How Do Tissues Respond and Adapt to Stresses Around a Prosthesis? A Primer on Finite Element Stress Analysis for Orthopaedic Surgeons

Abstract

Joint implant design clearly affects long-term outcome. While many implant designs have been empirically-based, finite element analysis has the potential to identify beneficial and deleterious features prior to clinical trials. Finite element analysis is a powerful analytic tool allowing computation of the stress and strain distribution throughout an implant construct. Whether it is useful depends upon many assumptions and details of the model. Since ultimate failure is related to biological factors in addition to mechanical, and since the mechanical causes of failure are related to load history, rather than a few loading conditions, chief among them is whether the stresses or strains under limited loading conditions relate to outcome. Newer approaches can minimize this and the many other model limitations. If the surgeon is to critically and properly interpret the results in scientific articles and sales literature, he or she must have a fundamental understanding of finite element analysis. We outline here the major capabilities of finite element analysis, as well as the assumptions and limitations.

INTRODUCTION

The relationship between implant design and clinical outcome is unquestioned owing to excellent long term results with some designs and poor results with others. Quite obviously, the outcome of total joint arthroplasty depends upon many patient-related and biological factors independent of the "mechanical environment" in the immediate vicinity of the implant. Equally obvious, not all mechanical phenomenon (e.g., wear) and their biological consequences (e.g., osteolysis) relate directly to that mechanical environment. However, with good evidence, most investigators believe outcome differences (and in particular, aseptic loosening) relate to long-term adaptation of tissues to the mechanical environment surrounding an implant.

What can we know about the mechanical environment, and what do we need to know? First, we can measure or estimate the time-varying load magnitudes and directions on an implant using a variety of approaches. Since experimental measurements and theoretical analyses of joint loads yield similar results, we have reason to believe current estimates are reasonably accurate.⁴ Second, we can measure initial bone implant motions (i.e., stability, mobility) using either in vitro or in vivo (roentgenstereophotogrammetry^*) methods, and we can estimate late implant stability using the latter approach.^14,35,42 Third, we can readily estimate how the loads are distributed at the organ (whole bone), and small macroscopic levels, that is, estimate stresses and strains. The question is whether this is what we need to know: mechanistically it is not the load, motion, and stress or strain magnitudes at a given time or under given load conditions which relate to local matrix microdamage or cell and tissue remodeling causing loosening. Rather, local (i.e., microscopic, if not ultrastructural) deformation histories ultimately cause loosening. This latter point raises a question of what structural analysis can provide.

Finite element analysis (FEA), is a powerful computational tool for estimating stress and strain magnitudes and studying the mechanical interactions between bones and implants. While implant design certainly affects stresses in bone, we do not yet know how bone stresses or strains relate to tissue adaptation. This chapter will explore the capabilities and limitations of current FEA in predicting tissue adaptation, and will suggest some new directions in an effort to improve the clinical applicability of this tool. Some issues are philosophical in nature, and while we do not focus on those issues, neither can we avoid them and adequately communicate the capabilities and limitations of FEA.

FINITE ELEMENT ANALYSIS (FEA)

Finite element analysis is merely a way to compute stress and strain fields throughout a structure: that is, how loads and deformations are distributed. Long-known relatively simple (partial differential) equations of static equilibrium readily accomplish this task for simple, solid structures (e.g., beams and rods), where an exact solution can be found which satisfies the equilibrium conditions. However, the equations for geometrically and/or materially complex or non-solid structures are intractable. FEA, introduced in the late 1940s, provided a means to estimate stress and strain fields within more complex structures. Simply stated, a complex structure is modeled as a grid or mesh of many small simple structures ("finite elements"), each with its own typical material and geometric properties, and each connected to its neighbors at discrete nodal locations to insure the coherent displacements and stresses which would occur in the actual structure when loaded. The solution in these cases is approximate, and satisfied only in a weak sense,²⁴ but becomes more accurate with increasing mesh refinement. Subsequently, stress and strain throughout a more complex structure can be computed since the states vary from element to element.

Despite the conceptual simplicity, the practical solution to adequately complex FEAs awaited the development of powerful digital computers. Two of the early FEAs of bone were reported only in 1972.^6,47 Modest in number of elements and other properties, the varying thickness of the two-dimensional section was modeled by varying the stiffness of the elements. (Ironically, the stress trajectories predicted with beam theory analysis more closely approximated trabecular architecture than with early FEA!) Within the next decade, many investigators applied the method to bones and prostheses,²⁶ although the accuracy and validity^* of some of these early models was open to question in part owing to computing limitations and lack of approaches to accurately model interfaces (of differing materials or contacting surfaces) and properties of complex materials.

With modern computing power allowing high mesh resolution (many elements) and new approaches more accurately modeling interfaces and materials, FEA now affords stress analyses with greater biological and clinical implications. Because the approach offers substantial power to explore many aspects of implant design prior to actual construction, the surgeon interested in developing an implant or in understanding the structural implications of an implant should have at least a basic knowledge of the approach, including its capabilities and limitations.

Any FEA requires: 1.) a sufficiently-refined mesh reflecting the external shape and the geometry of any and all relevant structures; 2.) boundary conditions reflecting any and all relevant external constraints or loads surrounding the structure; 3.) material properties appropriately describing those within each element; 4.) proper modeling of interfaces of differing materials or non-connected structures.

These requirements are necessary, but not sufficient for confirming models. At a minimum, model confirmation requires: 1.) a convergence study demonstrating model results do not change if mesh resolution is increased. The greater number of elements, the greater the fidelity to shape and material characteristics of the modeled structure since the real structure may be viewed as one with an infinite number of elements. While the computed solution will change as one begins with few elements and increases mesh resolution, at a certain resolution the solution does not substantially change (i.e., it converges). Normally it is not necessary to further increase mesh resolution once one shows a convergent solution. 2.) a reasonable comparison of model results with independent observations (e.g., laboratory strain analysis, analytic solutions). Reports not demonstrating convergence and not containing compelling confirmation (e.g., laboratory strain analysis, analytic solution, natural observations, animal or experimental observations) should be viewed with great caution not only for absolute values, but even relative comparisons.

With current technology and appropriate confirmation, such analyses have the capability to provide reasonable estimates of stress and strain in bone^* (as an organ) and implant materials. These models provide a powerful capability not readily achievable (and perhaps impossible to achieve) in any observational or experimental approach: the potential to vary only one parameter or feature and no other. In a clinical study, for example, one can use implants of two designs, but despite the best controls (e.g., matched patient groups, single surgeon, same approach, same post-operative regimen) the biological differences and other variables (e.g., biological activity of the tissues, patient activity levels) could obscure the effect of design variables despite a large study (see Frost¹⁶ for a discussion of biological and mechanical interactions). (A very large study with radically different designs and very long-term followup might be able to ascertain differences with reasonable power, but such studies are usually impractical for a variety of reasons.). The same argument applies to the typically better controlled bench study (e.g., cadaveric study). In a FEA, however, all other variables are absolutely controlled, providing an unexcelled opportunity to examine relative differences between implants or a single implant with design differences. Such investigations vary a single parameter (e.g., stem thickness or stiffness) keeping others constant and are accordingly termed, "parametric" studies. These studies, in addition to precision of strain computation at the macroscopic level, is perhaps the most powerful capability of most current FEA implant studies.

Thus, FEA does afford the opportunity to identify high and low stress or strain regions in bone. As will be obvious from subsequent discussion, contemporary FEA computes stresses and strains over some averaged region, and not the extremes. Nonetheless, given sufficient mesh resolution, averages of small regions would give some indication if the stresses and strains exceeded the static or fatigue limits of bone strength, and how design changes may relieve or exacerbate these limits.

GENERAL ASSUMPTIONS AND LIMITATIONS OF FEA

Effective Continuum Assumption

FEA arises out of continuum mechanics, the science on which the very concepts of stress and strain are in fact based. "Continuum" implies a material is continuous or solid and its properties (e.g., elastic modulus, Poisson's ratio, yield criteria, volumetric density) are locally homogeneous (i.e., do not vary). Quite obviously, at an atomic level, no materials are "solid," although from a practical point of view they may be so considered at a macroscopic level. In any continuum model, a specific length scale of interest is usually implied by the dimensions of the model. Material properties at the length scale of a continuum model are typically just averages of the material properties at smaller length scales. Consequently, information pertaining to stresses and strains at small length scales is usually not provided by the continuum model. For example, in the design of an implant, for most purposes (but importantly, not all) cortical bone could be quite reasonably treated as a homogenous continuum, since the heterogeneities associated with osteons, vascular porosity, lacunae, and canaliculi exist on much small length scales. However, in the design of an implant in trabecular bone, it is more questionable to model the bone as a homogeneous continuum, since the length scale of the heterogeneities within the bone (length of individual trabecula and the associate pores) approaches that of the implant dimensions. Architectural features such as osteons effectively create "stress risers" or features which substantially influence local tissue stresses or strains. One should then ask: At what scale is the assumption of a homogeneous material valid? In an attempt "to establish the length scales over which the continuum assumptions is valid in cancellous bone," Harrigan et al., concluded, "Within three to five trabeculae of an (implant) interface a continuum model is suspect."²⁰ This means the average strain calculations do not apply at the trabecular (let alone cell) level, owing to variations from the continuum-assumed averages. We do not currently know the range of stresses or strains created by these stress risers, although the estimates range from 2-3 times^46,51 to 10 times,⁴¹ or even 300²³ times the continuum or average level. The biological (i.e., mechanistic), if not predictive importance of this point will become apparent later.

For the reasons stated above, the continuum assumption is most likely valid when modeling prosthesis implantation in the cortical bone of a cadaveric femur or when modeling the initial state in a living femur immediately after implantation. In fact, an FEA is often confirmed ("validated") using a strain-gauged^** cadaveric experiment. Thus, despite the continuum assumption, FEA still has the potential to be powerful in predicting, although not mechanistically explaining, implant performance in a population. In such a situation, biological mechanism need not be incorporated into a model, although no model should contradict known biological mechanisms or clinical observation. However, most published FEA studies do not adequately address strain at the trabecular level, either because that architecture in the vicinity of the implant is not included, or when it is modeled, the variations in material properties at that level are simply not known.

Design Criteria

Designing "better" implants requires some "design criterion." That criterion usually minimizes or maximizes - i.e., "optimizes"-stress or strain, either globally or locally. That is, one identifies a design which reduces or increases some feature which will have beneficial effects. Given a design criterion, one parametrically varies a given design feature to determine which design best achieves that criterion. A typical criterion might be to minimize peak strain in bone cement. A parametric study is conducted, examining a single design change (say, stem thickness), then searching all cement elements for strain with each modeled thickness. One chooses that thickness which best achieves the criterion.

Design Features Not Directly Related to Stress and Strain

Many important or critical aspects of design, such as choice of implant material and surface finish are not directly addressed by FEA. While the differing material properties of an implant material are considered, FEA does not account for material differences affecting the system in biological ways not related to strain. For example, some implant materials (e.g., titanium) are believed to promote or tolerate bone ongrowth better than others (e.g., stainless steel). Surface finish at the microscopic level relates to cell responsiveness. An FEA does not consider such design features, when in fact they could be critical.

Design Features Related to Stress and Strain

Minimizing strain in an implant material (ceramic, metal, high density polyethylene, polymethylmethacrylate) is relatively straightforward, because the continuum assumption is typically reasonable, and because we have a good notion of the relationship between stress or strain and material failure. Thus, if the questions relate to failure of an implant material, including wear, FEA provides a powerful approach. On the other hand, if the questions relate to tissue adaptation, the FEA approach is not straightforward since we do not know the relationship between stress or strain and bone adaptation. In the case of an implant, one might empirically reduce bone strain in high strain regions, and perhaps increase strain in low strain regions if one presumed the low level would lead to reduced bone mass (density).

BONE ADAPTATION TO THE DEFORMATION HISTORY

The relationship between the mechanical environment and connective tissue adaptation has been known or suspected for centuries. Galileo recognized the generally similar shapes of animals of widely differing sizes, yet each adapted in particular size and shape to the size and anatomy of the animal. The observation of regularity of trabecular architecture (from individual to individual in the same bone) during the 19th Century suggested description by mathematics (since mathematics can theoretically describe all regularities). As early as 1851, Wyman suggested, "the cancelli in all those bones that assist in sustaining the weight of the body, or in locomotion, are arranged in definite directions, the directions being those of the reception and transmission of force."⁵⁶ The notion that trabecular architecture could be described by the principle tensile and compressive stresses which would occur in a solid, homogeneous structure of the same shape in the 1860's³⁹ led to the postulate that these stresses governed bone remodeling.^46,55 Following Wolff's classic monograph, "Das Gesetz der Transformation der Knochen," published in 1892, the general relationship between the mechanical environment and tissue adaptation has been generally termed, "Wolff's Law."⁵⁵

Wolff specifically noted:

Es ist demnach unter dem Gesetze der Transformation der Knochen dasjenige Gesetz zu verstehen, nach welchem im Gefolge primärer Abänderungen der Form Inanspruchnahme, oder auch bloß der Inanspruchnahme der Knochen, bestimmte, nach mathematischen Regeln eintetrende Umwandlungen der innerer Architectur und ebenso bestimmte, denselben mathematischen Regeln folgenden secondäre Umwandlungen der äusseren Form der betreffenden Knochen sich vollziehen.

(It is therefore under the laws of transformation [remodeling] of bone, the one law to understand, after which in the wake of the primary changes of utilized form, or certainly just the use, after mathematical rules set in transformation of the inner architecture and just as certainly, following the same mathematical rules the secondary transformation which is carried out on the outer form of the bone in question.)

Or, more compactly:

The law of bone remodeling is that mathematical law according to which observed alterations in the internal architecture and external form of bone occur as a consequence of the change in shape and/or stressing of bone.

Quite obviously, Wolff and subsequent investigators presumed something about stress and/or strain "governed^*" bone adaptation. Two problems immediately arise: First is Wolff's implicit continuum assumption. Stress and strain concepts arise from continuum mechanics, in which solid, continuous materials are assumed at the level of interest. Bone and interfacial tissues are not solid, continuous materials, however. In fact, discontinuities observed at the trabecular macroscopic (not to mention those at the microscopic and ultrastructural) level undoubtedly result in substantial variations of "average" or continuum level stress or strain predictions^**. Second is Wolff's implicit concept of "governing laws." While Wolff's understanding and/ or philosophy of the meaning of mathematics in Nature are not known, he surely implied something inherent in mathematical "laws" "governed" bone remodeling in particular, and Nature in general. Historically, this notion reflects a school of thought believing mathematics inherent in Nature, and mathematical quantities "governed" or "controlled" natural processes. An opposing school of thought believes mathematics merely a manmade artifice, which despite its immense power, could be replaced by some alternative construct (e.g., Pythagorian versus Reimannian geometry) and in the best of circumstances merely described or predicted natural process. Newton reflected this latter school when in formulating the "laws" of gravity, he commented, "Hypotheses non fingo": "I propose no explanation." Thus, he recognized his "law" merely described Nature, not explained it.

This latter point is critical in interpreting modern formulations of tissue adaptation: some contemporary investigators apparently ignore the distinction between mathematics as a way to describe or predict, and mathematics as a way to explain. In the former case, empirically derived mathematical formulations may accurately predict tissue adaptation without consideration of mechanism, while in the latter, mechanism would be essential. Since the mechanisms of tissue adaptation are poorly understood, known "steps" in cascades of events are complex, and many steps undoubtedly remain to be identified, it makes little sense to artificially introduce mechanism into mathematics which at best describes and predicts. On the other hand, it makes equally little sense to introduce mathematical formulations contradictory to known mechanisms. Further, in contrast to the mechanics where most "mathematical laws" accurately predict at macroscopic levels, in biology, such laws only approximate outcome for a population (of individuals, tissues, cells). Thus, physical scientists and engineers need an understanding of biological constraints, while biologists and physicians need an understanding of computational constraints. Given this understanding, FEA offers the potential to describe or predict implant performance for a population.

CAN FEA PREDICT BONE ADAPTATION?

It is fair to state no current approach to numerically predicting bone adaptation to implants has been correlated with clinical outcome. Thus far, FEA cannot tell us whether a given implant will loosen owing to mechanically-driven bone adaptation. This is one of its greatest limitations. Why is this the case. First, continuum level average stresses or strains do not likely directly related to implant failure. Second, tissues "temporally process" deformation histories and this process is key to adaptation; the typical FEA uses static loads and does not account for either deformation histories or temporal processing. Third, the individual cells responsible for adaptation do not respond to stress or strain per se, although they might respond to some related quantity.

Continuum Level Averages Do Not Likely Directly Relate to Implant Failure

Why are the continuum averages computed by FEA perhaps less important than the ranges of stresses and strains^* on the microscale? The reasons are several fold: 1.) Very high levels of local strains may lead to fatigue fracture of dead or even living bone (microfractures) within weeks or months following prosthetic implantation. 2.) Very high levels of local strains may lead to bone resorption of living bone. 3.) Very low levels of local strains may lead to bone resorption of living bone (i.e., generalized disuse atrophy and localized "stress shielding"). In each of these cases, support lost at one small point, leads to load transfer elsewhere (presuming the loading is the same), perhaps overloading and fracturing or resorbing adjacent areas.

Several observations and one argument support these reasons: 1.) The presence of micromotion (albeit sometimes only in the range of 10-50 micrometers) between all implants and bone, both initially and subsequently. The presence of motion itself likely engenders levels of stresses and strains not computed by the typical FEA (unless it were a contact model). 2.) The roentgenstereophotogrammetry (RSA) documented initial settling of virtually all implants (again, in a small range for clinically successful large joint implants). The presence of near universal small but detectable initial settling suggests overloading of some initially supportive bone with fracture. However, given the long-term endurance of most implants, we must presume implants reach some point of stability where interfacial stresses at both global and local levels are tolerated by bone. The RSA evidence of continued settling eventually leading to clinically significant implant loosening suggests a point of stability is never reached in some implants. That is, local loss of support in one region causes overloading in previously appropriately loaded regions, then the new region fails, and so on. 3.) The higher levels of stresses and strains (i.e., above the continuum level computations) likely exceeds the breaking strain of bone. Various authors suggest the static breaking strain in the range of 14,000-35,000 microstrain^2,12 while the ultimate fatigue strain in laboratory cortical specimens is in the range of 4000-8000 microstrain ^**.²² Thus, presuming the range of continuum-predicted strains in trabecular bone are 10 times the average, it is likely some trabeculae fail around implants some time shortly after implantation. This argument is consistent with nearly universal RSA-observed settling.

Temporal Processing of Deformation History Is Key to Tissue Adaptation

Realizing the state of strain around implants is not constant, some investigators have utilized iterative FEA, in which one assumes extreme regions of strain will change their properties as a result of the biological responses.^34,53 In such models regions of low strain energy density become less stiff, which increases the energy density level, and regions of high strain energy density become stiffer, which decreases the energy density level^†. These approaches assume the tissue is attempting to reach some more ideal range of strain energy density. If new moduli are assigned to these regions, a new FEA solution may be obtained with a differing distribution of strain. The re-assignment of properties is continued until the solution converges (i.e., all elements of bone experience some specified range of strain energy density). These sorts of models can account for the clinically observed remodeling (changes in radiographic density) which occurs around implants.^{13,18,36,49,52}

Models accurately predicting bone density necessarily require adaptation rules. Current investigators select some strain (or strain-related) magnitude, presuming tissues and/or cells differentiate the various forms of stress or strain^††. In fact, we do not know which will most accurately predict bone response, and in fact we do not even know whether individual cells indeed differentiate say, compressive versus tensile strain. Thus, the predictive model presumes the response of cell population in a large tissue region is associated with a given stress or strain quantity, but one should not confuse individual cell responsiveness with population responsiveness. The magnitudes of some calculations of stress or strain (e.g., longitudinal tensile or compressive stresses) depend upon some more or less arbitrary reference frame, and it seems quite obvious these sorts of parameters may be irrelevant, since cells do not know the orientation of such frames. Other stress/strain parameters (e.g., strain energy density) do not depend upon a reference frame, and therefore seem more appropriate as candidate predictors. In attempting to identify particular aspects of loads (i.e., stress/strain parameters) relating to tissue adaptation, recent investigators assume tissues "seek" (i.e., remodel to achieve) some identifiable and "optimal" state of stress or strain (e.g., "attractor state"), and further implicitly or explicitly postulate the attractor state arises from peak loads,^9,27 "averaged" strains,^15,16 or "values which cause fatigue microdamage".^11,17

Understanding the relationship between tissue adaptation and loading must involve some "cumulative effects of loading," but not the time-averaged sort suggested by Cowin.¹¹ Carter⁹ and Whalen and Carter⁵⁴ formulated a remodeling rule based upon summing a number of discrete peak loads during some given number of occurrences of similar activities, assuming the entire loading history influenced bone maintenance and/or remodeling. One recent study, however, suggests stress/strain magnitude alone does not predict bone adaptation theoretically⁷ or experimentally,⁴⁰ while another suggests magnitude and cycle number, regardless of how the two are weighted, also do not predict bone adaptation.¹

Cowin noted, "The precise aspect of the strain history sensed by bone tissue is an open question."¹¹ The recognition that strain history, rather than merely some strain quantity at a given time, initiates remodeling arises from experimental evidence suggesting tissues account for temporal aspects of the stimulus; for example, bones respond very differently to static loads over time than similar magnitude dynamic loads.^10,32 Further, Gross et al. demonstrated the strain distribution in bone at the instant of peak strains differs from the distribution of strain at other times; therefore if submaximal strains initiate bone remodeling, the new bone distribution will differ substantially from that predicted from the location of peak strains.¹⁹ Perhaps more importantly, some regions of bone habitually experience low strains while others experience high strains (at both continuum and cell levels), yet maintain spatial concordance.³

To date, however, these remodeling rules ignore additional features inherent in any mechanical signal: duty cycle (i.e., distribution of events over some time frame termed "dose-fractionation"^{30, 31} or "partitioning"⁴⁵), the interlinked frequency and strain rate,^37,38 signal duration, and wave form. The biological ramifications of these features are likely interdependent in ways currently unknown. And although their importance is unquestionable, they are virtually ignored by current investigators using FEA to study bone adaptation.

An abundance of in vivo and in vitro studies suggest cells and tissues in fact "ignore" the majority of mechanical signal content, "selecting" and responding only to certain features.⁵ For example, rather than exhibiting a dose response to cycle number, one observes a trigger response in many systems. These sorts of observations led to the hypothesis that tissues "temporally process" mechanical signals, and do so in distinct ways: 1.) They respond in a trigger-like manner after a relatively few events or cycles of loading; 2.) They respond only to some window of strain magnitude; 3.) They exhibit a refractory period after a response; 4.) They have a memory for previous stimuli. These characteristics do not necessarily mirror distinct cellular phenomena but rather reflect typical features of experiments.

FEA, even those which iteratively predict changes in bone density, do not account for temporal processing features of tissue adaptation. Rather, they assume the changes occur primarily, if not exclusively as a result of some stress or strain magnitude. This is not to say they could not incorporate such features. Based on his own work and that of others, Turner recently noted three "fundamental rules" of bone adaptation: "(1) It is driven by dynamic, rather than static loading. (2) Only a short duration of mechanical loading is necessary to initiate an adaptive response. (3) Bone cells accommodate to a customary mechanical loading environment, making them less responsive to routine loading signals."⁵⁰ These and other sorts of arguments can be combined with iterative FEA magnitude predictions at a phenomenological level without considering mechanism.

The question is whether FEA needs to incorporate such adaptation rules to successfully predict loosening. As earlier noted, a predictive model need not incorporate "mechanistic rules" to be successful, although a mechanistic model obviously would. However, there are times when incorporation of basic mechanisms into predictive models is useful either for enhancing predictive or for heuristic reasons. Thus, the question is not answered at this time.

Individual Cells Do Not Respond to Stress or Strain

Whether or not FEA can accurately predict bone adaptation, the question arises as to whether individual cells respond to stress or strain. Abundant experimental evidence suggests many connective tissue cells respond to various alterations in the mechanical environment at both physiological and supraphysiological (at the continuum computed) levels of strain. Almost any sort of mechanical stimulus (hydrostatic pressure, stretching on a substrate, poking) elicits cell responses. While identical cell culture systems have not been systematically explored using differing stimuli, available evidence suggests it may not matter how the stimulus is applied. In turn, this suggests cell deformation of any sort may be the stimulus rather than a specific sort of stress or strain.

A number of experimental observations indicate osteoblast-like cells in culture respond to fluid flow.^{21,28,29,44,48} Since bone has pores at various scales and contains water, deformation will cause fluid flow, leading to the hypothesis that fluid flow, rather than or in addition to deformation causes mechanically-driven adaptation. These hypotheses could be explored using poroelastic FEA formulations which simulate fluid flow. (It should be added that under normal loading conditions, poroelastic strain solutions of bone will not differ much from elastic solutions, so the advantage of poroelasticity is to explore hypotheses, not to create more accurate strain predictions.)

Other arguments lead to the same conclusion. Connective tissue cells are not rigid structures and are not rigidly attached to their relatively stiff matrix. Thus, even though we might infer a pure stress state (say compression), one cannot suppose cells loosely connected to a matrix in lacunae at all sorts of orientations to the load experience pure "compression." What they likely experience is deformation, to which they unquestionably respond.

FEA could theoretically model cells in lacunae at differing orientations if one knew the geometry of the structure (cells are indeed complex structures), the material properties of each element in that structure (including all the proteins connecting the cell to the matrix) and the structure and properties of the surrounding matrix. In this case, one could ascertain how an individual cell deforms under some pure load. However, the material and structural property information required for such a model is not available.

For the time being, we must accept that we can construct finite element models which compute only average stresses and strains for a large region of cells, matrices, and discontinuities. Thus, any true (biologically) mechanistic model is impossible^40,43 That is not to say, however, some predictive model which considered only average stresses or strains might not accurately predict tissue adaptation to an implant.

In fact, a number of FEA studies incorporating some stress/strain magnitude and/or aspect of time do correlate with bone density, one aspect of tissue adaptation.^{25,33,49,52,53} Interestingly, different loading conditions produced similar bone density distributions in one such model.¹³ Bone density and in particular stress shielding, however, has not been shown to correlate with long term outcome.⁸ That is not to say some other sort of FEA might not accurately predict mechanically-caused long-term aseptic loosening.

CONCLUSIONS

We are unaware of anyone who has used FEA to accurately predict clinical outcome. The primary value of FEA in exploring design changes has thus far been heuristic; that is, parametric exploration of design changes in the same model provides substantial insight regarding the mechanical interactions between bones and implants. Quite clearly, we should avoid implant designs associated with regions of very high and very low stresses and strains. But while we have a good idea of the static and fatigue behavior of bone and thus fracture, we typically do not compute the range of trabecular stresses and strains leading to fractures, and we do not know what mechanical environment will stimulate beneficial or detrimental bone adaptation over time.

An incomplete understanding of the biological mechanisms of bone adaptation does not preclude, however, an FEA model which might prove predictive of implant loosening. We suspect, however, the most promising approaches will require incorporation of two features: 1.) hierarchical modeling; 2.) iterative, adaptive modeling. Hierarchical modeling simply means the tissues are coherently modeled at multiple levels from micro- to macroscopic. Thus, the investigator will more reasonably predict the range of local strains at the level of interest: those quantities which lead to trabecular fracture or relate to cell responses and tissue adaptation. Iterative modeling recognizes the initial effect of the implant on the immediate mechanical environment does not remain the same for very long. Within days or weeks of implantation, a portion of the interface will change as a result of injury-repair reactions, and when that interface changes, the stress distribution and small implant position changes. (RSA studies confirm such position changes actually occur, and even predict outcome.) Iterative modeling can account for the changes since the local stresses/strains are known through the hierarchical modeling. Further, the introduction of these approaches does not preclude identification of empirical remodeling rules predicting bone adaptation and loosening. These currently available refinements in the use of FEA should enhance their value in implant design.