Section 1 In vivo determination of the Achilles tendon moment arm in three-dimensions
2-1-1. Introduction
The mechanical advantage is defined as the ratio of the Achilles tendon moment arm and the moment arm of the GRF, and therefore the Achilles tendon moment arm must be determined accurately for determining the mechanical advantage accurately.
Previously, the center of rotation method was well used to determine the Achilles tendon moment arm in two dimensions (Fath et al., 2010; Fukunaga et al., 1996b; Maganaris et al., 1998; Rugg et al., 1990). This method assumes that the selected plane for capturing the bony configurations is orthogonal to the joint axis. The joint axis, however, may vary in its orientation among individuals and the plane orthogonal to it may not necessarily coincide with anatomical planes (Isman and Inman, 1969; Leitch et al., 2010). Reportedly, the error in the measurement of the Achilles tendon moment arm with the center of rotation method may reach 9 % due to the non-orthogonality of the sagittal plane to the talocrural joint axis (Isman and Inman, 1969; Sheehan 2010; van den Bogert et al., 1994). Achilles tendon moment arm should be determined with a three-dimensional approach so as to overcome the limitations associated with the two-dimensional approaches. The purpose of this study, therefore, was to develop a
Section 1 of Chapter 2
48
method for determining the Achilles tendon moment arm in three-dimensions.
2-1-2. Methods Subjects
Fifteen young males voluntarily participated in this study. Mean ± standard deviations (SDs) of their age, body height and body mass were 25.5 ± 3.1 years, 1.72 ± 0.05 m, 63.5 ± 6.8 kg, respectively. Informed consent had been obtained from each subject before the experiment. The study protocol was approved by the Ethics Committee of Human Research at Waseda University (approval number; 2009-138).
Data collection
A custom-made apparatus (VINE, Japan) was constructed with non-metal material to stabilize the foot and the lower leg at six ankle positions (Figure 2-1-1a). A MRI system (Signa HDxt 1.5T, GE Medical Systems, USA) was used to record a series of sagittal images of the right lower leg and foot for each ankle position with the following scan parameters: fast spin echo, 21.7 ms for time to echo, 1400 ms for repetition time, 1 mm of slice thickness, 0 mm of interspaced distance, 300 mm×300 mm of field of view, 512×512 pixels of matrix. The leg frame of the apparatus was aligned with the length of the bed and stabilized using Velcro tapes. The foot plate was rotated about the fixed axis by 10° increments to configure the apparatus at various ankle positions: -20°, -10° (dorsiflexed position), 0° (neutral position), +10°, +20°, and +30° (plantarflexed position). Each subject was, then, asked to lie on the bed in a supine
49
position with the knees fully-extended. The position of the right foot was carefully adjusted by the MRI operator to have the length of the foot aligned with the foot plate of the apparatus. The foot was then securely fastened to the foot plate using three non-elastic bands, so that the sole of the forefoot and the heel tightly contacted the foot plate. The right thigh and lower leg were aligned to the leg frame and the distal end of the right thigh together with the leg frame was fastened to the bed using Velcro tapes (Figure 2-1-1b). With this arrangement, the imaging planes of the MRI system were aligned with the sagittal plane of the subject and, thereby, the positions of any given pixels on the captured images could be expressed as the three-dimensional coordinates in the anatomical reference system. Prior to the scanning sessions, two reference markers were attached to the skin overlaying the anterior border of tibia, so that the long-axis of the tibia could be defined consistently across the different sets of MR images.
Data reduction
Selected bony landmarks of the tibia and the talus were manually identified from the obtained MR images and these coordinates were recorded by using the computer-aided diagnosis system (PLUTO, Nagoya University, Japan). The four points for tibia (a1- a4) (Figure 2-1-2) and the three points for talus (b1- b3) (Figure 2-1-3) were selected as the bony landmarks (Hashizume et al., 2012). The point a1 is the most proximal point of one reference marker attached to the tibia, the point a2 is the most distal point of the other reference marker attached to the tibia, the point a3 is the most
Section 1 of Chapter 2
50
distal tip of the medial malleolus, and the point a4 is the most distal point of the posterior bony projection forming the incisure fibularis. The point b1 is the most posterior tips of the lateral tubercle of the talus, the point b2 is the center of the posterior edge of the talus sulcus, and the point b3 is the most lateral tips of the lateral processus of talus. The local reference system embedded to each segment was defined as follows:
tibia tibia tibiatibia tibia tibia tibia tibia tibia tibia tibia
tibia y z u x y z A x y z
r r
r z r
r r
r
u r & & &
; ; ;
2 1
2
; 1 4 3
4 3
talus talus talustalus talus talus talus
talus talus talus
talus
talus z u y x y z A x y z
r r
r y r
r r
r
u r & & &
; ; ;
5 6
5
; 6 7 6
7 6
where r→i = the raw vector presenting the position of the bony landmark i in the anatomical reference system,
x
→
j, y
→
j, z
→
j = the unit vectors of three axes defining the orientation of the segment j,
u
Ѝ
j = a temporal unit vector used to define the anatomical reference system of the segment j,
Ak = the 3 × 3 matrix defining the local reference system embedded to the segment j.
The talocrural joint axis for a given ankle position (α) configured with the apparatus was computed as a finite helical axis, as follows:
(1) The orientations of the tibia relative to the talus were determined for two successive ankle positions (α+10° & α-10°):
51
10 10
10 / 10
10 10
/ ( ) ; ( ) tibia
T talus talus
tibia tibia
T talus talus
tibia A A A A A
A
(2) The angular displacement of the tibia relative to talus (∆Atibia/talusα±10
) was determined:
10 / 10
/ 10
/ ( )
tibiatalus
T talus tibia talus
tibia A A
A
(3) The finite helical axis for the ankle position α (u→) and the talocrural joint rotation (θ) were determined from the elements of ∆Atibia/talusα±10
(Spoor & Veldpaus, 1980):
) 1 2(
cos 1 33
10 22 /
10 11 /
10 /
1
Atibiatalus Atibiatalus Atibia talus
12 10 21 /
10 /
31 10 13 /
10 /
23 10 32 /
10 /
sin 2
1
talus tibia talus
tibia
talus tibia talus
tibia
talus tibia talus
tibia
A A
A A
A A
u
(4) The position of the point through which the helical axis passed was determined with the method described by Woltring et al (1985).
(5) Finally, the vectors defining the helical axis and the point through which the helical axis passed were re-expressed in the anatomical reference system.
The line of action of the Achilles tendon force was determined for each ankle position as a straight line passing through the centers of cross-sectional areas of the Achilles tendon at the proximal insertion site to the soleus and the distal insertion site to the calcaneus. The line of action of the Achilles tendon force was projected to the orthogonal plane of the talocrural joint axis, and the shortest distance from the projected line to the talocrural joint axis was determined as the moment arm (Krevolin et al., 2004; Pandy, 1999; Sheehan, 2007).
For reference, the moment arm was determined with the center of rotation method
Section 1 of Chapter 2
52
described by Rugg et al (1990). The single sagittal plane image passing through the mid-point of the medio-lateral width of the tibia was used to determine the talocrural joint center of rotation. The instantaneous talocrural joint center of rotation was determined with the Reuleaux method (Reuleaux, 1875) and the line of action of the Achilles tendon force was determined with the same method as the 3D determination.
The moment arm, defined as 2DMA, was determined as the shortest distance from the talocrural joint center of rotation to the line of action of the Achilles tendon force.
By an experienced examiner, the MR images taken for all subjects for all ankle positions were analyzed twice to determine within-examiner reliability of the visual identification of bony landmarks. In addition, the entire data collection process was repeated on two separate days with an interval of two to four weeks to test the day-to-day reliability. Three subjects were recruited and two sets of MR images for six ankle positions for each subject were obtained by an MRI operator. The experienced examiner analyzed the two sets of data on different days. To test the inter-examiner reliability of the visual identification of bony landmarks, another experienced examiner was recruited to analyze the same sets of MR images. This analysis was conducted on a separate day to eliminate a possibility of cross-talk between the two examiners.
Statistical analysis
A two-way analysis of variance (ANOVA) with repeated measures (2 methods × 4 ankle positions) was used for testing the effects of the methods and the ankle positions on the Achilles tendon moment arm values. When interaction between the two factors
53
was found to be significant, post-hoc tests with Bonferroni's correction were conducted to test the difference between 3DMA and 2DMA in each ankle position. The level of significance was set at 0.05.
2-1-3. Results
For the within-examiner reliability of visual identification, the coefficients of variation (CVs) were 2.1 ± 1.2 % for 3DMA and 3.1 ± 1.9 % for 2DMA, and the corresponding intra-class correlation coefficients (ICCs) (1, 1) were 0.970 and 0.933, respectively. The CVs for the day-to-day reliability were 2.2 ± 0.4 % for 3DMA and 3.3
± 1.0 % for 2DMA, and the corresponding ICCs (1, 1) were 0.982 and 0.962, respectively. The CV for the inter-examiner reliability and the corresponding ICC (2, 1) were 3.5 ± 1.8 % and 0.948, respectively.
The talocrural joint axis was not orthogonal to the sagittal plane, but deviated by 21.4 ± 20.7º on the transverse plane and 14.8 ± 22.6º on the coronal plane. The 3DMA values were 35 ± 4.6 mm, 41 ± 5.9 mm, 41 ± 4.4 mm and 40 ± 5.3 mm at -10°, 0°, +10° and +20° respectively, whereas the corresponding 2DMA values were 46 ± 3.0 mm, 49 ± 4.0 mm, 53 ± 4.3 mm and 56 ± 4.2 mm. There was a significant interaction between the methods and the ankle positions and the 3DMA values were found to be significantly smaller than the 2DMA at every ankle position (p < 0.01) (Figure 2-1-4).
2-1-4. Discussion
This study introduced a method for determining the Achilles tendon moment
Section 1 of Chapter 2
54
arm in three-dimensions. This method found to be highly reliability for within-examiner, day-to-day and inter-examiner. Then, the determined orientation of the talocrural joint axis in present study agreed good with the corresponding values reported in previous studies (Isman and Inman, 1969; Sheehan 2010; van den Bogert et al., 1994), and this good agreement validated the method of present study.
Present result showed that the 3DMA (40 mm) was significantly smaller by 11 mm than the corresponding 2DMA (51 mm) (Table 2-1-1). The difference between the 3DMA and 2DMA could be explained geometrically by three factors. The first factor was that the talocrural joint axis was not orthogonal to the sagittal plane (that is, the plane scanned by MRI system). The deviations of 14.8º from the transverse plane and 21.4º from the coronal plane account for overestimation of the 2DMA by 1.3 mm and 3.0 mm, respectively (Figure 2-1-5). The second factor was that the 2DMA might be influenced by the selection of the sagittal image used for the determination of 2DMA.
The sagittal image used for the 2D determinations (the image captured through the mid-point of the medio-lateral width of the tibia) was, on average, positioned medially by 9 mm relative to the geometric center of the Achilles tendon insertion to the calcaneus. As the talocrural joint axis is deviated laterally by 21.4º from the coronal plane, the use of the image taken at the medially shifted position induces an overestimation of the 2DMA by about 3.5 mm. The remaining difference might be accounted for by the random error associated with the visual identification of the bony configurations. The CVs were 2.1 % for 3DMA and 3.1 % for 2DMA, which might induce errors of 0.8 mm and 1.6 mm, respectively. The total error attributable to the
55
three factors was 10.2 mm, accounting for 93 % of the difference between 3D and 2DMA. Furthermore, the 2DMA might be contaminated by the inherent source of error associated with the two-dimensional determination. As bone moves out of the image plane, the shape of the bone depicted in the imaging plane changes (Shibanuma et al., 2005). This may be an essential factor inducing an error in the center of rotation and moment arm determination. These results indicate that the two-dimensional center of rotation method overestimates the Achilles tendon moment arm by 21% and this error was induced by the assumption of the center of rotation method that the selected plane for capturing the bony configurations is orthogonal to the joint axis. A three-dimensional method developed in this study, therefore, can determine the Achilles tendon moment arm accurately as compared with the traditional two-dimensional center of rotation method.
Section 1 of Chapter 2
56 2-1-5. Summary
This study developed a three-dimensional method determining the Achilles tendon moment arm accurately. This method found to be highly reliability for within-examiner, day-to-day and inter-examiner. Then, the determined orientation of the talocrural joint axis in present study agreed good with the corresponding values reported in previous studies (Isman and Inman, 1969; Sheehan 2010; van den Bogert et al., 1994), and this good agreement validated the method of present study. Present result showed that the two-dimensional center of rotation method overestimates the Achilles tendon moment arm by 21%. A three-dimensional method developed in this study, therefore, can determine the Achilles tendon moment arm accurately as compared with the traditional two-dimensional center of rotation method and should be used to determine the mechanical advantage.
57
Figure 2-1-1 The custom-made apparatus consisted of a rectangular foot plate and a leg frame (a). The foot and lower extremity were configured at various positions.
The angle between the foot plate and the leg frame was used to define the ankle position.
hinge joint
foot plate band hole
leg frame
250 mm
(a)
bed
foot plate
pad velcro tape
leg frame
(b)
Section 1 of Chapter 2
58
Figure 2-1-2 Magnetic resonance images of the sagittal view of an ankle (A) and the coronal views of the same ankle (B&C). The points indicated by the yellow dots are the reference points used to define the position and the orientation of the tibia.
59
Figure 2-1-3 Magnetic resonance images of the sagittal view of an ankle (A) and the coronal views of the same ankle (B&C). The points indicated by the yellow dots are the reference points used to define the position and the orientation of the talus.
Section 1 of Chapter 2
60
Figure 2-1-4 Relationship between ankle joint positions and Achilles tendon moment arm. The open squares represent the 3DMA and the closed squares represent the 2DMA. At every ankle joint angles, the 3DMA was significant lower than the 2DMA.
2DMA 3DMA
foot position (degree)
A ch ille s te n d o n m o m en t a rm (m m )
䠆p < 0.01
* * * *
Section 1 of Chapter 2 61
Table 2-1-1 The Achilles tendon moment arm reported in two-dimensions. All studies except one by Fath et al presented a graph indicating the relations between the moment arm and the ankle joint po graphs were scanned, enlarged, and measured the values directly
Achilles tendon moment arm in two dimensions (mm) Ankle joint position (degree)-20º-15º-10º0º10º15º20º Rugget al (1990) †4953555656 Maganariset al (1998) † †434751 Leardiniand O’Connor (2002) † † †49515253525048 Fathet al (2010)465255 Present study46495356
S
Section 2 of Chapter 2
62
Figure 2-1-5 Difference between 3DMA and 2DMA. The gray ellipse indicates the cross-section of the Achilles tendon and the line of action of the Achilles tendon force is orthogonal to this cross-section. The white circle indicates the intersection of the talocrural joint axis and the mid-sagittal plane of the foot often selected for determining the moment arm. The 3DMA represents the shortest distance between the talocrural joint axis to the line of action of the Achilles tendon force indicated by the white arrow, while the 2DMA determined on the sagittal plane represents the shortest distance between the intersection of the talocrural joint axis and the selected sagittal plane and the line of action of the Achilles tendon force indicated by the black arrow.
joint axis 3DMA 2DMA
21.4º
sagittal plane
63
Section 2 The contraction induced increase in the Achilles tendon moment arm:
A three-dimensional study
2-2-1. Introduction
The moment arm of a muscle-tendon force about a joint axis is completely determined by the positions and the orientations of the joint rotation axis and the line of action of the muscle-tendon force. Previously, the moment arm of various muscle-tendon forces was determined with a two-dimensional approach with which the lever was assumed to rotate in the sagittal plane about a point on the plane, called COR (Tsaopoulos et al., 2007; Maganaris et al., 1998; Rugg et al., 1990). With this method, researchers have found that the moment arm of a muscle-tendon force increased as the muscle contracts voluntarily (Akagi et al., 2012; Maganaris et al., 1998; 1999;
Tsaopoulos et al., 2007). Maganaris et al (1998; 1999) and Tsaopoulos et al (2007) identified two factors inducing the increase in moment arm due to contraction, as follows; (a) linear and angular displacements of the line of action of tendon force and (b) linear displacement of joint COR. In Section 1 of Chapter 2, it was found that non-orthogonality of the talocrural joint axis to the sagittal plane induced systematic error in the Achilles tendon moment arm determined with the two-dimensional COR method. Specifically, angular displacement and forward shift of the talocrural joint axis were found to induce the anterior displacement of the COR observable in the mid-sagittal plane image of the foot, which, in turn, resulted in an increased Achilles tendon moment arm value with the COR method. A question arises on the validity of the
S
Section 2 of Chapter 2
64
finding obtained with the COR method, that is, the contraction induces an increase in Achilles tendon moment arm. The additional analysis with a three-dimensional approach is needed to validate the finding of two-dimensional studies. Furthermore, it is difficult to determine the Achilles tendon moment arm in three-dimensions under muscle contraction condition for children and adolescents, because subjects are required to sustain a muscle contraction in long scan time. The examination of the relationship between the Achilles tendon moment arms determined in the rest and muscle contraction conditions should be need to clarify whether the Achilles tendon moment arm determined in the rest condition corresponds to that determined in the muscle contraction condition. The purposes of this study, therefore, were 1) to re-examine the influence of the isometric plantarflexors contraction on the Achilles tendon moment arm in three-dimensions and 2) to examine the relationship between the Achilles tendon moment arms determined in the rest and muscle contraction conditions.
2-2-2. Methods Subjects
Eight young males voluntarily participated in this study (Table 2-2-1). Before the experiment, subjects were well trained to perform the isometric plantarflexors contraction on the device used for the experiment. The study was approved by the Institutional Ethics Committee of Human Research at Waseda University (approval number; 2009-138) and informed consent was obtained from each subject before the experiment.
65 Data collection
The method described in Section 1 of Chapter 2 was applied to determine the Achilles tendon moment arm in two conditions; the condition in which the lower leg muscles were at rest and the condition in which plantarflexors were contracting. Each subject was asked to lie in a supine position on a bed of a MRI system (Signa HDxt 1.5T, GE medical systems, USA). The right lower extremity was fixed to a custom-made torque meter (Figure 2-2-1) at foot positions of 10° of dorsiflexion, neutral position and 10° of plantarflexion with the knee full-extended. The long-axes of the foot and the lower leg were carefully aligned to the long-axes of the foot plate and the leg frame of the torque meter, respectively, and fixed with non-elastic belts. The subject was, then, asked to contract plantarflexors isometrically at each foot position with the maximum effort. The change in the torque meter output representing the magnitude of the force exerted on the foot plate was amplified and recorded at 100 Hz.
The change in the torque meter output due to plantarflexors contraction with the maximum effort was used as the index representing isometric plantarflexion torque attainable with the MVC (Iwanuma et al., 2011a, b). By using this index, the magnitude of torque meter output representing the target intensity (30%MVC) was calculated. A head-mounted display (VisuaStim Digital's controller, Resonance Technology, USA) was used to provide the subject with the intensity of the isometric plantarflexion torque so that the subject was able to adjust and sustain the plantarflexion torque at the target intensity during MRI scanning (Figure 2-2-2).
S
Section 2 of Chapter 2
66 Data reduction
A series of ankle images were obtained for each foot position with the scan parameters listed in Table 2-2-2. The imaging plane was carefully aligned to the medio-lateral axis of the foot plate and the long-axis of the leg frame, so that the
"depth" of the series imaging planes was aligned to the long-axis of the foot plate. The position of any given voxel in the series of MR images, therefore, was assumed to represent three-dimensional coordinate in the orthogonal coordinate system embedded to the torque mater. The local reference systems embedded to the tibia and the talus were defined, respectively, from the four (a1-a4) and three (b1-b3) bony landmarks of each bone (Figure 2-1-2 & 2-1-3). The finite helical axis was computed from the elements of the angular displacement of the tibia-embedded reference system relative to the talus-embedded reference system as the foot position was changed from 10° of dorsiflexion to 10° of plantarflexion. A line passing through the centers of cross-sectional areas of the Achilles tendon at the proximal insertion to the soleus and the distal insertion to the calcaneus defined the line of action of the Achilles tendon force for the neutral foot position. The shortest distance between the talocrural joint axis to the line of action of the Achilles tendon force projected to the plane orthogonal to the talocrural joint axis was determined as the Achilles tendon moment arm for the foot position of 0°.
The orientations and the positions of the talocrural joint axis and the line of action of the Achilles tendon force were expressed in the anatomical reference system. The Z
67
axis of the anatomical reference system was defined as the z axis of the local reference system embedded to the tibia. The X axis of the anatomical reference system was defined as the cross product of the vector defining the antero-posterior axis of the torque-meter-embedded reference system and the Z axis of the anatomical reference system. The Y axis of the anatomical reference system was defined as the cross product of the Z axis and the X axis of the anatomical reference system. The mid-point of the line connecting the a3 and the a4 was defined as the origin of the anatomical reference system. The X-Y, Y-Z and Z-X planes of the anatomical reference system, therefore, represented the transverse, mid-sagittal and coronal planes, respectively. The orientation of the talocrural joint axis was expressed as two projection angles; the angles formulated by the X axis of the anatomical reference system and the projections of the talocrural joint axis on the transverse plane (θ ) and on the coronal plane (θ). The orientation of the line of action of the Achilles tendon force was also expressed as two projection angles; the angles formulated by the Z axis of the anatomical reference system and the projections of the line of action of the Achilles tendon force on the sagittal plane (φ) and on the coronal plane (φ). The three-dimensional coordinates of the point on the talocrural joint axis passing through the mid-sagittal plane were determined in the anatomical reference system to represent the position of the talocrural joint axis. The three-dimensional coordinates of the point of the distal insertion of the Achilles tendon were also determined in the anatomical reference system to represent the position of the line of action of the Achilles tendon force.
S
Section 2 of Chapter 2
68 Statistical analysis
Paired t-test was used to test the differences in the Achilles tendon moment arms between the rest and 30%MVC conditions. Pearson's product-moment correlation coefficient was used to examine the relationship between the Achilles tendon moment arms determined in the rest and 30%MVC conditions. Statistical significance was set at 0.05.
2-2-3. Results
The Achilles tendon moment arm determined in the 30%MVC condition was significantly greater by 8 ± 2.3 mm than that in the rest condition (p < 0.05). The differences in θ and θ between the two conditions were 0 ± 6.0º and 1 ± 9.0º, respectively. The differences in φ and φ between the two conditions were 2 ± 2.5º and 1 ± 2.2º, respectively. The line of action of the Achilles tendon force moved posteriorly by 5 ± 1.3 mm and medially by 2 ± 0.6 mm due to the muscle contraction. The talocrural joint axis also moved anteriorly by 3 ± 1.9 mm and distally by 2 ± 1.0 mm. The correlation between the Achilles tendon moment arms determined in the rest and 30%MVC conditions was significant (r = 0.968, p < 0.05) (Figure 2-2-3).
2-2-4. Discussion
The results showed a significant contraction-induced increase in the moment arm