## 1. Equilibrium of forces and torques

Within biomechanics the effects of mechanical forces (forces due to gravity, elasticity, and friction) on living mechanisms are considered. These forces determine the movement of human and animals which is, especially in vertebrates, enabled by a complex and interconnected network of muscles, tendons, and bones that act as a consistent kinematical chain. A living system is never static on the cellular level, however, as a whole, the body can attain certain positions which are taken to correspond to static equilibria. The body is in static equilibrium when the sum of all external forces acting upon it equals to zero and sum of all torques subject to these forces equals to zero. The first condition is expressed by equation

where

where

with

The cross product can be expressed by the matrix

with the result

In the description of the static equilibrium, the image of the body is divided into segments. These segments act one upon another which is expressed by means of intersegment forces. The segments are also subjected to attraction of the Earth. As these forces and their momentum arms in general attain different directions in space, all torque components have in general nonzero values. However, in certain situations the expressions are simplified, such as in the case where the balance consists of a dimensionless rigid rod supported in a certain point, with two vertical load forces *r**r***Figure 1**). Let the positive

There are three forces acting on the balance, the two load forces

The resultant force is not known; therefore, we will consider that it has three components,

To determine the momentum arms, a choice of the origin of the coordinate system must be made. It is convenient to choose it at the origin of the resultant force

however, in the case presented in **Figure 1**, the rod extends in the direction of

while the momentum arm of the force

The momentum arm of the resultant force

The torques of all three forces are

In general, the equilibrium of forces is given by three equations for three components,

Following Eqs. (17)–(19), the components of the force

and the resultant force can be given as

The equilibrium of torques is given by three equations for three components,

As the torque of the force

Considering also the expressions (14) and (15), we obtain

and finally

## 2. A two‐segment model for the resultant hip force in the one‐legged stance

In a simple model of a one‐legged stance (**Figure 2**), the body is divided into two segments: the loaded leg and the rest of the body (**Figure 2a**). The two segments are connected by the hip joint. **Figure 2b** presents an abstraction of the two segments (labeled I and II, respectively). For simplicity, the pelvis is taken to be leveled in the model. The sizes of the boxes correspond to approximate weight proportion of the two segments. Further, it is assumed that all the forces lie in the frontal plane of the body through the centers of both femoral heads (their components in the **Figure 2c**). It is taken that the muscle force acts in the direction of the line connecting both origins, expressed by the inclination angle

The model is based on equilibrium equations of forces and torques (Eqs. (1) and (2), respectively) acting on the segment I. Momentum arms of the weight of the segment I and of the effective muscle force can be determined from the geometry of the pelvis and proximal femur and the weight of the segment I can be determined from the body weight and an approximation that the leg weights about 1/7 of the entire body [1]. There are three unknown parameters in the model: the magnitude of the effective muscle force (

### 2.1. A primitive model for resultant hip force

In the model (**Figures 3** and **4**), we have chosen the origin of the coordinate system at the center of the hip joint (that coincides with the center of the femoral head and the center of the acetabular shell). The loading forces are the weight of the segment I,

with momentum arm

and the force of the effective muscle, which lies in the frontal plane through centers of the femoral heads,

with momentum arm

The origin of the weight of the segment I is taken at the center of mass of the segment. It is approximated that this point lies in the sagittal plane of the body through the midline. Note that the components of the forces

The respective torques are

and

as the momentum arm of the resultant hip force is zero due to the choice of the origin of the coordinate system.

Following the above procedure, in particular Eq. (26), which describes equilibrium of torques, we obtain

Rearranging the above equation yields for the unknown magnitude of the effective muscle force

Following Eqs. (20)–(22), we obtain for the components in the direction of the

and in the direction of the

Dividing Eq. (40) by Eq. (41) eliminates the unknown magnitude of the resultant hip force

By knowing

It is often convenient to present the results with respect to the body weight

the inclination of the resultant hip force

and the normalized resultant hip force

In a special case when the effective muscle force points in the vertical direction, i.e., **Figure 4**), the expressions (44)–(46) simplify into

Note that these expressions (Eqs. (47)–(49)) are the same as if obtained for a simple balance with the two loading forces

and

and respective momentum arms

and

Following Eqs. (29), (50), and (51), we obtain

or (by taking into account that

Following Eqs. (22), (50)–(51), and

or, normalized

Taking into account Eqs. (55) and (57) yields

It can be seen that Eqs. (47) and (55) are identical. Likewise, Eqs. (49) and (58) are identical. Although the effective muscle attachment point on the iliac bone, the center of the femoral head, and the center of mass of the body segment I do not lie in the same horizontal plane, the model of simple balance derived for a weightless rigid bar with all forces originating in the same horizontal plane, gives the same solution, owing to a special case that the forces lie in the vertical direction only. It should however be kept in mind that this is a consequence of the simplifications used in the model of the one‐legged stance and that in reality segment I has a characteristic shape that may impact the forces, which is not considered in the simple balance model. Some textbooks use a simple balance as an illustrative model to explain the principles of the effect of the muscle forces (the principles of different types of levers). It should be borne in mind that such approximations are valid only if all forces act in the same direction.

**Figure 5** shows the dependence of the magnitude of the resultant hip force **Figure 6** shows the dependence of the inclination of the resultant hip force with respect to vertical direction

### 2.2. HIPSTRESS model for resultant hip force

The primitive model and the simple balance approximation consider only one muscle acting in a hip in the one‐legged stance. Measurements however indicate that there are several muscles that are active in this body position. The static equilibrium requires that the resultant of all external forces acting on each segment is zero and that the resultant of all external torques acting on each segment is zero, therefore in a more realistic model, contributions of all active muscles should be taken into account. The equilibrium equation for forces acting on segment I is

where index

where

where

The forces and the torques have three dimensions, therefore the model consists of six equations (three for equilibrium of forces and three for equilibrium of torques). For known origin and insertion points of the muscles and known cross‐section areas, the unknown quantities are the muscle tensions and three components of the resultant hip force *R*. Since there are 9 effective muscles and 3 components of the force *R*, there are 12 unknowns and 6 equations. To solve this problem, a simplification was introduced by dividing the muscles into three groups (anterior, middle, and posterior) with respect to the position. It was assumed that the muscles in the same group have the same tension. This reduced the number of unknowns to six as required for solution of the complex of six equations. The muscle origin and insertion points and the muscle cross‐section were taken from Refs. [3] and [4], respectively. The geometry of the individual patient was taken into account by correction of muscle attachment points according to the geometrical parameters obtained from the standard anteroposterior radiograph, the distance from the center of the femoral head to the midline **Figure 7**). Results obtained with the HIPSTRESS model for resultant hip force showed that the force lies almost in the frontal plane of the body through both femoral heads [1]. To further simplify the calculations it was assumed in most clinical studies using HIPSTRESS model that the force lies in the frontal plane and is, like in the primitive model, represented by its magnitude

## 3. HIPSTRESS model for contact stress in the hip

Once we know what is the overall load

We neglect all other stresses but the contact hip stress acting perpendicularly to the spherical articular surface, by assuming that the joint is well lubricated. A surface is imagined that is a part of a sphere with radius

where

It is assumed that stress is proportional to strain due to the squeezing of the cartilage between the femoral head and the acetabulum [6], which yields

where

which simplifies into

by introducing the expressions

and

As

while its proper value can be calculated by multiplying the left side of Eq. (68) by

**Figures 8** and **9** show the dependence of the polar angle and stress at the pole (Eqs. (69) and (68), respectively), on parameter y. Clinical studies that have validated the HIPSTRESS method have used the parameter peak stress on the weight‐bearing area as the relevant quantity. Namely, the stress pole is an abstract point in which the respective spheres outlining the femoral head and the acetabulum most closely approach each other upon loading of the joint. The pole may therefore be located within the load‐bearing area of the joint or outside it. In the first case, the peak stress is identical to the value of stress at the pole

## 4. Comparison of the primitive model and the HIPSTRESS model

The primitive model and the HIPSTRESS model both use the same characteristic points on the iliac bone and on the greater trochanter (i.e., the highest and the most lateral points). In both models, the center‐edge angle and the radius of the articular surface (i.e., the radius of the femoral head) is needed to calculate stress distribution. Both models consider the center of mass and the corresponding momentum arm. There are however differences in parameters for the resultant hip force. The HIPSTRESS model includes more parameters (

For illustration we calculate the biomechanical parameters by using both models and also the simple balance approximation. **Figure 10** shows the measured geometrical parameters for the primitive model and **Figure 11** shows the measured parameters for the HIPSTRESS model.

To determine the magnitude and the inclination of the resultant hip force (**Table 1**. It can be seen that for the chosen hip and pelvis, the magnitude of the resultant hip force in the primitive model and in the HIPSTRESS model differ by only 9%, while in the simple balance approximation the result deviates by about 40%. The inclination of the resultant hip force **Figure 10**. Parameter **Figure 9**. The inset of the figure with the values corresponding to all three models is shown in **Figure 12**.

It can be seen that in the primitive model and in the HIPSTRESS model the pole lies within the load‐bearing area while in the simple balance approximation it falls outside the load‐bearing area (**Table 1**). The HIPSTRESS model in this case yields the lowest stress. Note that in the simple balance approximation the hip would according to the criteria of the HIPSTRESS [14, 15] be considered as dysplastic since it exhibits rapidly decreasing stress at the lateral acetabular rim. However, the center‐edge angle is 27° which is considered as a healthy hip. The simple balance model overestimates hip stress and is in most cases not suitable to give quantitative result regarding biomechanical parameters of the hip and pelvis.

The example that we have shown corresponds to a normal hip geometry. Also, the values of peak stress that were obtained by the primitive model and the HIPSTRESS model are within the values corresponding to hips that would remain without clinical problems up to about 85 years of age [16]. In this case, the primitive model proved successful in estimating biomechanical parameters. However, to see whether it has a predictive value, it should be validated by clinical studies. The advantage of the primitive model is that it is simpler and does not need special software. Determination of the resultant hip force with the primitive model is scale independent which is an advantage over the HIPSTRESS model. Namely, the HIPSTRESS model uses three‐dimensional coordinates of the muscle attachment points of a reference hip and pelvis but only the

We have used standard anteroposterior radiograms to measure geometrical parameters. Imaging with magnetic resonance has recently improved to enable determination of three‐dimensional positions of muscle attachment points for the needs of the HIPSTRESS method, but has not yet been used for the determination of biomechanical parameters by this method. This would be a major improvement over using radiograms, as the direct data on the muscle attachment points could be used and there would be no need for rescaling of the reference geometry. In considering the three‐dimensional data the primitive model could not do justice to the system as its assumptions are bounded to the simplification to two dimensions. However, the primitive model (but not the simple balance approximation) merits to minimal standards to be used for understanding of the principles of the equilibrium of forces and torques in the one‐legged stance, and can in certain cases (such as the one shown here) also yield a valid quantitative estimation of the biomechanical parameters.