# Biomechanical Gain in Joint Excursion from the Curvature of the Achilles Tendon: Role of the Geometrical Arrangement of Inflection Point, Center of Rotation, and Calcaneus

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## Abstract

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_{AT}) in the proximal displacement of the calcaneus compared to the change in the Achilles tendon length. Here, we empirically validate and extend our previous modeling study by investigating the effects of a broad range of obstruction locations on gain

_{AT}. The largest gain

_{AT}could be achieved when the obstruction was located on the most ventral and distal sides within the physiological range of the Achilles tendon, irrespective of the ankle position.

## 1. Introduction

_{AT}), which is defined as the ratio of the vertical displacement of the calcaneus to the change in the Achilles tendon length, varies depending on both the inflection point with respect to the ankle center of rotation (COR) and the ankle position. In fact, Hodgson et al. [3] evaluated the gain

_{AT}at three locations of the inflection point (Supplementary Materials Figure S1) given by coordinates (x, y; horizontal and vertical axes of a system of coordinates, in cm) and obtained the largest gain

_{AT}for an inflection point located at (3, 0) from ankle COR at (0, 0). This was followed by a decreasing order of gain

_{AT}observed at inflection points located at (4, 0) and (3, 2.5) during ankle plantarflexion. In contrast, the gain

_{AT}decreased as the ankle dorsiflexed. Several significant issues posed in this previous report include the fact that the gain

_{AT}is unknown at locations other than the three locations mentioned above, and the geometrical structure of the actual human musculoskeletal system is neglected. Further, the mechanisms determining the gain

_{AT}are still not well understood, and the location resulting in the largest gain

_{AT}remains unknown.

_{AT}during ankle rotation at the two positions. The present study conducted a modeling investigation, because it was difficult to identify the location of the inflection point within the Achilles tendon that exhibits the largest gain

_{AT}within the physiological range of the Achilles tendon using an in vivo human experimental study.

## 2. Methods

_{AT}over a range of ankle positions.

_{AT}is defined as the reciprocal of the change in the Achilles tendon length ($\Delta {l}_{AT}$) when the calcaneus is displaced by 1 mm, as follows:

## 3. Results

_{AT}always exceeded unity when the inflection point was located within the physiological range for the two ankle positions (Figure 3, top). This indicates that the calcaneus displacement has a gain corresponding to the change in the Achilles tendon length. The largest gain

_{AT}was realized when the inflection point is located on the most ventral and distal sides, irrespective of ankle position. The ratio of the change in the Achilles tendon length to the moment arm remains nearly constant (Figure 3, middle). The product of the ratio of the change in the Achilles tendon length to the moment arm and the x-coordinate of the calcaneus position remains nearly constant within the physiological range of the Achilles tendon. This value equaled 0.990 and 0.996 when the ankle remained in the plantarflexed and dorsiflexed positions, respectively (Figure 3, bottom). The ankle angle and Achilles tendon insertion on the calcaneus (x, y) coordinates were measured as 20.7 ± 1.5° and (x = 67.8 ± 7.1, y = −34.4 ± 4.2) mm at the plantarflexed position, and −9.3 ± 0.8° and (x = 52.7 ± 9.9, y = −55.0 ± 2.6) mm at the dorsiflexed position.

_{AT}can be estimated using the moment arm and the x-position of the calcaneus ($\mathrm{calcaneus}\_\mathrm{x}$) as follows:

_{AT}calculated considering the change in the Achilles tendon length (Equation (1)) and that calculated using the moment arm (Equation (6)) equals 0.38% and 0.98% in the plantarflexed and dorsiflexed positions, respectively.

## 4. Discussion

_{AT}, during ankle rotation at two ankle positions. The largest gain

_{AT}was observed when the inflection point is located on the most ventral and distal sides within the physiological range. However, unexpectedly, this was independent of the ankle position. The former result is observed because as the gain

_{AT}increases and the inflection point is closer to the ankle COR. These results are not consistent with those presented by Hodgson et al. [3], who demonstrated that as the gain

_{AT}increases, the inflection point is closer to the ankle COR. This difference might be partly due to the different models of the inflection point and ankle position employed. While our model relied on human data obtained empirically from physiological experiments, Hodgson et al. [3] established their model on logical speculation; thus, it was not based on human data. The results of our model are partially supported by the findings of the experimental investigation conducted by Csapo et al. [5], who demonstrated that there was no significant difference in the gain

_{AT}among passive ankle rotations from 10° to 20°, 0° to 10°, and −10° to 0°.

_{AT}is not determined solely by the distance from the ankle COR to the inflection point but rather by the geometrical arrangement of the inflection point, myotendinous junction, and ankle position relative to the ankle COR. Although the geometry of the ankle COR and ankle position is determined by an intrinsic arrangement, the location of the inflection point can be variable. Our previous experimental and computational investigations using a multi-modality approach revealed that the inflection point of the Achilles tendon is due to neither the nature of the tissue deformations surrounding the Achilles tendon nor its physical properties [4]. Instead, we concluded that the inflection point results from the geometric architecture of the Achilles tendon and its configuration relative to the surrounding tissues, such as the Kager’s fat pad. This pad is a mass comprised of adipose tissue, the synovial membrane located in the superior tuberosity of the calcaneal bone inferiorly, and the Achilles tendon posteriorly [6]. As the ankle plantarflexes, the calcaneus moves proximally and dorsally, and a force is produced in the ventral direction around the inflection point location. These configurations appear to deform the Kager’s fat pad, the degree of which is determined by the stiffness of the Kager’s fat pad. Because this pad is comprised of adipose cells [7], such deformation is assumed to minimize pressure change during ankle flexion [8]. Surgery [9], aging [10], and diabetic disease [11] alter the stiffness of the fat pad due to fibrosis, resulting in changes in the location of the inflection point, and ultimately, in the gain

_{AT}.

_{AT}, the largest gain

_{AT}can be realized when the inflection point is located on the most ventral and distal sides within the physiological range of the Achilles tendon regardless of the ankle position. Thus, this study affords a better understanding of the possible factors influencing the gain

_{AT}, the evaluation of which might be essential to accurately predict the joint kinematics generated by large muscle-tendon complexes.

## Supplementary Materials

_{AT}with respect to the three positions of the inflection point relative to the ankle center of rotation (Hodgson et al. 2006).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Maganaris, C.N.; Baltzopoulos, V.; Sargeant, A.J. In vivo measurement-based estimations of the human Achilles tendon moment arm. Eur. J. Appl. Physiol.
**2000**, 83, 363–369. [Google Scholar] [CrossRef] [PubMed] - Gans, C. Fiber architecture and muscle function. Exerc. Sport Sci. Rev.
**1982**, 10, 160–207. [Google Scholar] [CrossRef] [PubMed] - Hodgson, J.A.; Finni, T.; Lai, A.M.; Edgerton, V.R.; Sinha, S. Influence of structure on the tissue dynamics of the human soleus muscle observed in MRI studies during isometric contractions. J. Morphol.
**2006**, 267, 584–601. [Google Scholar] [CrossRef] [PubMed] - Kinugasa, R.; Taniguchi, K.; Yamamura, N.; Fujimiya, M.; Katayose, M.; Takagi, S.; Edgerton, V.R.; Sinha, S. A multi-modality approach towards elucidation of the mechanism for human Achilles tendon bending during passive ankle rotation. Sci. Rep.
**2018**, 8, 4319. [Google Scholar] [CrossRef] [PubMed] - Csapo, R.; Hodgson, J.; Kinugasa, R.; Edgerton, V.R.; Sinha, S. Ankle morphology amplifies calcaneus movement relative to triceps surae muscle shortening. J. Appl. Physiol.
**2013**, 115, 468–473. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ly, J.Q.; Bui-Mansfield, L.T. Anatomy of and abnormalities associated with Kager’s fat pad. Am. J. Roentgenol.
**2004**, 182, 147–154. [Google Scholar] [CrossRef] [PubMed] - Theobald, P.; Bydder, G.; Dent, C.; Nokes, L.; Pugh, N.; Benjamin, M. The functional anatomy of Kager’s fat pad in relation to retrocalcaneal problems and other hindfoot disorders. J. Anat.
**2006**, 208, 91–97. [Google Scholar] [CrossRef] [PubMed] - Ghazzawi, A.; Theobald, P.; Pugh, N.; Byrne, C.; Nokes, L. Quantifying the motion of Kager’s fat pad. J. Orthop. Res.
**2009**, 27, 1457–1460. [Google Scholar] [CrossRef] [PubMed] - Kitagawa, T.; Nakase, J.; Takata, Y.; Shimozaki, K.; Asai, K.; Tsuchiya, H. Histopathological study of the infrapatellar fat pad in the rat model of patellar tendinopathy: A basic study. Knee
**2019**, 26, 14–19. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kinoshita, H.; Francis, P.R.; Murase, T.; Kawai, S.; Ogawa, T. The mechanical properties of the heel pad in elderly adults. Eur. J. Appl. Physiol. Occup. Physiol.
**1996**, 73, 404–409. [Google Scholar] [CrossRef] [PubMed] - Hsu, T.C.; Wang, C.L.; Shau, Y.W.; Tang, F.T.; Li, K.L.; Chen, C.Y. Altered heel-pad mechanical properties in patients with Type 2 diabetes mellitus. Diabet. Med.
**2000**, 17, 854–859. [Google Scholar] [CrossRef] [PubMed] - Carrier, D.R.; Heglund, N.C.; Earls, K.D. Variable gearing during locomotion in the human musculoskeletal system. Science
**1994**, 265, 651–653. [Google Scholar] [CrossRef] [PubMed][Green Version] - Sinha, S.; Kinugasa, R. Imaging Studies of the Mechanical and Architectural Characteristics of the Human Achilles Tendon in Normal, Unloaded and Rehabilitating Conditions. In Achilles Tendon; Cretnik, A., Ed.; IntechOpen: London, UK, 2012; pp. 1–22. [Google Scholar]
- Hashizume, S.; Iwanuma, S.; Akagi, R.; Kanehisa, H.; Kawakami, Y.; Yanai, T. In vivo determination of the Achilles tendon moment arm in three-dimensions. J. Biomech.
**2012**, 45, 409–413. [Google Scholar] [CrossRef] [PubMed] - Shibanuma, N.; Sheehan, F.T.; Stanhope, S.J. Limb positioning is critical for defining patellofemoral alignment and femoral shape. Clin. Orthop. Relat. Res.
**2005**, 434, 198–206. [Google Scholar] [CrossRef] [PubMed] - Patel, N.N.; Labib, S.A. The Achilles Tendon in Healthy Subjects: An Anthropometric and Ultrasound Mapping Study. J. Foot Ankle Surg.
**2018**, 57, 285–288. [Google Scholar] [CrossRef] [PubMed] - Fukashiro, S.; Abe, T.; Shibayama, A.; Brechue, W.F. Comparison of viscoelastic characteristics in triceps surae between Black and White athletes. Acta Physiol. Scand.
**2002**, 175, 183–187. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Simplified ankle model with the presence of an inflection point. The inflection point prevents the Achilles tendon from moving dorsally. The arc indicates the trajectory of the calcaneus during ankle joint rotation. Radius M is the distance from the ankle center of rotation to the insertion of the Achilles tendon on the calcaneus (i.e., inferior calcaneal tuberosity, grey cross symbol). Distance ${d}_{dis}$ is the length of the segment from the inflection point to the insertion of the Achilles tendon on the calcaneus, whereas ${d}_{prox}$ is the length of the segment from the myotendinous junction to the inflection point. The Achilles tendon length ${l}_{AT}$ is defined as the sum of the distances from the myotendinous junction to the inflection point and from the inflection point to the calcaneus.

**Figure 2.**Representation of the physiological range of the inflection point within the Achilles tendon where the gain

_{AT}is calculated. The positions of the start (i.e., myotendinous junction) and end (i.e., inferior calcaneal tuberosity) points of the Achilles tendon, and ankle center of rotation are shown in a typical sagittal-plane MR image from one subject. The ankle angle was set at approximately –10° (ankle dorsiflexed position). The physiological range of the inflection point is defined as the area within the Achilles tendon and is represented as the area (gray color) of a quadrilateral comprising four vertices (points; indicated by white circle symbols).

**Figure 3.**Effects of inflection point locations and ankle positions on gain

_{AT}(

**top**), ratio of change in the Achilles tendon length to moment arm (

**middle**), and product of the ratio of change in the Achilles tendon length to moment arm and the x-coordinate of calcaneus position (

**bottom**), for 1 mm calcaneus displacement. We used a model where the position of the inflection point is varied, whereas the displacements of the myotendinous junction with respect to the calcaneus remains constant. Three-dimensional surface plots showing the change in the Achilles tendon length (Z axis) with respect to the location of the inflection point along the x (X axis) and y (Y axis) directions. This change is shown for two ankle positions: plantarflexed (

**left**) and dorsiflexed (

**right**) positions.

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**MDPI and ACS Style**

Kinugasa, R.; Yamamura, N.; Takagi, S.; Sinha, S.
Biomechanical Gain in Joint Excursion from the Curvature of the Achilles Tendon: Role of the Geometrical Arrangement of Inflection Point, Center of Rotation, and Calcaneus. *Diagnostics* **2021**, *11*, 2097.
https://doi.org/10.3390/diagnostics11112097

**AMA Style**

Kinugasa R, Yamamura N, Takagi S, Sinha S.
Biomechanical Gain in Joint Excursion from the Curvature of the Achilles Tendon: Role of the Geometrical Arrangement of Inflection Point, Center of Rotation, and Calcaneus. *Diagnostics*. 2021; 11(11):2097.
https://doi.org/10.3390/diagnostics11112097

**Chicago/Turabian Style**

Kinugasa, Ryuta, Naoto Yamamura, Shu Takagi, and Shantanu Sinha.
2021. "Biomechanical Gain in Joint Excursion from the Curvature of the Achilles Tendon: Role of the Geometrical Arrangement of Inflection Point, Center of Rotation, and Calcaneus" *Diagnostics* 11, no. 11: 2097.
https://doi.org/10.3390/diagnostics11112097