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Research Papers

Introduction to Force-Dependent Kinematics: Theory and Application to Mandible Modeling

[+] Author and Article Information
Michael Skipper Andersen

Department of Materials and Production,
Aalborg University,
Fibigerstraede 16, Aalborg East,
Aalborg DK-9220, Denmark
e-mail: msa@m-tech.aau.dk

Mark de Zee

Department of Health Science and Technology,
Aalborg University,
Fredrik Bajers Vej 7, Aalborg East,
Aalborg DK-9220, Denmark
e-mail: mdz@hst.aau.dk

Michael Damsgaard

AnyBody Technology A/S,
Niels Jernes Vej 10, Aalborg East,
Aalborg DK-9220, Denmark
e-mail: md@anybodytech.com

Daniel Nolte

Department of Bioengineering,
Imperial College London,
London SW7 2AZ, UK
e-mail: d.nolte@imperial.ac.uk

John Rasmussen

Department of Materials and Production,
Aalborg University,
Fibigerstraede 16, Aalborg East,
Aalborg DK-9220, Denmark
e-mail: jr@m-tech.aau.dk

1Corresponding author.

Manuscript received April 26, 2016; final manuscript received June 11, 2017; published online July 7, 2017. Assoc. Editor: Silvia Blemker.

J Biomech Eng 139(9), 091001 (Jul 07, 2017) (14 pages) Paper No: BIO-16-1172; doi: 10.1115/1.4037100 History: Received April 26, 2016; Revised June 11, 2017

Knowledge of the muscle, ligament, and joint forces is important when planning orthopedic surgeries. Since these quantities cannot be measured in vivo under normal circumstances, the best alternative is to estimate them using musculoskeletal models. These models typically assume idealized joints, which are sufficient for general investigations but insufficient if the joint in focus is far from an idealized joint. The purpose of this study was to provide the mathematical details of a novel musculoskeletal modeling approach, called force-dependent kinematics (FDK), capable of simultaneously computing muscle, ligament, and joint forces as well as internal joint displacements governed by contact surfaces and ligament structures. The method was implemented into the anybody modeling system and used to develop a subject-specific mandible model, which was compared to a point-on-plane (POP) model and validated against joint kinematics measured with a custom-built brace during unloaded emulated chewing, open and close, and protrusion movements. Generally, both joint models estimated the joint kinematics well with the POP model performing slightly better (root-mean-square-deviation (RMSD) of less than 0.75 mm for the POP model and 1.7 mm for the FDK model). However, substantial differences were observed when comparing the estimated joint forces (RMSD up to 24.7 N), demonstrating the dependency on the joint model. Although the presented mandible model still contains room for improvements, this study shows the capabilities of the FDK methodology for creating joint models that take the geometry and joint elasticity into account.

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Figures

Grahic Jump Location
Fig. 1

The FDK analysis framework. The block on the left illustrates the nonlinear equation solver based on a Newton–Raphson scheme augmented with a golden section line search. Throughout the execution of this solver, the inverse dynamic analysis model is executed with the FDK DOFs, α(FDK), as input from which the FDK residual forces, f(FDK), are computed. In the figure, the kinematic analysis is illustrated by the equations solved at position level but the equations subsequently solved to compute the velocities and accelerations are not shown in the figure. These can be seen in Eqs. (2) and (3) in the case of determinate kinematics and Eqs. (9) and (10) in case of over-determinate kinematics. For a description of the terms, see the Method section and the Nomenclature section.

Grahic Jump Location
Fig. 2

(a) and (b) The subject sitting in the reference position while wearing the brace and with the skin markers attached. (c) The laser scanned brace components registered to the CT scan. (d) and (e) The musculoskeletal model and the defined anatomical reference frame. (f) The subject-specific braces (see color figure online).

Grahic Jump Location
Fig. 3

Illustration of the POP model without muscles. (a) A right side view of the model with the plane that the most superior point of the condyle was constrained to stay on. (b) Zoom of the TMJ with the constraint of the most superior point of the condyle (shown as a green dot) is restricted to only slide and rotate in that plane. Note that the constraint shown only applies in the viewing plane. The same constraint is applied on the left-hand side. (c) A front view of the model with the planes of the left and right TMJs included. (d) A top view of the mandible. The skull anatomical reference frame is indicated in black, and the origin of the mandible anatomical reference frame is indicated as a green dot. The origin of the mandible anatomical frame is constrained such that it cannot move medial or lateral relative to the skull anatomical reference (see color figure online).

Grahic Jump Location
Fig. 4

Illustration of the FDK model without muscles. (a) Right side view of the model and (b) a zoom of the TMJ. The three white lines illustrate the elements of the TMJ ligament and the small black arrows illustrate the contact forces for each triangle in the STL files. (c) and (d) The contact force of the right TMJ applied by the skull onto the mandible. Note that the skull and mandible were positioned with excessive penetration of the contacting surfaces into each other to clearly illustrate the contact forces.

Grahic Jump Location
Fig. 5

Simulation results for the chewing task. The two top rows show a comparison of the TMJ kinematics measured with the brace (red) and predicted with the POP model (green) and the FDK model (blue). The two bottom rows show the estimated TMJ forces with the POP model (green) and FDK model (blue). The gray area shows the results of the parameter study. The shaded areas indicate ±1 standard deviation. Note that the gray areas largely overlap with the other areas (see color figure online).

Grahic Jump Location
Fig. 6

Simulation results for the open–close task. The two top rows show a comparison of the TMJ kinematics measured with the brace (red) and predicted with the POP model (green) and the FDK model (blue). The two bottom rows show the estimated TMJ forces with the POP model (green) and FDK model (blue). The gray area shows the results of the parameter study. The shaded areas indicate ±1 standard deviation. Note that the gray areas largely overlap with the other areas (see color figure online).

Grahic Jump Location
Fig. 7

Simulation results for the protrusion task. The two top rows show a comparison of the TMJ kinematics measured with the brace (red) and predicted with the POP model (green) and the FDK model (blue). The two bottom rows show the estimated TMJ forces with the POP model (green) and FDK model (blue). The gray area shows the results of the parameter study. The shaded areas indicate ±1 standard deviation. Note that the gray areas largely overlap with the other areas (see color figure online).

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