Chapter 5. Finger Model: Extensor and Flexor mechanisms 5.1 INTRODUCTION

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1 Chapter 5 Finger Model: Extensor and Flexor mechanisms 5.1 INTRODUCTION Different postures of the human hand present a different range of parameters, i.e. level of force/effort, available range of motion, and sensory data (Iberall, 1997). Active control of the hand is the result of a coordinated effort of extrinsic and intrinsic musculature. The contribution of extrinsic extensor and flexor muscles is prominent in changing the shape of the hand by manipulating the digit rays (fingers), while intrinsic muscles are responsible for maintaining the different configurations of the hand, i.e. flattening or cupping the palm to grasp objects of different sizes and shapes. A complex tendinous network known as the extensor hood controls the movement of joints during extension, while a well developed flexor system facilitates smooth and stable flexion of finger joints (Nordin et al., 21). Although, CNS plays a major role in achieving same standard postures, repeatedly for the routine tasks, its role is beyond the scope of this work. An interesting feature of hand articulation is that phalanges are designed to work in flexion. The palmer tendinous apparatus (flexor tendons) is twice as strong as the dorsal extensor apparatus. The skin on the dorsal (back) of hand is different from that of palmer side. Dorsal skin is thin and mobile to facilitate extension and flexion. Skin on the palmer side is thick and comparatively less elastic as compared to dorsal skin, but plays a crucial role in perception of touch, protection and safety (Nordin, et al., 21). A bond graph model of the complete finger has been presented, which is an extension of the already formulated finger model consisting of three phalanges and the extensor mechanism. The metacarpal is attached to the proximal phalanx of existing model as the first element of the digit ray. It helps to position the MCP joint at different inclinations with respect to the metacarpal bone. Flexors (FDP and FDS) have also been attached to respective phalanges. The bond graph of the existing model has been further augmented and extended to incorporate the above modifications. These additions to the model help in investigating the functioning of the flexor tendons, the distribution pattern

2 of the tensions in different tendon bands of tendinous network and effect of variation in the joint angles on the tensions in tendon bands. Minute details of the various subsystems as building blocks of the finger model have also been covered in (Vaz, Singh & Dauphin-Tanguy, 215). Fundamentals of the flexor mechanism are elaborated in section 5.2. Finger models proposed by various researchers have been summarized in section 5.3. A bond graph model of the musculoskeletal actuation system for a finger is discussed in section 5.4. Simulation results have been presented in section 5.5. Results and discussions along with various plots are given in section FLEXOR MECHANISM The index finger model consists of a chain of four rigid bodies (the metacarpal and three phalanges). The DIP and PIP joints have been considered as hinges with one DOF (rotation about the Z-axis), simulating the flexion and the extension motion. The DIP and PIP joints have been considered to be hinged joints because of the shape of the articulating bone surfaces and the arrangement of the concerned ligaments and tendons around these joints. It has been assumed that these joints have rotational motion about only one axis. The MCP joint has been modeled as a universal joint with two DOFs in the Y and Z-axes, simulating adduction and abduction, and flexion and extension motions, respectively (Schöffl et al., 212; Roloff et al., 26). Major modification in existing model is the addition of metacarpal and the flexor apparatus on the palmer side of the finger. The flexor apparatus is a functional unit of tendons, tendon sheath and pulleys. Pulley system, itself has three sub-systems, the transverse carpal ligament, the palmar aponeurosis pulley, and the digital flexor pulley sub-system. Both flexor tendons, the flexor digitorum profundus (FDP) and flexor digitorum superficialis (FDS) pass commonly through the carpal tunnel and pulleys and subsequently intersect at the chiasm. The FDS tendons are superficial (less deep) as compared to the FDP tendons. FDS bifurcates and allows the FDP to pass through it at the base of the finger. Two slips as a result of bifurcation rejoin and insert on the middle phalanx and it acts as prime flexor for the PIP joint (Zidel, 27). 96

3 A well developed pulley system is in place in the flexor apparatus to stabilize the flexor tendons on the palmer side of the phalanges. Main function of pulley system is, to maintain the proximity of flexor tendons to the phalanges, thus converting translation and force developed in the flexor muscle-tendon unit into rotation and torque at the joints between the phalanges. The loss of any of the pulleys can cause bowstringing of the flexor tendons, which leads to inefficient and inconsistent moment of arm, significant reduction in the strength and a decreased range of motion. Absence or injury to the pulley results in higher than normal moment arm and more tendon excursion is needed to produce the same range of motion. But the tendon excursion is limited by many other factors and is directly proportional to muscle fiber length. Hence, normal tendon excursion should be maintained with the help of the effectiveness of the pulley systems at various places along the phalanges (Doyle, 21). In the proposed model hooks emulate the pulley system and act as restraints to the deflection and slack of the tendons. Lumbericals have peculiar topography as they are attached to the other tendons instead of the phalanges. They do not have direct impact on the movement of phalanges. So lumbericals are not modeled. 5.3 FINGER MODELS Sufficient detail of the finger models proposed by various researchers has been covered in chapters two and four. Mathematical models of fingers are helpful in studying the various aspects of the hand functions for biomedical as well as ergonomic purpose. Most of the models found in literature are planar models considering flexion and extension only. Such models provide insufficient information about the dynamics of the system and are not helpful in reconstructive hand surgery or prosthesis design (Sancho-Bru et al., 21). Both static and dynamic models have been proposed by the various researchers. Most of the models reported are either static models or planar ones. Static models were used to estimate finer forces during isometric functions (Valero-Cuevas et al., 1998). The majority of the dynamic models use inverse dynamics i.e. tendon forces were estimated for given specification of finger motions (Sancho-Bru et al. 21). 97

4 The mathematical model proposed by Hume et al. (1991) was used by Roloff et al. (26) to evaluate the forces acting on the pulley system and the effect of other factors on these forces. It has been observed that the tendon gets deflected at point of contact between the tendon and the pulley. Hence the force acting on the pulley is a function of the angle between the tendon and the pulley. The forces acting on the pulley due to change in angle between the tendon bands and the pulley, are accounted for in the proposed model by formulating the dynamics of the hook-string sub-system. Efforts generated by the tendons on both sides of the corresponding hook act on the hook in the direction of tendons. This has been achieved by multiplying the scalar efforts by corresponding unit vectors. The modulated transformers in the hook-string WBGO represent these dynamically changing unit vectors. Effort on hook will be less in case (a) as compared to the case (b) of Fig Floating mass Hook point ˆ H rfl 4 46 ˆ H rfl 4 24 FL 46 H 4 Hook point FL 46 ˆ H r FL 4 24 H 4 ˆ H r FL 4 46 FL 46 Floating mass FL 46 Floating mass Floating mass (a) (b) Fig 5.1 Effort acting on the hook was a function of the angle of deflection of the tendon bands at the hook. 5.4 BOND GRAPH MODEL OF THE FINGER Topology of the extensor mechanism (Winslow's rhombus) and associated terminology was elaborated in detail in (Vaz et al., 215). Some aspects have been reproduced for the sake of continuity. MCP joint which was constrained in previous three-phalanx model, is now providing two DOF as possessed in real life. Metacarpal has been added as the 98

5 fourth link to the existing model. This will help in studying the effect of finger posture on the distribution of tensions in various tendon bands. In this extended work emphasis has been laid on modeling of the flexor tendons. The tendon topologies of the extensor mechanism as well as the flexor apparatus are modeled on the basis of the hook string mechanism. Tendons are considered as strings passing through hooks fixed on phalanges. The tendons are modeled based on the Hill's four-element muscle model. The Winslow s tendinous network is depicted in the Fig. 5.2 and flexor tendons are shown in Fig The difference between FDS and FDP tendons is exaggerated for clarity. The floating point between hooks H x and H y is represented as FL x,y. FL x,y,z represents the floating point between hooks H x, H y and H z and floating points at the ends of the tendons and slips are mentioned in single subscript i.e. FL x (e.g. FL 31, FL IN41 etc.). P xx are points of insertion of tendons on bones. O 11, O 21, O 31 and O 41 are the origin points of moving frames {1},{2},{3} and {4} respectively. FL IN44 and FL IN45 are the end points of FDP and FDS tendons respectively. During simulation, motion has been imparted to these floating points to move the flexor tendons, which in turn move the finger joints in flexion (towards the palm). The hook points on the phalanges emulate the pulley system of the flexor mechanism and serve the same purpose. P 32 represents slip point, which was termination of FDP tendon on the palmer side of the distal phalanx. Similarly P 34 denotes slip point on the palmer side of the middle phalanx (the termination point of FDS). FDS will apply flexor torque at PIP joint and FDP will apply flexor torque at DIP joint. The finger model presented in this work is an open chain of four interconnected digits, which are treated as rigid bodies. Contraction of the flexor tendon fibers results in the flexion motion of the phalanges. The flexor tendons, by the virtue of the presence of various pulley systems, remain close to the surface of bones, for transmission of constant moment of arms to joints. A simulation run has been performed on the new model. Both the flexors are pulled smoothly through 1.5mm in 1s. The phalanges move in flexion as expected. The model reflects the correct dynamics of the system as intended. 99

6 O 33 P 31 X3 Terminal slip (TE) O 31,O 22 FL 31 Z3 DIP joint (flexion-extension) Central band of long extensor tendon Central band of interosseous tendon H 3 D 16,P 16 FL 23,25 H 23 FL 3,5 H 1 H 4 H 2 FL 1,4 FL 2,6 H 5 H 6 H 9 H 7 H 21 FL 21,22,23 H 22 X2 FL 22,24 P 33 H 25 H 24 FL FL 25,3 33 O 21,O 12 X1 H 8 H 1 FL 24,2 Z2 Extensor slip(es) PIP joint (flexion-extension) Lateral band of interosseous tendon Metacarpophalangeal joint (flexion-extension and abduction-adduction) FL 5,13 FL 13,43 FL IN43 LUM FL 4,11 H 11 FL 11,41 FL 6,12 H 13 H 12 O 42,O 11 H 43 EIP H 41 FL & IN41 EDC EIP & EDC X,4 PI FL 12,42 H 42 FL IN42 Z1 Carpometacarpal joint (Constrained) O 41 Z,Z4 Fig.5.2 The graphical representation of the extensor mechanism model showing the Winslow s tendinous rhombus (Dorsal view). 1

7 O 3 X 3 P 32 FDP slip O 31,O 2 FL 32 Z DIP joint (flexion-extension) FL 26,27 6 H 2 7 FL O 21,O 12 H 1 H 2 X P 34 FL 34 H 16 Z FDS slip PIP joint (flexionextension) 4 FL FL 16,17 14,15 H 15 H 1 FL 15,44 O 42,O 11 X 1 7 FL 17,45 H 4 EIP H & 45 FL EDFL C IN45 PI Metacarpophalangeal joint (flexion-extension and abduction-adduction) ) Z X,4 FDS Carpometacarpal joint (Constrained) ) O 41 Z,Z FDP Fig.5.3 The graphical representation of the flexor mechanism (Palmer view). 5.5 SIMULATION The model is thoroughly modified and additional WBGOs for flexor mechanism were put in place. A revamping of the overall dynamics of the model has occurred due to the additional slips, hooks and the floating points, etc. All the joints were kept at zero flexion 11

8 angle, i.e. phalanges were kept in a straight line at the beginning of the simulation run. The only carpometacarpal joint has been fully constrained, the other joints are free to move as per designed degrees of freedom under the efforts generated by the excursion of tendons. Smooth and jerk less flow, as applied to the previous three phalanx model, has been applied to the ends of FDP and FDS tendons. Both the tendons under consideration are pulled by 1.5mm over a trajectory interval of 1s. The density of the phalanges is assumed to be uniform throughout the phalanx. Circular area of cross section has been assumed for all the phalanges. Geometric data of phalanges based on Wu et al. (28) and Alexander and Viktor (21) is presented in Table No Table 5.1 Physical data of phalanges and metacarpal. Phalanx Mass(kg) Length(m) Radius(m) Metacarpal Proximal Middle Distal Articulation of FDS and FDP tendons with respect to synovial sheath and pulleys has been assumed to be frictionless, because the friction between the tendons and synovial sheath is negligibly small in a normal finger (Fung, 1993; Roloff et al., 26). Hence the corresponding damping values (R 7 and R 17 ) modeled to account for the frictional losses have been assigned zero values. The linear stiffness, K SE (K 15 ) and damping value (R 11 ) for the tendon and hook system have been taken as 25N/mm and 4N.s/m respectively, which are one and half times more than the corresponding values for the extensor tendons (Nordin, Lorenz & Campello, 21). The inertial coordinate frame {} was made to coincide with the frame {4} attached to the proximal end of the metacarpal as initial condition for the simulation run. Similarly end points of metacarpal-proximal, proximal middle and middle-distal phalanges were made to coincide to achieve the desired posture of the finger. The linear and angular motions of the metacarpal w.r.t. the inertial frame were constrained along all the axes by providing flow sources having zero value and nonlinear stiffness element C, and matching R elements of.5 U N.s/m in translational coupling and stiffness 12

9 element C with diagonal values 3 U N.m/rad and matching R elements of.5n.m.s/rad in rotational coupling. Relative rotation between the middle-distal (DIP joint) and middle-proximal (PIP joint) phalanges about axes X and Y was constrained using very stiff C elements i.e. K= N.m/rad. and the resulting transient oscillations about the these axes were damped out using R =.5N.m.s/rad. Free relative rotation was permitted about the Z axis by providing zero value of stiffness C element about this axis. Relative rotation between the metacarpal-proximal phalanges (MCP joint) about X axis was constrained using very stiff C elements, i.e. K= N.m/rad. and the resulting transient oscillations about these axes were damped out using R =.5N.m.s/rad. Free relative rotation was permitted about the Y and Z axes by providing zero value of stiffness C element about these axes. Relative translation between phalanges was constrained using nonlinear stiffness element C, and matching R elements of.5 U N.s/m. Flexion of was provided at PIP, DIP and MCP joints as initial condition. Very high stiffness (15 N/mm) is taken, while considering the normal reaction between phalanges and floating points. The coefficient of friction is taken as RESULTS AND DISCUSSION The 3-D plots of the centers of mass all the four digits of the ray, along with their profiles are depicted in Fig Rotational and translational motions of the metacarpal were constrained with respect to the inertial frame, hence the center of mass of metacarpal (C 4 ) shows negligible movement. Movement of the proximal phalanx is also negligibly small because no flexor tendon terminates on this phalanx and the flexors are not imparting any direct effort to it. FDS and FDP produce direct flexor torque on the DIP and PIP joints respectively. Both the joints are hinged ones, hence the phalanges moved in a circular arc about their respective Z-axis. Flexion of middle phalanx seems less as compared to the distal one, but it is a visual illusion. This is due to the compounding effect of the flexion of both the joints (DIP and PIP) on the distal phalanx. In contrast the flexion of the distal phalanx in its own frame {3} is less as compared to the middle one {2}. This is because there are more number of hooks crossed by the FDP, before the termination on the distal 13

10 z coordinate (m) phalanx, as compared to the FDS. Hence net effort generated at the FDP slip is less as compared to the FDS one. Hooks and the floating points on the distal phalanx are shown in Fig. 5.5 (2-D) and Fig. 5.6 (3-D). Floating points and hook points belonging to extensor mechanism are not plotted to avoid clutter in plots. Those points have already been plotted and discussed in the previous chapter. The distal phalanx moves in flexion under the flexor torque exerted by the FDS. Time trajectories of center of mass (C 3 ) and the slip point (P 32 ) are circular arcs due to their fixed positions on the phalanx. The trajectory of the floating point (FL 32 ) is a bit elongated arc due to its movement under the influence of the applied flow. Initial position of finger Centre of mass (Distal Phalanx) Centre of mass (Middle Phalanx) x x 1-3 y coordinate (m) x coordinate (m) Fig. 5.4 Three dimensional time trajectories of centers of mass of all the digits along with their profiles. 14

11 z coordinate(m) y coordinate(m) x 1-3 Planer motion of the distal phalanx FL32 Centre of Mass Intial position of the phalanx (Zero flexion) P x coordinate(m) Fig. 5.5 Position trajectories of the center of mass of the distal phalanx, the slip point and floating point on the tendon grazing on it. Spatial motion of the distal phalanx x 1-3 Centre of mass 2-2 P x FL y coordinate(m) x coordinate(m) Fig. 5.6 Three dimensional position trajectories of center of mass of the distal phalanx, the slip point and floating point on the tendon grazing on it. 15

12 y coordinate(m) All the floating points and hook points on middle phalanx, belonging to flexor apparatus are plotted against time in Fig. 5.7 (2-D) and Fig. 5.8 (3-D). The center of mass (C 2 ), slip point (P 34 ) and hook points (H 26 and H 27 ) on the phalanx are integral part of the rigid body (the phalanx) hence these points showed no translational movement with respect to the phalanx. They move along with the phalanx in a proper circular arc. The distances between these points remain constant throughout the movement. Time trajectories of the floating points of flexor mechanism are easily predictable as compared to their counterparts on the extensor hood, due to absence of multiple tendon connections to any single floating point. x 1-3 Planer motion of the middle phalanx Centre of mass -2-4 H27 P34 H FL2714 FL FL x coordinate(m) Fig. 5.7 Position trajectories of the center of mass, the slip point, the hook points on the middle phalanx and floating points on the tendons grazing on it. 16

13 z coordinate(m) Spatial motion of the middle phalanx Centre of mass H x 1-3 FL2714 H27 P34 FL x 1-3 FL y coordinate(m) -1.1 x coordinate(m) Fig. 5.8 Three dimensional position trajectories of the center of mass, the slip point, the hook points on the middle phalanx and floating points on the tendons grazing on it. Time trajectories of the various points on the proximal phalanx are shown in Fig. 5.9 (2-D) and Fig. 5.1 (3-D). As discussed earlier, the center of mass and the hook points are part of the phalanx hence their motion is similar to that of the phalanx. The rotational motion of the proximal phalanx is negligible as none of the tendons terminated on this phalanx and exerted moment of arm on it. Only moments exerted on the phalanx are due to the interaction of the flexor tendons with the hook points on the phalanx. Floating points graze along the profile of the proximal phalanx in accordance with the applied pulls. The efficacy of the normal reaction and friction sub-model is also visible from the trajectory of the floating points. 17

14 z coordinate(m) y coordinate(m) Planer motion the proximal phalanx Centre of mass FL15,44 & FL17, H15 & H17 FL14,15 & FL16,17 H14 & H x coordinate(m) Fig. 5.9 Position trajectories of the center of mass, the hook points on the proximal phalanx and floating points on the tendons grazing on it. Spatial motion the proximal phalanx H14 & H16 Centre of mass FL14,15 & FL16,17.95 x y coordinate(m) x 1-3 FL15,44 & FL17, H15 & H17.65 Fig. 5.1 Three dimensional position trajectories of the center of mass, the hook points on the proximal phalanx and floating points on the tendons grazing on it x coordinate(m)

15 effort(n) Fig shows the time history of the scalar forces developed across the termination points of the FDP tendon on the distal phalanx (slip point, P 32 ) and termination of the FDS tendons on the middle phalanx (slip point, P 34 ) due to the motion applied to the tendon ends. The patterns of time histories of development of forces are in accordance with the applied pull. Magnitude of effort built at the slip point P 34 is more than the effort at the slip point P 32. This variation is the result of the difference of the lengths of FDP and FDS tendons and number of hook points each flexor crosses. These scalar forces get multiplied by respective unit vectors and act along the direction of the tendon bands. 2.5 x FDP slip (P 32 ) FDS slip(p 34 ) time (s) Fig 5.11 Time histories of the scalar efforts developed at the flexor slips. Fig depicts the magnitude of efforts developed at the hook points (H 44, H 14 and H 26 ). This combination of the hook points is considered on purpose. These hook points are on the same flexor (FDP), but on the different phalanges (Fig. 5.3). Maximum effort has been developed on H 44, as it is closer to the point of application of external effort. This effort is reduced at hook H 14 because the flexor tendon crosses another hook point 19

16 effort(n) on the way before passing through H 14. Further reduction in the effort takes place at H 26 due to the same reason x 1-3 H44 H14 H CLOSURE time Fig Time histories of the scalar efforts developed at various hook points. A bond graph model of the complete finger (the extensor mechanism, the flexor apparatus, three phalanges and metacarpal) has been developed. The stiffness of flexor tendons is more than that of the extensor ones. The flexor tendons were pulled during the simulation run. The Hill's four element muscle model has been used for the extensor and flexor tendon modeling. Hooks emulate the working of well developed pulley system of the flexor mechanism and achieve the intended grazing action of soft tendons along hard phalanges. The motion of the floating points reveals the motion of the flexor tendons through the hook points. Simulation plots depict the effectiveness of the bond graph model in capturing the dynamics of the complete finger system. 11

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