Anatomically Based Modelling of the Human Skull and Jaw

A 3D Model of the Jaw Applied to Paediatric Dentistry

As education and knowledge are adapted to new education systems, as per the Bologna Plan, new technologies are required for educational support. In dentistry, the creation of virtual simulators can advance understanding in areas like anatomy. With this aim, a three-dimensional virtual model of the maxilo-mandibular system was created, based on a real infantile specimen. Once this model was developed, we applied this virtual structure to a teaching tool in a dentistry subject.The main objective of this project is the creation of a virtual model of the jaw, based on a real and infantile subject that serves as an educative tool in the scope of Paediatric Dentistry Anatomy. The secondary aims were to evaluate this model as an educational tool in a paediatric dentistry subject. For the main objective we obtained tomographic cuts of the craniofacial skeleton of a three-year-old girl that were transcribed to the program Amira 5.3.0. On the cuts, we segmented and named all the mandibular structures: jaw, temporary teeth, and permanent teeth. For the secondary aims we developed virtual clinical cases based on this mandibular model and gave a questionnaire to 29 dentistry students in order to evaluate the tool. A total of 512 cuts were obtained in sagittal and coronal planes and 309 in the cross-sectional plane of a thickness of 0.625 mm. In different colours, we segmented the total 25 structures to generate a three-dimensional mandibular model. For the questionnaire, the results of the students’ satisfaction of the tool were high, with an overall score of 8.5 out of 10. The educative system based on the Bologna Plan is a reality. The self-training based on test and error, is a strategy of extreme utility for the student. With an interactive model, the student is able to value his knowledge instantaneously, and the presence of a professor is not essential at all times. Through this real model, we have described the anatomical study of temporary teething, as well as its interactions with the developing permanent dentition, in a three-dimensional form. The students’ satisfaction of the teaching tool was high.

Construction of virtual intercuspal occlusion: Considering tooth displacement

Common impressions cannot accurately duplicate the dental occlusion under occlusal force due to tooth displacement and mandibular deformation. To establish new methods to construct virtual intercuspal occlusion and assess their construction accuracy. The intraoral occlusal contacts of posterior teeth of 15 subjects were recorded with 8 μm and 100 μm articulating paper, respectively, and the marked teeth and buccal bite data were scanned with an intraoral scanner. The virtual dental occlusions were separately determined by buccal bite registration (BBR) method, and 3 new methods, namely segmented tooth registration (STR), occlusal contact areas (marked by 8 μm articulating paper) registration (OCR) and mixing registration (MR) methods. With the intraoral contact areas marked by 100 μm articulating paper set as reference and contact areas of the 4 virtual occlusions as tests, sensitivity, positive predictive value (PPV) and the ratio of overlapping areas were calculated. Kruskal-Wallis test or 1-way ANOVA was used to analyse the difference among groups. The sensitivity ranged from 0.69 to 0.94 and the PPV from 0.67 to 0.90. Sensitivity of OCR group and PPV of STR and OCR groups were different from that of BBR group at overlapping threshold of 50% (P = .028, .028 and .006). There was statistical difference of the ratio of overlapping areas over reference areas, and the values of STR and OCR groups were higher than that of BBR group (P = .045 and .021). The ability of STR and OCR methods to construct virtual intercuspal occlusion was better than BBR method.

Comparison between Occlusal Errors of Single Posterior Crowns Adjusted Using Patient Specific Motion or Conventional Methods

Recently, digital technology has been used in dentistry to enhance accuracy and to reduce operative time. Due to advances in digital technology, the integration of individual mandibular motion into the mapping of the occlusal surface is being attempted. The Patient Specific Motion (PSM) is one such method. However, it is not clear whether the occlusal design that is adjusted using PSM could clinically show reduced occlusal error compared to conventional methods based on static occlusion. In this clinical comparative study including fifteen patients with a single posterior zirconia crown treatment, the occlusal surface after a clinical adjustment was compared to no adjustment (NA; design based on static occlusion), PSM (adjusted using PSM), and adjustment using a semi-adjustable articulator (SA) for the assessment of occlusal error. The root mean square (RMS; μm), average deviation value (±AVG; μm), and proportion inside the tolerance (in Tol; %) were calculated using the entire, subdivided occlusal surface and the out of tolerance area. Using a one-way ANOVA, the RMS and +AVG from the out of tolerance area showed a statistical difference between PSM (202.3 ± 39.8 for RMS, 173.1 ± 31.3 for +AVG) and NA (257.0 ± 73.9 for RMS, 210.9 ± 48.6 for +AVG). For the entire and subdivided occlusal surfaces, there were no significant differences. In the color-coded map analysis, PSM demonstrated a reduced occlusal error compared to NA. In conclusion, adjustment occlusal design using PSM is a simple and effective method for reducing occlusal errors that are difficult to identify in a current computer-aided design (CAD) workflow with static occlusion.

Finite Element Analysis of the Septal Cartilage L-Strut

Importance: Septoplasty is one of the most commonly performed operations in the head and neck. However, the reasons for septoplasty failure and the additional stress of performing a chondrotomy on the septal cartilage are not well understood. Design, Setting, and Participants: A finite element model of the nasal septum was created using a microcomputed tomography scan of the nasoseptal complex that was reconstructed into a three-dimensional model in silico. Testing included four common chondrotomy designs: traditional L-strut, double-cornered chondrotomy (DCC), curved L-strut, and the C-curve. Tip displacement was applied in a vector parallel to the caudal strut to simulate nasal tip palpation. Main Outcomes and Measures: With finite element analysis, the maximum principal stress (MPS), von Mises stress (VMS), harvested cartilage volume, and surface area were recorded. Results: The highest MPS for the L-strut, DCC, curved L-strut, and C-curve was identified at the corner of the chondrotomy. The MPS at the corner of the chondrotomy was reduced 44% when comparing the C-curve with the traditional L-strut. The VMS patterns showed compressive stress along the caudal septum in all models, but at the corner, the stresses were highest in the chondrotomies designed with sharp-angled corners. The VMS showed a 76% decrease when comparing the C-curve with the traditional L-strut. The stress across the anterior septal angle is also higher in models with sharp-angled corners. Cartilage harvest volumetric and surface area assessments did not show meaningful differences between shapes. Conclusions and Relevance: The highest area of stress is near the transition of the dorsal to caudal septum in all models. Stresses are relatively higher in chondrotomy shapes that contain sharp-angled corners. The relative reduction in MPS and VMS utilizing a C-curve instead of an L-strut may decrease the likelihood that the septum will deform or fail in this region. The volume and surface area of the C-curve are similar to that of the L-strut technique. Avoiding sharp-angled corners reduces the stresses at the corner of the chondrotomy and across the anterior septal angle. Using a C-curve may be an improved septoplasty design.

Occlusal load modelling significantly impacts the predicted tooth stress response during biting: a simulation study

Computational models of the masticatory system can provide estimates of occlusal loading during (static) biting or (dynamic) chewing and therefore can be used to evaluate and optimize functional performance of prosthodontic devices and guide dental surgery planning. The modelling assumptions, however, need to be chosen carefully in order to obtain meaningful predictions. The objectives of this study were two-fold: (i) develop a computational model to calculate the stress response of the first molar during biting of a rubber sample and (ii) evaluate the influence of different occlusal load models on the stress response of dental structures. A three-dimensional finite element model was developed comprising the mandible, first molar, associated dental structures, and the articular fossa and discs. Simulations of a maximum force bite on a rubber sample were performed by applying muscle forces as boundary conditions on the mandible and computing the contact between the rubber and molars (GS case). The molar occlusal force was then modelled as a single point force (CF1 case), four point forces (CF2 case), and as a sphere compressing against the occlusal surface (SL case). The peak enamel stress for the GS case was 110 MPa and 677 MPa, 270 MPa and 305 MPa for the CF1, CF2 and SL cases, respectively. Peak dentin stress for the GS case was 44 MPa and 46 MPa, 50 MPa and 63 MPa for the CF1, CF2 and SL cases, respectively. Furthermore, the enamel stress distribution was also strongly correlated to the occlusal load model. The way in which occlusal load is modelled has a substantial influence on the stress response of enamel during biting, but has relatively little impact on the behavior of dentin. The use of point forces or sphere contact to model occlusal loading during mastication overestimates enamel stress magnitude and also influences enamel stress distribution.

Relationship between alveolar bone thickness, tooth root morphology, and sagittal skeletal pattern : A cone beam computed tomography study

PURPOSE

The goal of this work was to examine the relationship between sagittal facial pattern and thickness of alveolar bone in conjunction with root morphology of teeth by using cone beam computed tomography (CBCT).

METHODS

The study was carried out on the CBCT scans from 3 group of patients (n = 20 in each group). The first group involved skeletal class 1, the second group involved skeletal class 2, and the third group involved skeletal class 3 patients. In all, 14 permanent teeth and interdental regions in the maxilla and mandible were evaluated. Root length and root width were measured on each tooth. Buccal cortical bone thickness, cancellous bone thickness, and lingual cortical bone thicknesses were measured in each interdental region. Analysis of variance, Kruskall-Wallis H and Mann-Whitney U tests were used for statistical comparisons.

RESULTS

No significant difference was found between the groups for root length, root width, buccal cortical bone and lingual cortical bone thickness. A significant difference was observed between the groups for cancellous bone thickness as it was thicker in skeletal class 2 group. Cortical bone was thicker in the mandible compared to maxilla on both buccal and lingual sides and it was thicker in the posterior region compared to the anterior region on the buccal side.

CONCLUSIONS

Differences in cancellous bone thickness between different sagittal facial patterns and differences in cortical bone thickness between different alveolar regions should be taken into consideration when planning orthodontic tooth movements and anchorage mechanics.

Modelling of the forces acting on the human stomatognathic system during dynamic symmetric incisal biting of foodstuffs

A major stage in the preparation of a computational model of the human stomatognathic system is the determination of the values of the forces for the adopted loading configuration. In physiological conditions, food is a factor having a significant effect on the values of the loads acting on the stomatognathic system. Considering that the act of mastication is a complex process, this research undertook to determine the forces (bite forces, muscular forces and temporomandibular joint reaction forces) acting on the stomatognathic system during the dynamic symmetric incisal biting of selected foodstuffs. The investigations were divided into two stages: (1) experimental tests and (2) numerical simulations. In the first stage, classic force-displacement characteristic curves (F_i-Δh) were determined for the food while in the second stage, the curves were used as a dynamic stomatognathic system model load function. One of the most important results of this research is that the food characteristic in the form of a force-displacement function has been shown to have a significant effect not only on the values of the muscular forces and the temporomandibular joint reaction forces, but also on their curves during the dynamic loading of the stomatognathic system. The analysis of the results indicates that F_i-Δh has an effect on not only the (active and passive) forces, but also on other parameters, such as stress, deformation, displacement, and probably the rigidity of the muscles.

Resorptive potential of impacted mandibular third molars: 3D simulation by finite element analysis

OBJECTIVES

Previous studies have suggested a relationship between resorption in second molars and pressure from the eruptive force of the third molar. The aim of this study was to simulate functional forces in a mandible model by means of finite element analysis and then assess the biomechanical response produced by impacted third molars on the roots of the second molar.

MATERIALS AND METHODS

A cone beam computed tomography scan presenting an impacted mandibular third molar was segmented (Mimics V17 software). The modeling process was performed using the reverse engineering technique provided by the Rhinoceros 3D 5.0 software. The third molar position was changed in order to produce different inclinations of the impacted tooth. Bite forces were simulated to evaluate total deformation, the equivalent von Mises stress, minimum principal stress on hard tissue, and equivalent elastic strain on soft tissue.

RESULTS

Areas of high energy dissipation and compression stress were detected in the second molar root, independently of the inclination of the impacted third molar. In general, the horizontal position was the situation in which major stress and the amount of deformation occurred in the second and third molar regions.

CONCLUSION

Impacted third molars in close proximity with the adjacent tooth can generate areas of compression concentrated at the site of contact, which suggests an involvement of mechanical factors in the triggering of resorption lesions.

CLINICAL RELEVANCE

The results of these computational experiments contribute to the understanding of the triggering and progression of resorptive lesions in the adjacent second molar.

Viscoelastic computational modeling of the human head-neck system: Eigenfrequencies and time-dependent analysis

A subject-specific 3-dimensional viscoelastic finite element model of the human head-neck system is presented and investigated based on computed tomography and magnetic resonance biomedical images. Ad hoc imaging processing tools are developed for the reconstruction of the simulation domain geometry and the internal distribution of bone and soft tissues. Material viscoelastic properties are characterized point-wise through an image-based interpolating function used then for assigning the constitutive prescriptions of a heterogenous viscoelastic continuum model. The numerical study is conducted both for modal and time-dependent analyses, compared with similar studies and validated against experimental evidences. Spatiotemporal analyses are performed upon different exponential swept-sine wave-localized stimulations. The modeling approach proposes a generalized, patient-specific investigation of sound wave transmission and attenuation within the human head-neck system comprising skull and brain tissues. Model extensions and applications are finally discussed.

Dynamic finite element simulation of dental prostheses during chewing using muscle equivalent force and trajectory approaches

The long-term application of dental prostheses inside the bone has a narrow relation to its biomechanical performance. Chewing is the most complicated function of a dental implant as it implements different forces to the implant at various directions. Therefore, a suitable holistic modelling of the jaw bone, implant, food, muscles, and their forces would be deemed significant to figure out the durability as well as functionality of a dental implant while chewing. So far, two approaches have been proposed to employ the muscle forces into the Finite Element (FE) models, i.e. Muscle Equivalent Force (MEF) and trajectory. This study aimed at propounding a new three-dimensional dynamic FE model based on two muscle forces modelling approaches in order to investigate the stresses and deformations in the dental prosthesis as well as maxillary bone during the time of chewing a cornflakes bio. The results revealed that both contact and the maximum von Mises stress in the implant and bones for trajectory approach considerably exceed those of the MEF. The maximum stresses, moreover, are located around the neck of implant which should be both clinically and structurally strong enough to functionally maintain the bone-implant interface. In addition, a higher displacement due to compressive load is observed for the implant head in trajectory approach. The results suggest the benefits provided by trajectory approach since MEF approach would significantly underestimate the stresses and deformations in both the dental prosthesis and bones.

Finite element analysis of stress distribution on the mandible and condylar fracture osteosynthesis during various clenching tasks

PURPOSE

The purpose of the study was to evaluate the effect of clenching tasks on the stress and strain of condylar osteosynthesis screws and plates, as well as on the stress, strain distribution and displacement on the whole mandible and bone surrounding screws.

METHODS

Three-dimensional finite element models of the mandible, two straight four-hole plates and eight screws were established. Six static clenching tasks were simulated in this study: incisal clench (INC), intercuspal position (ICP), right unilateral molar clench (RMOL), left unilateral molar clench (LMOL), right group function (RGF) and left group function (LGF).

RESULTS

Based on the simulation of the six clenching tasks, none of the inserted screws and plates were broken or bended. For the whole mandibular bone, the maximum von Mises stress and von Mises strain observed were yielded by the ICP. For the bone surrounding the inserted screws, the maximum von Mises stress and von Mises strain were yielded by the LMOL (49.2 MPa and 3795.1 μ).

CONCLUSION

Clenching tasks had significant effects on the stress distribution on the condylar osteosynthesis and the bone surrounding screws. Contralateral occlusion task (LMOL) had the maximal results of von Mises stress and strain and healing problems could be occur, this result confirms the importance of soft diet after surgery.

A Finite Element Model to Simulate Formation of the Inverted-V Deformity

IMPORTANCE

Computational modeling can be used to mimic the forces acting on the nasal framework that lead to the inverted-V deformity (IVD) after surgery and potentially determine long-range outcomes.

OBJECTIVE

To demonstrate the use of the finite element method (FEM) to predict the formation of the IVD after separation of the upper lateral cartilages (ULCs) from the nasal septum.

DESIGN, SETTING, AND PARTICIPANTS

A computer model of a nose was derived from human computed tomographic data. The septum and upper and lower lateral cartilages were designed to fit within the soft-tissue envelope using computer-aided design software. Mechanical properties were obtained from the literature. The 3 simulations created included (1) partial fusion of the ULCs to the septum, (2) separation of the ULCs from the septum, and (3) a fully connected model to serve as a control. Forces caused by wound healing were prescribed at the junction of the disarticulated ULCs and septum. Using FEM software, equilibrium stress and strain were calculated. Displacement of the soft tissue along the nasal dorsum was measured and evaluated for evidence of morphologic change consistent with the IVD.

MAIN OUTCOME AND MEASURES

Morphologic changes on the computer models in response to each simulation.

RESULTS

When a posteroinferior force vector was applied along the nasal dorsum, the areas of highest stress were along the medial edge of the ULCs and at the junction of the ULCs and the nasal bones. With full detachment of ULCs and the dorsal septum, the characteristic IVD was observed. Both separation FEMs produced a peak depression of 0.3 mm along the nasal dorsum.

CONCLUSIONS AND RELEVANCE

The FEM can be used to simulate the long-term structural complications of a surgical maneuver in rhinoplasty, such as the IVD. When applied to other rhinoplasty maneuvers, the use of FEMs may be useful to simulate the long-term outcomes, particularly when long-term clinical results are not available. In the future, use of FEMs may simulate rhinoplasty results beyond simply morphing the outer contours of the nose and allow estimation of potentially long-term clinical outcomes that may not be readily apparent.

LEVEL OF EVIDENCE

NA.

Generating Facial Expressions Using an Anatomically Accurate Biomechanical Model

This paper presents a computational framework for modelling the biomechanics of human facial expressions. A detailed high-order (Cubic-Hermite) finite element model of the human head was constructed using anatomical data segmented from magnetic resonance images. The model includes a superficial soft-tissue continuum consisting of skin, the subcutaneous layer and the superficial Musculo-Aponeurotic system. Embedded within this continuum mesh, are 20 pairs of facial muscles which drive facial expressions. These muscles were treated as transversely-isotropic and their anatomical geometries and fibre orientations were accurately depicted. In order to capture the relative composition of muscles and fat, material heterogeneity was also introduced into the model. Complex contact interactions between the lips, eyelids, and between superficial soft tissue continuum and deep rigid skeletal bones were also computed. In addition, this paper investigates the impact of incorporating material heterogeneity and contact interactions, which are often neglected in similar studies. Four facial expressions were simulated using the developed model and the results were compared with surface data obtained from a 3D structured-light scanner. Predicted expressions showed good agreement with the experimental data.

Nasal tip support: a finite element analysis of the role of the caudal septum during tip depression

OBJECTIVES/HYPOTHESIS

Although minor and major tip support mechanisms have been described in detail, no quantitative models exist to provide support for the relative contributions of the structural properties of the major alar cartilage, the fibrous attachments to surrounding structures, and the rigid support structures in an objective manner.

STUDY DESIGN

The finite element method was used to compute the stress distribution in the nose during simple tip compression, and then identify the specific anatomic structures that resist deformation and thus contribute to tip support. Additionally, the impact of caudal septal resection on nasal tip support was examined.

METHODS

The computer models consisted of three tissue components with anatomically correct geometries for skin and bone derived from computed tomographic data. Septum, upper lateral cartilages, and major alar cartilages were fitted within the model using three-dimensional computer-aided design software. Five-millimeter nasal tip compression was performed on the models with caudal septal resection (3 and 5 mm) and without resection to simulate palpation, then the resulting spatial distribution of stress and displacement was calculated.

RESULTS

The von Mises stress in the normal model was primarily concentrated along the medial crural angle. As caudal septum length was reduced, stress was redistributed to adjacent soft tissue and bone, resulting in less force acting on the septum. In all models, displacement was greatest near the intermediate crura.

CONCLUSIONS

These models are the first step in the comprehensive mechanical analysis of nasal tip dynamics. Our model supports the concept of the caudal septum and major alar cartilage providing the majority of critical load-bearing support.

Cortical bone and ridge thickness of hyperdivergent and hypodivergent adults

INTRODUCTION

The purpose of this study was to assess differences in dentoalvolar cortical bone thickness between hyperdivergent and hypodivergent young adults.

METHODS

Pretreatment cone-beam computed tomography images of 57 patients, including 30 hypodivergent subjects (22 women, 8 men) and 27 hyperdivergent subjects (20 women, 7 men), were analyzed. The data were imported into imaging software (version 10.5; Dolphin Imaging Systems, Chatsworth, Calif); standardized orientations were used to measure buccal and lingual cortical bone thicknesses at 16 interradicular sites of the maxilla and the mandible. Total alveolar ridge thickness and medullary space thickness were measured at the same sites.

RESULTS

T tests showed significant (P <0.05) group differences, with hypodivergent subjects having significantly thicker buccal cortices. The lingual cortex of the maxilla was also significantly thicker in the hypodivergent than in the hyperdivergent subjects. Alveolar ridge thickness was significantly greater at all sites of the hypodivergent mandible and at the anterior 2 sites of the hypodivergent maxilla. Medullary thickness was significantly greater only in the hypodivergent mandibles between the first molars and the second premolars, and between the first and second premolars. Buccal cortical bone was significantly thicker than lingual cortical bone in the mandible; lingual bone was significantly thicker in the maxilla.

CONCLUSIONS

Cortical bone tends to be thicker in hypodivergent than in hyperdivergent subjects. This explains the concomitant differences in alveolar ridge thickness. Medullary space thickness is largely unaffected by facial divergence.

Current computational modelling trends in craniomandibular biomechanics and their clinical implications

Computational models of interactions in the craniomandibular apparatus are used with increasing frequency to study biomechanics in normal and abnormal masticatory systems. Methods and assumptions in these models can be difficult to assess by those unfamiliar with current practices in this field; health professionals are often faced with evaluating the appropriateness, validity and significance of models which are perhaps more familiar to the engineering community. This selective review offers a foundation for assessing the strength and implications of a craniomandibular modelling study. It explores different models used in general science and engineering and focuses on current best practices in biomechanics. The problem of validation is considered at some length, because this is not always fully realisable in living subjects. Rigid-body, finite element and combined approaches are discussed, with examples of their application to basic and clinically relevant problems. Some advanced software platforms currently available for modelling craniomandibular systems are mentioned. Recent studies of the face, masticatory muscles, tongue, craniomandibular skeleton, temporomandibular joint, dentition and dental implants are reviewed, and the significance of non-linear and non-isotropic material properties is emphasised. The unique challenges in clinical application are discussed, and the review concludes by posing some questions which one might reasonably expect to find answered in plausible modelling studies of the masticatory apparatus.

A call to action for bioengineers and dental professionals: directives for the future of TMJ bioengineering

The world's first TMJ Bioengineering Conference was held May 25-27, 2006, in Broomfield, Colorado. Presentations were given by 34 invited speakers representing industry, academics, government agencies such as NIH, and private practice, which included surgeons, engineers, biomedical scientists, and patient advocacy leaders. Other attendees included documentary film makers and FDA officials. The impetus for the conference was that the field of TMJ research has been lacking continuity, with no open forum available for surgeons, scientists, and bioengineers to exchange scientific and clinical ideas and identify common goals, strengths, and capabilities. The goal was thus to plant the seeds for establishing a forum for multidisciplinary and interdisciplinary interactions. The collective wisdom and interactions brought about by a melting pot of these diverse individuals has been pooled and is disseminated in this article, which offers specific directives to bioengineers, basic scientists, and medical and dental professionals including oral and maxillofacial surgeons, pain specialists, orthodontists, prosthodontists, endocrinologists, rheumatologists, immunologists, radiologists, neurologists, and orthopaedic surgeons. A primary goal of this article was to attract researchers across a breadth of research areas to lend their expertise to a significant clinical problem with a dire need for new talent. For example, researchers with expertise in finite element modeling will find an extensive list of clinically significant problems. Specific suggestions for TMJ research were presented by the leading organizations for TMJ surgeons and TMJ patients, and further research needs were identified in a series of group discussions. The specific needs identified at the conference and presented here will be essential for those who endeavor to engage in TMJ research, especially in the areas of tissue engineering and biomechanics. Collectively, it is our hope that many of the questions and directives presented here find their way into the proposals of multidisciplinary teams across the world with new and promising approaches to diagnose, prevent and treat TMJ disorders.

Three-dimensional finite element modelling of muscle forces during mastication

This paper presents a three-dimensional finite element model of human mastication. Specifically, an anatomically realistic model of the masseter muscles and associated bones is used to investigate the dynamics of chewing. A motion capture system is used to track the jaw motion of a subject chewing standard foods. The three-dimensional nonlinear deformation of the masseter muscles are calculated via the finite element method, using the jaw motion data as boundary conditions. Motion-driven muscle activation patterns and a transversely isotropic material law, defined in a muscle-fibre coordinate system, are used in the calculations. Time-force relationships are presented and analysed with respect to different tasks during mastication, e.g. opening, closing, and biting, and are also compared to a more traditional one-dimensional model. The results strongly suggest that, due to the complex arrangement of muscle force directions, modelling skeletal muscles as conventional one-dimensional lines of action might introduce a significant source of error.