Short communication Impact of material and morphological parameters on the mechanical response of the lumbar spine – A finite element sensitivity study Thomas Zander a,n , Marcel Dreischarf a , Anne-Katrin Timm a , Wolfgang W. Baumann b , Hendrik Schmidt a a Julius Wolff Institute, Charité – Universitätsmedizin Berlin, Germany b Zuse Institute Berlin (ZIB), Germany article info Article history: Accepted 11 December 2016 Keywords: Finite element modelling Lumbar spine Sensitivity analysis Model validation Coefficient of determination abstract Finite element models are frequently used to study lumbar spinal biomechanics. Deterministic models are used to reflect a certain configuration, including the means of geometrical and material properties, while probabilistic models account for the inherent variability in the population. Because model para- meters are generally uncertain, their predictive power is frequently questioned. In the present study, we determined the sensitivities of spinal forces and motions to material parameters of intervertebral discs, vertebrae, and ligaments and to lumbar morphology. We performed 1200 model simulations using a generic model of the human lumbar spine loaded under pure moments. Coefficients of determination and of variation were determined for all parameter and response combinations. Material properties of the vertebrae displayed the least impact on results, whereas those of the discs and morphology impacted most. The most affected results were the axial compression forces in the vertebral body and in several ligaments during flexion and the facet-joint forces during extension. Intervertebral rotations were considerably affected only when several parameters were varied simultaneously. Results can be used to decide which model parameters require careful consideration in deterministic models and which parameters might be omitted in probabilistic studies. Findings allow quantitative estimation of a model's precision. & 2017 Elsevier Ltd. All rights reserved. 1. Introduction Finite element (FE) simulations are frequently used in lumbar spinal biomechanics. However, simulations always require sim- plifications and estimations, because model parameters are numerous, often only vaguely known and too complex to imple- ment. Their impact on responses is a priori unknown and complete validation is mostly impossible (Oreskes, 1998). Therefore, the need for sensitivity analyses has been indicated (Anderson et al., 2007; Ayturk and Puttlitz, 2011; Viceconti et al., 2005). The smaller the impact of uncertain parameters on model responses, the more reliable it can be considered. In contrast, parameters with high impact on responses require careful consideration, particularly when creating models representing an average or even an indi- vidual situation. Knowledge about a model's sensitivities is furthermore advan- tageous in probabilistic models with a high number of parameters, which are varied according to their population-based natural probability and combined with each other. For each combination in the design space, a simulation must be performed. The results of all simulations provide the requested result distribution. Parameters with a marginal impact do not need to be varied, thereby reducing the dimensionality of the design space and thus the modelling, computational, and evaluation efforts. Occasionally, model calibration is required (e.g., Garo et al., 2011; Malandrino et al., 2009; Schmidt et al., 2006), with the model being fitted to measurements. This parameter identification by inverse modelling can only be solved if the parameters influ- encing the model responses are known. When more “responsible” parameters have to be determined than experimental results exist, the problem is under-determined and can generally not be uniquely solved. Finally, sensitivity analyses provide an insight into the robust- ness of deterministic FE studies that do not consider probability distributions, rather use a fixed configuration. Such analyses show the results’ robustness with respect to model parameter variations. Due to the complexity of probabilistic approaches, deterministic models are still most commonly used. In 2008, Jones and Wilcox systematically summarised existing sensitivity studies. Current Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com Journal of Biomechanics http://dx.doi.org/10.1016/j.jbiomech.2016.12.014 0021-9290/& 2017 Elsevier Ltd. All rights reserved. n Correspondence to: Julius Wolff Institute Charité – Universitätsmedizin Berlin Augustenburger Platz 113353 Berlin, Germany. Fax: þ4930/209 34 60 01. E-mail address: [email protected](T. Zander). Journal of Biomechanics 53 (2017) 185–190
Impact of material and morphological parameters on the mechanicalresponse of the lumbar spine – A finite element sensitivity study
Thomas Zander a,n, Marcel Dreischarf a, Anne-Katrin Timma, Wolfgang W. Baumann b,Hendrik Schmidt a
a Julius Wolff Institute, Charité – Universitätsmedizin Berlin, Germanyb Zuse Institute Berlin (ZIB), Germany
a r t i c l e i n f o
Article history:
Accepted 11 December 2016
Finite element models are frequently used to study lumbar spinal biomechanics. Deterministic modelsare used to reflect a certain configuration, including the means of geometrical and material properties,
Keywords:Finite element modellingLumbar spineSensitivity analysisModel validationCoefficient of determination
x.doi.org/10.1016/j.jbiomech.2016.12.01490/& 2017 Elsevier Ltd. All rights reserved.
espondence to: Julius Wolff Institute Chariténburger Platz 113353 Berlin, Germany. Fax: þail address: [email protected] (T. Zan
a b s t r a c t
while probabilistic models account for the inherent variability in the population. Because model para-meters are generally uncertain, their predictive power is frequently questioned. In the present study, wedetermined the sensitivities of spinal forces and motions to material parameters of intervertebral discs,vertebrae, and ligaments and to lumbar morphology. We performed 1200 model simulations using ageneric model of the human lumbar spine loaded under pure moments. Coefficients of determinationand of variation were determined for all parameter and response combinations. Material properties ofthe vertebrae displayed the least impact on results, whereas those of the discs and morphology impactedmost. The most affected results were the axial compression forces in the vertebral body and in severalligaments during flexion and the facet-joint forces during extension. Intervertebral rotations wereconsiderably affected only when several parameters were varied simultaneously. Results can be used todecide which model parameters require careful consideration in deterministic models and whichparameters might be omitted in probabilistic studies. Findings allow quantitative estimation of a model'sprecision.
& 2017 Elsevier Ltd. All rights reserved.
1. Introduction
Finite element (FE) simulations are frequently used in lumbarspinal biomechanics. However, simulations always require sim-plifications and estimations, because model parameters arenumerous, often only vaguely known and too complex to imple-ment. Their impact on responses is a priori unknown and completevalidation is mostly impossible (Oreskes, 1998). Therefore, theneed for sensitivity analyses has been indicated (Anderson et al.,2007; Ayturk and Puttlitz, 2011; Viceconti et al., 2005). The smallerthe impact of uncertain parameters on model responses, the morereliable it can be considered. In contrast, parameters with highimpact on responses require careful consideration, particularlywhen creating models representing an average or even an indi-vidual situation.
Knowledge about a model's sensitivities is furthermore advan-tageous in probabilistic models with a high number of parameters,
which are varied according to their population-based naturalprobability and combined with each other. For each combination inthe design space, a simulation must be performed. The results of allsimulations provide the requested result distribution. Parameterswith a marginal impact do not need to be varied, thereby reducingthe dimensionality of the design space and thus the modelling,computational, and evaluation efforts.
Occasionally, model calibration is required (e.g., Garo et al.,2011; Malandrino et al., 2009; Schmidt et al., 2006), with themodel being fitted to measurements. This parameter identificationby inverse modelling can only be solved if the parameters influ-encing the model responses are known. When more “responsible”parameters have to be determined than experimental results exist,the problem is under-determined and can generally not beuniquely solved.
Finally, sensitivity analyses provide an insight into the robust-ness of deterministic FE studies that do not consider probabilitydistributions, rather use a fixed configuration. Such analyses showthe results’ robustness with respect to model parameter variations.Due to the complexity of probabilistic approaches, deterministicmodels are still most commonly used. In 2008, Jones and Wilcoxsystematically summarised existing sensitivity studies. Current
Morphological parameters (MOR)Spinal shape According to Roussouly et al. (2005) “Type” Default 1, 2, 3, 4Joint gap distance Exponential pressure-clearance law Gap distance 0.0 mm 0 … 0.5 mm
T. Zander et al. / Journal of Biomechanics 53 (2017) 185–190186
pre-processors simplify geometrical variations, which have sincebeen performed by considering the dimensions of vertebrae, discs,and ligaments (Cappetti et al., 2016; Meijer et al., 2011; Niemeyeret al., 2012) or by sophistically creating a set of shape modes(Campbell and Petrella, 2016). In addition to systematicallyinvestigating certain geometrical parameters, there are studiesinvestigating several distinct spinal shapes (Campbell et al., 2016;Naserkhaki et al., 2016; Zanjani-Pour et al., 2016).
The present study aims to investigate and compare the modelsensitivities of disc, bone, and ligament material properties as wellas of distinct spinal morphologies between and among themselvesfor the intervertebral rotations, intradiscal pressures, axial-sectionforces, and forces in the ligaments and facet joints.
2. Material and methods
2.1. Finite element model
A previously developed FE model of the lumbar spine for L1–L5 served as thebasis for all analyses. The model was validated using in vitro data for both intra-segmental and intralumbar motions, for intradiscal pressure, and for facet-jointforces (partly shown in Fig. 3). Furthermore, the model delivers results comparablewith those of other research groups. Further details are provided elsewhere(Dreischarf et al., 2014; Zander et al., 2009; Zander et al., 2007).
The basic model contains five vertebrae, four intervertebral discs, and sevenligaments. The vertebrae consist of the posterior structures, cortical shell, andspongious bone in the vertebral bodies. The seven lumbar ligaments are modelledas one-dimensional force elements (Rohlmann et al., 2006). The discs consist of thenucleus, annulus ground substance, annulus fibres, and cartilaginous endplates.
T. Zander et al. / Journal of Biomechanics 53 (2017) 185–190 187
2.2. Sensitivity analyses
The default model was made generic to allow for sensitivity analyses ofintervertebral-disc, bone, and ligament parameters and for morphological varia-tions (Fig. 1). Table 1 provides an overview of all the investigated parameters, theconstitutive material laws, and the ranges covered, and shows to which of the fourindicated groups the varied parameters were assigned. The ranges were chosenbetween 50% and 150% of the default to represent a reasonable area around thevalidated configuration. Morphological variations reflect the four lumbar shapesdescribed by Roussouly et al. (2005) and the gap distance between the facet-jointsurfaces. Variations principally refer to the sacral slope, lumbar lordosis, and apexof lordosis. Morphometric details are provided elsewhere (Dreischarf et al., 2015).In total, 1200 FE analyses were performed to obtain a confidence interval of o0.1for a coefficient of determination (CoD) of 0.7 at a 95% confidence level (Bucher,2009). Latin hypercube sampling was performed to reduce the necessary number ofsamples. In addition to evaluating CoDs for linear correlations, those for quadraticcorrelations were analysed to identify non-linear behaviour. The coefficient ofvariation was used to determine the relevance of the response range. Table 2 liststhe load cases taken into account to calculate the mean responses used as referencevalues for the coefficients of variation (non-existing facet-joint forces during
Table 2Reference values and load cases used to calculate the coefficient of variation. FL,flexion; EX, extension; LB, lateral bending; AR, axial rotation.
Response Reference value Relevant load cases
Intradiscal pressure 0.12 MPa FL, EX, LB, ARAxial section force 41 N FL, EX, LB, ARFacet joint force 59 N EX, LB, ARALL force 250 N EXPLL force 25 N FLITL force 39 N LBLFL force 18 N FLFCL force 43 N ARISL force 6 N FLSSL force 36 N FLIntervertebral rotation 3.6° FL, EX, LB, AR
Fig. 2. Influence of the parameter gro
flexion were, for example, not taken into account). Sampling and statistical eva-luation were performed using OptiSLang 4.2.2 (Dynardo GmbH, Germany).
2.3. Model simulations
The nonlinear, implicit FE analyses were performed using Abaqus/Standard6.12.4 (Dassault Systèmes Simulia GmbH, Germany). All models were subjected topure moments of 7.5 Nm in flexion, extension, lateral bending, and axial rotation(Fig. 1).
The most frequent model outputs of spinal FE investigations were considered:intradiscal pressure, axial force in the vertebral body, contact force in the facetjoints, forces in the ligaments, and intervertebral rotations at level L4 or L4/L5,respectively. The total range of results is provided for all four load cases and all fourparameter groups for the whole range of loading between 0 and 7.5 Nm. Further-more, the CoDs are provided for the maximum load (7.5 Nm) for all investigatedparameters and responses.
3. Results
3.1. Influence of parameter groups
The amount of loading influenced the impact of the studiedparameter groups (Fig. 2). In general, the intradiscal pressure wasstrongly affected by the intervertebral-disc parameters. Theinfluence of bone and ligament parameters was, in general, small.The morphology also displayed a strong influence, except forflexion, where ligaments played a greater role (Figs. 2 and 3).Similar results were obtained for the axial-section force. In exten-sion, the morphology had the strongest effect. Extension, axialrotation, and small lateral-bending moments resulted in tensilesection forces. Facet-joint forces were dominated by the morphol-ogy. They were largest in axial rotation, zero in flexion and,depending on the initial joint-gap distance, zero in lateral bendingand extension. The effect of other parameter groups was relatively
ups during the course of loading.
Fig. 3. Maximum range of responses for the different parameter groups and load cases. In vitro values in grey: intradiscal pressure and intervertebral-rotation ranges fromHeuer et al. (2007); facet-joint forces (mean7SD) from Zhu et al. (2008).
T. Zander et al. / Journal of Biomechanics 53 (2017) 185–190188
small. Intervertebral rotations were little affected, particularly bybone parameters. The largest effect was caused by disc parameters,particularly for lateral bending and small moments during flexion,followed by morphological parameters. Anterior longitudinalligament forces were predicted mainly during extension (Fig. 3).The response ranges were similar for all parameter groups. Thecontra-lateral intertransverse ligament mainly resisted lateralbending. Disc-parameter variation led to the largest responserange. The facet capsular ligament was the only ligament active forall load cases, with maximum loads during axial rotation andmostly influenced by disc and morphological parameters. Poster-ior longitudinal, flaval, inter-, and superspinous ligaments wereonly tense during flexion and mostly affected by ligament andmorphological parameters.
3.2. Coefficients of determination for maximum load
Overall, only a few responses were markedly correlated to theparameters, and none of the quadratic was significantly superior tothe linear approximations: 80% of all combinations displayed aCoDo0.2 (Fig. 4) and several of the combinations with a high CoDshowed a small effect. The posterior bone modulus of elasticity, forexample, was strongly correlated to the facet-joint forces duringlateral bending (CoD¼0.91) but showed a range of forces over only2 N (44–46 N). In the following, the only combinations describedare those which showed a CoD40.5 and a coefficient of variation410% (Fig. 4). For disc parameters, only the annulus coefficient C10was relevant. A decrease substantially increased the intradiscalpressure for all load cases, except for axial rotation. C10 furtherincreased the axial force in the vertebra for lateral bending and
flexion, and influenced some ligament forces for certain loaddirections. Variation of bone material properties displayed in gen-eral the least influence on investigated model responses. Ligamentstiffnesses mostly influenced the forces within themselves. A weakanterior longitudinal ligament additionally considerably reducedthe intradiscal pressure during extension.
The morphology essentially affected the intradiscal pressureduring lateral bending and extension, the axial-section force dur-ing lateral bending and flexion, the contact force, and the force inthe intertransverse ligament during lateral bending. A larger joint-gap distance markedly reduced the axial tensile force and thefacet-joint force during extension as well as the latter during axialrotation.
4. Discussion
By definition, models contain assumptions whose uncertaintiesreduce predictive power by an unknown degree. This study aimedto provide answers to what degree imprecise (with respect to anindividual) and variable assumptions of a static lumbar-spine FEmodel (with respect to a population) influence predictions. Severalmaterial and morphological parameters were investigated insensitivity analyses with variations lying centrally around thedefault configuration of a validated FE model. Bone properties,particularly those of vertebral bodies, appear to be of less impor-tance for the studied responses because they are magnitudesstiffer than discs and ligaments. The largest impact of studied boneproperties was determined for the posterior structures because oftheir small cross-sectional area and large bending moments
Fig. 4. Linear coefficients of determination (CoDs) for all studied model parameters and responses. Bordered results mark coefficients of variation of 410% with referencevalues according to Table 2. ALL, anterior longitudinal ligament; PLL, posterior longitudinal ligament; ITL, inter-transverse ligament; LFL, flaval ligament; FCL, facet capsularligament; ISL, inter-spinous ligament; SSL, supra-spinous ligament. Colours represent CoDs in the hot/cold spectrum, upper-right square for flexion, lower-left for extension,upper-left for lateral bending, and lower-right for axial rotation. (For interpretation of the references to color in this figure legend, the reader is referred to the web version ofthis article.)
T. Zander et al. / Journal of Biomechanics 53 (2017) 185–190 189
caused by ligament and facet-joint forces. Once the ligaments’slack lengths are calibrated to fit experimental intervertebralrotations, scaling of the ligament forces mainly influences theforce they have to exert. Their influence on other parameters wasgenerally not very pronounced. This might be due to the flat slopeat the beginning of their exponentially increasing force straincharacteristic and is in accordance with the findings of Heuer et al.(2007). For larger loads and different slack lengths, ligament forcescaling could have a greater impact. Disc and morphological para-meters displayed the greatest influence. This is in accordance withseveral studies, which reported the strong effect of facet-jointmorphology (Heuer et al., 2007; Holzapfel and Stadler, 2006;Maurel et al., 1997; Naserkhaki et al., 2016; Noailly et al., 2007).From this point of view, so-called patient-specific models, which isa term often used as a synonym for patient-specific morphologies,are reasonable. However, overall disc stiffness depends on materialas well as on morphology, and it appears likely that there areinterdependencies (Cappetti et al., 2016; Maquer et al., 2014).Patient-specific modelling would additionally require knowledgeabout individual loading. Preloading of the spine, for example,shows a remarkable influence on kinematics (Renner et al., 2007;Weisse et al., 2012). In the current study, we maintained constantloading. Adapting the loading to keep rotations constant, forexample, would lead to different results. However, loading is fre-quently not considered a model parameter. If investigated inbroader detail, load variations should always be studied in
combination with other parameter variations, as was performedhere only for the different load directions.
With respect to morphology, the current study focused onfacet-joint distances and five distinct spinal shapes. More detailedvariations in morphologies, as investigated by Niemeyer et al.(2012) and Cappetti et al. (2016), require a new mesh generationfor each geometry and a subsequent mesh-density convergenceanalysis. This is seldom performed (Ayturk and Puttlitz, 2011) andwas beyond the scope of the present study. Due to that limitationthe sensitivity of the results to other morphological changes isnot known.
The number and possible combinations of model parameterseven when limited to statics are tremendous. This reflects thecomplexity of the geometry in addition to the material behaviour,which might even be simulated by different material laws.Therefore, the present study comprised some parameters, whichcan readily be adapted in future FE models. Only pure momentswere studied, as in many in vitro studies. Other loads, includingcompression and shear forces, would alter the predicted responsesand sensitivities. Results were evaluated at L4/L5 level. Due toother geometrical dimensions and orientations somewhat differ-ent results can be expected at other lumbar levels. Only CoDs40.5and CoVs410% were discussed. Even lower limits or referencevalues (Table 2) would indicate more results as relevant.
Within the considered results, loads, and indicated limitations,excluding bone elasticity did not markedly alter the predictions.
T. Zander et al. / Journal of Biomechanics 53 (2017) 185–190190
Morphology, particularly facet-joint gap distance, and inter-vertebral discs should be modelled as precisely as possibleaccording to the investigated specimens, because they stronglyinfluence results. Considerable variations in intersegmental rota-tions can only be expected when simultaneously varying severalmodel parameters of discs, ligaments, and morphology.
Conflict of interest statement
All authors declare that there are no financial or personalrelationships with other persons or organisations that could haveinappropriately influenced this study.
Acknowledgements
This study was financially supported by the German ResearchFoundation (DFG, SCHM 2572/4-1). We thank A. Grosche fromDynardo GmbH, Weimar, Germany for supporting us in the statis-tical evaluation and acknowledge the North-German Super-computing Alliance (HLRN) for providing computational resourcesthat have contributed to the research results reported in this paper.
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