Internal Mechanics of a Subject-Specific Wrist in the Sagittal versus Dart-Throwing Motion Plane in Adult and Elder Models: Finite Element Analyses

Introduction: Most of the wrist motions occur in a diagonal plane of motion, termed the dart-throwing motion (DTM) plane; it is thought to be more stable compared with movement in the sagittal plane. However, the effect of the altered carpus motion during DTM on the stress distribution at the radiocarpal joint has yet to be explored. Aim: To calculate and compare the stresses between the radius and two carpal bones (the scaphoid and the lunate) in two wrist positions, extension and radial extension (position in DTM), and between an adult and an elder model. Methods: A healthy wrist of a 40-year-old female was scanned using Magnetic Resonance Imaging in two wrist positions (extension, radial extension). The scans were transformed into three-dimensional models and meshed. Finite element (FE) analyses in each position of the wrist were conducted for both adult and elder models, which were differentiated by the mechanical properties of the ligaments. The distal surfaces of the carpal bones articulating with the metacarpals were loaded by physically accurate tendon forces for each wrist position. Results: The von Mises, shear stresses and contact stresses were higher in the extension model compared with the radial-extension model and were higher for the radius-scaphoid interface in the adult model compared with the elder model. In the radius-scaphoid interface, the stress differences between the two wrist positions were smaller in the elder model (11.5% to 22.5%) compared with the adult model (33.6–41.5%). During radial extension, the contact area at the radius-lunate interface was increased, more so in the adult model (222.2%) compared with the elder model (127.9%), while the contact area at the radius-scaphoid was not affected by the position of the wrist in the adult model (100.9%) but decreased in the elder model (50.2%) during radial extension. Conclusion: The reduced stresses during radial extension might provide an explanation to our frequent use of this movement pattern, as the reduced stresses decrease the risk of overuse injury. Our results suggest that this conclusion is relevant to both adults and elder individuals.


Background
Most of the wrist motions occur in a diagonal plane of motion, termed the dartthrowing motion (DTM) plane [1], i.e., a wrist plane that ranges from radial extension to ulnar flexion. During DTM, the proximal carpal row undergoes minor sagittal translations and is therefore thought to be more stable [2]. This plane of motion is often exercised during daily activities performed at different heights [3] by both dominant and nondominant upper limbs [4]. Since the DTM is our natural wrist motion, occupational therapists strive to incorporate it in the rehabilitation of the wrist [5][6][7]. However, although their interest in DTM has grown in the last decade and carpal kinematics have been documented, the stress distribution at the radiocarpal joint has yet to be explored, and 2 of 11 thus a complete biomechanical analysis is yet to be reported. Additionally, since ligament mechanical properties might be influenced by aging [8,9], this might be an important factor for consideration in the interpretation of biomechanical analyses of DTM.
The biomechanics of the wrist was previously explored using finite element (FE) analyses to compare stresses during wrist flexion, extension, radial and ulnar deviation [10], for different purposes, e.g., to explain the mechanism of wrist injury [11] and to test the effects of surgical reconstruction techniques [12,13]. These studies showed that the load distribution changed according to the posture of the wrist; however, no data exist regarding loads during DTM.
The aim of this study was to calculate and compare the stresses between the radius and two carpal bones (the scaphoid and the lunate) in two wrist positions, extension and radial extension (position in DTM), and between an adult and an elder model.

Methods
Our work included four FE analyses of a healthy human wrist in two different positions of the wrist: sagittal plane (extension) and DTM plane (radial extension). The geometry of the wrist was obtained using magnetic resonance imaging (MRI) in the aforementioned wrist positions. The FE models included the distal parts of the radius and ulna, the carpal bones, cartilage, and main ligaments. Each position was analyzed once with ligament mechanical properties of a healthy adult and again with properties pertaining to age-related decline of structural properties. The stresses in the articulating surfaces of the radius bone and the two bones in the first carpal row, the scaphoid and lunate, were calculated.

Wrist Geometry
A Helsinki Committee approval was obtained pretrial (approval number 2943-16-SMC). The geometry of the right wrist of a healthy 40-year-old female was obtained using a 3T MRI (VB17A Magnetom Trio, a Total Imaging Matrix System; Siemens Healthcare, Erlangen, Germany). The distal forearm of the subject was fixed using a Three-Dimensional (3D) printed 'arm constrainer' (Figure 1). The arm was placed between the vertical walls, which were tied using Velcro straps in order to maintain the arm in mid-position and prevent elbow pronation and supination. The wrist was scanned twice: in extension and maintaining the same extension but with added radial deviation. The scan during wrist extension included 20 images with a slice distance of 3.3 mm (total length of 66 mm) and transverse resolution of 256 × 256. The scan during radial extension included 24 images with a slice distance of 3.3 mm (total length of 79.2 mm) and transverse resolution of 320 × 320. The scanning properties were chosen so that both wrist scans included the distal ends of the radius and the ulna, eight carpal bones and five proximal ends of the metacarpal bones. interest in DTM has grown in the last decade and carpal kinematics have been documented, the stress distribution at the radiocarpal joint has yet to be explored, and thus a complete biomechanical analysis is yet to be reported. Additionally, since ligament mechanical properties might be influenced by aging [8,9], this might be an important factor for consideration in the interpretation of biomechanical analyses of DTM. The biomechanics of the wrist was previously explored using finite element (FE) analyses to compare stresses during wrist flexion, extension, radial and ulnar deviation [10], for different purposes, e.g., to explain the mechanism of wrist injury [11] and to test the effects of surgical reconstruction techniques [12,13]. These studies showed that the load distribution changed according to the posture of the wrist; however, no data exist regarding loads during DTM.
The aim of this study was to calculate and compare the stresses between the radius and two carpal bones (the scaphoid and the lunate) in two wrist positions, extension and radial extension (position in DTM), and between an adult and an elder model.

Methods
Our work included four FE analyses of a healthy human wrist in two different positions of the wrist: sagittal plane (extension) and DTM plane (radial extension). The geometry of the wrist was obtained using magnetic resonance imaging (MRI) in the aforementioned wrist positions. The FE models included the distal parts of the radius and ulna, the carpal bones, cartilage, and main ligaments. Each position was analyzed once with ligament mechanical properties of a healthy adult and again with properties pertaining to age-related decline of structural properties. The stresses in the articulating surfaces of the radius bone and the two bones in the first carpal row, the scaphoid and lunate, were calculated.

Wrist Geometry
A Helsinki Committee approval was obtained pretrial (approval number 2943-16-SMC). The geometry of the right wrist of a healthy 40-year-old female was obtained using a 3T MRI (VB17A Magnetom Trio, a Total Imaging Matrix System; Siemens Healthcare, Erlangen, Germany). The distal forearm of the subject was fixed using a Three-Dimensional (3D) printed 'arm constrainer' (Figure 1). The arm was placed between the vertical walls, which were tied using Velcro straps in order to maintain the arm in mid-position and prevent elbow pronation and supination. The wrist was scanned twice: in extension and maintaining the same extension but with added radial deviation. The scan during wrist extension included 20 images with a slice distance of 3.3 mm (total length of 66 mm) and transverse resolution of 256 × 256. The scan during radial extension included 24 images with a slice distance of 3.3 mm (total length of 79.2 mm) and transverse resolution of 320 × 320. The scanning properties were chosen so that both wrist scans included the distal ends of the radius and the ulna, eight carpal bones and five proximal ends of the metacarpal bones. The arm fixation apparatus. The distance between the walls was adjusted according to the width of the forearm and secured using two Velcro straps.

Modeling and Meshing
The MRI scans were used to create a Solid 3D model using ScanIP software (Simpleware Ltd., Exeter, UK). The bones were identified using a Radiology Atlas in each slice [14]. Then, the pixels of each bone were identified in each slice ('masking technique') and each bone was formed into 3D geometry by layering up the masks. Touching surfaces were smoothed by 1 mm to allow a minimum gap for cartilage modeling. In order to minimize noise, a Recursive Gaussian filter was used, which smoothed the layers. The geometries of the bones were then meshed into 3D tetrahedral elements (Tetra 10).
The 3D model was imported into Patran 2018 software (MSC Software Corporation, CA, USA). The cartilage was modeled using wedge elements, as suggested by Gíslason and Nash in the modeling of wrist cartilage [15]. It was extruded using half of the minimum distance of the articulating surfaces [16] (The cartilage thickness between the scaphoid and the radius in the radial-extension model was smaller than half of the distance in some regions due to convexities). Contact was set to surface-to-surface (touch contact). The ligaments were modeled as multi-parallel linear springs (8 CELAS elements), based on similar wrist FE studies [16,17]. The two FE models are shown in Figure 2. The total number of elements in the extension model was 770,682 and 206,645 nodes. The total number of elements in the radial-extension model was 836,842 and 221,408 nodes.
Appl. Sci. 2021, 11, x FOR PEER REVIEW 3 o Figure 1. The arm fixation apparatus. The distance between the walls was adjusted according to width of the forearm and secured using two Velcro straps.

Modeling and Meshing
The MRI scans were used to create a Solid 3D model using ScanIP software (Sim ware Ltd., Exeter, UK). The bones were identified using a Radiology Atlas in each s [14]. Then, the pixels of each bone were identified in each slice ('masking technique') each bone was formed into 3D geometry by layering up the masks. Touching surfa were smoothed by 1 mm to allow a minimum gap for cartilage modeling. In order to m imize noise, a Recursive Gaussian filter was used, which smoothed the layers. The ge etries of the bones were then meshed into 3D tetrahedral elements (Tetra 10).
The 3D model was imported into Patran 2018 software (MSC Software Corporat CA, USA). The cartilage was modeled using wedge elements, as suggested by Giśla and Nash in the modeling of wrist cartilage [15]. It was extruded using half of the m mum distance of the articulating surfaces [16] (The cartilage thickness between the sca oid and the radius in the radial-extension model was smaller than half of the distanc some regions due to convexities). Contact was set to surface-to-surface (touch conta The ligaments were modeled as multi-parallel linear springs (8 CELAS elements), ba on similar wrist FE studies [16,17]. The two FE models are shown in Figure 2. The t number of elements in the extension model was 770,682 and 206,645 nodes. The total nu ber of elements in the radial-extension model was 836,842 and 221,408 nodes.

Mechanical Properties
The bones were modeled as a solid bulk, with linear isotropic mechanical properties [13,18]. Specifically, Young's modulus of the bone was defined as E = 18GPa with a Poisson's ratio of ν = 0.2 [15][16][17]19]. Although the cortical bone undergoes biomechanical Appl. Sci. 2021, 11, 5275 4 of 11 changes with age, studies showed a negligible decrease in Young's modulus value [20]. Hence, we did not apply change in Young's modulus value in the elder model.
The cartilage was defined with hyperplastic properties since it undergoes large deformations [12,17,19,21], using a two-parameter Mooney-Rivlin model [22], with the following coefficients: C10 = 4.1 MPa and C01 = 0.41 MPa. The Mooney-Rivlin strain energy density potential equation is: where µ = 2(C 10 + C 01 ), µ is the material shear modulus [23] and the Green-Lagrange strain tensor is E = 1 2 (C − I), where I is the second order identity tensor [23]. The principal invariants of C are calculated by: Table A1 (see Appendix A) lists the ligament's stiffness properties as well as their origin of insertion. The ligament's insertions were determined using an anatomic visualization software (McGrouther DA HP, Interactive Hand-Anatomy, Primal Pictures). The stiffness of the proximal scapho-lunate interosseous ligament (SLIL) was assumed to be similar to that of the scaphoid-trapezium-trapezoid ligament because there are no available data regarding its stiffness and thickness reported as variables [24]. Ligament decline in stiffness in the elder model was defined using a 20% stiffness reduction (compared with the adult model) [9]. The value presented in the Table for the transverse carpal ligament (TCL) refers to the stiffness of elder females [25]. We did not alter the cartilage thickness in the elder model since there is no clear documentation of cartilage thickness changes in the wrist with age, e.g., as shown by [26]. Furthermore, we did not alter the applied forces to the tendons because we simulated two simple and natural wrist positions, with no external forces applied to the hand. Although there is a documented decline of hand-grip force with age [27], the movements chosen for this study require a small percentage of the maximal muscle force, so that both an elderly individual and younger populations could produce these forces.
The forces (Table 1) were applied to the distal surfaces of the carpal bones articulating with the metacarpals (Figure 2c). The tendons' loads were distributed on the surfaces of the bones. Each force was applied in the direction of a vector from the insertion point in the metacarpal relevant bone to the bone in the distal carpal row. The tendon forces reflect a wrist position of 20 • in extension. Unfortunately, the radial deviation angle during the radial-extension position studied in [28] was not measured. Table 1. Forces applied to the tendons of the two models during the two conditions of extension and radial extension [28]. Note that the total force applied for each wrist position is 267 N.

Boundary Conditions
The proximal ends of the radius and the ulna were constrained for all translations and rotations (Figure 2c). The contact was defined as deformable-to-deformable with no friction [10,17].

FE Solver and Parameters
The analyses were processed by MSC Nastran software (MSC Software Corporation, Irvine, CA, USA). Implicit non-linear solver (Pure Newton Raphson) was used, and the convergence criterion was set to load equilibrium error. Separation of contacting surfaces was based on absolute stress (extrapolating integration point stresses).
The average stresses were calculated by dividing the summation of vector forces (in each grid point of the contacting surface) by the contact areas. Minor negative forces (tension) in the peripheral of the contact areas were ignored and referred to as side effects caused by mid-side nodes of the contact patch.

Results
The bone displacements and stresses are presented below. The forces in the ligaments relevant to the scaphoid, lunate, and radius are depicted in Appendix B.

Displacements
The displacements of the proximal carpal row are depicted in Figure 3.

Boundary Conditions
The proximal ends of the radius and the ulna were constrained for all translations and rotations (Figure 2c). The contact was defined as deformable-to-deformable with no friction [10,17].

FE Solver and Parameters
The analyses were processed by MSC Nastran software (MSC Software Corporation, Irvine, CA, USA). Implicit non-linear solver (Pure Newton Raphson) was used, and the convergence criterion was set to load equilibrium error. Separation of contacting surfaces was based on absolute stress (extrapolating integration point stresses).
The average stresses were calculated by dividing the summation of vector forces (in each grid point of the contacting surface) by the contact areas. Minor negative forces (tension) in the peripheral of the contact areas were ignored and referred to as side effects caused by mid-side nodes of the contact patch.

Results
The bone displacements and stresses are presented below. The forces in the ligaments relevant to the scaphoid, lunate, and radius are depicted in Appendix B.

Displacements
The displacements of the proximal carpal row are depicted in Figure 3. The pitch angles (rotation in the sagittal plane) between the lunate and scaphoid were 27.0° for the extension model and 17.2° for the for radial-extension model. The pitch angles (rotation in the sagittal plane) between the lunate and scaphoid were 27.0 • for the extension model and 17.2 • for the for radial-extension model.

Stresses
The von Mises and maximum shear stresses in the cartilage of the radius during extension and radial extension are depicted in Figure 4. Moreover, the maximal von Mises stresses, maximal shear stresses, maximal and average contact stresses and contact area in extension and radial-extension models in the adult and elder simulations are presented in Table 2.
lunate interface, the stress differences between the two wrist positions were smaller in the elder model compared with the adult model. During radial extension, the contact area at the radius-lunate interface was increased, more so in the adult model compared with the elder model. The contact area at the radius-scaphoid was not affected by the position of the wrist in the adult model, but decreased in the elder model during radial extension.    The von Mises, shear stresses and contact stresses were higher in the extension model compared with the radial-extension model and were higher for the radius-scaphoid interface in the adult model compared with the elder model (Table 2, Figure 4). In the radius-lunate interface, the stress differences between the two wrist positions were smaller in the elder model compared with the adult model. During radial extension, the contact area at the radius-lunate interface was increased, more so in the adult model compared with the elder model. The contact area at the radius-scaphoid was not affected by the position of the wrist in the adult model, but decreased in the elder model during radial extension.

Discussions
In this study, we aimed to compare the stresses between the radius and two carpal bones (the scaphoid and the lunate) in two wrist positions: extension and radial extension (position in DTM) and between an adult and elder model. Our finding shows that the von Mises, shear stresses and contact stresses were higher in the extension model compared with the radial-extension model and were higher for the radius-scaphoid interface in the adult model compared with the elder model. In the radius-scaphoid interface, the stress differences between the two wrist positions were smaller in the elder model (11.5% to 22.5%) compared with the adult model (33.6-41.5%). During radial extension, the contact area at the radius-lunate interface was increased, more so in the adult model (222.2%) compared with the elder model (127.9%), while the contact area at the radius-scaphoid was not affected by the position of the wrist in the adult model (100.9%) but decreased in the elder model (50.2%) during radial extension.
The altered stress distribution at the radiocarpal interface during radial extension compared with extension most likely originates from the differences in loading dynamics. As depicted in Table 1, approximately 24% of the force applied by the FCU during extension is divided between the other muscles during radial extension. Consequently, kinematics of the carpal bones is altered, as previously reported by [29], which showed that during flexion-extension, the scaphoid contributes mostly in the radiocarpal, whereas during DTM, it contributes almost equally in the radiocarpal and midcarpal joints. The lunate shows slightly different kinematics, as it contributes almost equally in both joints during flexion-extension motion, but is more prominent in the midcarpal joint during DTM [29]. We surmise that due to the differences in extrinsic wrist muscle forces between the two postures, the carpus kinematics was altered, thereby affecting the stress distribution at the radiocarpal joint.
Previous 3D computation analyses suggested that ligament laxity affects carpal bone motion of the proximal row throughout radial and ulnar deviation motions [30]. By reducing the ligament stiffness in our elder model, we showed that a larger range of motion was enabled, in both extension and radial-extension positions ( Figure 3). Furthermore, due to lower stresses at the radius-scaphoid interface during radial extension compared with the adult model and in the radius-lunate interface, the stress differences between the two wrist positions were smaller in the elder model compared with the adult model. The differences in contact area and stresses found in the elder model compared with the adult model may help to explain symptoms in the aging wrist. For example, in a prospective study with 96 patients with symptomatic dorsal wrist ganglions and 96 individuals without dorsal wrist ganglions, the symptoms were associated with hyperlaxity of the ligaments and a positive scaphoid shift test [31].
The magnitude of the stresses calculated in our study is comparable to similar FE analyses of wrist loading in the literature. An FE study, where 100N axial compressive force was applied to the virtual wrist, showed peak contact stresses of 5.2 MPa and 2.9 MPa at the radius-scaphoid and radius-lunate articulations, respectively, during neutral position [13]. Their model differed from our model mainly in the isotropic elastic properties attributed to the cartilage, as well as wrist position. Consequently, in the FE analyses presented here, a higher compressive force was applied (a total force of 267 N), with a peak contact stress values of 29.7 MPa and 15.5 MPa at the radius-scaphoid and radius-lunate articulations, respectively, in extension. Another study presented a peak contact pressure in the radiusscaphoid interface of 2.9 MPa during extension (maximum extension with a total 7 Kg force applied on the tendons) [10]. The FE model did not include ligaments but used almost full contact areas between the bones to distribute the loads. The FE modeling included the bones as rigid bodies and the cartilage. The contact layer of each bone was defined as initially having mutual contacting surfaces. The contact stress results here (relative to the loads) were higher than presented in the aforementioned study, since their loads were distributed over vast contacting areas due to the modeling assumptions.
There are several limitations to the study. First, the analyses presented refer to a healthy adult female, so that the results are not conclusive for males or individuals with arthritis or other pathologies that might cause bone deformities or ligament impairments. This limitation could be addressed in future studies by using statistical shape models that could allow extending results to a whole set of geometries reproducing the actual variability in population [32]. Second, several assumptions were used to simplify the model and adjust it to the computational capabilities. We assumed that cartilage thickness may have affected the cartilage contact stresses, as previously suggested for hip models [33]. Additionally, the properties of the proximal SLIL were assumed. Third, the elder was defined by altered properties of the ligaments, whereas bone and cartilage properties were not adjusted. In matters of resembling the cartilage degradation by thickness change, a former FE study depicted rheumatoid arthritis by removing all the cartilage from the wrist model [34]. The collagen network and proteoglycans that bond to water allow the cartilage to resist compressive forces [35]. Changes in these matrix components might influence the mechanical properties of the cartilage [35]. Finally, since we used the actual wrist positions, as carried out in [28], which reported the tendon forces necessary to hold specified static positions of a cadaveric wrist, segmentation uncertainties might have affected the stress distribution in the comparison between the two positions.
In summary, this is the first study to compare the wrist FE models of adults and elders in the extension and radial-extension positions. The reduced stresses during radial extension might provide an explanation for our frequent use of this movement pattern, as the reduced stresses decrease the risk of overuse injury [36]. Further work should evaluate the effect of the DTM posture on wrist stresses in individuals with different pathologies or after fractures and ligament tears. Additionally, estimating the tendon forces through a multibody model, as simulated in [37], might advance our understanding of the biomechanics of different wrist positions under different loading conditions.

Acknowledgments:
We appreciate the help of Amit Gefen with the segmentation and meshing of the models and David Barlam for reviewing and consulting regarding the construction of the FE models. This work is based on the M.Sc. thesis of Vered Mahpari.

Conflicts of Interest:
The authors declare no conflict of interest.