# A Planar Model of an Ankle Joint with Optimized Material Parameters and Hertzian Contact Pairs

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## Abstract

**:**

## 1. Introduction

#### 1.1. Modeling the Ankle

#### 1.2. Dimensionality of the Models

#### 1.3. The Aim of the Study and Its Novel Aspects

## 2. Materials and Methods

#### 2.1. The Model

- Two rigid bodies: the basis - tibia-fibula segment (TFS), the moving body - talus-calcaneus segment (TCS);
- Six nonlinear planar cables, representing ATT, TC, PTT, ATF, CF, and PTF;
- Two symmetrical sphere–sphere Hertzian contact pairs representing the cartilage between the tibia and the talus;
- To specify the location of the TFS with regard to the TCS, three variables were used;
- One angular coordinate θ, used to compute the rotation matrix
**R**from the TCS to TFS:$$\mathit{R}\left(\theta \right)=\left[\begin{array}{cc}\mathrm{cos}\theta & -\mathrm{sin}\theta \\ \mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right];$$ - Two linear coordinates, which formed the position vector between the frames
**p**:$$\mathit{p}={\left[\begin{array}{cc}{p}_{x}& {p}_{y}\end{array}\right]}^{T}.$$

#### 2.2. The Ligaments

**p**, the length l

_{i}of the cable at the location was obtained as follows:

**a**(

_{i}**b**)—the position vector of the i-th cable’s attachment to the TCS (TFS), i ∈ {ATF, ATT, TC, CF, PTT, PTF}.

_{i}**R**—the rotation matrix from the TCS to TFS, and

**p**—the position vector between the frames.

_{i}, B

_{i}—the stiffness coefficients of the ligament i, ε

_{i}—the strain of the ligament i:

_{slack,i}—the slack length of the ligament i (computed at the neutral location of the TCS).

_{i}:

#### 2.3. Optimizing the Material Parameters of The Ligaments

**F**—using Equation (8)—and the strain of its planar counterpart ε

_{3D}_{2D}—using Equation (7). The three-dimensional force

**F**was then projected onto the sagittal plane and the magnitude of this projection F

_{3D}_{3D/sag}was obtained. Finally, we assumed that the planar substitute under ε

_{2D}should generate the force of equal value to the one generated by the three-dimensional link when projected onto sagittal plane F

_{3D/sag}under all of the considered displacements. This allowed us to formulate the following set of residual equations:

_{i}, B

_{i}—the stiffness coefficients of the planar cable i representing ligament i, ε

_{2D/disp j}—the strain of the planar cable i while the segments are under displacement j, and F

_{3D/sag/disp j}—the force value generated by the three-dimensional ligament i when projected onto sagittal plane while the segments are under displacement j.

#### 2.4. The Cartilage

_{c,i}—the value of the contact force generated by the contact pair i, K

_{i}—the generalized stiffness coefficient for the contact pair i, and δ

_{i}—the relative penetration of the bodies in the contact pair i.

_{a,i}, r

_{b,i}—the radii of the spheres in the contact pair i (the radii were assumed to be of positive values for the male and the female sphere), ν

_{i}—Poisson’s ratio for the elements in the contact pair i, and E

_{i}—Young’s modulus for the elements in the contact pair i (both spheres in the contact pairs were assumed to be of the same material).

_{i}of the spheres was obtained as follows:

**O**,

_{a,i}**O**—the position vectors of sphere centers with regard to the TFS frame.

_{b,i}**O**had to be obtained through a reference frame transformation:

_{b,i}#### 2.5. Solving Elastostatic Problems

**R**, and two linear coordinates that form the position vector

**p**), in which the sums of the forces and the moments acting on the TCS are equal to 0. This condition was specified as the following equilibrium equations [38,39]:

**F**(

_{i}**M**)—the forces (moments) generated by the nonlinear cables representing the ligaments,

_{i}**F**(

_{c,i}**M**)—the forces (the moments) generated by the spheres in contact,

_{c,i}**F**(

_{ext}**M**)—the external force (moment) acting on the TCS, and n (m)—the number of the cables (contact pairs).

_{ext}**F**,

_{x}**F**and

_{y}**M**) was less than 1.0 × 10

^{−10}.

## 3. Results

#### 3.1. The Input Dataset

- Simulation #1:
**M**= −0.20:0.20 Nm in 51 steps;_{ext} - Simulation #2:
**M**= −5.00:5.00 Nm in 51 steps;_{ext} - Simulation #3:
**M**= −5.00:5.00 Nm in 51 steps the geometry of PTT was modified to assess the sensitivity of the model._{ext}

**M**corresponded to dorsiflexion. Conversely, the positively-valued moments caused plantarflexion.

_{ext}#### 3.2. Simulation #1

**M**were presented in Figure 4. In both cases, the displacement function was highly nonlinear. An initial low-stiffness region was followed by a significant increase in stiffness in the latter part of the simulation. The total angular displacement of the model, under the assumed loading conditions, was 30.65 deg–10.79 deg in dorsiflexion and 19.86 deg in plantarflexion.

_{ext}#### 3.3. Simulation #2

**M**was increased to 5.00 Nm. As seen in Figure 7, after 0.20 Nm, the angular stiffness of the model continued to increase. Again, higher stiffness was observed in dorsiflexion, which resulted in a significantly smaller range of motion in this phase of the simulation. The total displacement of the model, under the assumed loading conditions, was 59.24 deg–23.54 deg in dorsiflexion and 35.70 deg in plantarflexion.

_{ext}#### 3.4. Simulation #3—Elements of a Sensitivity Analysis

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The proposed planar model of the ankle joint in the sagittal plane and a schematic representation of its spherical contact pairs in the frontal plane. TFS, basis-tibia-fibula segment; TCS, moving body-talus-calcaneus segment; ATT, anterior tibiotalar ligament; TC, tibiocalcaneal ligament; PTT, posterior tibiotalar ligament; ATF, anterior talofibular ligament; CF, calcaneofibular ligament; PTF, posterior talofibular ligament.

**Figure 2.**(

**a**) The ligament system of the ankle as seen in the frontal plane in two configurations: initial and with the segments displaced by 4 mm along the y-axis; (

**b**) PTF and its projection onto the sagittal plane in two configurations: initial and with the segments displaced by 4 mm along the y-axis.

**Figure 3.**The graph of one of the considered displacements versus the value of the force generated by the following: a three-dimensional PTT ligament with unaltered stiffness parameters from the work of [37] in the sagittal plane (green), its planar counterpart with unaltered stiffness parameters from the work of [37] (red), and a planar counterpart with fine-tuned material parameters using the proposed custom approach (yellow).

**Figure 5.**The strains within the ligaments as seen in the sagittal plane versus the external moment

**M**in simulation #1 (negative [positive] strain corresponded to an inactive [active] ligament).

_{ext}**Figure 6.**The values of the forces generated by the ligaments versus the external moment

**M**in simulation #1, where "con." is the magnitude of the contact force generated by the two contact pairs.

_{ext}**Figure 8.**The strains within the ligaments as seen in the sagittal plane versus the external moment

**M**in simulation #2 (negative [positive] strain corresponded to an inactive [active] ligament).

_{ext}**Figure 9.**The values of the forces generated by the ligaments versus the external moment

**M**in simulation #2, where "con." is the magnitude of the contact force generated by the two contact pairs.

_{ext}**Figure 10.**The model (as seen in the prepared software) (

**a**) used in simulations #1 and #2; (

**b**) with a modified PTT, where the grey (black) platform corresponds to TCS (TFS) and the colors of the links correspond to the ones used in previous figures.

**Figure 12.**The values of the forces generated by the ligaments versus the external moment

**M**in simulation #3, where "con." is the magnitude of the contact force generated by the two contact pairs.

_{ext}© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Borucka, A.; Ciszkiewicz, A.
A Planar Model of an Ankle Joint with Optimized Material Parameters and Hertzian Contact Pairs. *Materials* **2019**, *12*, 2621.
https://doi.org/10.3390/ma12162621

**AMA Style**

Borucka A, Ciszkiewicz A.
A Planar Model of an Ankle Joint with Optimized Material Parameters and Hertzian Contact Pairs. *Materials*. 2019; 12(16):2621.
https://doi.org/10.3390/ma12162621

**Chicago/Turabian Style**

Borucka, Aleksandra, and Adam Ciszkiewicz.
2019. "A Planar Model of an Ankle Joint with Optimized Material Parameters and Hertzian Contact Pairs" *Materials* 12, no. 16: 2621.
https://doi.org/10.3390/ma12162621