Journal of Biomechanics 47 (2014) 1241–1250

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Journal of Biomechanics

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Perspective article Biological variability in biomechanical research: Significance and meta-analysis of current modeling practices

Douglas Cook a,n, Margaret Julias a, Eric Nauman b a New York University, Abu Dhabi, United Arab Emirates b Purdue University, School of , Weldon School of , Department of Basic Medical Sciences, West Lafayette, IN, USA article info abstract

Article history: Biological systems are characterized by high levels of variability, which can affect the results of Accepted 22 January 2014 biomechanical analyses. As a review of this topic, we first surveyed levels of variation in materials relevant to biomechanics, and compared these values to standard engineered materials. As expected, we Keywords: found significantly higher levels of variation in biological materials. A meta-analysis was then performed Biological variation based on thorough reviews of 60 research studies from the field of biomechanics to assess the methods Sensitivity analysis and manner in which biological variation is currently handled in our field. The results of our meta- Modeling analysis revealed interesting trends in modeling practices, and suggest a need for more biomechanical studies that fully incorporate biological variation in biomechanical models and analyses. Finally, we provide some case study example of how biological variability may provide valuable insights or lead to surprising results. The purpose of this study is to promote the advancement of biomechanics research by encouraging broader treatment of biological variability in biomechanical modeling. & 2014 Elsevier Ltd. All rights reserved.

1. Introduction loading and calcium , elucidate the physiological mechanisms necessary to permit the rational design of engineered Biomechanical models have been used to great success in a tissues, and predict the efficacy of therapeutic interventions with- variety of biomedically relevant applications including the design out the need to perform extensive human trials. of advanced prosthetics (2007), orthopedic implants (Keaveny and The efficacy of computational models to predict whether or not Bartel, 1993), cardiac modeling (Humphrey and Yin, 1987; a medical intervention will be successful often depends on subtle Chen and Humphrey, 1998), drug delivery methods (Schuff et al., factors operating at the level of unique individuals. While “subject- 2012, 2013), as well as helping to elucidate evolutionary processes specific” are useful in some cases, we typically are more interested (Darwin, 1859; Weiner, 1995). These achievements were based on in trends that can be reliably predicted across a . impressive advances in computational (Cowin and However, the ability to predict such behavior is hampered by Hegedus, 1976; Hart et al., 1984; Huiskes and Nunamaker, 1984), significant levels of variability that are present in all aspects of the application of mixture theory models to problems in tissue human biomechanics, including dimensions and material proper- mechanics (Mow et al., 1980; Weinbaum et al., 1994; Cowin et al., ties (Saulgozis et al., 1974; van Geemen et al., 2011), stature 1995, Dickerson et al., 2008), development of elegant physical (Daubes, 1887; Visscher, 2008), (Brutsaert and Parra, models (Liang and Mahadevan, 2011), and the integration of model- 2006; Tahmoush and Silvious, 2010), and pathological conditions ing with micro and nanoscale experimental methods (Raman et al., (Drumm et al., 2012). Consequently, much of our future success as 2011). biomechanical engineers depends on our ability to quantify and Future endeavors will eventually integrate muscle biomecha- integrate physiological variation into our modeling processes, nics with metabolic load and neural control, provide an under- considering not just an “average” model, but creating models that standing of remodeling in the context of both mechanical predict distributions of possible outcomes. The purpose of this approach is to stimulate discussion and reflection among the biomechanics research on the topic of biological

n variation. Correspondence to: PO Box 129188, Abu Dhabi, United Arab Emirates. Tel.: þ971 2 6284 192. We therefore have examined biological variation in three ways: E-mail address: [email protected] (D. Cook). first by quantifying the general levels of uncertainty in biomechanics http://dx.doi.org/10.1016/j.jbiomech.2014.01.040 0021-9290 & 2014 Elsevier Ltd. All rights reserved. 1242 D. Cook et al. / Journal of Biomechanics 47 (2014) 1241–1250 and comparing these with levels of uncertainty in standard engineer- These articles are indicated in the references section by the dagger symbol ing materials; second, by assessing the manner in which biological (Aissaoui et al. (2011); Al-Jumaily et al. (2011); Alastruey et al. (2011); Bonnet et al. (2011); Bruijn et al. (2011); Carnelli et al. (2011); Chaichana et al. (2011); variability is currently being considered in the biomechanical Chong et al. (2011); Cox et al. (2011); de Tullio et al. (2011); de Vaal et al. (2011); Di research community; and third, by providing examples from the Martino et al. (2011a, 2011b); Drury et al (2011); Dvinskikh et al. (2011); Ferrara literature illustrating ways in which biological variation has affected et al. (2011); Gonçalves Coelho et al. (2011); Henak et al. (2011); Henderson et al. research results. We conclude with a broad discussion of these inter- (2011); Johnson et al. (2011); Khalil et al. (2011); Kociolek and Keir (2011); Konala et al. (2011); Labrosse et al. (2011); Landsberg et al. (2011); Lin, C.-J. et al. (2011); related issues. Lin, C.-L. et al. (2011); Liu et al. (2011); Manda et al. (2011); Martelli et al. (2011); Mihaescu et al. (2011); Morbiducci et al. (2011); Olgac et al. (2011); Pahlevan and Gharib (2011); Rahbar and Moore (2011); Rankin et al. (2011); Renders et al. (2011); 2. Methods Roddy et al. (2011); Rothstock et al. (2011); Scheys et al. (2011); Speelman et al. (2011); Stevanella et al. (2011); Tovar-Lopez et al. (2011); Trabelsi et al. (2011); Tse 2.1. Quantifying levels of variation et al. (2011); Turnbull et al. (2011); van der Giessen et al. (2011); Varghese et al. (2011); Vavourakis et al. (2011); Vetter et al. (2011); Waanders et al. (2011); Wang To quantify levels of variation in biomechanics, and to contrast these levels and Li (2011); Weaver et al. (2011); Willemet et al. (2011); Wilson et al. (2011); with those in traditional engineering, we compiled coefficient of variation values Winkel and Schleichardt (2011); Wong and Tang (2011); et al. (2011); Yu for several materials. The coefficient of variation (CV) is commonly used to quantify et al. (2011)). variation, and is defined as the ratio of the standard deviation (s) to the mean (m): Each article selected for inclusion was reviewed thoroughly and corresponding data were recorded in a database. The database fields were chosen based on a s criterion of objectivity: only features which could be objectively determined were CV ¼ μ ð1Þ included in this study. Data was collected for each paper in aspects such as: the Low values of CV (e.g. CV¼0.05) indicate a narrow or “tight” distribution, while number of reported parameters in the modeled system; the statistical distribution values greater than 0.5 indicate a broad distribution. Because the coefficient of of each parameter; the number of parameters varied (and held constant); the variation is non-dimensional, comparisons can be made between different properties technique(s) used for varying model parameters; the source(s) of parameter values; of a given material, as well as between different materials. Six common materials were the total number of performed; and the type of validation performed, selected for analysis, three of which are engineering materials (aluminum, concrete, just to name a few. fi and steel), and three of which have biomechanical relevance (bone, , and Variation techniques were classi ed into two broad categories: parametric wood). Numerous research sources (Ellingwood, 1980; Martens et al., 1980; variation (i.e. one parameter varied while all others held constant); and simulta- Buckwalter et al., 1994; Myers et al., 1995; Langton et al., 1996, Rho et al., 1997; neous variation of two or more parameters. In both cases, the number of varied Wimmer et al., 1997; Hou et al., 1998; Ladd et al., 1998; Stammberger et al., 1999; parameters was also recorded. Where relevant, all information was parsed into Turner et al., 1999; Zysset et al., 1999; Niebur et al., 2000; Morgan and Keaveny, 2001; aspects of geometry, material, and boundary conditions. The resulting database fi fi fi Hess et al., 2002; Bayraktar et al., 2004; Schriefer et al., 2005) were utilized, and consisted of 35 elds: 25 numeric elds and 10 nominal (i.e. typographic) elds. coefficient of variation values were collected for multiple properties of each material. An example of the type and structure of data collected is shown in Fig. 1. This figure depicts the most relevant type of data collected from one of the sampled articles. Note that each model was decomposed into aspects of geometry, material, 2.2. Assessing the role of biological variation in biomechanics research and boundary conditions, and parameter counts in each area were recorded along with number of simulations. Additional information is provided in the supplemen- An extensive meta-analysis was performed based on research articles pub- tary material associated with this article (available online). lished in Journal of Biomechanics in the year 2011. This year included 16 issues In presenting meta-analysis data, relative frequency histograms are used comprising a total of 354 research articles. Other article types (perspective, review, extensively to summarize trends. For ease of interpretation and to facilitate letter to editor, and short communications) were excluded from our analysis. comparison between charts, all histogram results were normalized by the total Of the 354 total research articles, 158 articles involved computational modeling to number of studies (60 studies) and the term “overall relative frequency” was used some degree. Articles were sampled randomly (n¼60) from these 158 articles. to describe this approach.

Fig. 1. A sampling of data collected from a study involving the human mandible (Gröning et al., 2011), and an illustration of how this data was organized according to model aspect. D. Cook et al. / Journal of Biomechanics 47 (2014) 1241–1250 1243

Table 1 Descriptive statistics for the number of model parameters used to define biome- chanical models, separated into major aspects of geometry, material, and boundary conditions.

Min Median Mean (sd) Max n

Idealized geometry 2 6 27.04 (38.81) 135 25 Image-based geometry – – ––33 Material 1 6.5 8.39 (6.75) 37 54 Boundary conditions 1 5 8.85 (12.4) 72 53

Fig. 2. Average (bar) and distribution (dots) of coefficient of variation for various experimentally determined properties of engineered and biological materials (Ellingwood, 1980; Martens et al., 1980; Buckwalter et al., 1994; Myers et al., 1995; Langton et al., 1996; Rho et al., 1997; Hou et al., 1998; Stammberger et al., 1999; Turner et al., 1999; Zysset et al., 1999; Niebur et al., 2000; Ellingwood, 2001; Morgan and Keaveny, 2001, Hess et al., 2002; Bayraktar et al., 2004; Schriefer et al., 2005). Engineering materials and structures typically have coefficients of variation less than 20% (dashed line) (Ellingwood, 1980, 2001; Hess et al. 2002). Measured parameters include apparent density, compressive modulus, shear strength, tensile yield stress, ultimate strength, etc.

2.3. Examples of variation affecting research results

To show the significance of biological variation on research results, three examples of biomechanics research are presented in Section 4. The purpose of the discussion was not to compare one research case to another but merely to illustrate that biological variation can lead to both interesting and surprising results in biomechanics research. Fig. 3. Relative histogram of the number of varied model aspects, with each column segmented by its constituents: G – geometry, M – material properties, and BC – boundary conditions. Most of the studies held at least one of the model 3. Results aspects fixed throughout the study. Only 8% of studies varied all model aspects.

3.1. Levels of biological variation geometric features, but do not consist of a definite number of A dramatic difference was observed between the levels of parameters. As a result, the geometry aspect was split into two variation present in biological materials as compared to engi- components: image-based geometries and idealized geometries. neered materials. Fig. 2 compares coefficient of variation values for The statistics for the number of parameters used to define numerous material properties of six common materials. All data biomechanical models are provided in Table 1. from each category were combined to produce an average value for Both the number of parameters used to define biomechanical each material (bars). As seen in Fig. 2, engineered systems typically models, and the number of parameters used to define model have a coefficient of variation less than 20%, while biological aspects (geometry, material, and boundary conditions), exhibited a systems regularly exhibit coefficients of variation significantly above roughly log-normal distribution. The amount of variation between 20%. For example, the aggregate coefficient of variation for steel is models, as well as the non-normal distributions in each category 7.5%. Assuming a normal distribution, 95% of steel samples can thus makes a description of an “average” biomechanical model difficult. be expected between 85% and 115% of the mean value. On the other Indeed, the quantitative description of a hypothetical “average” hand, the aggregate coefficient of variation for cartilage is 49.2%, model is perhaps misleading since in cases involving log-normal indicating that 95% of measured values may lie between 1.8% and distributions, the interpretation of a mean value is not as straight- 198% of the mean value, a range that spans two orders of forward as it is for normally distributed quantities. For this reason, magnitude, similar to the ranges that have been reported for data regarding the number of model parameters is presented in cartilage permeability (Sander and Nauman, 2003). Table 1 in terms of range, median, mean, and standard deviation. Of these descriptive statistics, median is the most representative 3.2. Meta-analysis of biomechanics models value, and is thus emphasized in Table 1. The high degree of variation between models is evident by noticing that the standard 3.2.1. General observations deviation is generally greater than the mean for aspects described A meta-analysis of 60 biomechanics research articles was in Table 1. conducted and analyzed. A tremendous variety in biomechanical models was observed. Models ranged from very simple, idealized descriptions to exceptionally complex models involving over 200 3.2.2. Variation by model aspects (geometry, material, and boundary model parameters. In tallying the number of parameters used to conditions) define each model, one challenge was the fact that medical images Examination of the meta-data revealed several interesting (MRI, CT, optical, etc.) were used as the basis of approximately half trends in biomechanical modeling practices. The most common (55%) of the studies. Such images provide detailed descriptions of method of variation in model parameters was to vary parameters 1244 D. Cook et al. / Journal of Biomechanics 47 (2014) 1241–1250 within one aspect while keeping other aspects fixed throughout studies appears to be exponentially distributed. In contrast, the the study. Approximately half of the studies used this approach number of model parameters is log-normally distributed (as (Fig. 3). We considered that a model aspect is varied when a shown by the line plots in Fig. 4A, C, and D). As a result, the variation of at least one parameter was performed within the median value for parameters varied is much lower than the model aspect. Variation of all model aspects was relatively rare; median value of number of actual model parameters. This suggests approximately 8% of studies used this approach while the remain- that biomechanics studies often utilize an experimental design ing studies held all the features of one or more model aspects fixed similar to those used in in vitro medical studies: a select number of throughout the study. Geometric aspects were found to be varied parameters are varied in a controlled fashion while the remaining most commonly (in 57% of studies). Boundary conditions were parameters are held fixed. varied in 40% of studies. The least varied model aspect was the material property (in 32% of studies). 3.2.4. Variation type In addition to the number of parameters varied, the type of 3.2.3. Parameter variation within model aspects variation utilized was also of interest. Although many methods To obtain a more detailed description of model variation, exist for experimental design, these can be classified into two variation patterns within each aspect were examined. Relative broad categories: parametric variation in which one parameter is frequency histograms were used to tabulate the distribution of varied while others are held constant; and simultaneous variation, parameter variation within each aspect, and these distributions in which two or more parameters are varied simultaneously. These were compared to the number of parameters in each aspect. As types of variation were tabulated in terms of relative frequency of mentioned previously, the geometry aspect was segmented into occurrence, and segmented by model aspect (Fig. 5). idealized and image-based subcategories. Some studies primarily Most studies that used parametric variation varied only a small utilized medical images, but then proceeded to parametrically vary number of parameters. More specifically, 80% of studies in this certain aspects of the geometry. This approach is evident in category varied only one or two parameters (as indicated by the histogram 4B, which shows parameter variation in conjunction non-shaded region under “parametric” variation in Fig. 5). How- with medical image-based geometries. ever, the studies in the simultaneous category exhibited the For aspects of idealized geometry, material properties, and opposite trend; 74% of studies in this category varied more than boundary conditions, the number of parameters varied within two parameters (as indicated by the shaded region under

Fig. 4. Relative distribution histograms (bars) depicting the number of varied parameters in each aspect. Curves indicate the log-normal distributions for the number of parameters for each corresponding aspect of individual models. Vertical lines indicate the median value for each distribution. Medians at zero indicate that 50% or more of data is within the zero parameter bins. Curve omitted for the image-based geometry because no corresponding distribution is available (see Table 1). D. Cook et al. / Journal of Biomechanics 47 (2014) 1241–1250 1245

the number of model parameters (Np): N S ¼ s ð2Þ Np The ratio allows computational studies to be assessed in relation to the number of model parameters. Overall, 38% of analyzed studies had a simulation ratio higher than 1. The median simulation rate was found to be 0.43 (Fig. 6).

3.2.6. Biological parameters as random variables Two approaches were used to determine the degree to which biological parameters were treated as random variables. First, we tallied the number of parameters that were referred to in terms of probability distributions or random variables. The 60 reviewed articles utilized a total of 1598 discrete model parameters. Of these, only seven parameters (0.4%) were described in terms of biological variation. In each of these seven cases, parameters were described in terms of biological range (min/max), but the type of distribution was not specified. Second, we considered the number of cases in which biological variability was incorporated into the study by virtue of the study Fig. 5. Rates and types of variation utilized in analyzed studies, decomposed by design. Many studies incorporated the essential features of random category. Relative frequency is defined by category. Shaded regions indicate the proportion of each subcategory in which more than two parameters were varied. variables through the use of image-based geometries. Approxi- mately half of all the studies utilized medical images, and of the image-based models, half again utilized multiple samples in the study design (see the “entire image” column in Fig. 4b). Overall, 30% of all studies incorporated biological variability in this way.

3.2.7. Justifications for fixed parameters As illustrated in Fig. 4, the most common approach in biome- chanics research is to focus on a small number of key parameters, with the remaining parameters fixed throughout the study. We examined the reasons stated for using this approach. The presence or absence of justification for this approach was tabulated for all analyzed studies. We observed that reasons for varying certain parameters were almost always provided, but justification for holding other parameters fixed was uncommon. Overall, 59 out of 60 studies have at least one parameter that was fixed through- out the study. In 90% of studies (53 out of 59), no justification was given for parameter fixation.

4. Discussion

Levels of biological variation are never constant and cannot Fig. 6. Relative frequency histogram of the simulation ratio to model parameters. currently be predicted. Some parameters have narrow distribu- Only 38% of studies has simulation ratio greater than 1 indicating the number of tions (the density of for example), while others have very fi simulations performed was signi cantly lower than the number of model broad distributions (e.g., cartilage stiffness or bone strength). parameters. Furthermore, while there are some constants in engineering (specific of water, charge of an electron, etc.), biological “simultaneous” variation, Fig. 5). This was almost exclusively due constants have proven to be elusive. In fact, the existence of to the use of multiple medical images in deriving the geometry of “biological constants” is currently an area of active research and the structure. Relatively smaller percentage of studies varied more debate (Dhar and Giuliani, 2010). Because of the ubiquity of than two parameters in the aspects of material and boundary variation in biological systems, it is useful to conceptualize model condition. parameters in terms of variability. Every biological parameter can be thought of as a random variable with a distinct (though perhaps unknown) probability distribution. In our analysis of biomechanics 3.2.5. Number of simulations research studies, the majority of model parameters were fixed The number of simulations performed in each study was also throughout the study, while a select number were varied, usually tabulated. The number of simulations ranged from 1 to 400, with a parametrically. However, the practice of viewing model para- median of 11 simulations, a mean of 48.1, and a standard deviation meters as random variables was not observed in our sampled of 103.1. The mode of this data was 1 (22% of all studies involved studies. one simulation). Because computational biomechanics research seeks to answer It is informative to compare the number of simulations to the many of the same types of questions as traditional medical total number of model parameters. To this end, we define the research, we can conceptualize biomechanics studies using termi- simulation ratio (S) as the ratio of number of simulations (Ns)to nology from and . Of all possible experimental 1246 D. Cook et al. / Journal of Biomechanics 47 (2014) 1241–1250 designs, there are a few major types: case studies, in vitro studies, vocal fold model in which all relevant geometric and materials ex vivo studies, and in vivo studies. The level of experimental properties were varied using Monte Carlo and other techniques control is highest for in vitro studies, and lowest in in vivo studies. (Cook, 2009; Cook et al., 2009). This study revealed that the As a consequence, the in vitro studies provide the advantage to longitudinal shear modulus is one of the most influential factors examine relationships between variables in the absence of other affecting the vibratory characteristics of the vocal folds. Due to lack effects, but these results cannot be directly generalized at the of experimental data on this property, a single value for the system level, due to the restrictions imposed by experimental longitudinal shear modulus had been used in all prior modeling control. On the other hand, in vivo studies provide results that are studies (Alipour et al., 2011). The inclusion of biological variation more easily generalized, but the relationships between parameters in a vocal fold model, combined with powerful computational may be extremely difficult to detect due to simultaneous variation of tools, led to the identification of an important feature that had many inter-related parameters. Naturally, ex vivo studies lie between been previously obscured because of the lack of experimental data the other two methods in terms of control and generalizability. and biological variation. Based on our analysis, the experimental approach most com- Another example illustrates the use of sensitivity analysis to mon in the field of biomechanics is the in vitro type, as illustrated develop patient-specific calibration tools. The success of chemother- in Figs. 4 and 5. Studies in our field are focused on key parameters apeutic treatments and the potential efficacy of nanoparticle-based and relationships. They seek to provide understanding in areas therapies depend on a number of factors including geometric where little information and knowledge exists, and in these (capillary length, diameter, density, and plasma skimming layer situations, in vitro studies are the most appropriate choice. How- thickness), material properties (tissue permeability, fluid conductiv- ever, advances in computational power now provide the oppor- ity through the vessels, plasma viscosity, and the like), and bound- tunity to use experimental designs that mimic ex vivo, and in vivo ary conditions (arteriole pressure, venule pressure, osmotic pressure studies. This approach is exciting and has tremendous potential gradients, and tissue pressure). Previous studies examined a handful but is not commonly applied in biomechanics research studies. of variables at a time (Jain and Baxter, 1988), leading to the To encourage this approach, we suggest the use of terminology conclusion that the tissue pressure was one of the most important that supports the concept of biological variation in computational variables, but a complete assessment of all the aforementioned modeling: “instances”, “virtual sample” and “virtual individuals.” variables was daunting due to a lack of data for calibration. Schuff An instance is any fully defined computational model (i.e., all et al. (2012, 2013) demonstrated one of the benefits of statistical model parameters have been assigned as discrete numerical sensitivity analysis, by characterizing which variables have the value), though the manner in which they are assigned may be biggest effect on the ultimate response of the computational model. controlled and limited. A virtual sample is formed as a collection of Their data demonstrated that the penetration of small instances of a computational model. Finally, a virtual individual (or into the extravascular space depends not only on the tissue virtual human subject (Cook, 2009)) is a special case in which an pressure, but also the capillary density, pressure difference between instance of a model is obtained by allowing each parameter to take the arteriole and venule sides, and the properties of the vascular on values that reflect patterns of natural biological variation. In wall. More importantly, incorporating biological variability into the other words, no parameters are explicitly fixed, but are allowed to model and assessing the effects of the critical variables allowed vary according to their respective probability distributions. calibration of the model to individual patients. Because very few parameters are directly controlled, this approach As a final example, Lee and Piazza (2009) observed that the foot mirrors in vivo research methods, as suggested by the name itself. and ankle geometry of sprinters (track athletes) was noticeably There are practical reasons why the in vitro approach is most different from that of the general population. As compared to the common. One limiting factor in biomechanics research is the lack general population, sprinters were found to have a significantly of distribution data for many biological parameters. While mean shorter calcaneus, longer toes, and longer fascicles of the lateral and standard deviation data is available for many biological gastrocnemius. Rather than providing a mechanical advantage to parameters, this information alone is incomplete unless the sprinters, these factors actually produce a mechanical disadvan- relevant studies have specifically investigated the distribution of tage, in terms of force and leverage. However, a complete analysis this parameter to ensure that it is actually normally distributed. of the entire system (geometry, muscular contraction rates, and Furthermore, many biological parameters have not yet been dynamic interactions between these factors), revealed that parti- measured. Probability distributions for these parameters are thus cular geometric configurations enable faster , due to unknown. This however does not preclude the application of advantages in terms of speed and transfer. The authors stochastic methods which can be used even in the absence of concluded that, to some extent, an individual's potential to experimental data. In the following section, we highlight three become a world-class sprinter is determined by biological varia- studies in which biological variation has revealed surprising or tion. While running ability can be refined by training, a person insightful results in biomechanics research. without the “sprinter's heel” may never be an outstanding runner, regardless of training. 4.1. Examples 4.2. Paradigm shift The first example illustrates how the consideration of biological variation can be used to gain important insight, even in the The above examples illustrate ways in which the inclusion of absence of experimental data. The vibratory characteristics of the biological variation can provide insights in biomechanics research. vocal folds have been a subject of study for many decades (Titze, Looking outside of the biomechanics community, we observed 2006). Because the vocal folds are a complex structure consisting that high levels of variation in other fields have initiated important of multiple layers of transversely isotropic tissue, few material paradigm shifts and new techniques have been developed to deal measurements are available. As a result, models of this structure with the consequences of variation. In the past 10–20 years, are often based on assumptions or estimates of material proper- several engineering fields have shifted from a deterministic design ties. This practice is common in biomechanics owing to the methodology to a more sophisticated approach that considers the difficulty in measuring mechanical properties of various tissues. effects of random variation. For example, steel construction build- Even though not all material properties of this system have been ing codes were previously based on an extensive set of safety measured, stochastic sensitivity analysis was applied to a detailed factors to account for uncontrolled variation in material, geometry, D. Cook et al. / Journal of Biomechanics 47 (2014) 1241–1250 1247 and loading (1972). Revised standards (Ellingwood, 2001) are above, we noticed that while the in vitro approach is most based on stochastic methods which can be used to quantify risk, common, the corresponding limitations of this approach are which is not possible by using safety factors. Second, the rapidly seldom addressed. In fact, conclusions are often stated in general changing field of computer chip manufacturing has also recently terms even though the majority of model parameters were fixed undergone a similar transition. For many years, computer chips throughout the study. Experimental designs that incorporate only were designed using a methodology known as static timing a small number of parameters (in vitro) can obviously obscure analysis. This method was used to compute the performance of important features. That certain relationships exist in a controlled chip designs, assuming constants for the rates of various internal situation cannot be argued, but whether or not such relationships processes (Kirkpatrick and Clark, 1966; Hitchcock et al., 1982). As persist and are detectable when all other variables are considered the size of integrated circuits has become progressively smaller, cannot be concluded from an in vitro study; these questions can be however, variations in production have become more influential. answered by simultaneous variation of all model parameters. This has required the development of statistical static timing analysis, which considers statistical distributions of internal chip processes (Orshansky and Keutzer, 2002). Thus, in both steel 5. Conclusions construction and chip design, a consideration of variation has led to engineering methods which provide more accurate design Levels of biological variation were quantified, a meta-analysis methodologies for the overall system. Biomechanics as a field of biomechanics research studies was performed, and examples has not yet undergone a transition from a deterministic to a illustrating the effects of biological variation in biomechanics statistical paradigm. Such a transition will likely be needed if research were discussed. The meta-analysis provided a high- biomechanical models are to continue to increase in accuracy and level overview of modeling practices in biomechanics, with an relevance to the scientific and healthcare fields, especially regula- emphasis on biological variation. Each study was analyzed to tory bodies. The need for this transition is illustrated by a determine the number of model parameters in each essential comparison between levels of variation in engineered and biolo- aspect (geometry, material properties, and boundary conditions). gical systems (Fig. 2), as well as the understanding that in biology, In addition, data was collected regarding study design, variation all variables are inextricably connected. type, definition of variables, etc. The conclusions of this study are Numerous methods have been developed for assessing the as follows: effects of variation on model behavior. These include sensitivity analysis techniques (Saltelli et al., 2009), reliability analysis 1. The dominant approach in performing biomechanical research (Haldar and Mahadevan, 2000a), and stochastic modeling using biomechanical models is similar to in vitro studies: a few (Haldar and Mahadevan, 2000a, 2000b; Buzzard and Xiu, 2011). select model parameters are typically varied while the majority Among these, Monte Carlo analysis may be the most popular, due of model parameters are held fixed. Overall, 92% of all studies to its ease of implementation and flexibility. For examples of this are based on an experimental design in which at least one technique applied in biomechanics, see Gentle (2003), Kalos and major aspect (geometry, material properties, or boundary Whitlock (2008) and Ackland et al. (2012), among others. conditions) is held constant throughout the study. We now introduce a new conceptual framework for evaluating 2. The parameters of biomechanical models are not often biomechanical models. It is evident from the meta-analysis data conceptualized as random variables. Consideration of model that many biomechanical models are sufficiently complex that a parameters as random variables may be useful in shifting the thorough analysis involving all model parameters may beyond the paradigm in biomechanics towards a perspective that more scope of a single research paper. Modern computational tools comprehensively addresses biological variation. support the creation of extremely large, highly complex models. 3. Biomedical imaging is currently the most successful avenue for These models are powerful tools if we understand the many including natural biological variation in biomechanical simula- interactions that take place within the model. The results of our tions. Multiple medical images were used in 30% of biomecha- analysis show that typical biomechanics studies involve approxi- nical modeling studies. We anticipate that this figure will mately 10 simulations for every 18 model parameters. Because an continue to rise with the improvement of image processing n-parameter model potentially exhibits ðn2=2nÞ interactions software and image acquisition tools. between pairs of parameters, a typical 18-parameter model may 4. Biomechanical models may currently have a tendency toward have 144 such interactions. However, most studies in the biome- over-development. Most models rely upon fewer simulations chanics literature rely on fewer simulations than the number of than the number of model parameters, and are focused on model parameters, with the typical (median) number of simula- varying a minority of model parameters. tions being 10. It is clear that 10 simulations are insufficient to 5. Consideration of biological variation has been shown (anecdo- understand either the primary effects of each of the 18 model tally) to lead to useful, sometimes counter-intuitive, or unex- parameters or the 144 potential interactions between model pected results. This approach is not currently common in parameters. The quadratic relationship between interactions and biomechanics research. the number of parameters is a tremendous problem, especially for 6. We recommend the increased use and application of methods models involving a large number of parameters. As a consequence, for incorporating biological variation in biomechanical models, we suggest that many biomechanics models may be “over-devel- including sensitivity analysis techniques such as Monte Carlo oped.” In other words, these models are so complex, and require a methods, etc. computational expense that precludes the quantification of each parameter's effect on the model. The authors feel that the issue of It is the authors' hope that the data and analysis included in model development merits attention because the tendency in all this study will encourage the biomechanics research community areas of research is to build upon previous efforts. The introduc- to broaden the current scope of modeling practices. While in vitro tion of a single over-developed model could to lead to additional experimental designs will always be an essential component of over-developed models as the new model is expanded upon, biomechanics research, we feel that more attention can and combined with other models, etc. should be paid to the issues of biological variation, biomechanical Finally, we consider the conclusions that are drawn from our parameters as random variables, and interactions between model biomechanical models. In performing the meta-analysis described parameters. As has occurred in other research fields, this approach 1248 D. 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