CC BY
48
THE CALCULATION OF LOADS ACTING ON THE FEMUR DURING
NORMAL HUMAN WALKING
A.V. Sotin, Yu.V. Akulich, R.M. Podgayets
Department of Theoretical Mechanics, Perm State Technical University, 29a, Komsomolsky Prospect, 614600, Perm, Russia
Abstract; For the construction of biomechanical model of the femur it is necessary to know all external forces acting in the neighborhood of the hip, A three-dimensional model of the human lower limb is presented, it allows to calculate the forces produced by muscles acting on the femur, and the reaction force in the hip joint during normal gait. The statically indeterminable problem is solved by introduction of some criterion of finding the optimum solution. The algorithm of calculations uses the data on variable intersegmental angles in sagittal and frontal planes, restrictions on maximal isometric muscle force, and takes into account the passive resistance of ligaments surrounding and crossing the hip joint, and time variations of coordinates of the point of application of the foot-ground reaction force.
Key words: muscular force, loads during the gait, optimum criterion
Introduction
In the biomechanical simulation of the femur it is necessary to know external forces acting on the lower limb. In different studies concerned the stresses and strains in the femoral head (Akulich et al. [10], Ueo et al. [28]) or the influence of the implant on the cortical bone (Cheal et al. [14], Denisov et al [19], Kalidindi and Ahmad [22], Weinans et al. [29]), not only the contact force in the hip joint is taken into account, but also the muscular forces. As walking is the main human locomotive act, the loads on the femur during walking are of great interest. In most cases experimental methods employing the measurement of electric activity of muscles allow to estimate the muscular forces only qualitatively, so for their determination the methods of biomechanical simulation are required. The construction of more exact models is encouraged by intensive development of methods for registration (of different parameters of walking. Detailed studies concerned the biomechanical structure of normal gait are presented in the works by Vitenzon [4], Farber et al. [8]. Some authors studied in addition the passive resistance moment of hip joint ligaments (Davy and Audu [18], Yoon and Mansour [30]), the motion of bone pelvis and femur in frontal plane (Vaganova et al. [2], Pokatilov and Sanin [7], Kawate et al. [23]), and the time variations of the point of application of the foot-ground reaction force (Dushin and Svechkopal [5], Suto and Kawamura [27]). The papers by Brand et al. [12], Dostal and Andrews [20], Sotin et al. [26] provide detailed anatomic data concerning the hip musculature required for the development of biomechanical model of lower extremity. The investigation of loads acting on the femur is the subject of a big number of papers [6, 9, 11 -13, 15-17, 21, 25], these studies employ different problem statements, different number of muscles and leg segments involved into calculations, and different objective functions.
In the given paper the three-dimensional quasi-static formulation of the problem concerning the determination of the reaction force in the hip joint and calculation of muscular forces during gait has been used. The model takes into account the passive resistance of ligaments, variations of the point of application of the foot-ground reaction force as well as
the restrictions on maximal isometric forces produced by muscles. Three linear objective functions have been used for the solving the optimization problem.
Materials and Methods
For the calculation of loads acting on the hip joint and muscular efforts during walking, the four-segmental biomechanical model of the leg consisted of thigh, shank, foot and toe was constructed. Mutual location of the segments during walking has been determined by intersegmental angles in the hip, knee, ankle and metatarsophalangial joints. Mass, length and position of mass center of each segment were calculated by relative (to the height and weight of human body) values provided in the works by Vinogradov [3], Farber et al. [8]. The anthropometric data used in the model are presented in Table 1.
Table 1. Mass-inertial characteristics of the leg.
Sex Male
Height, cm 175
Body weight, kg 68
Segment Thigh Shank Foot
Relative mass, % of body weight 11.58 4.75 1.65
Distance from proximal joint to mass center, % of height 10.32 10.25 4.32
Relative length, % of height 23.80 24.10 13.80
The walking has been simulated as a quasi-static process. The whole cycle of double step was divided into 20 equal time intervals. The data on the supporting reactions and intersegmental angles during normal gait (Fig. 1), and the time variations of the point of application of the foot-ground reaction force during the stance phase of gait were taken from the works by Vitenzon [4], Dushin and Svecbkopal [5].
The moment of resistance of hip joint ligaments in sagittal plane has been calculated by the following formula provided in the paper by Davy and Audu [18]:
MUg (cp>= 2.6*exp(-5.8-(cp~0,1744)8.7- exp(-L3-(0.95~cp))? (1)
where cp is the inter-segmental angle between bone pelvis and femur.
The model takes into account 27 hip and pelvic muscles. Each muscle had been simulated by the external force directed across the straight line connecting pelvic and femoral muscle attachment points (the coordinates of these points had been taken from the paper by Dostal and Andrews [20]).
Three reference frames have been used in calculations (Fig. 2): the stationary (laboratory) reference frame OXYZ and two movable reference frames OX'Y'Z1 and OX'VZ" embedded in the pelvis and femur, respectively. The origin O is situated in the center of the femoral head, The hip joint has been modeled by the spherical joint. The effect of pelvis on the femur has been changed by the hip joint reaction force.
In each phase of step, the current configuration of the leg has been determined by the known intersegmental angles in sagittal plane and the inclinations of pelvis and femur to the vertical in frontal plane (in accordance with the studies by Vitenzon [4], Pokatilov and Sanin [7]). For each leg configuration at different step phases, the coordinates of pelvic and femoral points of muscular attachment have been calculated, and the following equilibrium equations have been written;
Fig. 1. Segments of the lower limb (above in the left), the intersegmental angles in the hip, knee, ankle and metatarsophalangial (MTF) joints vs time (above in the right), the components of foot-ground reaction force Nxt Ny/ Nz vs time (middle in the right), variations of the point of its application to the foot (below in the left) during the gait cycle, phases of gait cycle for one leg and their duration in percentage of the total cycle time (below in the right).
The podogram notations are: h.s. - heel strike, f.f. - foot flat, h.o. - heel off, to. ~ toe off
m
£F/+£P/f+R + N = 0; /=1 k= 1
¿Mo(F;.) + |>o(P/c) + M%+Mo(N) = 0; ¡=1 fc=i
Ft > 0;
■max?
(3)
Ft < Fifi
where n is the number of muscles; m is the number of segments; R is the hip joint reaction force; N is the foot-ground reaction force; F} is the force produced by the /th muscle; Pk is the weight of the kih body segment; Mug is the resistance moment of ligaments; Fimax is the
maximal isometric force produced by the /th muscle (according to data from the paper by Sotin et al. [26]).
As the number of unknown values is more than the number of equations, the problem is statically indeterminate. To resolve the problem we have to select the actual solution of the big number of possible solutions by a certain optimum criterion, or an objective function. We
Y Y\Y"
Adductor Longus
Biceps Femoris (Long Head)
Fig. 2. The reference frames and lines of action of some muscular forces. OXYZ is a laboratory reference frame, OX'Y'Z' is a reference frame embedded in the pelvis, and QX"Y"Z" is a
reference frame embedded in the femur.
considered three linear objective functions. The first objective function is the sum of absolute values of all muscular forces involved in the model:
n
J\ ~ min >
1=1
the second is the sum of relative values of the muscular forces, i.e. the sum of the muscular forces divided by their peak values:
n
J2 = Z^'/^max ™in >
and the third objective function is the sum of the absolute values of the work done by all the muscles:
n
J3 = At min ,
where Ai is the work done by the /th muscle during gait. The muscular work has been
calculated as a product of the force exerted by the muscle into the increment of the muscle's length in the given step phase. The work is positive under contraction of the muscle and negative under it extension,
Results and Discussion
In Fig. 3 the time dependence of the hip joint reaction force during the gait cycle is shown for three objective functions, these curves have two peaks corresponding to front and back pushes, and the magnitude of the first maximum depends on the chosen objective
Fig. 3. The time dependence of hip joint reaction Fig. 4. The moments of muscular forces with
force R divided by body weight (BW) for different respect to axes of the laboratory reference frame vs optimum criteria. For the podogram explanation time during the step cycle, see Fig. 1.
Table 2. The muscular work calculated under different optimum criteria.
Objective function Muscular work, J
positive negative algebraic sum sum of modules
Ji 14.456 -9.746 4.710 24.202
Ji 15.151 -9.880 5.271 25.031
J3 13.310 -8.895 4.414 22.205
function. Fig. 4 shows the development in time of the components of the moment of muscular forces. The time dependence of the muscular moment with respect to the sagittal axis Mz also has two pronounced peaks coinciding in phase with front and back pushes.
In Table 2 the values of muscular work; calculated under different optimum criteria are presented. The absolute value of the work for different criteria changes insignificantly (within 10%), but the value of joint reaction force at the instant of the front push differs in two times. Thereby the increase of the hip joint load is significant but the advantage in the expended work is very small.
As mentioned above, 27 muscles have been taken into account in the model (see Table 3). The solution of the problem with each particular objective function gives us different number of the muscles involved into the motion, i.e. the muscles with non-zero forces. These muscles have been marked in Table 3 by the sign "+".
The general pictures of the muscular efforts calculated with different objective function have been presented in Figs 5, 6, 7. Figs 8, 9 show the active muscles at the moments of front and back pushes which coincide in time with the first and second peak values of the foot-ground reaction force. As can be seen from Figs 8 and 9, the activity of the musculus gluteus medius ensuring maintenance of vertical posture in the frontal plane, does not depend on the choice of the objective function but the onward movement is ensured by different ways.
When the first criterion is employed, the motion is realized by the back group of muscles (musculus biceps femoris, musculus semitendinousus). Under the second criterion, the motion is performed by the large muscles (musculus gluteus maximus), it results in increasing load on the joint. When the third objective function is employed, the motion is realized by many muscles of the internal group (due to minimal variations of their lengths during walking), it also brings about significant increasing the load on the joint.
Table 3. The hip area muscles involved into the calculations.
No Muscle Objective function
Ji Ji Ji
1 Iliopsoas + +
2 Pectineus
3 Sartorius + +
4 Rectus femioris + + +
5 Adductor longus + + +
6 Adductor brevis
7 Adductor minimus
8 Adductor magnus (med) +
9 Adductor magnus (post) +
10 Gracilis +
11 Gluteus maximus + + +
12 Gluteus medius (ant) + + +
13 Gluteus medius (med) + +
14 Gluteus medius (post) + +
15 Gluteus minimus (ant)
16 Gluteus minimus (med) +
17 Gluteus minimus (post) +
18 Tensor fascia latae + +
19 Piriformis +
20 Obturator internus
21 Gemellus superior +
22 Gemellus inferior
23 Quadratus femoris +
24 Obturator externus +
25 Biceps femoris (long) +
26 Semitendinousus + +
27 Semimembranosus
%s
Percent cycle
r 500
400
-300 z a;"
-200 o G LL
100
-0
Iliopsoas
Adductor minimus
Gluteus medius (med)
Piriformis
Biceps femoris (long)
Fig. 5. The muscular forces in the different phases of step calculated with the first optimum criterion.
54
Percent cycle
800 z
600 0)" o
400 o
u_
-200
0
Iliopsoas
Adductor magnus (med)
Gluteus minimus (ant)
Gemellus inferior
Fig. 6. The muscular forces in the different phases of step calculated with the second optimum criterion
0
25 50 75
Percent cycle ioo
Y 800
1600 z
¡400 CD a
200 o ll
0
Biceps Piriformis femoris Gluteus (long)
Adductor medius 1
Iliopsoas minimus (med)
Fig. 7. The muscular forces in the different phases of step calculated with the third optimum criterion.
Further we investigated how the solution of the problem was .influenced by used restrictions on the maximal isometric force produced by the muscle Fmax. According to the study by Sotin et al. [26], the value of Fmax is directly proportional to muscular cross-sectional area, with the conversion factor K = 40 N/cm2. Since the factor K can depend on physical training of the test subject, the value of the conversion factor in our study varied from 20 N/cm2 to 60 N/cm2.
When the first objective function is employed, the solution of the problem slightly depends on the factor K, and its reduction results in the musculus semimembranosus being included in the work of the back group of muscles. At solution with the second optimum criterion the result of the factor K reduction is the addition of the work of the back group
with the first objective function during the front push (left) and back push (right).
are the same during the front and back pushes.
muscles to the work of the musculus gluteus maximus, but the joint load is nearly unchanged, When the third objective function is used, a reduction of AT leads to a 25% decrease of the hip joint load during the front push, and to small increase of the work due to the back group of muscles. Application of the value K = 60 N/cm2 results in a 25% increase of the joint load during the front push against a small reduction of the total muscular work. Thus when the calculations have been performed with the third optimum criterion, the solution of the problem is sufficiently dependent on the restrictions imposed on the peak muscular forces. As the application of the second optimum criterion at calculations brings about the increase of the joint load and the increase of the muscular work, so it is expected that the first objective function allows to find a solution the most exactly describing the biomechanics of walking.
To validate the muscular forces calculated with the first optimum criterion, they were compared with the data by Vitenzon [4] concerning the electric activity of muscles in different phases of step (Fig. 10). As can be seen from these graphs, the time advances of the peaks of muscular electric activity relative to the calculated peaks of muscular forces are from 10% to 30% of the gait cycle (at the normal gait velocity this is approximately from 100 to
Force, kg 60-1
m.Biceps femoris
100o/o
EMG, mv Force, kg r 60 60-1
40 40 -
m .Semitendinosus
EMG, mv 60
20 20
m.Gluteus médius EMG, mv Force, kg r 60 60 t
40 40
m.Tensor fascia tatae
EMG, mv ■60
- 20 20
100%
Fig. 10. Time dependence of calculated forces in muscles and electrical activity of these muscles during the gait cycle (the solid line is the solution, and the dotted line is the EMG signal).
300 msec). In the literature (e.g. in the textbook on biophysics by Antonov et al. [1]) the existence of a latent period between maximum values of the electric activity and the force of isometric muscular contraction is described. Since the length of muscles in each phase of step changes insignificantly, so it is possible to consider the muscle contraction to be isometric. The need of taking into account the time advance at the calculations has been noted in the paper by Morecki [24].
The graphs related to the muscles of the back muscular group have an additional maximum of muscular activity coincide with the back push but this maximum has no electromyographic verification. This can be caused by the fact that the averaged data collected from the studies by different authors were used as the model input specifications.
To evaluate how the solution is effected by the use of data and findings by different authors, an additional calculations were performed. The works by Davy and Audu [18] and Yoon and Mansour [30] provide two different formulae for the calculation of the moment of the passive resistance in the joint Miig.
In Fig. 11 it is shown how the moment of the passive resistance depends on which of the formulae is employed. As can be seen from the graphs in Fig. 12, the application of the formula from the work [30] brings about significant increasing the load on the joint during the second push.
The comparison of the solutions calculated with the different input data concerning the coordinates of the point of application of the foot-ground reaction force in the supporting phase of step, taken from the studies by Dushin and Svechkopal [5], and Suto and Kawamura [27] (Fig. 13), is presented in Fig. 14. It can be seen that which of two peaks is the biggest depends on the choice of the input data. The picture of forces in the muscles stays identical
Kg, Nm 80-] *
70
R/BW
Mz, Nm
6.0
5.0
4.0
3.0
2.0
... 1.0
0.0
\
\ \ -1.0
"2.0
T--—r
20 30 40 angle, deg
Fig. 11. The dependence of the moment of the passive resistance Mug on the angle between pelvis and femur in the hip joint. The moments were calculated by two different formulae in accordance with the works [IS] (solid line) and [30] (dotted line).
Fig. 12. The time dependence of the hip joint reaction force and the moment of muscular forces with respect to sagittal axis, calculated with two different formulae for the moment of passive resistance of the hip joint ligaments according to the works [18] (solid line) and [30] (dotted line).
Table 4. The muscular work calculated with the different input data by the coordinates of the
Source of information Muscular work, J
positive negative algebraic sum sum of modules
Dushin and Svechkopal [5] 1.4456 -0.9746 0.4710 2.4202
Suto and Kawamura [2:7] 1.5151 -0.9880 0.5271 2.5031
R/BW
M7, Nm
-2.0
- -1.0
-2.0
Fig. 13. The variations of coordinates of the center of foot-ground reaction force in the supporting phase of step, taken from the studies by Dushin and Svechkopal [5] (left) and Suto and Kawamura [27] (right).
Fig. 14. The time dependence of the relative hip joint reaction force and the moment of muscular forces with respect to sagittal axis, calculated in accordance with different input data by the coordinates of the center of foot-ground reaction force from the works [5] (solid line) and [27] (dotted line).
but the activity of the back group of muscles increases at an initial period of the supporting phase coinciding in time with the front push, and the work of the muscles over a gait cycle increases too (see Table 4).
Conclusions
In given work a biomechanical model of the lower limb has been constructed, and it has been employed for calculations of loads acting on the hip joint under normal walking. The
model takes into account the operations of 27 hip and pelvic muscles, and the data concerning time variations of intersegmental angles in sagittal and frontal planes. The solutions of the statically indeterminate problem have been found with three different linear objective functions. The data on the maximum muscular force have been used in the model as the restrictions. The resistance force of hip joint ligaments has been considered as an external force. Three components of foot-ground reaction force and the coordinates of the point of its application to the foot in different phases of step are the model input specifications. The loads acting on the hip joint under normal gait have been calculated by the given model. This solution is in a good agreement with the solutions obtained by the other authors. The forces exerted by the muscles and the hip joint reaction force provided in given work, may be used in the finite element investigation of stresses and strains in the proximal part of the femur. The constructed model may be employed to analyse how the loads generated in the hip joint during walking depend on the variations in the kinematic structure of gait.
References
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РАСЧЕТ НАГРУЗОК НА БЕДРО ПРИ ХОДЬБЕ
A.B. Сотин, Ю.В. Акулич, P.M. Подгаец (Пермь, Россия)
Для численного анализа напряженно-деформированного состояния проксимального отдела бедра необходимо знать усилия мышц тазобедренного сустава. Б работах, посвященных этому вопросу, обычно учитывается действие не более трех мышц, что не позволяет судить об адекватности численных результатов расчета. В связи с этим проблема определения усилий мышц при нормальной ходьбе остается актуальной. Имеющиеся исследования в данном направлении не учитывают ряд важных физиологических факторов, что не позволяет оценить точность найденных решений.
В данной работе рассчитываются мышечные усилия и реакция в тазобедренном суставе при ходьбе. Используется трехмерная квазистатическая постановка задачи при заданной кинематике межзвенных углов, Дополнительно учитываются движения таза во фронтальной плоскости, пассивное сопротивление связок тазобедренного сустава, изменение координат точки приложения опорной реакции к стопе. Модель учитывает влияние 27 мышц бедра и таза. Статически неопределимая задача решалась с помощью введения критерия поиска оптимального решения. Исследовано решение для нескольких линейных целевых функций:
п п п
j\ = YjFi min ' Jl Ä HFi/Fi,max mm > = Ai И min >
i=l Ы ' /=1
где n - число мышц, F, - сила /-ой мышцы, Fh max - максимальная изометрическая сила /-ой мышцы, А\ - работа совершаемая /-ой мышцей, При расчетах учитывались ограничения на максимальную силу мышцы.
Показано, что решение, полученное при ислюльзовании критерия Jh наиболее точно описывает биомеханику ходьбы. Сравнение результатов расчетов с данными
электромиографических исследований выявило наличие латентного периода между максимальной электрической активностью и максимальной силой, развиваемой мышцей. Приведенные в данной работе значения мышечных усилий и реакции в суставе могут быть использованы при расчете напряженно-деформированного состояния бедра. Предложенную модель можно использовать для исследования того, как изменения в кинематической структуре ходьбы влияют на нагрузки, возникающие в тазобедренном суставе.
Ключевые слова: силы мышц, нагрузки при ходьбе, критерий оптимальности
Received 23 February 2000
