a validated thoracolumbar spine model, measured trunk extension strength, and subject-specific muscle morphology based on CT scans. This MMS is considered ‘apparent’ MMS since in addition to the muscle itself it may capture external factors that affect in vivo muscle strength, such as decreased neuromuscular activation or pain limitations. The range of MMS values across all subjects in this study (40–130 N/cm²) compares favorably with values reported in a prior study of 10 young men where trunk MMS ranged from 29–108 N/cm²,³⁶ and also overlaps the MMS of 100 N/cm² used in the base model.⁴² On the other hand, our mean trunk MMS value of 78 N/cm² for elderly adults with no back pain is higher than the commonly used value of 46 N/cm².³³ For this estimate, Bogduk et al.³³ created a lumbar spine model based on anatomic descriptions, radiographs, and cadaver studies of men and women. They then calculated the maximum extensor moment about L5/S1 for erector spinae and multifidus fascicles in terms of MMS, and equated this to the average moment of 145 Nm reported by McNeill et al.⁴⁹ for a group of healthy men and women (age=42 years). Using the approach as Bogduk et al.³³ with the OpenSim model of Bruno et al.⁴² used in this study produces a similarly low MMS estimate of 37 N/cm². This suggests that these lower MMS estimates are an artifact of the methodology, which does not require balanced forces across the whole spine. Simulating trunk extension tests using a customizable subject-specific model with static optimization, as in this study, does require balanced forces across the whole spine and thus should be considered more realistic. While our results suggest wide variation in MMS between individuals, very little of this is explained by the independent variables we examined in our primary analysis. Thus, we have provided regressions in Supplement 1 (Table S1) for investigators seeking to estimate apparent MMS based on direct measurements of trunk extension strength.

Overall, our data showed no association between apparent MMS and age or sex, but we did see a negative correlation with muscle attenuation. A few studies have reported an indirect negative association between age and MMS, due to a larger decrease in muscle strength than CSA,¹³ and an increase in intra-muscular adipose and connective tissue in the total muscle volume,¹² however it’s difficult to state that these apparent decreases in MMS weren’t due to other factors, such as pain or neuromuscular activation changes. Our study was limited to older adults with mobility problems, with an age range of 67 to 96 years. This sample of older adults may have limited our ability to detect correlations between age and MMS leading to the lack of support for our hypothesis regarding MMS decreasing with increasing age. We didn’t detect differences in MMS between sexes. This finding agrees with most available data in both young and old men and women.^15,17,50 The negative correlation of MMS with muscle attenuation was counter to our hypothesis, but arises from the fact that attenuation is associated with muscle size as well as strength. While we previously reported in this same group of subjects that attenuation was associated with strength independent of muscle size,⁴¹ muscle CSA and attenuation are also positively associated. In fact, the association between attenuation and CSA (r=0.69, p <0.0001) is stronger than the association between attenuation and strength (r =0.34, p = 0.02), meaning that since MMS can be thought of as strength/CSA, an increase in attenuation would decrease this ratio.

Subjects that self-reported moderate or severe low back pain had significantly lower MMS values than subjects with no or minimal low back pain, suggesting that these individuals produced less muscle force than their muscle size indicates they are capable of. This is consistent with studies reporting reduced trunk muscle strength in patients with low back pain.^24,51 However, it is not fully known if this difference reflects an actual reduction in muscle function, or an apparent reduction because subjects do not utilize the full strength capabilities of their muscles due to pain during strength testing. The latter seems likely, as a reliability study reported that acute development of moderate back pain in two participants led to a 4 standard deviation reduction in their measured extension strength between baseline and retesting.² Additionally, patients with low back pain may have increased co-contraction of abdominal muscles,^52,53 which would act in opposition to trunk extension, reducing their overall extension strength. Given that low back pain may inhibit extensor muscle activation due to pain and increase co-contraction of abdominal muscles, estimates of MMS from these patients could be artificially lower, albeit reflecting the true functional capacity of the individual. Based on the observations in this study, functional limitations due to pain should be considered in future modeling studies, as they may modify the capabilities suggested by the morphology of the musculoskeletal system.

Our study had a number of limitations that could introduce errors or uncertainty into estimates of MMS. One limitation of our study was that the differences between measured and model-predicted trunk extension strength were assumed to be entirely due to MMS. This is a reasonable assumption because it is known that MMS has a high inter-subject variability,³⁶ and will therefore have an important role in this demanding maximum trunk extension strength activity. Due to the extension exercise modeled, the estimated MMS primarily applies to trunk extensor muscles and may not be applicable to abdominal muscles or other trunk muscle groups. One concern that persists across all studies investigating MMS is that a subject’s measured trunk strength might not actually be their maximum trunk strength. Measured trunk extension strength underestimation might have resulted from muscle fatigue due to repeated exertions while testing for maximal strength. Furthermore, the slight concentric movement during trunk extension might have reduced the strength measurements from that of a completely isometric contraction. Thus measured strength may be below actual isometric strength, in which case the estimated MMS would be reduced as well. In biomechanical modeling, trunk strength depends on body positioning. For instance, lumbar lordosis and pelvic orientation are substantially altered from upright standing to a seated position. This is true of trunk extension strength as well, for instance trunk extension strength is significantly different when sitting or kneeling compared to standing.^54,55 Our OpenSim musculoskeletal model allowed replication of the testing set-up by placing the model in a seated position with arms crossed at the chest. However, a limitation is that our model assumed a similar spinal and pelvic position in each subject, while this could have varied between subjects during actual testing. On the other hand, the variations in how subjects of different body size encountered the machine were calculated and modeled accordingly. We also used a cost function that minimized the sum of cubed muscle activations, a standard approach that seeks to minimize fatigue and maximize endurance. It is not well studied whether a different cost function would be more appropriate for a maximum output situation such as this, although in theory any cost function should converge to a similar maximum output. Nonetheless, it’s possible that a different optimization approach could have resulted in slightly higher estimated strength values, which would have reduced estimated MMS. Lastly, while respectable, the sample size of the study limited our ability to investigate all possible MMS covariates. Overall, the level of subject specificity in the models is a great strength of this study, and our 49 unique subject-specific models were a large sample size for a modeling-based study.

In summary, this is the first study to estimate trunk muscle MMS in a large group of individuals using measured trunk extension strength, muscle morphology from lumbar spine QCT images, and subject-specific thoracolumbar musculoskeletal models. Our musculoskeletal models over-predicted trunk extension strength when using a constant MMS of 100 N/cm². We found that using an MMS of 78.3 N/cm² in the musculoskeletal models led to the best predictions of measured trunk strength for both men and women who are mobility limited, have minimal low back pain, and are over 65 years of age. Additional studies are needed to determine if MMS values are similar in younger healthy subjects. Trunk MMS was not correlated with sex or age in this study, but was significantly lower in those with moderate to severe low back pain. This finding warrants further investigation to discover if intrinsic MMS is actually decreasing with pain, or if subjects in pain were just unable to exert their true maximal strength, lowering their apparent MMS.