CC BY
41
sci^ Russian ¡T) Journal A J of Biomechanics
www.biomech.ac.ru
STUDIES ON THE BIOMECHANICS OF PROXIMAL FEMUR
Y.V. Akulich*, A.S. Denisov**, Y.I. Nyashin*, R.M. Podgaets*, V.L. Scryabin**, Ä.V. Sotin*, A.Y. Akulich**
* Department of Theoretical Mechanics, Perm State Technical University, 29a, Komsomolsky Prospect, 614600, Perm, Russia, e-mail: [email protected]
** Department of Orthopedics and Traumatology, Perm State Medical Academy, 39, Kuybishev Street, 614000, Perm, Russia
Abstract: The article presents a survey of authors' studies over the recent five years on some topical problems of the proximal femur biomechanics: the evolution of a necrotic area in the hip head and the possibility of its replacement by the composite implant; the mechanical interaction between the composite implant and the femur bone; the numerical analysis of muscular forces acting on the hip joint during walking; the construction of phenomenoiogical kinetic equation for bone tissue internal remodeling; the influence of the scheme of therapeutic loading on the recovering of the bone tissue elastic modulus in human proximal femur after its long enforced immobilization; the computer simulation of bone tissue remodeling in proximal femur under the variations of hip joint daily loading history. The analysis of the main findings is presented, and the prospects of further investigations are outlined.
Key words: proximal femur, bone tissue, necrosis, composite material, implants, muscular forces, adaptive remodeling, rehabilitation
The biomechanical simulation of stresses and strains in the proximal femur is a vital task because of medical problems connected with traumatism, aseptic necrosis, endoprosthetics, and therapeutic recovering of the bone tissue functions. The proportion of elderly and senile femur neck fractures in Russia makes about 20% out of the total number of bone fractures. Only about 40% sufferers come back to normal life whereas the others cannot cope with the everyday duties without outside aid [1]. The statistics shows that 25% patients who have come through the femur neck fracture die during the first 6 months after the trauma. Surgical treatment of the femur neck fractures decreases the death rate down to 15% while under the conservative treatment it increases up to 40% [2]. Active surgical tactics for fractures of this localization is commonly acknowledged, anyway the results of operations are not optimistic: in post-operational period 25% of patients develop serious arthrosis, and other 18% develop aseptic necrosis of the hip head and femur neck. According to data of Perm Regional Health Care Department, the regional annual demand for hip endoprostheses is from 200 to 250. These data signify the high priority of the proximal femur biomechanical problems in general, and in particular a need for individual computer prognosis of surgical treatment of hip head aseptic necrosis and the total endoprosthetics of hip head and femur neck, and for improvement the existing methods of post-operational rehabilitation with the aim to assign such individual loading regime in the rehabilitation period that does not result in local fractures of bone tissue.
Introduction
Fig. 2, The final stage of necrosis spreading of the cases: a - without substitution of the necrotic zone for connective tissue; b - with substitution the necrotic zone for material with elastic modulus E{= 70 MPa; c - Ef= 94 MPa;. - substituted area, - necrotical area.
2. Simulation of the necrotic zones replacement
Follow to two-dimensional finite element method simulation of the necrosis development in hip head (see Ueo et al. [3]), in paper [4] we studied the biomechanical possibility of replacement of the necrotic bone tissue by the carbon composite implant. The investigated area, loads and kinematics boundary conditions are shown in Fig. 1. The external loads on the joint were the same as in [3] (the weight was 600 N, and abductor, adductor and iliacus muscle's forces were 1500 N, 100 N and 300 N, respectively).
The simulation of the necrotic zone spreading was performed as follows. The finite elements in which von Mises stress (stress intensity) a, reached the critical value (for healthy spongy bone ct( = 5 MPa) were considered as destroyed, i.e. they turned into necrotic area. The outcome of the necrosis spreading is shown in Fig. 2.
These results are obtained under two assumptions: i) the process of replacement of the necrotic area is absent; ii) the necrotic area is substituted by a material with the various Young's modulus. The performed investigations confirm the technical possibility of substitution the affected bone tissue for composite implant, if the Young modulus of the implant material is not less than 94 MPa. The surgical possibilities of substitution the necrotic area in the hip head for the artificial implant do not studied at present, however the development of such operational technique would permit to avoid the autoplastics and retard the occurrence of more dangerous injuries of the hip joint. The further biomechanical studies in this area are connected with regard for the processes of damage and adaptation.
3. Simulation of the bone-implant mechanical action
The results of two-dimensional finite element method simulation of mechanical interaction between carbon femoral implant and the bone are present in paper [4]. The investigated area, loads and kinematics boundary conditions were the same as in paper [3] (see Fig. 3).
The aim of the investigation was to fit a design and mechanical properties of carbon composite implant. The selection criterion was the treshold value of the implant stem thrust on the bone channel wall of 2.0-2.2 MPa. The pressure higher than the critical value causes bone resorption at his position. The four alternatives of implant stem design were considered. All the designs had identical forms but various on their length Young's moduli as shown in Fig. 4. In the A-variant the homogeneous stem with elastic modulus of 70 GPa was considered. The carbon composite material stem in the B-variant consisted of two parts; Young's modulus was 70 GPa in proximal part and 10 GPa in distal part. In the case C, the stem contained three parts with elastic moduli of 70 GPa, 20 GPa and 1 GPa in proximal, middle and distal parts, respectively. The stem in the D-variant consisted of rigid core with Young's modulus of 70 GPa and a softer cover with thickness of 1 mm and modulus of 1 GPa.
Fig. 3. The geometry of proximal prosthetic femur and typical loads. Muscle's forces: abductor - 1500 N, adductor - 100 N, iliacus - 300 N, pelvic reaction - 2440 N.
Fig. 4. Studied variants of the stem design: A - homogeneous, B - two parts; C - three parts; D - two parts.
Fig. 5. The medial and lateral distributions of normal stresses an calculated for the design D.
The comparison of these alternatives of implant's design showed that the implant with a softer stem cover is the most acceptable (Fig. 4d, Fig. 5).
The typical curves of the normal stress distributions placed in adjacent to stem bone's strata and calculated for the A-variant are shown in Fig. 6. Here the carbon femoral implant stem elastic modulus is equal to 70 GPa.
From the comparison of these stress distributions it can be seen that the D-design implant stem transfers the stress to the bone channel wall more uniformly, and the stress does
not reach the critical value. On the contrary, the installation of the A-design implant results in advent of two resorptional areas. Therefore, the application of the D-design implant would permit to avoid the bone resorption around implant's stem, and would prolong its service life. This is essential importance for the middle age patients. Further we intend to carry out the fatigue tests of the endoprosthesis and the investigation of the bone behaviour taking into account its adaptive characteristics.
4. Constitutive equation of the elastic adaptive media
We started to study the problem of mathematical simulation of bone tissue adaptive remodeling in 1998. In paper by Akulich and Podgaets [5] the technique of the construction of bone tissue constitutive equation was worked out. This constitutive equation was based on the kinetic equation of adaptive remodeling (1) connecting the rate of Young's modulus change with the strain stimulus of bone cells activity:
hom .
£(x,o = c ||s(x,o||
(X)
X £ Q , (1)
where s, shom are Cauchy strain tensors during the remodeling and at homeostatic equilibrium, respectively; j| • || is an appropriate norm of strain tensor; C is a remodeling rate
parameter; x is a bone partial site vector in a considered area Q.
During the adaptive process, stress and strain in the bone tissue and its mechanical parameters are continuously and monotonously varied. Hence, the equation of Hook's law contains continuous time functions, and we can differentiate it with respect to time. In one-dimensional problem it may be written as
g = Es + Ee ^ (2)
where a, E, e are stress, Young's modulus and strain, respectively. Substitution of relation (1) in (2) gives us desired constitutive equation of the elastic adaptive media:
a = C(s-shom)e + £s. (3)
The computer solution of the equation (3) used for the one-dimensional test task about the elastic cylinder adaptable under one step increase in twice of the axial compression run monotonously to the homeostatic value. This is qualitative justification of the construction of bone tissue constitutive equation method. The kinetic equation of adaptive remodeling form (1) has important valuation as it makes possible (under some hypothesis about bone cells activity as function of the strain stimulus) to calculate the remodeling rate parameter C. Assuming the active bone cells number as linear function of the strain stimulus the calculation algorithm has presented in paper [6]. The absolute value of C of femur cortical bone tissue calculated by this algorithm is equal to 40.9MPa/(day-%s). Further we intend to spread this method on the anisotropic elastic properties of femur bone tissue in three-dimensional case.
4. Prediction of the recovering the bone tissue mechanical properties in the post-
operational rehabilitation period
Under different kinds of human physical activity different loads act on the hip joint and proximal femur, and they are dependent on the external forces and the forces generated by the muscles attached to the femur. The loads are cyclic, and during the study of stresses and strains in the femur their maximal magnitudes (or amplitudes in symmetric cycles) were used. In our studies [7-10] the investigation of stresses and strains in the proximal femur was carried out by quasi-two-dimensional finite element model with side cortical plates (see Svesnsson et al. [11]). We employed simplified two-dimensional model of adult proximal
femur with only two forces: the joint reaction force and the hip adductor force. In this model the joint reaction force was presented as a pressure distributed on an arc about a quarter of the hip head circle by the cosine law, with the resultant force Fi directed at angle a with the vertical. The hip adductor force was considered as a uniformly distributed pressure acting on the greater trochanter surface, with the resultant force F2 inclined at angle p to the vertical. Depending on the character of human physical activities the magnitudes and directions of these forces for the twenty-four hours are different, and the bone tissue architecture and mechanical characteristics are dependent on their overall effect. According to Carter et al. [12], the following three load cases were considered (see Fig. 7). The first load case with maximal magnitudes of forces acting on the proximal femur corresponds to one-limb-stance phase of gait, and two other load cases are selected to represent extreme ranges of human motion with reduced force levels. As a normal physiological load per day we assumed 6000 cycles of load applications for the first load case and 2000 loading cycles each for the other two cases.
In order to study the bone tissue remodeling process by the kinetic equation (1), we have to know:
i) the initial pattern of bone tissue elastic modulus and pattern of homeostatic strain intensity in the proximal femur under the above mentioned daily physiological loading history;
ii) the current distribution of the strain intensity s ¡(x,t) for the multiple loading by forces
with different magnitudes and directions.
The first problem was solved in our paper [7] regarding the bone tissue adaptation during the morphogenesis, with the energy remodeling stimulus proposed by Fyhrie and Carter [13]. The obtained pattern of elastic modulus corresponds well enough with the experimentally determined elastic modulus field in the proximal femur (Brown et al. [14, 15]). To answer the second question, three single-loading-direction solutions for each load case are calculated for each time instant, and in local kinetic equation (1) for each finite
element the current strain intensity '¿¡(x,l) is meant as an equivalent strain intensity
calculated by averaging the strain intensities referred to all the load cases, with taking into account their numbers of loading cycles.
Fig. 7. The central layer of the proximal femur finite element model under three loads cases:
a) the first load case: F, = 2317 N; a = 24°; F2 = 702 N; (3 = 28°;
b) the second load case: F, = 1158 N; a = -15°; F2 = 351 N; p = -8°;
c) the third load case: F, = 1548 N; a = 56°; F2 = 459 N; p = 35°.
In order to take into account the difference between remodeling rate factors in weakened and strengthened bone, it was assumed that the remodeling rate factor under
weakening the bone tissue (C ) was 1.33 times larger than the factor under bone
strengthening (C+). That is in a good agreement with medical expertise, saying that the resorption of a bone tissue is a faster process than its reposition. To bring the remodeling rate factors to the real time scale, their specification was carried out, where the time unit was one day. The calculations were performed with the following values of the remodeling rate
factors: C+ =960, C~ =1280.
In papers by Akulich et al. [7, 8] a numerical experiments were performed for prediction of the changes in mechanical characteristics of sponge bone tissue in the hip head and femur neck during the post-operational rehabilitation period. The main feature of this period is a gradual recovering the bone tissue elastic modulus after the long period of low load in the conditions of bed regimen that takes place due to the special system of therapeutic motions.
In our computations of the bone rehabilitation process we considered the stress, strain and elastic modulus patterns under the normal physiological load as an initial state. The first stage was a model of the joint immobilization after the trauma or surgical intervention; here the load fell abruptly down to 5% of the normal. The duration of this stage was 60 days. In the next stages, from the second to the seventh ones, the load has been monthly increasing up to 10%, 25%, 45%, 70%, 95% and 100% of the physiological load, respectively. The assumption was made that the time variations of therapeutic loading consisted only in proportional change of force magnitudes, while their inclinations to the vertical and a number of each type loading cycles remained unchanged. The time variations of the bone tissue elastic modulus and the strain intensity in the same finite element at the lateral side of the femur neck are plotted in Fig. 8 (see Akulich et al. [8]). It can be seen that elastic modulus at each stage tends to some magnitude, equilibrated for the given level of load, and the strain intensity tends to the homeostatic one. The final distribution of the elastic modulus was nearly the same as the initial one.
0.012 -■
Fig. 8. The elastic modulus of the bone tissue in a finite element at the lateral side of the femur neck (left) and the strain intensity in the same finite element (right) during the process of
post-operational rehabilitation.
800
600
400
200
-1—I—'—I—'—I ' I ' I
60 120 180 240 300 360 Time, days
0.010
2? 0.008 '</> c <D
1 0.006
c
2
CO 0.004
0.002 0.000
homeostatic strain intensity
I—1—1—1—1—'—I—r~l 1 I 60 120 180 240 300 360 Time, days
In order to take into account the difference between remodeling rate factors in weakened and strengthened bone, it was assumed that the remodeling rate factor under
weakening the bone tissue (C~) was 1.33 times larger than the factor under bone
strengthening (C+). That is in a good agreement with medical expertise, saying that the resorption of a bone tissue is a faster process than its reposition. To bring the remodeling rate factors to the real time scale, their specification was carried out, where the time unit was one day. The calculations were performed with the following values of the remodeling rate
factors: C+ =960, C" =1280.
In papers by Akulich et al. [7, 8] a numerical experiments were performed for prediction of the changes in mechanical characteristics of sponge bone tissue in the hip head and femur neck during the post-operational rehabilitation period. The main feature of this period is a gradual recovering the bone tissue elastic modulus after the long period of low load in the conditions of bed regimen that takes place due to the special system of therapeutic motions.
In our computations of the bone rehabilitation process we considered the stress, strain and elastic modulus patterns under the normal physiological load as an initial state. The first stage was a model of the joint immobilization after the trauma or surgical intervention; here the load fell abruptly down to 5% of the normal. The duration of this stage was 60 days. In the next stages, from the second to the seventh ones, the load has been monthly increasing up to 10%, 25%, 45%, 70%, 95% and 100% of the physiological load, respectively. The assumption was made that the time variations of therapeutic loading consisted only in proportional change of force magnitudes, while their inclinations to the vertical and a number of each type loading cycles remained unchanged. The time variations of the bone tissue elastic modulus and the strain intensity in the same finite element at the lateral side of the femur neck are plotted in Fig. 8 (see Akulich et al. [8]). It can be seen that elastic modulus at each stage tends to some magnitude, equilibrated for the given level of load, and the strain intensity tends to the homeostatic one. The final distribution of the elastic modulus was nearly the same as the initial one.
ro Q.
</> =j
T3
o E
o «
ro LLI
1000-
800
600-
400-
200
I-1-1-'-1-'-r~
60 120 180 240 Time, days
0.012
0.010
¿> 0.008 V) c a>
1 0.006
c
«5
W 0.004 0.002 0.000
300 360
U
homeostatic strain intensity
60 120 180 240 Time, days
~ 1 I 300 360
Fig. 8. The elastic modulus of the bone tissue in a finite element at the lateral side of the femur neck (left) and the strain intensity in the same finite element (right) during the process of
post-operational rehabilitation.
Another scheme was considered in paper by Akulich et al. [8]. According to that scheme the joint load after two-months immobilization was increased up to the same levels step by step, though not monthly, but every five days. In order to estimate the safety of both schemes of changing the daily loading history of the hip joint, the bone tissue fracture criterion was considered. In paper by Brown et al. [16] it is presumed that such a criterion is stress-to-strength ratio or safety factor (reciprocal of stress-to-strength ratio). If stress-to-strength ratio approaches to unity, probability of bone tissue collapse critically increases. We accepted the stress intensity as a measure of a stress state and used a hypothesis that bone tissue collapse occurred in a location corresponding to some finite element, when the stress intensity in this element reached the strength limit. It was shown in the experiments by Brown et al. [14] that the strength limits of bone tissue samples from different parts of proximal femur were proportional to their local elastic modules. It was supposed that this proportion remained valid also under variations of local elastic modulus in consequence of bone tissue remodeling. The proportionality factor has been found from the condition that average sponge bone elastic modulus of 500 MPa corresponds to strength limit of 5.5 MPa (Ueo, Tsutsumi et al [3]). By this approach it was shown that the obviously unreal scheme with the load increase every five days is actually dangerous because the stress intensities in most elements at lateral and medial sides of the femur neck exceed corresponding strength limits.
In paper by Akulich et al. [9] a variation of the hip joint loading history was considered as some hypothetical modification of human activity that causes changes in relative contribution of each load case, with unchanged magnitudes and directions of the forces in these load cases. With the aim of simulation of the process of bone tissue remodeling we considered a situation when a person changes abruptly his or her physical activities over a period of three months, and then returns to customary loads. As a changed human physical activity we assumed daily loading history consisted of 500 loading cycles each for the first and second loads cases and 9000 cycles of load applications for the third load case. Under normal physiological multiple-direction loading, when the first load case is prevailing as in force magnitudes as in number of loading cycles, the current distribution of strain intensities coincides with the homeostatic one, and any remodeling does not proceed, i.e. the elastic modulus pattern remains unchanged. If the third load case becomes prevailing in number of loading cycles, it causes the deviations of the current strain intensities in finite elements from the homeostatic pattern, and due to this strain stimulus, the remodeling process of the bone tissue elastic modulus is initiated.
T ine - O
1800.0
m io,
1640.O 140O.O 1320.0 1160.0 lOOO.O 640.0 680.0 520.0 360.0 200.0
Fig. 9. The elastic modulus pattern in the hip head and femur neck under normal physiological loading condition (in MPa).
Til« = 90
lOOO.O
■ io,
913.6
837.2
740.7
654.3 367.9
481.5 393. 1
308.6 222.2
135.8
Fig. 10. The elastic modulus pattern in the hip head and femur neck in 90 days after the daily loading history was changed (in MPa).
As the initial state we used elastic modulus pattern under the prescribed above normal physiological loading history (Fig. 9), and Fig. 10 shows the elastic modulus distribution after 90 days of changed loading condition. It can be seen that change of the prevailing direction of forces acting on the hip head causes appreciable modification of the bone tissue architecture, and specifically the bone tissue weakening at the medial surface of femur neck. After the customary physiological load was restored, the elastic modulus pattern was recovering step by step, and in six months it was nearly the same as the original one.
In paper [9] the problem of the bone tissue safety factor under the multiple-direction loading has been discussed. While the bone tissue remodeling is dependent on the averaged action of all the load cases calculated with taking into account their numbers of loading cycles, the safety factor has to be calculated for each load case separately, and the actual safety factor is the minimal one.
In the considered papers [8-10] a step in the direction of a more exact quantitative description of the bone remodeling process has been made. The analysis of computer simulations proves that the phenomenological model used in the present study describes the bone tissue remodeling qualitatively correct not only when the force amplitudes of different load cases are changed but also under the modification of the character of human physical activity which results in variations of cycle numbers of these load cases with different magnitudes and directions of forces acting on the proximal femur. The obtained data show that the architecture of the bone tissue of the hip head and femur neck is changing under the modification of the daily loading history but after the restoration of usual loading regime it gradually recovers itself.
Meanwhile many problems of the hip joint loading and bone tissue remodeling require further theoretical and clinical investigations, e.g. refinement of the forces acting on the hip joint during different kinds of human physical activity, experimental determination of bone remodeling rate factors in different parts of the proximal femur, the bone tissue mechanical properties and strength criteria, etc.
Conclusions
The over adduced survey shows necessary of application the three-dimensional finite element method simulation to the proximal femur stress-strain state, of accounting the anisotropic bone tissue elastic properties, of theoretical development and verification the constitutive equation (3) in three-dimensional case. The experimental evaluations of used numerical models adequacy and the hip carbon femoral implant durability, and also calculation of the recovery bone tissue individual programs used by special trainer are quite important.
References
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3. UEO Т., TSUTSUMIS., YAMAMOTOT., OKIMURA H„ SHIMIZIA., NAKAMURAT. Biomechanical aspects of the development of aseptic necrosis of the femoral head. Arch Orthop Trauma Surg, 104: 145-149, 1985.
4. DENISOV A.S., NYASHIN Yu.l., AKULICH Yu.V., ZMEEV Yu.A., OSORGINYu.K PODGAYETS R.M., SCRYABIN V.L., SOTIN A.V, Some aspects of carbon composite material in human hip joint prosthesis. Russian Journal of Biomechanics, 1 (1-2): 12-24, 1997.
5. AKULICH Yu.V., PODGAYETS R.M. The phenomenological model of adaptable adult spongy bone tissue. Russian Journal of Biomechanics, 2 (1-2): 53-57, 1998.
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9. AKULICH Yu.V., DENISOV A.S., NYASHIN Yu.l., PODGAYETS R.M., AKULICH A.Yu. The influence of the scheme of loading variations on the recovering of the bone tissue elastic modulus. Russian Journal of Biomechanics, 3 (3): 63-72, 1999.
10. AKULICH Yu.V., NYASHIN Yu.l., PODGAYETS R.M., DMITRENOK I.A. The bone tissue remodeling in the proximal femur under the variations of hip joint dally loading history. Russian Journal of Biomechanics, 5(1): 12-23, 2001
11. SVESNSSON N.L., VALLIAPAN S., WOOD R.D. Stress analysis of human femur with implanted charnley prosthesis. J Biomech., 10: 581-588, 1977.
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ИССЛЕДОВАНИЯ ПО БИОМЕХАНИКЕ ПРОКСИМАЛЬНОГО ОТДЕЛА
БЕДРА
Ю.В. Акулич, A.C. Денисов, Ю.И. Няшин, P.M. Подгаец, В.Л. Скрябин, A.B. Сотин, А.Ю. Акулич (Пермь, Россия)
В статье представлен обзор результатов исследований, выполненных авторами за последние пять лет по ряду актуальных проблем биомеханики проксимального отдела бедра: развитие некротической области в головке бедра и возможность её замещения имплантатом из композиционного материала; механическое взаимодействие имплантата из композиционного материала и бедренной кости; построение кинетического и определяющего соотношений упругой приспосабливающейся к изменяющимся нагрузкам среды, реабилитационное восстановление механических свойств губчатой костной ткани головки и шейки бедра после вынужденной иммобилизации конечности; компьютерное моделирование процесса перестройки костной ткани проксимального отдела бедра при изменении характера нагрузок на тазобедренный сустав. Приведен анализ результатов и намечены перспективы дальнейших исследований. Библ. 16.
Ключевые слова: проксимальный отдел бедра, костная ткань, некроз, композитный материал, имплантаты, усилия мышц, адаптивная перестройка, реабилитация
Received 15 June 2001
