Pressure Force in the Upper Ankle Joint

Abstract

Background: This paper concerns the study of forces acting on the upper ankle joint of a human in static and quasi-dynamic positions. This paper aimed to determine the pressure forces on the axis of the upper ankle joint in the position of the body tilting forward and backward, as well as in a neutral position. Methods: A model with designated centres of gravity (including and excluding the weight of the platform imitating the foot) and the point of gravity imitating the proximal insertion of the triceps surae and tibialis anterior muscles was developed for this study. The forces and the weight of the tilted object were measured using dynamometers. A method for determining the arms of gravitational forces and the angle of inclination of an object is presented. The function describing the distribution of gravitational loading along its tilting part was described. Next, all measurements and calculations were referred to the human body. Results: Measurements of muscle force, body gravity, the arms of these forces, and the angles of the object’s inclination on the axis of rotation are presented. A methodology for determining the pressure force on the human upper ankle joint axis is presented. The distribution of the value of the pressure force and its components from the maximal forward, through the vertical body position, up to the maximal backward position of the body tilt, is provided. Conclusions: The ankle joint pressure force is the vector sum of the force of gravity and the force of the muscle counteracting the body tilt. This force is the smallest in the vertical body position and increases with the body tilt. It reaches 5.23 times the weight of the tilting part of the body when the body is tilted to its maximum forward position, and 3.57 times the weight when the body tilts backward. Regardless of the direction of the body tilt, the joint pressure vector always runs through the axis of the upper ankle joint.

1. Introduction

In the statics and dynamics of the human body, the analysis of forces acting on the upper ankle joint has long been a focus of attention of biomechanical researchers for medicine and sports [1,2,3,4,5]. In this context, muscular force, gravity, and pressure force (described by intra-articular pressure) are mentioned, but without causal mechanisms between them [2,6,7,8,9]. Currently, the professional literature contains many reports on the distribution of pressure in the upper ankle joint in various foot positions [7,8,10,11] and pathological states [12,13]. However, there is a lack of biomechanical load models of the observed phenomena [14,15,16,17]. This topic is addressed in this study, indicating that in the analysis of these phenomena, it is important to notice that the body is tilted in the upper ankle joint and is supported on the platform, which is the foot [18].

The key to describing the biomechanics of a leaning body is to understand the anatomy of the joint and the “play” of muscular and gravitational torques in the living human body, as well as the fact that the human foot, adhering to the ground, with a hinge connection to the rest of the body, constitutes a support platform. In this context, the foot is like an element of the ground [18,19,20] and should not be included in the weight of the rest of the body leaning on the upper ankle joint. Meanwhile, the weight of the foot based on the ground is included in the weight of the leaning body [21,22,23], which is methodologically inaccurate. In addition, joint pressure force is commonly tested, excluding muscle work, because cadaveric testing does not account for active muscular contraction [7,8].

Currently, pressure force in the joint tests is determined by assessing intra-articular pressure, and, on this basis, conclusions are drawn about the pressure force and the distribution of the contact area in the tested joint [7,8,10,11]. The pressure in the upper ankle joint increases from the heel strike and reaches a maximum during toe-off. After 20% of the support phase, the pressure reaches the first plateau level, and the pressure peaks around 70% of the support phase [11]. As the weight-bearing increases, the pressure does not increase in proportion to the pressure force because the joint contact area increases [8]. This is because the upper ankle joint has a relatively high weight-bearing surface of 11 to 13 cm², which causes lower stresses in this joint compared to, for example, the knee or hip joint [10]. The largest contact area occurs in the dorsiflexion of the foot. In plantar flexion, the contact area is smaller, meaning the pressure force per unit area is greater than in dorsiflexion and neutral flexion (higher pressure is observed) [8]. With dorsiflexion, both medial and lateral ankle–talar forces increase with a tendency to increase the contact area, mainly of the anterolateral articular surface. Increased ankle joint stability in dorsiflexion allows it to withstand pressure forces of up to 450% of body weight [10,24]. This is possible because the upper ankle joint is exceptionally stable, resulting from the talus bone structure, which has the shape of a truncated cone, 4.2 mm wider in the front than in the back part. This anatomy ensures that during dorsiflexion, the anterior portion of the talus stretches the elastic tibiofibular ligament, causing the talus to be compressed between the tibia and fibula. Thanks to this, the upper ankle joint becomes “closely packed” and therefore maximally stabilised under dorsiflexion [22]. The main stresses in the ankle joint are located in the anterolateral part [11] and occur when the foot is positioned dorsally and partially neutral. The transition towards plantar flexion increases the contact area of the tibiotalar surface in the posteromedial zone [7]. The anterolateral contact area is larger than the posteromedial contact area; hence, the ankle joint is less stable and more susceptible to overload in plantar flexion (the loose-packed position), and therefore, the risk of ankle sprain is greater.

It has been experimentally confirmed that the joint pressure force depends not only on the weight but also on the activity of the external muscles of the foot. These muscles influence the contact of articular surfaces and the size and distribution of load in the upper ankle joint. The production of muscular force (extrinsic muscles: peroneals, tibialis anterior, tibialis posterior, and triceps surae) can potentially change the pressure distribution in the upper ankle joint (modify joint load, influencing the pressure force value and location) [25]. This paper presents the results of model tests demonstrating the essence of the joint pressure force, which is the result of gravity and muscle force. A methodology for determining and calculating the upper ankle joint pressure force depending on the angle of the centre of gravity inclination in the sagittal plane is presented. The described results provide the basis for developing tools for objective assessment of the biomechanics of the foot and upper ankle joint in humans. It can help assess body posture (statics) as well as gait analysis (dynamics) and can be used to precisely plan and carry out kinesiotherapy procedures, as well as sports training regimens.

2. Materials and Methods

2.1. The Main Biomechanical Parameters Related to Tilting the Centre of Gravity at the Upper Ankle Joint

The biomechanical system of the human feet is affected by two centres of gravity, CoG₁ and CoG₂. CoG₂ is a point that concentrates the weight of only the part that is tilted on the axis of rotation of the upper ankle joint. The weight of the support platform is an element of the ground and is not part of the weight of the tilting part of the object. The location of CoG₂ and the value of the body weight at this point, W_CoG2, are necessary for calculating the muscle and gravitational moments that balance each other and that exert the pressure on the upper ankle joint. CoG₁ (overall centre of gravity) is the point where the weight of the entire object is concentrated, including the weight of the support platform (W_CoG1). CoG₁ is necessary to determine the limit of the maximum forward and backward tilt of the body while maintaining balance.

The value of the weight at the CoG₁ point affects the direct pressure of the foot on the ground. The body remains in balance until the maximum tilt of CoG₁ occurs when the projection of—CoG₁ falls on the front (A-point) (Figure 1A,C) or rear (B-point) (Figure 1B,D) support point. At the maximum forward inclination of the object, the CoG₂ projection falls in front of the anterior fulcrum (A-point) and behind the posterior fulcrum (B-point) when the object is in the maximum backward inclination position. This difference is significant because the further forward projection of the CoG₂ (in comparison to CoG₁) has a larger gravitational arm, and therefore a larger gravitational moment forcing the generation of a correspondingly larger muscle moment (in fact, the rotational capacity of the muscle is larger because the arm of the muscle force is anatomically constant, r_TRI and r_TA = constant) (r_p > r_met). The analysis of the tilted body was based on the principle of conservation of gravitational moments [18], which states that in a tilted body, the moments of gravity for all points in the body at a given tilt angle are the same.

2.2. Models of a Tilting Object on the Axis of Rotation

2.2.1. Flat Bar Model

The flat bar model of the triceps surae muscle (TRI) action, balancing the moment of gravity, was used. It consisted of the following:

-

A metal flat bar (l₂ = 2 m, width: 0.03 m) with a total weight of 2.972 kG (2.648 kG without support platform), the height of which, together with the support platform, was l₁ = 2.06 m.

-

The centre of gravity for the leaning part of the model (CoG₂) (without the support platform) is located at a height of 1.1 m (h₂) from the axis of rotation, and 1.16 m (h_CoG2) from the ground (h₂ + 0.06 m (height of the platform with a hinge)). On the other hand, the overall centre of gravity for the entire model (CoG₁), together with the support platform and the hinge, is 0.01 m lower, i.e., at a height of 1.15 m (h_CoG1) measured from the ground, and measured from the axis of rotation at a height of 1.09 m (h₁). The distance from CoG₁ to CoG₂ for this model was 0.01 m (c).

-

The point imitating the place of the insertion of the Achilles tendon (TRI-point) and tibialis anterior muscle (TA-point) is located halfway between CoG₂ and the axis of rotation.

-

The part imitating the head (a threaded rod with nuts added to the flat bar).

-

A string indicating the line of action of gravity (vertical).

-

A wooden support platform imitating a human foot.

-

A hinge imitating the upper ankle joint connecting the flat bar with the support platform (Ra).

-

An electronic dynamometer (Ruhhy^® 0.005 kg–40 kg) mounted on a cable imitating the triceps surae muscle (Figure 2A), measuring its force (F_TRI). One end of the cable was attached to the flat bar at the TRI-point of gravity, and the other end was attached to the support platform (on “calcaneal tuberosity in the foot”). The steel cable was attached in such a way that the shortest distance from it to the hinge rotation axis was 0.04 m (which imitates the arm of the force of the triceps surae muscle). The Ruhhy^® electronic dynamometer (HDWR, Środa Wielkopolska, Poland) was calibrated using certified standard weights across its measurement range. The device was first zeroed, then incremental loads were applied to verify and adjust readings. Accuracy was confirmed with independent reference weights.

2.2.2. A Model of a Leaning Human Body on the Axis of the Upper Ankle Joint

The calculations were made for a human with a total body weight of 94.2 kg, a height of 1.82 m, and a distance from the axis of the upper ankle joint to the top of the head of 1.73 m. The centre of gravity of the tilting part of the human body (CoG₂) was calculated based on the principle of unilateral leverage [26]. The CoG₁ of the human body was located at a height of 1.064 m (h_CoG1) from the plantar side of the feet, and at a height of 0.974 m (h₁) from the axis of rotation. To calculate the CoG₂ of the leaning body, the total weight of the human was reduced by the weight of the feet [27], which was finally 91.6 kg. The CoG₂ of the leaning part of the human body was located at a height of 1.077 m (h_CoG2) and a height of 0.987 m (h₂) from the axis of rotation.

Calculations regarding the pressure on the upper ankle joint of a human were calculated for the intersection point of two axes of the upper ankle joint, i.e., the plantarflexion ankle axis and the dorsiflexion ankle axis. Since these two axes are slightly deviated from each other, the concept of one resultant axis of the upper ankle joint was used to simplify the biomechanical modelling.

2.3. Determination of Triceps Surae (TRI) and Tibialis Anterior (TA) Muscle Forces Depending on the Forward or Backward Body Tilt

The forward tilt of the CoG₂ (object without the weight of the support platform) is balanced by the imitation of the triceps surae muscle (TRI) in the model and the triceps surae muscle in humans. With the known anatomical arm of TRI force (r_TRI), the known weight of the tilted object (W_CoG2), and the distance of the projection of the centre of gravity of CoG₂ from the axis of rotation (r_CoG2), the value of the F_TRI force (1) can be determined (Figure 1A,C).

$$r_{CoG2}\ast W_{CoG2}=r_{TRI}\ast F_{TRI}$$

$$F_{TRI}={\displaystyle \frac{r_{CoG2}\ast W_{CoG2} }{r_{TRI}}}$$

(1)

A similar situation occurs when an object is tilted backward, with the difference being that the counterbalancing muscle is the tibialis anterior (TA) (2) (Figure 1B,D).

$$r_{CoG2}\ast W_{CoG2}=r_{TA}\ast F_{TA}$$

$$F_{TA}={\displaystyle \frac{r_{CoG2}\ast W_{CoG2} }{r_{TA}}}$$

(2)

2.4. A Method for Calculating the Gravitational Arms and Weight at a Given Point on a Leaning Part of an Object

Measurements of the model’s gravity arms were performed three times at each position, from neutral to maximum anterior and posterior deflection. At vertical alignment, the gravitational arm is zero; any deflection at angle β generates a gravitational moment. Knowing the value of the arm of gravity and the height of the point of gravity, the inclination angle of the object can be calculated (Figure 3A).

Knowing the angle of the tilted point of gravity, the arm of gravity for CoG₂ (3), and the TRI-point location in the leaning part of the human body, then r_{W_TRI} can be calculated (4) (Figure 3A).

$$r_{CoG2}=0.13 \mathrm{m}, h_2=0.987 \mathrm{m}$$

(3)

$$\cos (\alpha )={\displaystyle \frac{r_{CoG2}}{h_2}}={\displaystyle \frac{0.13 \mathrm{m}}{0.987 \mathrm{m}}}=0.1317$$

$$\alpha =\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}(0.1317)$$

$$\alpha =82.43° \left(1.439 \mathrm{r}\mathrm{a}\mathrm{d}\right), \beta =(90°-\alpha )=7.57° (0.13212 \mathrm{r}\mathrm{a}\mathrm{d})$$

$$r_{W\_TRI}= \cos (\alpha ) \ast {{\displaystyle \frac{1}{2}}h}_2=0.1317\ast 0.493 \mathrm{m}=0.0649 \mathrm{m}$$

(4)

Having the values of the r_CoG2 and r_{W_TRI} gravity arms and the value of the weight of the tilted body W_CoG2, one can determine the value of the weight at the TRI-point based on the principle of conservation of torques. The weight at the point of proximal insertion of the triceps surae muscle, i.e., TRI-point of gravity, is constant regardless of the degree of body tilt [18] (5) (Figure 3B).

$$W_{TRI}\ast r_{W\_TRI}=W_{CoG2}\ast r_{CoG1}$$

$$W_{TRI}={\displaystyle \frac{W_{CoG2}\ast r_{CoG2}}{r_{W\_TRI}}}={\displaystyle \frac{91.6 \ast 0.13}{\mathrm{c}\mathrm{o}\mathrm{s}(\alpha )\ast {\frac{1}{2}h}_2}}={\displaystyle \frac{11.9}{0.1317\ast 0.493}}==183.2 \mathrm{k}\mathrm{G} (1796.99 \mathrm{N})$$

(5)

where

W_TRI—body weight at the TRI-point of gravity;

W_CoG2—centre of gravity of the leaning part of the body;

r_CoG2—arm of the body’s gravity force for CoG₂;

r_{W_TRI}—arm of the body’s gravity force for the TRI-point of gravity.

The value of the weight on the object does not depend on the angle of inclination (at β > 0) [18] but depends on the distance of a given point of the object from the axis of rotation on which the object tilts. This relation is described by the homographic function (6), whose domain and counter-domain are the set R \{0}. In the case of a human used as a model, a > 0 and applies to the interval (0, +infinity) (Figure 3B).

$$y={\displaystyle \frac{a}{x}}$$

(6)

where

y—distance from the axis of rotation;

x—weight at a given point of the object;

a—directional coefficient.

Knowing the position of the centre of gravity of the leaning body and its position on the object, the directional coefficient (a) can be calculated, which enables obtaining a formula that allows calculating the gravity at each point of the object and the exact distance of a given point on the object from the axis of rotation. For the human model, the function formula is as follows (7) (Figure 3B):

$$y={\displaystyle \frac{90.4}{x}}$$

(7)

On this basis, the weight at the TRI-point (W_TRI) was calculated (8) as follows:

$$x={\displaystyle \frac{90.4}{\frac{h_2}{2}}}= {\displaystyle \frac{90.4}{0.4935}}=183.18 \approx 183.2 \mathrm{k}\mathrm{G} W_{TRI}=183.2 \mathrm{k}\mathrm{G} (1796.8 \mathrm{N})$$

(8)

The principle of calculating the weight in the TRI-point (W_TRI) is the same for the TA-Point (W_TA) because the proximal attachment points of the triceps, calf, and tibialis anterior are at the same height from the axis of rotation of the upper ankle joint. Hence, the weight values in the TA-point are the same as in the TRI-point.

3. Results

3.1. Joint Pressure Force–Flat Bar Model Tests

Every standing position generates a pressure force on the axis of the upper ankle joint. This force increases as the body tilts from the vertical. This is the result of an increasing moment of gravity, which is counteracted by a growing moment of muscular force. In practice, (F_J) is the vector sum of the gravitational force acting on a given point of the object and the counteracting “muscular” force (Fm) applied to this point. To demonstrate the above relationship, the arms of gravity were determined for the CoG₂ (r_CoG2) and TRI-point (r_{W_TRI}). Then, the value of the triceps calf muscle force (F_TRI) and the gravity force at the TRI-point were measured, which are a necessary condition for calculating the value of the pressure force (F_J) on the joint, i.e., the vector sum of W_TRI and F_TRI (Figure 4A–E). The experiments were performed in a neutral position, as well as with the object tilted forward and backward (Figure 5).

3.1.1. Pressure Force on the Axis of Rotation in the Neutral Position of the Object

In a vertical position (neutral), when the CoG₂ falls on the axis of rotation, the cable, imitating the action of the TRI muscle, does not generate any force because the gravity moment is equal to zero. Therefore, the pressing force (F_J) is equal to the weight of the flat bar with a “head” (9). The weight of the tilting part of the object (without including the weight of the support platform) concentrated in the centre of gravity (CoG₂) falls on the axis of rotation (Figure 5—neutral position).

$$F_J=W_{CoG2}=2.648 \mathrm{k}\mathrm{G} (25.97 \mathrm{N})$$

(9)

3.1.2. The Pressure Force on the Axis of Rotation When the Object Is Tilted Forward

The values of W_CoG2, F_TRI, r_{W_TRI}, and r_CoG2 were measured (Figure 4A–D). The relationship between the forces acting on the upper ankle joint is shown in Figure 4E. Knowing that the torques at each point on the flat bar tilted at a given angle are the same [18], and the W_TRI is twice as much as W_CoG2, the pressing force (F_J) was calculated.

Sample calculations are presented below. The flat bar tilted with the centre of gravity (CoG₂) projected 0.10097 m forward (Figure 4C) from the axis of rotation occurring at a beta angle of 4.32° (10) (Figure 3A). This slight angle of tilting causes the pressure force to reach a value of F_J = 11.976 kg (14) and is 4.52 times (11.976/2.648) the gravity force at CoG₂ (W_CoG2). For r_CoG1 = 0.10097 m, h₁ =1.09 m, c = 0.01 m (distance from CoG₁ to CoG₂ for the flat bar model), and r_TRI = constant = 0.04 m, the calculation is as follows:

$$\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathsf{\alpha }\right)={\displaystyle \frac{r_{CoG1}}{h_1}}={\displaystyle \frac{0.10097}{1.09}}= 0.09263, \alpha =arccos(0.09263)$$

α = 85.68°,

β = 90 − 85.68 = 4.32°

(10)

By calculating the value of angle α, the arm of the gravitational force applied to the CoG₂ point (r_CoG2) (11) (Figure 1C,D) was calculated, followed by the force of gravity at the TRI-point (W_TRI) (12) and the force of “the triceps calf muscle” (F_TRI) in the flat bar model (13).

$$f=\cos \left(\alpha \right)\ast c=9.2633\times {10}^{-4}$$

$$r_{CoG2}=\left(r_{CoG1}+f\right)=0.10097+0.00119=0.1019\hspace{0.5em}\mathrm{m}$$

(11)

$$W_{CoG2}\ast r_{CoG2}=W_{TRI}\ast r_{W\_TRI}$$

$$W_{TRI}={\displaystyle \frac{W_{CoG2}\ast r_{CoG2}}{r_{W_{TRI}}}}={\displaystyle \frac{2.648\ast 0.10097}{0.0504}}={\displaystyle \frac{0.267}{0.0504}}=5.297 \mathrm{k}\mathrm{G} (51.95 \mathrm{N})$$

(12)

$$W_{CoG2}\ast r_{CoG2}=F_{TRI}\ast r_{TRI}$$

$$F_{TRI}={\displaystyle \frac{W_{CoG2}\ast r_{CoG2}}{r_{TRI}}}={\displaystyle \frac{2.648\ast 0.10097}{0.04}}={\displaystyle \frac{0.26736}{0.04}}=6.68 \mathrm{k}\mathrm{G} (65.49 \mathrm{N})$$

(13)

Given the value of gravity at the TRI-point (W_TRI) (12) and the value of the triceps calf muscle force (F_TRI) (13), the vector sum of these forces, i.e., the pressure force F_J (14), was calculated.

$$\stackrel{\rightharpoonup }{F_J}=\stackrel{\rightharpoonup }{F_{TRI}}+ \stackrel{\rightharpoonup }{W_{TRI}}=11.976 \mathrm{k}\mathrm{G} (117.43 \mathrm{N})$$

(14)

The calculations of the “TRI muscle force” (F_TRI) performed based on the principle of the equality of torques match the force measurement determined using a dynamometer and are F_TRI = 6.68 kG (13) and 6.675 kG, respectively (Figure 4A). The values of the gravity force W_TRI calculated from the principle of the equality of torques or from the principle that twice the approach to the axis of rotation doubles the weight at that point are numerically equivalent.

At the maximum tilting of the flat bar, while maintaining balance (CoG₁ projects to A-point in the support platform, while CoG₂ projects to P-point), the angle is 6.85° (15) and the CoG₁ projection falls at a distance of 0.13 m from the rotation axis. r_CoG2 is equal to 0.1312 m (16), W_TRI is equal to 5.297 kG, which is the same regardless of the angle of inclination (12), and F_TRI is equal to 8.685 kG (17). Finally, F_J is 13.99 kG (18). The ratio F_J to W_CoG2 increases to 5.28.

For r_CoG1 = 0.13 m, h₁ =1.09 m, c = 0.01 m, and r_TRI = constants = 0.04 m, the calculations are as follows:

$$\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathsf{\alpha }\right)={\displaystyle \frac{r_{CoG1}}{h_1}}={\displaystyle \frac{0.13}{1.09}}= 0.1192 \alpha =arccos(0.1192)$$

α = 83.5°,

β = 90 − 83.15 = 6.85°

(15)

$$f=\cos \left(\alpha \right)\ast c={\displaystyle \frac{r_{CoG1}}{h_1}}\ast c=0.00119$$

$$r_{CoG2}=\left(r_{CoG1}+f\right)=0.13+0.00119=0.1312 \mathrm{m}$$

(16)

$$F_{TRI}={\displaystyle \frac{W_{CoG2}\ast r_{CoG2}}{r_{TRI}}}={\displaystyle \frac{2.648\ast 0.1312}{0.04}}={\displaystyle \frac{0.34424}{0.04}}=8.685 \mathrm{k}\mathrm{G} (85.18 \mathrm{N})$$

(17)

$$\stackrel{\rightharpoonup }{F_J}=\stackrel{\rightharpoonup }{F_{TRI}}+ \stackrel{\rightharpoonup }{W_{TRI}}=13.99 \mathrm{k}\mathrm{G} (137.21 \mathrm{N})$$

(18)

3.1.3. The Pressure Force on the Axis When Tilting the Object Backward

When the flat bar is tilted backward to the maximum, the tilt angle is 2.4° (19), the gravitational force arm of the tilted flat bar r_CoG2 is 0.0454 (20), the force of the tibialis anterior muscle (F_TA) in such a system is approximately 4 kG (21), and the force of gravity at the TA-Point has the same value as the TRI-point, i.e., 5.297 kG (22).

For r_CoG1 = 0.045 m, h₁ =1.09 m, c = 0.01 m, and r_TA = constants = 0.03 m, the calculations are as follows:

$$\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathsf{\alpha }\right)={\displaystyle \frac{r_{CoG1}}{h_1}}={\displaystyle \frac{0.045}{1.09}}= 0.04128 \alpha =arccos(0.04128)$$

α = 87.6°,

β = 90 − α = 2.4°

(19)

$$f=\cos \left(\alpha \right)\ast c={\displaystyle \frac{r_{CoG1}}{h_1}}\ast c=4.128\times {10}^{-4}=0.0004123$$

$$r_{CoG2}=\left(r_{CoG1}+f\right)=0.045+4.128\times {10}^{-4}=0.045413 \mathrm{m}$$

(20)

$$F_{TA}={\displaystyle \frac{W_{CoG2}\ast r_{CoG2}}{r_{TA}}}={\displaystyle \frac{2.648\ast 0.045413}{0.03}}={\displaystyle \frac{0.12025}{0.04}}=4.008 \mathrm{k}\mathrm{G} (39.29 \mathrm{N})$$

(21)

$$W_{TA}=W_{TRI}=2\ast W_{CoG2}=5.297 \mathrm{k}\mathrm{G} (51.95 \mathrm{N})$$

(22)

Based on the above calculations, the pressure force on the axis of rotation was calculated when the flat bar was tilted to the rearmost. This F_J value was 9.3041 kG (23) and was 3.51 times greater than the weight of the tilted part of the model, i.e., the flat bar with the “head” part without a support platform.

$$\stackrel{\rightharpoonup }{F_J}=\stackrel{\rightharpoonup }{F_{TRI}}+ \stackrel{\rightharpoonup }{W_{TRI}}=9.3041 \mathrm{k}\mathrm{g} (91.28 \mathrm{N})$$

(23)

3.2. The Pressure Force on the Axis of the Human Upper Ankle Joint

In the unloaded foot, the sum of gravity and muscle force due to the low value of resting muscle tension can be omitted; therefore, it can be said that there is no observed joint pressure force (Figure 6A). This situation was not considered in further study. In this case, the joint pressure is the effect of the resting tension of the antagonist muscles, which creates the joint adhesive force.

There are three main positions in a weight-bearing foot: neutral, forward, and backward. In the neutral position, no muscle activity is observed, only the force of gravity (W_CoG2), which in this case is equal to the force of pressure (F_J) (Figure 5). When the body is tilted forward or backward, a pressure force occurs on the upper ankle joint, which is the vector sum of the force of gravity and the muscle force (Figure 6C,D).

Assuming that the attachments of the TRI and TA muscles are located halfway between the axis of the upper ankle joint and the centre of gravity of the tilted body (CoG₂), it was assumed that the weight at these points is twice the weight of the W_CoG2 [18]. The joint pressure force (F_J) affects the flattening of the foot arch because, regardless of the tilt (front–back or neutral position), it always passes through the axis of the upper ankle joint. Flattening occurs primarily when the body tilts forward, to a lesser extent when tilted backward, and to the least extent in the neutral (straight) position when the CoG₂ projection falls on the axis of the upper ankle joint (Figure 6B–D).

3.2.1. Pressure Force on the Upper Ankle Joint in a Neutral Position

In a human’s neutral, upright position, the projection of the body’s centre of gravity, both CoG₁ and CoG₂, passes through the axis of the upper and lower ankle joints [21] at the so-called Basic Balance Point (BBP). In such a situation, the arm of gravity (r_CoG2) is zero. Therefore, the moment of gravity is zero (Figure 6B). In the neutral position, it is equal only to the weight of the body without a support platform, which is the feet (W_CoG2). For a person weighing 94.2 kG, the body weight without the weight of the feet is 91.6 kG, so the joint pressure force is equal to 91.6 kG (24). This is the case when standing on one leg. When standing on both legs, the value is half the body weight without the weight of the feet, i.e., 45.8 kG.

$$F_J=F_m+ W_{CoG2} only in neutral { W}_{TRI}=W_{TA}=W_{CoG2}$$

$$F_J=0+ 91.6 \mathrm{k}\mathrm{G}=91.6 \mathrm{k}\mathrm{G} (898.6 \mathrm{N})$$

(24)

Based on the calculations, the neutral position generates the least amount of pressure on the upper ankle joint (24) (Figure 7).

3.2.2. Joint Pressure Force at Maximum Forward Body Tilting

During the maximum forward tilting of the body while maintaining balance, the projection of the body’s overall centre of gravity (CoG₁) falls on the first head of the metatarsal bone (A-point) (exceeding this point results in loss of balance). On the other hand, the projection of the centre of gravity of the leaning part of the body (CoG₂) falls slightly further forward, i.e., at the height of the proximal phalanx of the big toe (P-point) (Figure 1C and Figure 6C). The forward-leaning of the body is balanced by the moment of the triceps calf muscle. For a leaning body weight of 91.6 kg (W_CoG2), the value of the weight at the TRI-point of gravity is 183.2 kg (W_TRI) (5) (8). The gravitational moment at the TRI-point is balanced by the triceps calf muscle moment (M_TRI) (25) (Figure 3A,D).

$$M_{CoG1}=M_{CoG2}=M_{W\_TRI}=M_{TRI}$$

(25)

$$W_{CoG1}\ast r_{CoG1}=W_{CoG2}\ast r_{CoG2}=W_{TRI}\ast r_{W\_TRI}=F_{TRI}\ast r_{TRI}$$

Knowing the weight of the leaning body (W_CoG2) and the arm of gravity (r_CoG2), as well as the constant, anatomically determined arm of the triceps calf muscle (r_TRI), the value of the triceps calf muscle force (F_TRI) at the maximum body tilt while maintaining balance (CoG₁) (tilted to A-point, CoG₂ tilted to P-point) can be calculated. For a human with a leaning body weight of 91.6 kG, a distance from the axis of rotation of the upper ankle joint to P-point of 0.143 m (r_CoG2) (Figure 1C and Figure 6C), and “c” (the difference in distance between CoG₂ and CoG₁) equal to 0.013 m [26], the force of the triceps calf muscle is 301.67 kg (26).

$$r_{TRI}=constans=0.04 \mathrm{m}, h_1=0.974 \mathrm{m},r_{CoG1}=0.13 \mathrm{m}$$

$$f=\cos \left(\alpha \right)\ast c={\displaystyle \frac{r_{CoG1}}{h_1}}\ast c=0.1335\ast 0.013=0.00173 \mathrm{m}$$

$$r_{CoG2}=\left(r_{CoG1}+f\right)=0.13+0.00173=0.1317 \mathrm{m}$$

$$W_{CoG2}\ast r_{CoG2}=F_{TRI}\ast r_{TRI}$$

$$91.6 \mathrm{k}\mathrm{G}\ast 0.1317 \mathrm{m}=F_{TRI}\ast 0.04 \mathrm{m}$$

$${ F}_{TRI}=301.67 \mathrm{k}\mathrm{G} (2959 \mathrm{N})$$

(26)

In the position of maximum forward body leaning (r_CoG1 = 0.13 m), the joint pressure force (F_J), which is the vector sum of the triceps calf force (F_TRI = 301.67 kg) (25), and the body weight at the proximal insertion point of the triceps calf muscle (W_TRI = 183.2 kg) (5) is 484.472 kg (27), which is 5.28 times greater than the force of gravity of the body leaning forward (W_CoG2) (Figure 6C).

$$\stackrel{\rightharpoonup }{F_J}=\stackrel{\rightharpoonup }{F_{TRI}}+ \stackrel{\rightharpoonup }{W_{TRI}}=484.792 \mathrm{k}\mathrm{G} (4754.35 \mathrm{N})$$

(27)

3.2.3. Joint Pressure Force at Maximum Backward Body Tilting

The weight at the proximal attachment point of the tibialis anterior muscle is 183.2 kg (W_TA) regardless of the degree of body tilt backward (Figure 6D). According to the principle of conservation of gravitational moments in a tilted body, a doubling of the weight at the point (W_TA) is associated with a double reduction in the length of the gravity arm (r_{W_TA}). So the gravitational moments are constant and are balanced by the muscle moment coming from the tibialis anterior muscle (M_TA) (28).

$$M_{CoG1}=M_{CoG2}=M_{W\_TA}=M_{TA}$$

(28)

$$W_{CoG1}\ast r_{CoG1}=W_{CoG2}\ast r_{CoG2}=W_{TA}\ast r_{W\_TA}=F_{TA}\ast r_{TA}$$

In the position of maximum body tilt backward (CoG₁ tilted to B-point, CoG₂ tilted to P-point) with the gravity arm equal to 0.045 m (r_CoG1—distance from the axis of rotation to point B), the joint pressure force (F_J), which is the vector sum of the tibialis anterior muscle force, is equal to 139.23 kG (F_TA) (29) and the weight of the tilted body at the TA-point (W_TA) is equal to 183.2 kG. The weight in the TA-point has approximately the same value as the weight in the TRI-point because the insert of this TA muscle is at the same distance from the axis of rotation of the upper ankle joint as the proximal insert of the TRI muscle (W_TRI = W_TA = 183.2 kG (5) and (8)). Therefore, the vector sum of the weight at the TA-point (W_TA), the TA muscle force (F_TA), and the joint pressure force (F_J) equals 322.431 kG (30), and is therefore 3.52 times greater than the force of gravity of the body tilted backward (W_CoG2) (Figure 1D and Figure 6D).

$$r_{TA}=constans=0.03 \mathrm{m},h_1=0.974 \mathrm{m},r_{CoG1}=0.045 \mathrm{m}$$

$$f=\cos \left(\alpha \right)\ast c={\displaystyle \frac{r_{CoG1}}{h_1}}\ast c$$

$$r_{CoG2}=\left(r_{CoG1}+f\right)=(r_{CoG1}+({\displaystyle \frac{r_{CoG1}}{h_1}}\ast c))$$

$$W_{CoG2}\ast r_{CoG2}=F_{TA}\ast r_{TA}$$

$$91.6 \mathrm{kG}\ast \left(0.045+\left(0.0462\ast 0.013\right)\right)\mathrm{m}=F_{TA}\ast 0.03 \mathrm{m}$$

$${ F}_{TA}=139.23 \mathrm{k}\mathrm{G} (1365.82 \mathrm{N})$$

(29)

$$\stackrel{\rightharpoonup }{F_J}=\stackrel{\rightharpoonup }{F_{TA}}+ \stackrel{\rightharpoonup }{W_{TA}}=322.431 \mathrm{k}\mathrm{G} (3161.64 \mathrm{N})$$

(30)

3.2.4. Pressure Force on the Axis Upper Ankle Joint at Different Degrees of Inclination of the Object Forward and Backward

The values of the triceps surae muscle force F_TRI and the joint pressure force F_J were calculated depending on inclination of the object, starting from the vertical, neutral position (with the arm of gravity equal to zero) to the maximum tilting forward position (the arm of CoG₁ equal to 0.13 m and the arm of CoG₂ equal to 0.13 + f m) and to the maximum tilting backward position (the arm of CoG1 equal to 0.045 m and the arm of CoG2 equal to 0.045 + f m) (Figure 1C,D) (

Table 1. Dependence of the pressure force (F_J) on the axis of rotation on the degree of inclination (forward <--, backward -->) of the tilting part of the object while maintaining balance for the flat bar model and the human body. W_TA and W_TRI—weight of the tilted object at the TA-point and TRI-point, respectively; W_CoG1 and W_CoG2—weight at CoG₁ and CoG₂, respectively; r_CoG1 and r_CoG2—gravity arms for W_CoG1 and W_CoG2; h₁—distance from the axis of rotation to the overall centre of gravity of the CoG₁ object; c—distance of CoG₂ from CoG₁; r_TA and r_TRI—arms of the tibialis anterior and triceps calf muscles, respectively; F_TA and F_TRI—forces of the tibialis anterior and triceps calf muscles, respectively; M_CoG2, M_TRI, and M_TA—torques for the tilted part of the object, the triceps calf muscle, and the tibialis anterior muscle, respectively; α—angle between the tilted part of the object and the ground; μ—mean value; and SD—standard deviation.

	Flat Bar Model							Human Body
Tilting	<--	<--	<--	<--	Neutral	-->	-->	<--	<--	<--	<--	Neutral	-->	-->
WA in A-Point WB in B-Point	100% 0%	75% 25%	50% 50%	40% PBP 60%	26% BBP 74%	10% 90%	0% 100%	100% 0%	75% 25%	50% 50%	40% PBP 60%	26% BBP 74%	10% 90%	0% 100%
WTA [kG] [N]	5.296 51.9	5.296 51.9	5.296 51.9	5.296 51.9	2.648 26	5.296 51.9	5.296 51.9	183.2 1797.2	183.2 1797.2	183.2 1797.2	183.2 1797.2	91.6 898.6	183.2 1797.2	183.2 1797.2
WTRI [kG] [N]	5.296 51.9	5.296 51.9	5.296 51.9	5.296 51.9	2.648 26	5.296 51.9	5.296 51.9	183.2 1797.2	183.2 1797.2	183.2 1797.2	183.2 1797.2	91.6 898.6	183.2 1797.2	183.2 1797.2
WCoG1 [kG] [N]	2.972 29.1	2.972 29.1	2.972 29.1	2.972 29.1	2.972 29.1	2.972 29.1	2.972 29.1	94.2 923.2	94.2 923.2	94.2 923.2	94.2 923.2	94.2 923.2	94.2 923.2	94.2 923.2
WCoG2 [kG] [N]	2.648 26	2.648 26	2.648 26	2.648 26	2.648 26	2.648 26	2.648 26	91.6 898.6	91.6 898.6	91.6 898.6	91.6 898.6	91.6 898.6	91.6 898.6	91.6 898.6
rCoG1 μ [m] SD [m]	0.13 0.024	0.077 0.01	0.042 0.007	0.025 0.005	00	0.027 0.003	0.045 0.008	0.13 0.024	0.077 0.01	0.042 0.007	0.025 0.005	00	0.027 0.003	0.045 0.008
rCoG2 μ [m] SD [m]	0.131 0.025	0.078 0.015	0.043 0.006	0.025 0.005	00	0.028 0.003	0.045 0.007	0.132 0.025	0.078 0.015	0.043 0.006	0.025 0.005	00	0.028 0.003	0.046 0.007
cos(alpha) = rCoG1/h1	0.1335	0.0711	0.0390	0.0229	0	0.0252	0.0413	0.1335	0.0796	0.0436	0.0257	0	0.0282	0.0462
f = cos(alpha) * c	0.0013	0.0007	0.0004	0.0002	0	0.0003	0.0004	0.0017	0.0010	0.0006	0.0003	0	0.0004	0.0006
rTRI [m]	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04
rTA [m]	0.03	0.03	0.03	0.03	0.03	0.03	0.03	0.03	0.03	0.03	0.03	0.03	0.03	0.03
FTRI [kg] [N]	8.694 85.3	5.177 50.8	2.839 27.8	1.67 16.4	00	00	00	301.673 2959.4	179.843 1764.3	98.624 967.5	58.0141 569.3	00	00	00
FTA [kg] [N]	00	00	00	00	00	2.449 24	4.008 29.3	00	00	00	00	00	85.087 834.4	139.234 1365.9
MCoG2 [kGm] [Nm]	0.34 8 3.4	0.207 2	0.1136 1.11	0.067 0.66	00	0.0735 0.72	0.12 1.2	12.067 118.4	7.194 70.59	3.945 38.7	2.32 22.7	00	2.553 25	4.177 41
MTRI [kGm] [Nm]	0.348 3.4	0.2071 2	0.113 1.11	0.067 1.11	00	00	00	12.067 118.4	7.194 70.59	3.945 38.7	2.32 22.7	00	00	00
MTA [kGm] [Nm]	00	00	00	00	00	0.068 0.67	0.1820 1.79	00	00	00	00	00	2.371 23.26	6.349 62.25
FJ [kG] [Nm]	13.99 137.2	10.472 102.7	8.127 79.7	6.9658 68.3	2.648 26	7.745 75.99	9.3041 91.3	484.792 4754.35	363.01 3561.6	281.813 2763.6	241.21 2365.5	91.6 898.6	268.28 2631.1	322.431 3161.66

).

From full forward-leaning, when 100% of the weight “falls” on the front support point (A-point) to full backward-leaning, when the front support point is unloaded (0% of the weight on A-point), and the entire weight of the leaned body part falls on the rear support point (B-point), changes in the values of the gravity arm (r_CoG2), muscle forces (F_TA and F_TRI), and pressure force (F_J) are observed. The values of gravity at a given point of the body (W_TRI, W_TA, W_CoG2, and W_CoG1) and the anatomical arms of the muscle forces (r_TRI, r_TA) do not change regardless of the angle of leaning (for angle > 0°). During the leaning of the body, the arms of gravity (r_{W_TRI}, r_{W_TA}, r_CoG2, and r_CoG1), as well as the muscle forces, are changing (F_TRI, F_TA) (Table 1).

It has been shown that muscle strength (F_TRI) and joint pressure force (F_J) decrease as one approaches the vertical. The gravitational moment for CoG₂ (M_CoG2) is balanced at each titling level by muscle moments (M_TRI or M_TA). These balancing moments on either side of the axis of rotation (gravitational and muscle moments) decrease as the body approaches the neutral (vertical) position until they reach zero when the gravitational arm is zero (Table 1) (Figure 7).

4. Discussion

According to some authors, the maximum load on the talus trochlea occurs in the upright (neutral) position because all the weight concentrated in the overall centre of gravity in an upright position falls on the talus trochlea. When the body tilts forward, it is “thrown down” onto the arch of the foot and flattens it, thus relieving the upper ankle joint [28,29]. This is an opinion repeated without experimental justification. From their point of view, the force of gravity is the primary force exerting pressure on the upper ankle joint, causing the foot arch to flatten [22,28,30]. Our results (F_J = 484.8 kg at forward tilt) clearly exceed the body’s weight alone. According to our research, gravity is only one component, next to muscle strength. In addition, our calculations show that when the body tilts forward or backward from a neutral position, the upper ankle joint becomes more loaded as the body tilt increases. This occurs because the greater the body tilt, the greater the counteracting muscular force required to balance the torques. If the pressure force (F_J) is the vector sum of the constant force of gravity (W_TRI or W_TA) and the increasing muscle force (F_TRI or F_TA) with body tilt, this means that the pressure force (F_J) must increase with the tilt.

In other scientific reports [10,22], the authors indicate that the main compressive force on the upper ankle joint when the body tilts (also during walking) is the force generated by the contraction of the triceps calf muscle. As proven, this force’s value in the initial support phase is less than 20% of the body weight. In the late support phase of walking, the compressive force generated only by the triceps calf muscle reaches a value equal to four times the body weight [24], and according to others, even five times [10]. During running, the compression forces in the upper ankle joint can exceed even 13 times the body weight [31]. In reality, the force acting on the loaded upper ankle joint is the sum of the muscle force (of the triceps calf when tilted forward, and tibialis anterior when tilted backward) and the force of gravity. The transfer of the human body weight from the axis of the upper ankle joint forward creates a gravitational moment, which is balanced by the moment of the triceps calf muscle [19]; the greater the weight, the greater the gravitational arm. According to our findings, at the maximum forward tilting of the body, the pressure force is more than five times greater than the weight of the tilting body. A similar effect, although to a lesser extent, occurs when the body leans backward. Then, the force of the tibialis anterior muscle, which takes part in balancing the body leaning backward, adds up to the force of the body’s gravity and gives a value more than three and a half times greater than the weight of the leaning body.

The intuition of some researchers suggests that the force of pressure is the force coming from the triceps calf muscle [10,22], and others indicate that the body weight [28] was half-accurate because both the muscle force and the body weight are responsible for generating the pressure force in the upper ankle joint. Our research shows that in the vertical position of the body, the load on the upper ankle joint is the smallest and increases with the body tilt. The upright position is the most beneficial for humans because the muscle and gravitational moment are equal to zero, and therefore the joint pressure is the smallest and equal to the weight of the body without the feet. Based on our observations and calculations, the results show that tilting the body is not energetically beneficial. Therefore, the body strives to limit the gravitational moments by bringing the projection of the centre of gravity (CoG₂) as close as possible to the axis of rotation (minimising the length of the gravity arm), and thus reducing muscle force. In this way, the body strives to achieve an optimal state of energy expenditure (Electromyography (EMG) or oxygen consumption studies could clearly confirm our conclusion). Tilting the body increases the value of muscle force, which, together with the constant value of gravity, causes an increase in the pressure force on the upper ankle joint, resulting in increased energy expenditure. The greatest load is in the position of the maximum forward body tilting because the joint pressure force is the greatest. Muscle force increases with the body tilting, which results from the fact that the constant arm of the muscle force and the constant weight of the tilted body.

Any increase in the gravitational arm must be compensated by an increase in muscle strength (and thus an increase in muscle torque) to balance the gravitational torque. The consequence of this is an increase in joint pressure, which, in extreme situations, repeated many times, can probably lead to damage to the joint cartilage and degenerative changes. Too much body weight adversely affects the upper ankle joint because the increased weight forces increased work of the triceps calf muscle, thanks to which the joint pressure force also increases, affecting the risk of degenerative changes in this joint. The physiological protection against this process is the increase in the joint pressure area during forward tilt (dorsiflexion). Then, the trapezoid-shaped joint trochlea enters with its wider base into the fork of the upper ankle joint, stretching it, and thus increasing the fit of the joint elements and their stability. In this situation, the joint pressure force vector falls on the wider part of the talus trochlea. It should be emphasised that regardless of the direction of body tilt, the joint pressure vector always runs on the axis of the upper ankle joint.

5. Conclusions

• The ankle joint pressure force is the vector sum of gravitational and muscular forces.
• A greater force of gravity of the body, and consequently a greater muscle force, generates a greater pressure force in the joint when leaning the body.
• When the body leans forward concerning the upper ankle joint, a moment of gravity is generated (the value of the body weight multiplied by the arm of gravity), which is balanced by the moment of the triceps calf muscle (muscle force multiplied by the constant value of the arm of the muscle force) and, as a consequence, the joint pressure force increases.
• In the vertical position of the body, the joint load is the smallest (lack of muscle moment) and increases with the body tilting.
• After conducting computational studies with human data, knowing that regardless of the angle of inclination, the weight of the tilting part of the body remains unchanged, and the anatomical muscle arms of the triceps and the tibialis anterior are constant, the following was shown:

  -

  The pressure force in the upper ankle joint at maximum tilting of the body is the highest and is 5.23 times greater than the weight of the tilting part of the body (without feet). When the body is tilted backward, this ratio is 3.57. This applies to the load on one foot. Standing on both feet, the value is distributed to 50% on each foot.

  -

  The lowest value of the pressure force was obtained in the neutral position. It equals the weight of the tilting part of the body. When standing on one leg, the ankle joint is loaded with the entire body weight (without the weight of one foot), and when bipedal standing, the weight (without feet) is divided in half between two ankle joints, and each weight-bearing foot is divided by half of the tilting part of the body.

  -

  Forward tilt requires over twice the muscle force compared to backward tilt, highlighting asymmetric loading patterns.

6. Study Limitations

This study provides a valuable theoretical insight into the mechanical relationships between body tilt, muscle force, and the resulting pressure force in the upper ankle joint. However, several limitations must be acknowledged when interpreting the findings.

• The results are based on experimental measurements and theoretical and computational modelling, especially for the human body, without direct in vivo or in vitro experimental validation. The absence of empirical data (e.g., force plate measurements, EMG recordings, or in situ pressure mapping) limits the ability to confirm whether the calculated forces accurately reflect real biomechanical conditions.
• The model assumes constant anatomical moment arms for the triceps surae and tibialis anterior muscles, while in reality these lengths vary with ankle joint angle and individual anatomy. Passive stabilising structures such as ligaments, joint capsule, and tendon elasticity were not included in the model, potentially affecting the accuracy of the calculated torques and joint loads.
• The analysis refers to quasi-static positions of the body (forward and backward tilt) and does not account for dynamic phases of gait, running, or balance recovery. In dynamic movement, inertial forces and segment accelerations could substantially alter joint loading patterns.
• The assumption that greater muscle force implies higher energy expenditure was not verified experimentally. Electromyographic (EMG) data or oxygen consumption (VO₂) measurements could provide direct evidence supporting or refuting this interpretation.
• The model treats the “tilting part of the body” as a uniform mass, not accounting for inter-individual variability in body segment proportions, mass distribution, or gender- and age-related differences. Therefore, the calculated pressure ratios may not be representative of all populations.
• The analysis focuses primarily on the vertical (compressive) component of joint pressure, without considering shear or torsional loads that also contribute to articular stress and potential degenerative changes.
• Muscle fatigue and time-dependent tissue properties were not taken into account. In prolonged or repetitive loading, the ability of muscles to generate compensatory torque may decrease, modifying the joint pressure profile over time.