Sensitivity Analysis and Optimal Shape Design for Bone-Prosthesis Interfaces in a Femoral Head Surface Replacement

Abstract

A numerical optimization procedure has been applied for the shape optimal design of a femoral head surface replacement. The failure modes of the prosthesis that were considered in the formulation of the objective functions concerned the interface stress magnitude and the bone remodelling activity beneath the implant. In order to find a compromising solution between different requirements demanded by the two objective functions, a two step optimization procedure has been developed. Through step I the minimization of interface stress was achieved, through step 2 the minimization of bone remodelling was achieved with constraints on interface stresses.

The results obtained provided an optimal design that generates limited bone remodelling activity with controlled interface stress distribution.

The computational procedure was based on the application of the finite element method, linked to a mathematical programming package and a design sensitivity analysis package.     

The physiological basis for a flexible condylar tibial plateau design

Knee resurfacing is a successful treatment for osteo- and rheumatoid arthritis in elderly patients. The application of this treatment to younger more active and obese persons has the potential to produce premature wear, loosening, and undesirable bone remodelling. A new generation of more physiologically compatible components is required for these situations. This paper discusses the design and analysis of a prototype tibial base plate aimed at physiological load transfer. Incorporated in the design are mechanisms to alleviate lift-off phenomena, bone stress concentrations, stress shielding, and micromotion at the bone-implant interface. The design requires viable cancellous bone stock, so that the bone may respond by remodelling to the dynamic loading during normal ambulatory activities.     

Acetopyruvate hydrolase production by Pseudomonas putida O1--optimization of batch and fed-batch fermentations

The main objective of this work was the optimization of the production of the beta-ketolase, acetopyruvate hydrolase, from Pseudomonas putida O1. Orcinol was used as an inducer for enzyme production. The growth medium was optimized in two steps. In the first step, screening for optimal glucose concentration was performed. In the second step, a central composite design was used to optimize carbon and nitrogen sources in the medium. After this optimization procedure, a medium was obtained which produced seven times more biomass than the initial medium. Acetopyruvate hydrolase enzyme production was optimized by determining the optimal time of feed and amount of orcinol, using statistical methods. In a subsequent step, the maximal orcinol-degradation rate was determined. The results obtained were used to find an optimal feeding profile for enzyme production. By using the optimized fed-batch process, acetopyruvate hydrolase activity was enhanced from 10 units l(-1)to 400 units l(-1), in comparison with previously reported fermentation experiments. Productivity could even be increased by a factor of 75, to a value of 20 units l(-1 )h(-1).     

Prediction of bone adaptation in the ulnar-osteotomized sheep's forelimb using an anatomical finite element model

A method for the prediction of the time-course of bone adaptation based on an alternative hypothesis of strength optimization has been previously investigated and developed by Prendergast and Taylor¹. This paper extends our work in the study of the effectiveness of this bone adaptation model in predicting similar bone remodelling to that observed in animal experiments. In particular the experimental work which has been modelled is that of Lanyon, Goodship, Pye and McFie². An anatomical finite element model of the sheep's forelimb has been generated for this purpose and is used to estimate stresses in the bone structure for the normal and osteotomized condition. The propensity for remodelling of the altered bone structure is predicted using the proposed remodelling law for the new stress field in the bone structure. The preliminary results indicate an initial bone adaptation pattern similar to that observed experimentally without the necessity to use arbitrarily different constants for the endosteal and periosteal surfaces. We therefore suggest that the remodelling law based on damage and repair gives a better predictive model of bone adaptation than previous models.     

Mathematical shape optimization of hip prosthesis design

The long-term success of artificial-joint replacement depends partly on the chances for acrylic cement failure and interface disruption. These chances can be diminished by an optimal load-transfer mechanism, whereby stress concentrations are avoided. The present paper introduces a method for numerical shape optimization, whereby the finite element method is used iteratively to determine optimal prosthetic designs, which minimize interface stresses. The method is first applied in a simplified one-dimensional model of a cemented femoral stem fixation, using acrylic cement. The results show that 30-70% cement and interface stress reductions can be obtained in principle with an optimized design. Although the actual optimal shape is susceptible to the characteristics of the joint load, the stem length, stem modulus, cement modulus and bone properties, its general geometrical characteristics are consistent, featuring proximal and distal tapers, and a belly-shaped middle region. These general characteristics are confirmed in a more realistic two-dimensional FEM model. It is concluded that this method of shape optimization can provide a meaningful basis for prosthetic design and analysis activities in general.     

Fluid-induced osteolysis: modelling and experiments

A model to calculate bone resorption driven by fluid flow at the bone–soft tissue interface is developed and used as a basis for computer calculations, which are compared to experiments where bone is subjected to fluid flow in a rat model. Previous models for bone remodelling calculations have been based on the state of stress, strain or energy density of the bone tissue as the stimulus for remodelling. We believe that there is experimental support for an additional pathway where an increase in the amount of the cells directly involved in bone removal, the osteoclasts, is caused by fluid pressure, flow velocity or other parameters related to fluid flow at the bone–soft tissue interface, resulting in bone resorption.     

Fluid-induced osteolysis: modelling and experiments

A model to calculate bone resorption driven by fluid flow at the bone-soft tissue interface is developed and used as a basis for computer calculations, which are compared to experiments where bone is subjected to fluid flow in a rat model. Previous models for bone remodelling calculations have been based on the state of stress, strain or energy density of the bone tissue as the stimulus for remodelling. We believe that there is experimental support for an additional pathway where an increase in the amount of the cells directly involved in bone removal, the osteoclasts, is caused by fluid pressure, flow velocity or other parameters related to fluid flow at the bone-soft tissue interface, resulting in bone resorption.     

The effect of three-dimensional shape optimization on the probabilistic response of a cemented femoral hip prosthesis

Probabilistic analyses allow the effect of uncertainty in system parameters on predicted model performance measures to be determined. Furthermore, using performance functions to describe a failure event, the probability of failure can be quantified. The effect of three-dimensional prosthesis shape optimization on the probabilistic response and failure probability of a cemented hip prosthesis system is investigated. Random variables include joint and muscle loading, cortical and cancellous bone and PMMA bone cement elastic properties, and strength parameters describing failure of the bone cement and the prosthesis-bone cement interface. Several performance functions describing the bone cement and prosthesis-cement interface are used to compute the probability of failure. When evaluated deterministically, most performance functions indicated a safe design, with the exception of interface tensile failure. However, when evaluated probabilistically, finite probabilities of failure were computed, some significant. The most likely mode of failure before shape optimization was prosthesis-bone cement interface tensile failure with a predicted probability of failure of 97.9%. Deterministic prosthesis shape optimization reduced the probability of failure for all performance functions and reduced prosthesis-bone cement interface tensile failure by 31.7%. Probability sensitivity factors indicate that the uncertainty in the joint loading, cement strength, and implant-cement interface strength have the greatest effect on the computed probability of failure. Implant shape optimization results in a more robust implant design that is less sensitive to uncertainties in joint loading, which cannot be easily controlled, and more sensitive to cement and interface properties, which are easier to modify.     

Implant–bone interface healing and adaptation in resurfacing hip replacement

Hip resurfacing demonstrates good survivorship as a treatment for young patients with osteoarthritis, but occasional implant loosening failures occur. On the femoral side there is radiographic evidence suggesting that the implant stem bears load, which is thought to lead to proximal stress shielding and adaptive bone remodelling. Previous attempts aimed at reproducing clinically observed bone adaptations in response to the implant have not recreated the full set of common radiographic changes, so a modified bone adaptation algorithm was developed in an attempt to replicate more closely the effects of the prosthesis on the host bone. The algorithm features combined implant–bone interface healing and continuum bone remodelling. It was observed that remodelling simulations that accounted for progressive gap filling at the implant–bone interface predicted the closest periprosthetic bone density changes to clinical X-rays and DEXA data. This model may contribute to improved understanding of clinical failure mechanisms with traditional hip resurfacing designs and enable more detailed pre-clinical analysis of new designs.     

Prediction of shape and internal structure of the proximal femur using a modified level set method for structural topology optimisation

A computational framework was developed to simulate the bone remodelling process as a structural topology optimisation problem. The mathematical formulation of the Level Set technique was extended and then implemented into a coronal plane model of the proximal femur to simulate the remodelling of internal structure and external geometry of bone into the optimal state. Results indicated that the proposed approach could reasonably mimic the major geometrical and material features of the natural bone. Simulation of the internal bone remodelling on the typical gross shape of the proximal femur, resulted in a density distribution pattern with good consistency with that of the natural bone. When both external and internal bone remodelling were simulated simultaneously, the initial rectangular design domain with a regularly distributed mass reduced gradually to an optimal state with external shape and internal structure similar to those of the natural proximal femur.     

Functional strain in bone tissue as an objective, and controlling stimulus for adaptive bone remodelling

The skeleton consists of a series of elements with a variety of functions. In locations where shape or protection are of prime importance the bone's architecture is achieved during growth under direct genetic control. In locations where resistance to repetitive loading is important only the general form of the bone will be achieved as a result of growth alone, the remaining characteristics result from functional adaptation. This mechanism ensures that bone architecture is modelled and remodelled until prevailing strains match those genetically prescribed for that location. For this match to be established, and subsequently maintained, bone cells must be able to 'assess' feedback derived directly or indirectly from the functional strains produced within the tissue. These strains are therefore the objective of functionally adaptive remodelling, and the stimulus for its control. Evans was the first person to refer to the recording of functional strains from gauges attached to bone in vivo. This technique has allowed quantitative investigations on bone's normal functional strain environment, and its adaptive response to changes in its state of strain. Recent investigations have extended to the immediate effects of dynamic strains on the structure of the bone matrix, and the biochemical behaviour of the resident bone cells. Such studies should reveal the mechanism by which strains within the matrix are transduced into the biochemical signals by which adaptive remodelling is controlled.     

Prediction of shape and internal structure of the proximal femur using a modified level set method for structural topology optimisation

A computational framework was developed to simulate the bone remodelling process as a structural topology optimisation problem. The mathematical formulation of the Level Set technique was extended and then implemented into a coronal plane model of the proximal femur to simulate the remodelling of internal structure and external geometry of bone into the optimal state. Results indicated that the proposed approach could reasonably mimic the major geometrical and material features of the natural bone. Simulation of the internal bone remodelling on the typical gross shape of the proximal femur, resulted in a density distribution pattern with good consistency with that of the natural bone. When both external and internal bone remodelling were simulated simultaneously, the initial rectangular design domain with a regularly distributed mass reduced gradually to an optimal state with external shape and internal structure similar to those of the natural proximal femur.     

Stress changes in the femoral head due to porous ingrowth surface replacement arthroplasty

Finite element stress analyses were conducted of the canine femoral head before and after implantation of various surface replacement-type components. The femoral head was replaced by four implant geometries; (a) shell, (b) shell with peg, (c) shell with rod, and (d) a new epiphyseal replacement design. All implants were modelled to simulate bony ingrowth along the underside of the shell and along the surfaces of the peg and rod. The results indicated that in the normal femur the forces are transferred from the articular surface through the femoral head cancellous bone to the inferior cortical shell of the femoral neck. After shell-type surface replacement, forces were transferred more distally at the rim of the shell and at the end of the peg or rod, thereby reducing the stresses in the proximal head cancellous bone. Computer simulation of bone remodelling due to proximal bone stress reduction was shown to accentuate the abnormality of the stress fields. Surface replacement with a lower modulus material created a less abnormal redistribution of bone stresses. The new epiphyseal replacement design resulted in stress distributions similar to those in the normal femoral head and minimal shear stresses at the implant/bone interface. These findings suggest that the epiphyseal replacement concept may provide better initial mechanical integrity and create a more benign milieu for adaptive bone remodelling than conventional, shell-type surface replacement components.     

Implant-bone interface healing and adaptation in resurfacing hip replacement

Hip resurfacing demonstrates good survivorship as a treatment for young patients with osteoarthritis, but occasional implant loosening failures occur. On the femoral side there is radiographic evidence suggesting that the implant stem bears load, which is thought to lead to proximal stress shielding and adaptive bone remodelling. Previous attempts aimed at reproducing clinically observed bone adaptations in response to the implant have not recreated the full set of common radiographic changes, so a modified bone adaptation algorithm was developed in an attempt to replicate more closely the effects of the prosthesis on the host bone. The algorithm features combined implant-bone interface healing and continuum bone remodelling. It was observed that remodelling simulations that accounted for progressive gap filling at the implant-bone interface predicted the closest periprosthetic bone density changes to clinical X-rays and DEXA data. This model may contribute to improved understanding of clinical failure mechanisms with traditional hip resurfacing designs and enable more detailed pre-clinical analysis of new designs.     

On the role of bone damage in calcium homeostasis

Bone serves as the reservoir of some minerals including calcium. If calcium is needed anywhere in the body, it can be removed from the bone matrix by resorption and put back into the blood flow. During bone remodelling the resorbed tissue is replaced by osteoid which gets mineralized very slowly. Then, calcium homeostasis is controlled by bone remodelling, among other processes: the more intense is the remodelling activity, the lower is the mineral content of bone matrix. Bone remodelling is initiated by the presence of microstructural damage. Some experimental evidences show that the fatigue properties of bone are degraded and more microdamage is accumulated due to the external load as the mineral content increases. That damage initiates bone remodelling and the mineral content is so reduced. Therefore, this process prevents the mineral content of bone matrix to reach very high (non-physiological) values. A bone remodelling model has been used to simulate this regulatory process. In this model, damage is an initiation factor for bone remodelling and is estimated through a fatigue algorithm, depending on the macroscopic strain level. Mineral content depends on bone remodelling and mineralization rate. Finally, the bone fatigue properties are defined as dependent on the mineral content, closing the interconnection between damage and mineral content. The remodelling model was applied to a simplified example consisting of a bar under tension with an initially heterogeneous mineral distribution. Considering the fatigue properties as dependent on the mineral content, the mineral distribution tends to be homogeneous with an ash fraction within the physiological range. If such dependance is not considered and fatigue properties are assumed constant, the homogenization is not always achieved and the mineral content may rise up to high non-physiological values. Thus, the interconnection between mineral content and fatigue properties is essential for the maintenance of bone's structural integrity as well as for the calcium homeostasis.     

Experimental design as a tool when evaluating stationary phases for the capillary electrochromatographic separation of basic peptides

Two different capillary electrochromatography (CEC) stationary phases, Hypersil phenyl and Hypersil C(18), have been characterised with respect to their ability to separate the four basic peptides H-Tyr-(D)Ala-Phe-Phe-NH(2) (TAPP), H-Tyr-(D)Ala-Phe-NH(2) (TAP), H-Phe-Phe-NH(2) (PP) and H-Phe-NH(2) (P). Optimal separation conditions were first established separately for the two phases by applying experimental design in a stepwise procedure. The first step comprised a study to acquire basic knowledge about the variables, their influence on the response and their respective experimental domains for each of the two stationary phases. The second step was screening the significant variables and the third step was an optimisation with response surface modelling (RSM) to locate the optimum separation conditions for each stationary phase. The experimental procedure was identical for both stationary phases, but their respective experimental domains were different. The response functions were peak resolution and peak efficiency. This procedure enables specific optimal experimental conditions to be identified for each of the two stationary phases. The optimal conditions identified for the separation on the phenyl stationary phase were to use 50% ACN, 20% 50 mM Tris(hydroxymethyl)aminomethane (TRIS) pH 7.5, 30% H(2)O as BGE, operating at 20 degrees C and 20 kV high voltage. For the C(18) stationary phase optimal separation was achieved using a BGE with 80% ACN, 20% 30 mM TRIS pH 8.5, again operating at 20 degrees C and 20 kV high voltage. Results show that the phenyl stationary phase is better suited for the separation of basic, hydrophilic peptides.     

On the Use of Bone Remodelling Models to Estimate the Density Distribution of Bones. Uniqueness of the Solution

Bone remodelling models are widely used in a phenomenological manner to estimate numerically the distribution of apparent density in bones from the loads they are daily subjected to. These simulations start from an arbitrary initial distribution, usually homogeneous, and the density changes locally until a bone remodelling equilibrium is achieved. The bone response to mechanical stimulus is traditionally formulated with a mathematical relation that considers the existence of a range of stimulus, called dead or lazy zone, for which no net bone mass change occurs. Implementing a relation like that leads to different solutions depending on the starting density. The non-uniqueness of the solution has been shown in this paper using two different bone remodelling models: one isotropic and another anisotropic. It has also been shown that the problem of non-uniqueness is only mitigated by removing the dead zone, but it is not completely solved unless the bone formation and bone resorption rates are limited to certain maximum values.     

On the optimal shape of hip implants

The success of a total hip arthroplasty is strongly related to the initial stability of the femoral component and to the stress shielding effect. In fact, for cementless stems, initial stability is essential to promote bone ingrowth into the stem coating. An inefficient primary stability is also a cause of thigh pain. In addition, the bone adaptation after the surgery can lead to an excessive bone loss and, consequently, can compromise the success of the implant. These factors depend on prosthesis design, namely on material, interface conditions and shape. Although, surgeons use stems with very different geometries, new computational tools using structural optimization methods have been used to achieve a better design in order to improve initial stability and therefore, the implant durability. In this work, a multi-criteria shape optimization process is developed to study the relationship between implants performance and geometry. The multi-criteria objective function takes into account the initial stability of the femoral stem and the effect of stress shielding on bone adaptation after the surgery. Then, the optimized stems are tested using a concurrent model for bone remodeling and osseointegration to evaluate long-term performance. Additionally, the sensitivity to misalignments is analyzed, since femoral stems are often placed in varus or valgus position. Results show that the different criteria are contradictory resulting in different characteristics for the hip stem. However, the multi-criteria formulation leads to compromise solutions, with a combination of the geometric characteristics obtained for each criterion separately.     

Influence of interface condition and implant design on bone remodelling and failure risk for the resurfaced femoral head

Resurfacing of the femur has experienced a revival, particularly in younger and more active patients. The implant is generally cemented onto the reamed trabecular bone and theoretical remodelling for this configuration, as well as uncemented variations, has been studied with relation to component positioning for the most common designs. The purpose of this study was to investigate the influence of different interface conditions, for alternative interior implant geometries, on bone strains in comparison to the native femur, and its consequent remodelling. A cylindrical interior geometry, two conical geometries and a spherical cortex-preserving design were compared with a standard implant (ASR, DePuy International, Ltd., UK), which has a 3° cone. Cemented as well as uncemented line to line and press-fit conditions were modelled for each geometry. A patient-specific finite element model of the proximal femur was used with simulated walking loads. Strain energy density was compared between the reference and resurfaced femur, and input into a remodelling algorithm to predict density changes post-operatively. The common cemented designs (cylindrical, slightly conical) had strain shielding in the superior femoral head (>35% reduction) as well as strain concentrations (strain>5%) in the neck regions near the implant rim. The cortex-preserving (spherical) and strongly conical designs showed less strain shielding. In contrast to the cemented implants, line to line implants showed a density decrease at the centre of the femoral head, while all press-fit versions showed a density increase (>100%) relative to the native femur, which suggests that uncemented press-fit implants could limit bone resorption.