journal of the mechanical behavior of biomedical materials 39 (2014) 270 –278
Available online at www.sciencedirect.com
www.elsevier.com/locate/jmbbm
Research Paper
An alternative measurement tool for the identification of hysteretic responses in biological joints Sean L. Borkowskia,b, Sophia N. Sangiorgiob, Edward Ebramzadehb,n, Sami F. Masric a
Biomedical Engineering IDP, University of California, Los Angeles, CA, United States Orthopaedic Biomechanics and Mechanobiology Lab, The J. Vernon Luck, Sr., M.D. Orthopaedic Research Center at the Orthopaedic Institute for Children and University of California, Los Angeles, 403 W. Adams Blvd. Los Angeles, CA 90007, United States c Sonny Astani Dept. of Civil and Environmental Eng., Viterbi School of Engineering, University of Southern California, 3620 S. Vermont Ave, KAP 210, MC 2531, Los Angeles, CA 90089, United States b
ar t ic l e in f o
abs tra ct
Article history:
In structural engineering, sophisticated multi-dimensional analysis techniques, such as
Received 2 May 2014
the Restoring Force Method (RFM), have been established for complex, nonlinear hysteretic
Received in revised form
systems. The purpose of the present study was to apply the RFM to quantify nonlinear
12 July 2014
spine hysteresis responses under applied moments. First, synthetic hysteretic spine
Accepted 30 July 2014
responses (n ¼50) were generated based on representative results from pure moment
Available online 8 August 2014
flexion–extension loading of a human cadaveric lumbar spine segment. Then, the RFM was
Keywords:
applied to each hysteresis response to describe the flexion–extension rotation as a function
Spine biomechanics
of applied moment and simulated axial displacement using a set of 16 unique coefficients.
Spine hysteresis
Range of motion, neutral zone, elastic zone, and stiffness were also measured. The RFM
Nonparametric identification
coefficient corresponding to the 1st-order linear dependence of rotation on applied
Restoring Force Method
moment was dominant, and paralleled changes in elastic zone. The remaining RFM coefficients were not captured from the traditional biomechanical analysis. Therefore, the RFM may potentially supplement the traditional analysis to develop a more comprehensive, quantitative description of spine hysteresis. The results suggest the potential for more thorough and specific characterization of spine kinematics, and may lead to future applications of such techniques in characterizing biological structures. & 2014 Elsevier Ltd. All rights reserved.
1. n
Corresponding author. Tel.: þ1 213 742 1378; fax: þ1 213 742 1365. E-mail addresses: [email protected] (S.L. Borkowski), [email protected] (S.N. Sangiorgio), [email protected] (E. Ebramzadeh), [email protected] (S.F. Masri). http://dx.doi.org/10.1016/j.jmbbm.2014.07.033 1751-6161/& 2014 Elsevier Ltd. All rights reserved.
Introduction
Until recently, many biomechanical studies have examined spine motion using parameters obtained from spine responses as a function of two variables, moment and rotation. For example, many previous studies have drawn conclusions by
journal of the mechanical behavior of biomedical materials 39 (2014) 270 –278
analyzing only range of motion (ROM) (Ames et al., 2005; Oda et al., 2002; Ponnappan et al., 2009; Rathonyi et al., 1998). This simplification has facilitated solving many biomechanical problems with reasonable accuracy. For example, when comparing motion of an injured spine to its motion after rigid fusion, for a given moment, the ranges of motion are very different (Ponnappan et al., 2009). However, with an increasing trend towards the use of motion-preservation implants, such a simple analysis may be insufficient in characterizing all of the differences among the various treatment options. Spine hysteresis curves obtained from measuring rotation as a function of applied moments are typical of many other nonlinear systems encountered in structural engineering and applied mechanics. Consequently, analysis techniques have been developed over the last four decades that allow quantitative characterization of these systems. Specifically, systematic nonparametric identification techniques have been established and utilized to comprehensively and quantitatively describe complex nonlinear hysteresis through a series of coefficients, each relating to distinct attributes of the system (Anderson et al., 1984; Masri et al., 1991, 2004, 2005; Worden and Tomlinson, 2001). However, to date, the spine biomechanics community has not adapted these techniques. If applied to spine hysteresis, rather than analyzing rotation solely as a function of applied moments, other variables, including axial displacement, intervertebral disc health, or disc strain, could be incorporated. With a multi-dimensional analysis, clinical diagnosis and treatment may improve, similar to other fields. For example, given both imaging and mechanical knee data, studies have distinguished between healthy and unhealthy knee conditions (LeBrun et al., 2012; Makihara et al., 2006). The future of spine research may be enhanced by developing similar multi-dimensional analyses to provide convenient methods for clinicians to synthesize patient data for improved diagnosis and treatment.
271
We have previously described the application of a multidimensional nonparametric identification technique to hysteretic responses of similar cadaveric lumbar spines under pure moment loading (Borkowski et al., 2014). In the present study, we generated synthetic hysteretic spine responses to determine whether the nonparametric identification technique is capable of distinguishing among the complex motions and provide information beyond that captured by traditional moment–rotation analyses.
2.
Materials and methods
2.1.
Experimental spine data
For the present study, we needed an assortment of hysteresis curves that systematically differed in certain variables, but kept others constant. Since measurements with different cadaveric spines vary in random and uncontrolled ways, we chose to create a synthetic set of hysteresis curves. In a previous study, intact cadaveric lumbar functional spine units (FSUs) were subjected to pure moment flexion– extension loading, from which, hysteresis curves were generated (Sangiorgio et al., 2011). From these results, a representative hysteresis curve was used (Fig. 1). Three variables had continuously been recorded as a function of time: the flexion–extension rotation, defined as the rotation in the sagittal plane of L4 with respect to L5 (θ); the axial displacement, defined as the motion of L4 with respect to L5 in the superior–inferior (cranial–caudal) direction (y); and the applied moment (x) (Fig. 2). In order to apply the multi-dimensional nonparametric identification technique, all three variables were needed. Based on the representative cadaveric lumbar spine results, 50 synthetic datasets were created, each consisting
Fig. 1 – (a) Applied moment as a function of time; (b) Flexion–extension rotation as a function of time; (c) Flexion–extension rotation as a function of applied moment.
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journal of the mechanical behavior of biomedical materials 39 (2014) 270 –278
Fig. 2 – The motion of the superior vertebra (top) relative to the inferior vertebra (bottom) is shown. In this example, the vertebrae are flexed with respect to each other as a result of the applied flexion moments, My and MY0 . Two local coordinate axes, xyz and X0 Y0 Z0 , are shown, each attached to a single vertebra. The pure moment loading, My and MY', are applied about the y and Y0 axes which are perpendicular to the page, with a positive direction coming out of the page. Rotation during flexion is defined as the Euler angle describing the rotation of X0 Y0 Z0 with respect to xyz about the y-axis. Axial displacement is defined as the translation of X0 Y0 Z0 with respect to xyz in the z-direction.
Table 1 – Fourier-series coefficients.
2.2. Nonparametric identification technique (Restoring Force Method)
Parameter
Value
Parameter
Value
a0 a1 a2 a3 a4 a5 a6 a7 a8
0.1363 8.999 0.3196 1.638 0.1523 0.5257 0.07491 0.1446 0.01429
W b1 b2 b3 b4 b5 b6 b7 b8
0.3135 1.513 0.1536 0.6893 0.1482 0.4044 0.1369 0.2756 0.08603
of estimated rotation as a function of time, applied moment as a function of time, and axial displacement as a function of time. To obtain the estimated rotation as a function of time, we modeled the flexion–extension rotation (θ), recorded during the cadaveric experiment, using the Fourier-series as a function of time (t), as follows: θðtÞ ¼
a0 þ 2
N
∑ ½an cos ðnωtÞ þ bn sin ðnωtÞ; NZ 0
n¼1
the response curve compared to other parameters, such as a2, which resulted in asymmetric changes. This enabled a controlled simulation in which to evaluate the applied nonparametric technique. To obtain the applied moment as a function of time, the moment profile from the representative cadaveric lumbar spine results was used. This moment profile was the same for each of the 50 synthetic datasets, defining the applied moment as an independent variable. Finally, to obtain simulated axial displacement as a function of time, a trigonometric calculation was performed. Based on the experimental cadaveric results, an approximated trigonometric relationship was determined between the measured flexion–extension rotation and axial displacement. This relationship enabled the calculation of axial displacement as a function of time, given a rotation profile. Each of the 50 rotation profiles generated from the Fourierseries variations was subjected to this trigonometric relationship, resulting in 50 axial displacement profiles, one for each of the 50 synthetic datasets. Each of the 50 datasets, consisting of simulated flexion– extension rotation, applied moment, and simulated axial displacement, were subsequently analyzed independently using both the nonparametric identification technique and a traditional biomechanical parametric method, as described below.
ð1Þ
An 8-term Fourier series was applied to minimize error of the fit, resulting in a set of 18 parameters governing the estimated rotation (Table 1). To vary the rotation, parameter a1 was incrementally changed by 0.5, starting at 4, and decreasing to 29 (Fig. 3), resulting in 50 different rotation profiles, one for each of the 50 synthetic datasets. With this range of a1 values, a large spectrum of responses with varying biomechanical parameter magnitudes was obtained. While each of the 18 parameters affected the simulated hysteresis curves, a1 was varied as it resulted in symmetric and systematic changes in
The Restoring Force Method (RFM) (Masri and Caughey, 1979) was applied to analyze the experimental hysteresis datasets. The RFM develops an approximated analytical function to represent the restoring force surface associated with nonlinear systems. In this study, this restoring surface was flexion–extension rotation as a function of applied moment and axial displacement. Specifically, the assumption is made that the restoring force, r, can be expressed in terms of a series in the form: imax jmax
r r^ ðx; yÞ ∑ ∑ Cij Ti ðx0 ÞTj ðy0 Þ i¼0j¼0
ð2Þ
Here, Tk is the Chebychev polynomial of order k, and (x0 ) and (y0 ) denote normalization in the range 71.0. The RFM estimates r^ , corresponding to the simulated flexion–extension rotation, as a function of x and y, corresponding to the applied moment and the axial displacement, respectively (Eq. (2)). While many other variables affect spine rotation, such as intervertebral disc health and intervertebral disc strain, the analysis in the present study was on spine rotation as a function of applied moment and axial displacement. Both variables were readily measured throughout physical biomechanics experiments, and could be subsequently modeled to generate the 50 synthetic datasets. Changes in axial displacement occurred in the actual pure moment cadaveric spine experiments during rotation due to the freedom of motion in the cranial–caudal direction, i.e. 0 N axial load, as well as deformation of disc and facet joint spaces. Due to the viscoelastic properties of the discs, ligaments and connective tissues, the relationship between rotation and axial displacement was hysteretic. Therefore,
journal of the mechanical behavior of biomedical materials 39 (2014) 270 –278
273
Fig. 3 – Simulated moment–rotation curves based on the Fourier-series model from iteration 1, i.e. index 1 (Fourier-series parameter a1 ¼ 4), iteration 25 (a1 ¼ 16), and iteration 50 (a1 ¼ 29). The 50 iterations (index 1 through 50) represent incremental changes to the Fourier-series parameter a1.
the RFM was used to characterize hysteretic rotational responses as a function of both applied moment and axial displacement. A similar analysis could be performed for any set of three variables. The RFM was applied to each of the simulated datasets, and the dominant features of each response were embedded in the normalized Chebychev series coefficients Cij. The set of Cij coefficients for each synthetic dataset provided a comprehensive description of each hysteresis response. The accuracy of the fit between the identified restoring force, r^ (the estimated flexion–extension rotation), and the reference variable (θ(t), the simulated flexion–extension rotation), was calculated as the normalized mean square error (Worden and Tomlinson, 2001) of the deviation between the two measures, defined as a percentage.
2.3.
Biomechanical parametric analysis
In addition to the RFM, each of the moment–rotation curves was analyzed with traditional techniques to obtain the routinely extracted biomechanical parameters (Wilke et al., 1998). Custom MATLAB (The MathWorks, Inc, Natick, MA) programs were used to extract, from each curve, the ROM, neutral zone (NZ), elastic zone (EZ), neutral zone stiffness (NZS), and elastic zone stiffness (EZS).
3.
Results
3.1.
Fourier series results
The 8-term Fourier-series simulating flexion–extension rotation accurately estimated the flexion–extension response from the cadaveric testing, with r2 ¼ 0.9997, and a rootmean-square error of 0.1161. The simulated flexion–extension rotation responses were then plotted against the applied
flexion–extension moment to create simulated hysteretic response curves.
3.2.
Nonparametric identification results
For each of the 50 simulations, the RFM estimated the simulated rotation as a function of applied moment and simulated axial displacement. In each case, the RFMestimated rotation represented the actual simulated rotation data, with normalized error decreasing from just over 22% for the first simulation (Fourier-series parameter a1 ¼ 4), to approximately 7% for the last simulation (parameter a1 ¼ 29) (Fig. 4). For each curve, a set of 16 RFM series coefficients were generated, corresponding to a 3rd order fit. The associated surface (estimated rotation vs. applied moment vs. simulated axial displacement) for the first simulation was highly nonlinear, as represented by the doubly-curved nature of the surface, whereas the nonlinearity decreased with decreases in the Fourier-series parameter a1 (Fig. 4). While a set of 16 coefficients was obtained for each simulation, the majority of the coefficients were negligible, with the exception of C11, C12, C21, C22, C31, and C41. The most dominant term was C21, corresponding to the term x1y0 from the RFM, where x corresponds to the applied moment and y corresponds to the axial displacement. Coefficient C21, or the 1st order dependence of rotation on the applied moment, increased linearly from approximately 6 to 32 (Fig. 5). In comparison, C22 and C41, corresponding to the terms x1y1 and x3y0 respectively, changed nonlinearly, with magnitudes ranging from approximately 0.4 to -1.4. C11, C12, and C31 also changed nonlinearly, but the coefficient magnitudes were smaller. Specifically, magnitudes ranged from approximately 0.07 to 0.14, 0.16 to 0.44, and 0.14 to 0.125 for coefficients C11, C12, and C31, respectively.
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journal of the mechanical behavior of biomedical materials 39 (2014) 270 –278
Fig. 4 – (a) RFM surface estimation for Fourier-series model iteration 1; (b) RFM surface estimation for Fourier-series model iteration 50; (c) Normalized error of estimated surface fit (RFM estimation vs. actual Fourier-series model data) as a function of n¼ 50 index data sets.
3.3.
Biomechanical parameters of Fourier series fit
With each incremental change in the simulated hysteresis curves, the biomechanical parameters changed linearly (Table 2). Specifically, ROM increased from 12.41 to 60.51, EZ increased from 4.91 to 30.01 in simulated flexion, and EZ increased from 2.61 to 26.71 in simulated extension. Additionally, the NZS more than doubled, while the EZS increased nearly 30-fold. On the other hand, changes in NZ were much smaller.
4.
Discussion
In recent years, pure moment testing has emerged as a widespread method for preclinical biomechanical testing of surgical interventions for the spine. Typically, pure moment
tests are conducted to quantify the structural properties of the spine before and after injury, prosthetic device implantation, or surgical release (Cunningham et al., 2003; Sangiorgio et al., 2011, 2013; Wilke et al., 2006). From the resultant moment–rotation hysteresis curves, several parameters are reduced, including range of motion (ROM), neutral zone (NZ), and stiffness. In many cases, this traditional analysis of rotation as a function of applied moment provides valuable insight into the mechanical changes induced by a given spine condition. For example, ROM may increase due to instability caused by a simulated disc injury (Sangiorgio et al., 2011). Similarly, in the presence of degeneration, the NZ may increase (Quint and Wilke, 2008). However, the moment– rotation curves may not always capture changes in the specimens. For example, in a previous study (Sangiorgio et al., 2011), an L2–L3 cadaveric specimen was tested under pure flexion–extension moments in its intact condition (prior
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journal of the mechanical behavior of biomedical materials 39 (2014) 270 –278
Fig. 5 – (a) Average magnitude of RFM generated coefficients (C11 through C44) for the n ¼ 50 Fourier-series model iterations; (b) Standard deviations of RFM generated coefficients for the n¼ 50 Fourier-series model iterations; (c) Histogram representation for C11; (d) Histogram representation for C21; (e) Histogram representation for C31; (f) Histogram representation for C41; (g) Histogram representation for C22; (h) Histogram representation for C12.
Table 2 – Biomechanical parameters of 50 simulated hysteresis curves. EZ (degrees)
NZS (degrees/Nm)
EZS (degrees/Nm)
Measure
ROM (degrees)
NZ (degrees)
þ
þ
þ
Mean Std dev Min Max
36.1 14.5 12.4 60.5
4.7 0.2 4.3 5.1
17.3 7.5 4.9 30.0
14.5 7.3 2.6 26.7
3.1 0.8 1.8 4.4
3.0 0.8 1.8 4.3
1.4 0.9 0.1 3.0
1.3 0.9 0.1 2.8
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journal of the mechanical behavior of biomedical materials 39 (2014) 270 –278
to any injury or surgical intervention). Following this measurement, a disc injury was simulated by cutting the intervertebral disc using a scalpel, creating an annular tear, and the same measurements were repeated under this relatively destabilized condition. A third condition was created by implantation of a prosthetic device called a ‘posterior dynamic stabilization device’ designed to limit extension motion of the spine, and therefore, stabilize the structure (Fig. 6). In this particular case, the moment–rotation responses from each condition appear to be very similar,
and in fact, a traditional biomechanical analysis would not be able to detect differences in ROM, or perhaps even other parameters (Fig. 7). In contrast, in such cases, the RFM may provide insight into the underlying physics of the specimen to explain why the specimen was unaffected by injury or implant. The main advantage of using the RFM is that it removes the necessity to select a class of models for surface fitting. Thus, specific nonlinear features in the system response do not need to be postulated in order to carry out the data
Fig. 6 – Photograph of the posterior dynamic stabilization device implanted in the cadaveric spine is shown. The implant is designed to be placed between adjacent posterior spinous processes to limit motion during extension. The photographs show the (a) entire L2–L3 functional spine unit with the dynamic stabilization device implanted between the adjacent spinous processes, and the (b) close-up view of the implanted dynamic stabilization device.
Fig. 7 – Moment–rotation curve from a representative L2–L3 cadaveric specimen tested under pure flexion–extension moments in the intact condition, the injured condition (simulated disc injury), and the implanted condition (posterior dynamic stabilization device) (Sangiorgio et al., 2011).
journal of the mechanical behavior of biomedical materials 39 (2014) 270 –278
analysis. Specifically, the contribution of specific details of a hysteretic system, such as the presence of inner loops and outer loops or memory effects, is automatically reflected in the identified RFM coefficients. Once identified, a comprehensive and quantitative description of the nonlinear behavior of the spine can be used as a constitutive model for future applications, such as developing nonlinear numerical or finite-element models to capture the complex multidimensional and multi-degree of freedom dynamic responses of the whole spine under a variety of dynamic loading conditions. Such a model could then be used to characterize more complex nonlinear responses of the spine, including damping effects. In the present study, the RFM was used to supplement the traditionally extracted parameters, and develop a more thorough quantitative description of the hysteresis curves. From the RFM results, changes in C21, the most dominant RFM coefficient, paralleled the changes in the flexion EZ. Intuitively, this parameter, C21, could be captured using the traditional moment– rotation analysis, as the term represents the 1st-order dependence of rotation on applied moment. While the other biomechanical parameters changed, including EZS and NZS, none corresponded to changes in the RFM coefficients. For example, changes in coefficient C22, corresponding to the dependence of rotation on the 1st-order applied moment term multiplied by the 1st-order axial displacement term, could not be captured from the traditional analysis of a typical moment–rotation curve. By adding another dimension to the analysis, in this case axial displacement, a more comprehensive description of the hysteretic response can be obtained. Specifically, the set of 16 coefficients provided by this multi-dimensional analysis can help elucidate differences in spine hysteresis which may, in the future, improve clinical diagnoses. In other orthopaedic fields, such as knee or shoulder biomechanics (LeBrun et al., 2012; Makihara et al., 2006; Tyler et al., 2005), instruments have been developed to obtain both kinematic and force data in patients to provide a more comprehensive understanding of each patient's biomechanical function when combined with imaging results. In the spine, this type of data has been limited to primarily in-vitro testing, with only few studies attempting to obtain spine biomechanical data in patients. Specifically, studies have assessed the kinematics of patients’ spines using ROM; (Bible et al., 2010; Mayer et al., 1984; Parks et al., 2003; Saur et al., 1996) however, moment response has not been routinely measured. The moment response has been measured intraoperatively in a variety of patient groups; (Brown et al., 2002; Reutlinger et al., 2012) however, these measurements are limited to the intraoperative environment and cannot be used as a diagnostic tool. Other studies have measured the moments acting through instrumented spine internal fixators; (Rohlmann et al., 1994; Wilke et al., 2001) however, this method clearly has major limitations including cost, surgical invasiveness, and major constraints on the type of data that can be captured. Another approach has been to measure the motion of the spine in vivo, and use the rotations to estimate the corresponding bending moments using simplified models ranging from free-body diagrams to numerical models (Adams and Dolan, 1991; Cholewicki and McGill, 1996; Shum et al., 2010).
277
Shum et al. (2010) assessed the moment–rotation relationship of lumbar spines in patients with and without back pain, using sensors to record the motion and a custom numerical model to estimate the moments. While the maximum moment was lower in symptomatic patients, likely due to pain and the inability to bend, there were significant increases in flexion moment during forward bending motion in symptomatic patients compared with asymptomatic patients. From this result, the authors concluded that it is important to analyze not just the end-points of the hysteresis response, i.e. ROM, but the entire moment–rotation profile as well. Their findings highlight the importance of studying the entire hysteresis response, as we have advocated in the present study. Furthermore, given enough subjects to develop statistical distributions of each RFM coefficient, the RFM could provide, for instance, a more comprehensive diagnostic technique to further distinguish between symptomatic and asymptomatic patients, or among spines treated using different procedures. The present study was based on synthetically derived curves that differed systematically by known parameters; however, these curves were based on physical experiments, and the Fourier-series accurately modeled the measured flexion–extension rotation-time curve recorded during the pure moment testing of an intact, L4–L5 cadaveric spine. Once the model was established, with each incremental change in the Fourier-series hysteresis curves, the typically extracted biomechanical parameters, including ROM and NZ, also changed. Therefore, in this study we have developed a useful method to generate and analyze synthetic hysteresis curves which mimic actual changes in spine hysteresis. In future studies using the RFM, spine rotation will be modeled not only as a function of applied moment and axial displacement, but other variables as well, including disc height, disc strain, and disc degeneration. Overall, the Restoring Force Method accurately estimated spine hysteresis responses (Fig. 4c), providing unique coefficients describing physical characteristics not captured from traditional biomechanical analyses. With further development and understanding of the coefficients as they relate to spine conditions, the RFM has potential for spine diagnostics. Our results demonstrate the potential of advanced analyses in preclinical spine research. With future work, the progression of preclinical testing methods and analyses will help to ultimately improve clinical outcomes and decision making. The multi-dimensional analysis applied in the present study represents one such step.
Acknowledgements No funding was received in support of this work. The authors report no conflicts of interest.
r e f e r e nc e s
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www.elsevier.com/locate/jmbbm
Research Paper
An alternative measurement tool for the identification of hysteretic responses in biological joints Sean L. Borkowskia,b, Sophia N. Sangiorgiob, Edward Ebramzadehb,n, Sami F. Masric a
Biomedical Engineering IDP, University of California, Los Angeles, CA, United States Orthopaedic Biomechanics and Mechanobiology Lab, The J. Vernon Luck, Sr., M.D. Orthopaedic Research Center at the Orthopaedic Institute for Children and University of California, Los Angeles, 403 W. Adams Blvd. Los Angeles, CA 90007, United States c Sonny Astani Dept. of Civil and Environmental Eng., Viterbi School of Engineering, University of Southern California, 3620 S. Vermont Ave, KAP 210, MC 2531, Los Angeles, CA 90089, United States b
ar t ic l e in f o
abs tra ct
Article history:
In structural engineering, sophisticated multi-dimensional analysis techniques, such as
Received 2 May 2014
the Restoring Force Method (RFM), have been established for complex, nonlinear hysteretic
Received in revised form
systems. The purpose of the present study was to apply the RFM to quantify nonlinear
12 July 2014
spine hysteresis responses under applied moments. First, synthetic hysteretic spine
Accepted 30 July 2014
responses (n ¼50) were generated based on representative results from pure moment
Available online 8 August 2014
flexion–extension loading of a human cadaveric lumbar spine segment. Then, the RFM was
Keywords:
applied to each hysteresis response to describe the flexion–extension rotation as a function
Spine biomechanics
of applied moment and simulated axial displacement using a set of 16 unique coefficients.
Spine hysteresis
Range of motion, neutral zone, elastic zone, and stiffness were also measured. The RFM
Nonparametric identification
coefficient corresponding to the 1st-order linear dependence of rotation on applied
Restoring Force Method
moment was dominant, and paralleled changes in elastic zone. The remaining RFM coefficients were not captured from the traditional biomechanical analysis. Therefore, the RFM may potentially supplement the traditional analysis to develop a more comprehensive, quantitative description of spine hysteresis. The results suggest the potential for more thorough and specific characterization of spine kinematics, and may lead to future applications of such techniques in characterizing biological structures. & 2014 Elsevier Ltd. All rights reserved.
1. n
Corresponding author. Tel.: þ1 213 742 1378; fax: þ1 213 742 1365. E-mail addresses: [email protected] (S.L. Borkowski), [email protected] (S.N. Sangiorgio), [email protected] (E. Ebramzadeh), [email protected] (S.F. Masri). http://dx.doi.org/10.1016/j.jmbbm.2014.07.033 1751-6161/& 2014 Elsevier Ltd. All rights reserved.
Introduction
Until recently, many biomechanical studies have examined spine motion using parameters obtained from spine responses as a function of two variables, moment and rotation. For example, many previous studies have drawn conclusions by
journal of the mechanical behavior of biomedical materials 39 (2014) 270 –278
analyzing only range of motion (ROM) (Ames et al., 2005; Oda et al., 2002; Ponnappan et al., 2009; Rathonyi et al., 1998). This simplification has facilitated solving many biomechanical problems with reasonable accuracy. For example, when comparing motion of an injured spine to its motion after rigid fusion, for a given moment, the ranges of motion are very different (Ponnappan et al., 2009). However, with an increasing trend towards the use of motion-preservation implants, such a simple analysis may be insufficient in characterizing all of the differences among the various treatment options. Spine hysteresis curves obtained from measuring rotation as a function of applied moments are typical of many other nonlinear systems encountered in structural engineering and applied mechanics. Consequently, analysis techniques have been developed over the last four decades that allow quantitative characterization of these systems. Specifically, systematic nonparametric identification techniques have been established and utilized to comprehensively and quantitatively describe complex nonlinear hysteresis through a series of coefficients, each relating to distinct attributes of the system (Anderson et al., 1984; Masri et al., 1991, 2004, 2005; Worden and Tomlinson, 2001). However, to date, the spine biomechanics community has not adapted these techniques. If applied to spine hysteresis, rather than analyzing rotation solely as a function of applied moments, other variables, including axial displacement, intervertebral disc health, or disc strain, could be incorporated. With a multi-dimensional analysis, clinical diagnosis and treatment may improve, similar to other fields. For example, given both imaging and mechanical knee data, studies have distinguished between healthy and unhealthy knee conditions (LeBrun et al., 2012; Makihara et al., 2006). The future of spine research may be enhanced by developing similar multi-dimensional analyses to provide convenient methods for clinicians to synthesize patient data for improved diagnosis and treatment.
271
We have previously described the application of a multidimensional nonparametric identification technique to hysteretic responses of similar cadaveric lumbar spines under pure moment loading (Borkowski et al., 2014). In the present study, we generated synthetic hysteretic spine responses to determine whether the nonparametric identification technique is capable of distinguishing among the complex motions and provide information beyond that captured by traditional moment–rotation analyses.
2.
Materials and methods
2.1.
Experimental spine data
For the present study, we needed an assortment of hysteresis curves that systematically differed in certain variables, but kept others constant. Since measurements with different cadaveric spines vary in random and uncontrolled ways, we chose to create a synthetic set of hysteresis curves. In a previous study, intact cadaveric lumbar functional spine units (FSUs) were subjected to pure moment flexion– extension loading, from which, hysteresis curves were generated (Sangiorgio et al., 2011). From these results, a representative hysteresis curve was used (Fig. 1). Three variables had continuously been recorded as a function of time: the flexion–extension rotation, defined as the rotation in the sagittal plane of L4 with respect to L5 (θ); the axial displacement, defined as the motion of L4 with respect to L5 in the superior–inferior (cranial–caudal) direction (y); and the applied moment (x) (Fig. 2). In order to apply the multi-dimensional nonparametric identification technique, all three variables were needed. Based on the representative cadaveric lumbar spine results, 50 synthetic datasets were created, each consisting
Fig. 1 – (a) Applied moment as a function of time; (b) Flexion–extension rotation as a function of time; (c) Flexion–extension rotation as a function of applied moment.
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Fig. 2 – The motion of the superior vertebra (top) relative to the inferior vertebra (bottom) is shown. In this example, the vertebrae are flexed with respect to each other as a result of the applied flexion moments, My and MY0 . Two local coordinate axes, xyz and X0 Y0 Z0 , are shown, each attached to a single vertebra. The pure moment loading, My and MY', are applied about the y and Y0 axes which are perpendicular to the page, with a positive direction coming out of the page. Rotation during flexion is defined as the Euler angle describing the rotation of X0 Y0 Z0 with respect to xyz about the y-axis. Axial displacement is defined as the translation of X0 Y0 Z0 with respect to xyz in the z-direction.
Table 1 – Fourier-series coefficients.
2.2. Nonparametric identification technique (Restoring Force Method)
Parameter
Value
Parameter
Value
a0 a1 a2 a3 a4 a5 a6 a7 a8
0.1363 8.999 0.3196 1.638 0.1523 0.5257 0.07491 0.1446 0.01429
W b1 b2 b3 b4 b5 b6 b7 b8
0.3135 1.513 0.1536 0.6893 0.1482 0.4044 0.1369 0.2756 0.08603
of estimated rotation as a function of time, applied moment as a function of time, and axial displacement as a function of time. To obtain the estimated rotation as a function of time, we modeled the flexion–extension rotation (θ), recorded during the cadaveric experiment, using the Fourier-series as a function of time (t), as follows: θðtÞ ¼
a0 þ 2
N
∑ ½an cos ðnωtÞ þ bn sin ðnωtÞ; NZ 0
n¼1
the response curve compared to other parameters, such as a2, which resulted in asymmetric changes. This enabled a controlled simulation in which to evaluate the applied nonparametric technique. To obtain the applied moment as a function of time, the moment profile from the representative cadaveric lumbar spine results was used. This moment profile was the same for each of the 50 synthetic datasets, defining the applied moment as an independent variable. Finally, to obtain simulated axial displacement as a function of time, a trigonometric calculation was performed. Based on the experimental cadaveric results, an approximated trigonometric relationship was determined between the measured flexion–extension rotation and axial displacement. This relationship enabled the calculation of axial displacement as a function of time, given a rotation profile. Each of the 50 rotation profiles generated from the Fourierseries variations was subjected to this trigonometric relationship, resulting in 50 axial displacement profiles, one for each of the 50 synthetic datasets. Each of the 50 datasets, consisting of simulated flexion– extension rotation, applied moment, and simulated axial displacement, were subsequently analyzed independently using both the nonparametric identification technique and a traditional biomechanical parametric method, as described below.
ð1Þ
An 8-term Fourier series was applied to minimize error of the fit, resulting in a set of 18 parameters governing the estimated rotation (Table 1). To vary the rotation, parameter a1 was incrementally changed by 0.5, starting at 4, and decreasing to 29 (Fig. 3), resulting in 50 different rotation profiles, one for each of the 50 synthetic datasets. With this range of a1 values, a large spectrum of responses with varying biomechanical parameter magnitudes was obtained. While each of the 18 parameters affected the simulated hysteresis curves, a1 was varied as it resulted in symmetric and systematic changes in
The Restoring Force Method (RFM) (Masri and Caughey, 1979) was applied to analyze the experimental hysteresis datasets. The RFM develops an approximated analytical function to represent the restoring force surface associated with nonlinear systems. In this study, this restoring surface was flexion–extension rotation as a function of applied moment and axial displacement. Specifically, the assumption is made that the restoring force, r, can be expressed in terms of a series in the form: imax jmax
r r^ ðx; yÞ ∑ ∑ Cij Ti ðx0 ÞTj ðy0 Þ i¼0j¼0
ð2Þ
Here, Tk is the Chebychev polynomial of order k, and (x0 ) and (y0 ) denote normalization in the range 71.0. The RFM estimates r^ , corresponding to the simulated flexion–extension rotation, as a function of x and y, corresponding to the applied moment and the axial displacement, respectively (Eq. (2)). While many other variables affect spine rotation, such as intervertebral disc health and intervertebral disc strain, the analysis in the present study was on spine rotation as a function of applied moment and axial displacement. Both variables were readily measured throughout physical biomechanics experiments, and could be subsequently modeled to generate the 50 synthetic datasets. Changes in axial displacement occurred in the actual pure moment cadaveric spine experiments during rotation due to the freedom of motion in the cranial–caudal direction, i.e. 0 N axial load, as well as deformation of disc and facet joint spaces. Due to the viscoelastic properties of the discs, ligaments and connective tissues, the relationship between rotation and axial displacement was hysteretic. Therefore,
journal of the mechanical behavior of biomedical materials 39 (2014) 270 –278
273
Fig. 3 – Simulated moment–rotation curves based on the Fourier-series model from iteration 1, i.e. index 1 (Fourier-series parameter a1 ¼ 4), iteration 25 (a1 ¼ 16), and iteration 50 (a1 ¼ 29). The 50 iterations (index 1 through 50) represent incremental changes to the Fourier-series parameter a1.
the RFM was used to characterize hysteretic rotational responses as a function of both applied moment and axial displacement. A similar analysis could be performed for any set of three variables. The RFM was applied to each of the simulated datasets, and the dominant features of each response were embedded in the normalized Chebychev series coefficients Cij. The set of Cij coefficients for each synthetic dataset provided a comprehensive description of each hysteresis response. The accuracy of the fit between the identified restoring force, r^ (the estimated flexion–extension rotation), and the reference variable (θ(t), the simulated flexion–extension rotation), was calculated as the normalized mean square error (Worden and Tomlinson, 2001) of the deviation between the two measures, defined as a percentage.
2.3.
Biomechanical parametric analysis
In addition to the RFM, each of the moment–rotation curves was analyzed with traditional techniques to obtain the routinely extracted biomechanical parameters (Wilke et al., 1998). Custom MATLAB (The MathWorks, Inc, Natick, MA) programs were used to extract, from each curve, the ROM, neutral zone (NZ), elastic zone (EZ), neutral zone stiffness (NZS), and elastic zone stiffness (EZS).
3.
Results
3.1.
Fourier series results
The 8-term Fourier-series simulating flexion–extension rotation accurately estimated the flexion–extension response from the cadaveric testing, with r2 ¼ 0.9997, and a rootmean-square error of 0.1161. The simulated flexion–extension rotation responses were then plotted against the applied
flexion–extension moment to create simulated hysteretic response curves.
3.2.
Nonparametric identification results
For each of the 50 simulations, the RFM estimated the simulated rotation as a function of applied moment and simulated axial displacement. In each case, the RFMestimated rotation represented the actual simulated rotation data, with normalized error decreasing from just over 22% for the first simulation (Fourier-series parameter a1 ¼ 4), to approximately 7% for the last simulation (parameter a1 ¼ 29) (Fig. 4). For each curve, a set of 16 RFM series coefficients were generated, corresponding to a 3rd order fit. The associated surface (estimated rotation vs. applied moment vs. simulated axial displacement) for the first simulation was highly nonlinear, as represented by the doubly-curved nature of the surface, whereas the nonlinearity decreased with decreases in the Fourier-series parameter a1 (Fig. 4). While a set of 16 coefficients was obtained for each simulation, the majority of the coefficients were negligible, with the exception of C11, C12, C21, C22, C31, and C41. The most dominant term was C21, corresponding to the term x1y0 from the RFM, where x corresponds to the applied moment and y corresponds to the axial displacement. Coefficient C21, or the 1st order dependence of rotation on the applied moment, increased linearly from approximately 6 to 32 (Fig. 5). In comparison, C22 and C41, corresponding to the terms x1y1 and x3y0 respectively, changed nonlinearly, with magnitudes ranging from approximately 0.4 to -1.4. C11, C12, and C31 also changed nonlinearly, but the coefficient magnitudes were smaller. Specifically, magnitudes ranged from approximately 0.07 to 0.14, 0.16 to 0.44, and 0.14 to 0.125 for coefficients C11, C12, and C31, respectively.
274
journal of the mechanical behavior of biomedical materials 39 (2014) 270 –278
Fig. 4 – (a) RFM surface estimation for Fourier-series model iteration 1; (b) RFM surface estimation for Fourier-series model iteration 50; (c) Normalized error of estimated surface fit (RFM estimation vs. actual Fourier-series model data) as a function of n¼ 50 index data sets.
3.3.
Biomechanical parameters of Fourier series fit
With each incremental change in the simulated hysteresis curves, the biomechanical parameters changed linearly (Table 2). Specifically, ROM increased from 12.41 to 60.51, EZ increased from 4.91 to 30.01 in simulated flexion, and EZ increased from 2.61 to 26.71 in simulated extension. Additionally, the NZS more than doubled, while the EZS increased nearly 30-fold. On the other hand, changes in NZ were much smaller.
4.
Discussion
In recent years, pure moment testing has emerged as a widespread method for preclinical biomechanical testing of surgical interventions for the spine. Typically, pure moment
tests are conducted to quantify the structural properties of the spine before and after injury, prosthetic device implantation, or surgical release (Cunningham et al., 2003; Sangiorgio et al., 2011, 2013; Wilke et al., 2006). From the resultant moment–rotation hysteresis curves, several parameters are reduced, including range of motion (ROM), neutral zone (NZ), and stiffness. In many cases, this traditional analysis of rotation as a function of applied moment provides valuable insight into the mechanical changes induced by a given spine condition. For example, ROM may increase due to instability caused by a simulated disc injury (Sangiorgio et al., 2011). Similarly, in the presence of degeneration, the NZ may increase (Quint and Wilke, 2008). However, the moment– rotation curves may not always capture changes in the specimens. For example, in a previous study (Sangiorgio et al., 2011), an L2–L3 cadaveric specimen was tested under pure flexion–extension moments in its intact condition (prior
275
journal of the mechanical behavior of biomedical materials 39 (2014) 270 –278
Fig. 5 – (a) Average magnitude of RFM generated coefficients (C11 through C44) for the n ¼ 50 Fourier-series model iterations; (b) Standard deviations of RFM generated coefficients for the n¼ 50 Fourier-series model iterations; (c) Histogram representation for C11; (d) Histogram representation for C21; (e) Histogram representation for C31; (f) Histogram representation for C41; (g) Histogram representation for C22; (h) Histogram representation for C12.
Table 2 – Biomechanical parameters of 50 simulated hysteresis curves. EZ (degrees)
NZS (degrees/Nm)
EZS (degrees/Nm)
Measure
ROM (degrees)
NZ (degrees)
þ
þ
þ
Mean Std dev Min Max
36.1 14.5 12.4 60.5
4.7 0.2 4.3 5.1
17.3 7.5 4.9 30.0
14.5 7.3 2.6 26.7
3.1 0.8 1.8 4.4
3.0 0.8 1.8 4.3
1.4 0.9 0.1 3.0
1.3 0.9 0.1 2.8
276
journal of the mechanical behavior of biomedical materials 39 (2014) 270 –278
to any injury or surgical intervention). Following this measurement, a disc injury was simulated by cutting the intervertebral disc using a scalpel, creating an annular tear, and the same measurements were repeated under this relatively destabilized condition. A third condition was created by implantation of a prosthetic device called a ‘posterior dynamic stabilization device’ designed to limit extension motion of the spine, and therefore, stabilize the structure (Fig. 6). In this particular case, the moment–rotation responses from each condition appear to be very similar,
and in fact, a traditional biomechanical analysis would not be able to detect differences in ROM, or perhaps even other parameters (Fig. 7). In contrast, in such cases, the RFM may provide insight into the underlying physics of the specimen to explain why the specimen was unaffected by injury or implant. The main advantage of using the RFM is that it removes the necessity to select a class of models for surface fitting. Thus, specific nonlinear features in the system response do not need to be postulated in order to carry out the data
Fig. 6 – Photograph of the posterior dynamic stabilization device implanted in the cadaveric spine is shown. The implant is designed to be placed between adjacent posterior spinous processes to limit motion during extension. The photographs show the (a) entire L2–L3 functional spine unit with the dynamic stabilization device implanted between the adjacent spinous processes, and the (b) close-up view of the implanted dynamic stabilization device.
Fig. 7 – Moment–rotation curve from a representative L2–L3 cadaveric specimen tested under pure flexion–extension moments in the intact condition, the injured condition (simulated disc injury), and the implanted condition (posterior dynamic stabilization device) (Sangiorgio et al., 2011).
journal of the mechanical behavior of biomedical materials 39 (2014) 270 –278
analysis. Specifically, the contribution of specific details of a hysteretic system, such as the presence of inner loops and outer loops or memory effects, is automatically reflected in the identified RFM coefficients. Once identified, a comprehensive and quantitative description of the nonlinear behavior of the spine can be used as a constitutive model for future applications, such as developing nonlinear numerical or finite-element models to capture the complex multidimensional and multi-degree of freedom dynamic responses of the whole spine under a variety of dynamic loading conditions. Such a model could then be used to characterize more complex nonlinear responses of the spine, including damping effects. In the present study, the RFM was used to supplement the traditionally extracted parameters, and develop a more thorough quantitative description of the hysteresis curves. From the RFM results, changes in C21, the most dominant RFM coefficient, paralleled the changes in the flexion EZ. Intuitively, this parameter, C21, could be captured using the traditional moment– rotation analysis, as the term represents the 1st-order dependence of rotation on applied moment. While the other biomechanical parameters changed, including EZS and NZS, none corresponded to changes in the RFM coefficients. For example, changes in coefficient C22, corresponding to the dependence of rotation on the 1st-order applied moment term multiplied by the 1st-order axial displacement term, could not be captured from the traditional analysis of a typical moment–rotation curve. By adding another dimension to the analysis, in this case axial displacement, a more comprehensive description of the hysteretic response can be obtained. Specifically, the set of 16 coefficients provided by this multi-dimensional analysis can help elucidate differences in spine hysteresis which may, in the future, improve clinical diagnoses. In other orthopaedic fields, such as knee or shoulder biomechanics (LeBrun et al., 2012; Makihara et al., 2006; Tyler et al., 2005), instruments have been developed to obtain both kinematic and force data in patients to provide a more comprehensive understanding of each patient's biomechanical function when combined with imaging results. In the spine, this type of data has been limited to primarily in-vitro testing, with only few studies attempting to obtain spine biomechanical data in patients. Specifically, studies have assessed the kinematics of patients’ spines using ROM; (Bible et al., 2010; Mayer et al., 1984; Parks et al., 2003; Saur et al., 1996) however, moment response has not been routinely measured. The moment response has been measured intraoperatively in a variety of patient groups; (Brown et al., 2002; Reutlinger et al., 2012) however, these measurements are limited to the intraoperative environment and cannot be used as a diagnostic tool. Other studies have measured the moments acting through instrumented spine internal fixators; (Rohlmann et al., 1994; Wilke et al., 2001) however, this method clearly has major limitations including cost, surgical invasiveness, and major constraints on the type of data that can be captured. Another approach has been to measure the motion of the spine in vivo, and use the rotations to estimate the corresponding bending moments using simplified models ranging from free-body diagrams to numerical models (Adams and Dolan, 1991; Cholewicki and McGill, 1996; Shum et al., 2010).
277
Shum et al. (2010) assessed the moment–rotation relationship of lumbar spines in patients with and without back pain, using sensors to record the motion and a custom numerical model to estimate the moments. While the maximum moment was lower in symptomatic patients, likely due to pain and the inability to bend, there were significant increases in flexion moment during forward bending motion in symptomatic patients compared with asymptomatic patients. From this result, the authors concluded that it is important to analyze not just the end-points of the hysteresis response, i.e. ROM, but the entire moment–rotation profile as well. Their findings highlight the importance of studying the entire hysteresis response, as we have advocated in the present study. Furthermore, given enough subjects to develop statistical distributions of each RFM coefficient, the RFM could provide, for instance, a more comprehensive diagnostic technique to further distinguish between symptomatic and asymptomatic patients, or among spines treated using different procedures. The present study was based on synthetically derived curves that differed systematically by known parameters; however, these curves were based on physical experiments, and the Fourier-series accurately modeled the measured flexion–extension rotation-time curve recorded during the pure moment testing of an intact, L4–L5 cadaveric spine. Once the model was established, with each incremental change in the Fourier-series hysteresis curves, the typically extracted biomechanical parameters, including ROM and NZ, also changed. Therefore, in this study we have developed a useful method to generate and analyze synthetic hysteresis curves which mimic actual changes in spine hysteresis. In future studies using the RFM, spine rotation will be modeled not only as a function of applied moment and axial displacement, but other variables as well, including disc height, disc strain, and disc degeneration. Overall, the Restoring Force Method accurately estimated spine hysteresis responses (Fig. 4c), providing unique coefficients describing physical characteristics not captured from traditional biomechanical analyses. With further development and understanding of the coefficients as they relate to spine conditions, the RFM has potential for spine diagnostics. Our results demonstrate the potential of advanced analyses in preclinical spine research. With future work, the progression of preclinical testing methods and analyses will help to ultimately improve clinical outcomes and decision making. The multi-dimensional analysis applied in the present study represents one such step.
Acknowledgements No funding was received in support of this work. The authors report no conflicts of interest.
r e f e r e nc e s
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