Elsevier

Journal of Biomechanics

Volume 43, Issue 3, 10 February 2010, Pages 387-396
Journal of Biomechanics

Review
Principles of determination and verification of muscle forces in the human musculoskeletal system: Muscle forces to minimise bending stress

https://doi.org/10.1016/j.jbiomech.2009.09.049Get rights and content

Abstract

While there are a growing number of increasingly complex methodologies available to model geometry and material properties of bones, these models still cannot accurately describe physical behaviour of the skeletal system unless the boundary conditions, especially muscular loading, are correct. Available in vivo measurements of muscle forces are mostly highly invasive and offer no practical way to validate the outcome of any computational model that predicts muscle forces. However, muscle forces can be verified indirectly using the fundamental property of living tissue to functional adaptation and finite element (FE) analysis. Even though the mechanisms of the functional adaptation are not fully understood, its result is clearly seen in the shape and inner structure of bones. The FE method provides a precise tool for analysis of the stress/strain distribution in the bone under given loading conditions. The present work sets principles for the determination of the muscle forces on the basis of the widely accepted view that biological systems are optimized light-weight structures with minimised amount of unloaded/underloaded material and hence evenly distributed loading throughout the structure. Bending loading of bones is avoided/compensated in bones under physiological loading. Thus, bending minimisation provides the basis for the determination of the musculoskeletal system loading. As a result of our approach, the muscle forces for a human femur during normal gait and sitting down (peak hip joint force) are obtained such that the bone is loaded predominantly in compression and the stress distribution in proximal and diaphyseal femur corresponds to the material distribution in bone.

Introduction

Using techniques available today one can create very impressive finite element (FE) models of any bone with high levels of geometrical accuracy and complex material property distributions (Pahr and Zysset, 2008; Schileo et al., 2007, Schileo et al., 2008; Taddei et al., 2006; Yosibash et al., 2007). However, such precise models will only deliver accurate stress and strain distributions through the bone if the applied boundary conditions are accurate. Muscular loading and a choice of constraints have great influence on the stress state of the bone (Duda et al., 1998; Speirs et al., 2007) and may not be casually simplified. At the same time, several studies (e.g. Baca et al., 2007; Taddei et al., 2006) showed that sufficient level of accuracy of FE analysis of long bones has already been reached if the bone geometry is generated using available imaging methods (magnetic resonance imaging—MRI, or computed tomography—CT) and if isotropic inhomogeneous material properties are assigned.

In order to make a fair estimate of the loading conditions of a bone, one should proceed from a consideration of the basic function of the bone. It is a widespread belief that along with some compression, long bones are subjected to large bending moments, which produce high tensile loading. Two well-known facts have contributed to the longevity of this common belief. The first is that cortical (compact) bone as a material can withstand tension almost as well as compression. The mean value of tensile strength of human cortical bone (for specimens in longitudinal direction) is 128–133 MPa, while the compressive strength is 182–205 MPa (Cowin, 1983; Martens, 1985; Reilly and Burstein, 1975; Savvidis and Stabrey, 1996). The second fact is that long bones are hollow, and in engineering such hollow structures are optimized for light-weight structures that carry bending (and torsion). The most obvious conclusion seems to be that the long bones are functionally adapted to carry bending loads (e.g. Currey, 2003). An additional contribution to the bending bone belief was made by the trajectorial theory early in the history of the biomechanics (Triepel, 1908), which evolved from the observation of similarities between the trabecular architecture in proximal femur and the principal stress trajectories in Culmann's crane (Wolff, 1870). Since then, the fine trabecular structures in cancellous bone are often interpreted as tension and compression patterns crossing at right angles, even though investigations carried out by Jansen as early as 1920 showed that the trabeculae may cross at different angles and hence do not correspond to the principal stress lines. Jansen (1920) stated that the bone formation occurs along the compressive stress patterns in bone in accordance with the weight-bearing function of bone. Many later anatomical studies (e.g. Skedros and Baucom, 2008) also found that the trabecular trajectories especially in the proximal femur are non-orthogonal. Furthermore, the analysis of the femoral neck fractures (Farkas et al., 1948) led the authors to the conclusion that “the presence of tensile stresses is doubtful”. Also Garden (1961) studied the inner structure of the proximal femur not only in the frontal but also in the sagittal plane and concluded that the forces acting on the proximal femur are compressive in nature. Based on the weight-bearing function of bone, he rejected the Ward's street-lamp bracket and Culmann's crane theories of trabecular architecture in the proximal femur.

Experimental research monitoring new bone formation often suggested that the normal loading of bone is bending based on the observation of bone remodelling, which occurs under tension as well as under compression. For example, in animal tests (Liskova and Hert, 1971) cyclic bending applied for certain limited time during the day resulted in extensive new bone formation on both tensile and compressive side of the bone. (Surprisingly, the noticeable bone atrophy reported by Liskova and Hert (1971) in the region of neutral axis of experimentally applied bending is disregarded.) In contrast to the time-limited experimentally applied dynamic bending (Rubin and Lanyon, 1984), experimentally applied constant compression and bending in the study produced bone atrophy, which was comparable to the unloading atrophy.

However, recent cell culture experiments show that osteoblast proliferation under tension alone is not an indication of new bone formation, since despite an increase of osteoblast activities related to matrix formation the osteoblast activities “relevant for matrix mineralization are decreased” under application of cyclic tensile strain (Kaspar et al., 2000). Moreover, Rath et al. (2008) showed that 10% cyclic compressive strains applied to cell-scaffolds induced the release of substances related not only to bone matrix formation but also to bone mineralisation.

According to Wolff's law, living structures that are not used must experience atrophy (Wolff, 1892). Bending loading of a bone poses a most unfortunate loading in terms of distributing of stress over a cross-section, because a large part of the cross-section suffers underloading, whereas another part of the cross-section may be overloaded. Pauwels (1965) described several ways, in which the musculoskeletal system deals with this problem and minimises bending. One of them, tension cord, also known as tension bracing, is a technical solution, which allows for the minimisation or ideally avoidance of bending. The technical tension cord is commonly used in the construction of bridges and cranes. Nature also has a number of means of applying tension cord to the design of light-weight and highly mobile musculoskeletal systems of animals (Rossmann et al., 2001; Witzel and Preuschoft, 2005) and humans. For example, bones can be protected from bending using ligaments as passive tension cords, usually during inactivity in a power saving mode, while muscles can be used as active tension cords usually during locomotion. Another way to avoid bending is the distribution of the material in a cross-section. Frost proposed general concepts of skeletal adaptations to mechanical usage (Frost, 1990) and described bone formation and resorption drifts in trabeculae and in diaphyseal bone. Cross-sectional material distribution in the whole bone facilitates the passage of the resultant cross-sectional force through the cross-sectional centre of gravity (Witzel, 1993).

Compensation for bending has been considered previously in a number of biomechanical analyses (Hert, 1994; Möser and Hein, 1992; Pussel, 2000; Rudman et al., 2006; Taylor et al., 1996; Witzel, 1993). Möser and Hein (1992) applied the technique of tension cording to a whole femur considering it as a crane. In a static analysis they determined the muscle forces acting on the femur, so that the compression of the bony structure was ensured. Rudman et al. (2006) investigated the role of the ligaments surrounding the femoral head and neck by means of FE analysis. The study showed that the inclusion of ligamentous bands in the model eliminated tensile stresses in the cranial side of the femoral neck, placing it instead under compression. An investigation of the whole femur during one-legged stance by Taylor et al. (1996) compared the stress–strain distributions in the bone under four different load cases. This study showed that including the forces arising from the iliotibial tract and iliopsoas muscle in the FE model, additionally to the hip reaction force and abductor muscle forces, could eliminate bending in the femoral shaft. As a result the strain energy density distribution along the diaphyseal femur became uniform and the deflection of the femoral head relative to the femoral condyles was reduced significantly. Both latter investigations were parametric studies. Finally, the compensation for bending and the view of bones as compressive structures led to the development of FE structure synthesis (FESS) and allowed a virtual synthesis of the form of a scull on the patterns of the compressive stresses (Witzel and Preuschoft, 2005; Witzel, 2007). According to FESS, a block of material with bone material properties is loaded with physiologically oriented muscle forces so that in each load case bending loading is compensated for. In an iterative procedure the unloaded material subject to physiological superposition of load cases is removed from the initial block and a structure is achieved, which replicates the investigated scull. Thus, the above biomechanical studies show that the approach of compensation of bending loading is beneficial in terms of even loading of the bone and can predict bony structures.

Currently, there is no connection between the biomechanical studies and mathematical prediction of the muscular loading. Most mathematical models that attempt to predict musculoskeletal loading solve an optimisation problem formulated based on the rigid body geometry and spatial considerations of the musculoskeletal system. Inevitably, the number of unknown parameters exceeds the number of available equations, and hence some simplifying assumptions and optimisation criteria are required in the mathematical model. Examples of optimization criteria include the minimisation of joint force, muscle forces, joint torques, total activation of the muscles, and endurance measures; (for a review, see e.g. Fischer et al., 1995 or Lenaerts et al., 2008). Thereby, the mechanical behaviour of the bone and response of the bone to the loading is disregarded. In vivo validation of the outcome of the mathematical models is restrained. Several techniques are currently available for in vivo tendon force measurements (Finni et al., 1998; Komi, 1990; Pourcelot et al., 2005). These methods are confined to the measurement of the tendon forces. Nevertheless, the available tendon force measurement techniques are not really feasible for validation of the outcome of mathematical models because of the invasiveness of the techniques.

However, the muscle forces can be validated indirectly by the means of FE analysis taking Wolff's law as a basis. In particular, the stress/strain distribution predicted by FE analysis in a loaded bone can be compared with the real bone mass distribution from bone imaging methods (CT, MRI). One of the first attempts to apply Wolff's law to the analysis of bone loading was undertaken by Fischer et al. (1995) in a so-called density-based load determination method. In a 2D FE model of an idealized bone, several “experimental” load cases were formulated, in which a force distribution (analogous to a joint force) was applied to the bone without muscular action. Using a bone remodelling algorithm based on a continuum-level effective stress, the bone density distribution was obtained for each of these load cases. For comparison, a so-called “standard” density distribution was also obtained numerically as a superposition of five “standard” load cases. The optimization procedure then selected from the “experimental” load cases to find the ones that matched the “standard” distribution. Unfortunately, a major drawback of the method consists in the bone remodelling algorithm that is based on positively defined strain energy density. Thus, the bone remodelling algorithm makes no distinction between tension and compression, and thereby assumes that the bone material has the same remodelling behaviour under tension as under compression. Thus, a 2D model with symmetric load cases producing bending will lead to equal remodelling on the tension and compression sides of the bone, and is predetermined to reproduce the bone density distributions comparable with a roentgenogram of a proximal bone.

The investigations described in the following are founded on the premise that all bones under physiological loading are predominantly compressive structures, which are designed for a low and nearly homogeneous compressive stress distribution in every cross-section. Here we make an important distinction between possible and physiological loading of bone. Possible loading could be of any type and magnitude. For example, bones of immobilized patients are extremely underloaded, while overloading of bone can take place with athletes who abruptly change training regime or as a result of injury and, in the best case, leads to hypertrophy or, in the worst case, to a fatigue fracture or traumatic injury (Noesberger and Eichenberger, 1997). Mild forms of unphysiological loading do not necessarily or immediately result in a trauma or a medical condition (Noesberger and Eichenberger, 1997). But any prolonged changes in loading conditions evoke an interplay of muscular adaptation and bone remodelling. We do not exclude bending from the range of possible loading conditions of bone, but we assume that in a long term it will be inevitably followed by muscular adaptation or bone remodelling or both, so that the bending is minimised. Therefore, in the present paper, we focus our attention on physiological loading of the bone: normal loading of the bone during everyday activities, which led to the present structure of bone. Physiological loading primarily ensures the function of the musculoskeletal system during activities, but at the same time it sustains the structure of the bone in accordance with Wolff's law of bone transformation (Wolff, 1892). The whole musculoskeletal system is viewed as a biotensegrity system, in which the function of the muscles and ligaments includes the elimination of bending in bones producing a light-weight mobile structure. The aim of this study is to apply the light-weight construction principle and determine the physiological muscle load cases that act on a human femur during normal everyday movements. In order to attain the physiological loading in the bone, the muscle loads must be formulated so that the internal bending moment in every cross-section is minimised.

Section snippets

Materials and methods

The FE model of the femur was analysed using the FE program ANSYS (ANSYS, Inc., Canonsburg, PA, USA). The model geometry was obtained from CT data of a male individual with the following scan parameters: pixel size 0.8398 mm; slice spacing 1.5 mm. Analysis of the CT slices provided the ranges of Hounsfield values (HU) listed in Table 1, which were used to define different material property values in different regions of the FE model. In total, the femur FE model consisted of 20 adjacent volumes

Results

Two different load cases, for gait and sitting down, were developed by the method described above. The first equilibrium calculations for the two load cases resulted in principal stress values in cortical bone lying in the range +15 to −25 MPa for one-legged stance during normal walking and +15 to −30 MPa during sitting down (not illustrated here). The bending minimisation procedure led to the muscle forces listed in Table 2, columns 6 and 7: MLC1 refers to the highest hip joint loading in

Discussion

In the present paper using the light-weight construction principle, we have developed a new method of estimating muscle loading, which is based on the assumption of a bending-minimised stress state for bone structures. The muscular load cases (MLCs) for one-legged stance during gait and for sitting down have been investigated using the above assumption. The MLCs have been developed using all available anatomical information, geometry of a real bone, measurements of the hip joint force and

Conflict of interest statement

The authors confirm that there is no conflict of interest.

Acknowledgements

The authors are greatly indebted to Prof. Dr. Michael J. Fagan, Centre for Medical Engineering and Technology, University of Hull for reading the draft of the manuscript, for his invaluable comments and recommendations of some references on the bone innervation. The submitted version of the manuscript was vastly improved thanks to comprehensive comments of two anonymous reviewers. We thank Rainer Gößling of the Ruhr-University of Bochum for the fruitful discussions. We also wish to thank

References (67)

  • D.B. Mach et al.

    Origins of skeletal pain: sensory and sympathetic innervation of the mouse femur

    Neuroscience

    (2002)
  • B. Noesberger et al.

    Overuse injuries of the hip and snapping hip syndrome

    Operative Techniques in Sports Medicine

    (1997)
  • P. Pourcelot et al.

    A non-invasive method of tendon force measurement

    Journal of Biomechanics

    (2005)
  • B. Rath et al.

    Compressive forces induce osteogenic gene expression in calvarial osteoblasts

    Journal of Biomechanics

    (2008)
  • D.T. Reilly et al.

    The elastic and ultimate properties of compact bone tissue

    Journal of Biomechanics

    (1975)
  • J.Y. Rho et al.

    Relation of mechanical properties to density and CT numbers in human bone

    Medical Engineering and Physics

    (1995)
  • E. Schileo et al.

    An accurate estimation of bone density improves the accuracy of subject-specific finite element models

    Journal of Biomechanics

    (2008)
  • E. Schileo et al.

    Subject-specific finite element models can accurately predict strain levels in long bones

    Journal of Biomechanics

    (2007)
  • A.D. Speirs et al.

    Physiologically based boundary conditions in finite element modeling

    Journal of Biomechanics

    (2007)
  • F. Taddei et al.

    Subject-specific finite element models of long bones: an in vitro evaluation of the overall accuracy

    Journal of Biomechanics

    (2006)
  • M.E. Taylor et al.

    Stress and strain distribution within the intact femur; compression or bending

    Medical Engineering and Physics

    (1996)
  • C.H. Turner et al.

    A uniform strain criterion for trabecular bone adaptation: do continuum-level strain gradients drive adaptation?

    Journal of Biomechanics

    (1997)
  • M.W. Whittle

    Clinical gait analysis: a review

    Human Movement Science

    (1996)
  • Z. Yosibash et al.

    Reliable simulations of the human proximal femur by high-order finite element analysis validated by experimental observations

    Journal of Biomechanics

    (2007)
  • V. Baca et al.

    Comparison of an inhomogeneous orthotropic and isotropic material models used for FE analyses

    Medical Engineering and Physics

    (2007)
  • G. Bergmann

    In vivo Messung der Belastung von Hüftimplantaten

    (1997)
  • T.M. Boyce et al.

    Damage and strain mode associations in human compact bone bending fatigue

    Journal of Orthopaedic Research

    (1998)
  • R.A. Brand et al.

    A model of lower extremity muscular anatomy

    Journal of Biomechanical Engineering

    (1982)
  • M.J. Ciarelli et al.

    Evaluation of orthogonal mechanical properties and density of human trabecular bone

    Journal of Orthopaedic Research

    (1991)
  • S.C. Cowin

    The mechanical and stress adaptive properties of bone

    Annals of Biomedical Engineering

    (1983)
  • A. Farkas et al.

    An anatomical study of the mechanics, pathology, and healing of fracture of femoral neck

    Journal of Bone and Joint Surgery

    (1948)
  • T. Finni et al.

    Achilles tendon loading during walking: application of a novel optic fiber technique

    European Journal of Applied Physiology and Occupational Physiology

    (1998)
  • H.M. Frost

    Skeletal structural adaptations to mechanical usage (SATMU): 1. Redefining Wolff's Law: the bone modeling problem

    The Anatomical Record

    (1990)
  • Cited by (72)

    View all citing articles on Scopus
    View full text