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00005768-201308000-0001300005768_2013_45_1524_naemi_mathematical_8miscellaneous-article< 208_0_24_10 >Medicine & Science in Sports & Exercise© 2013 American College of Sports MedicineVolume 45(8)August 2013p 1524–1533Mathematical Models to Assess Foot–Ground Interaction: An Overview[APPLIED SCIENCES]NAEMI, ROOZBEH; CHOCKALINGAM, NACHIAPPANFaculty of Health Sciences, Staffordshire University, Stoke-on-Trent, UNITED KINGDOMAddress for correspondence: Roozbeh Naemi, Ph.D., Faculty of Health Sciences, Staffordshire University, Leek Road, Stoke-on-Trent ST4 2DF, United Kingdom; E-mail: .Submitted for publication September 2012.Accepted for publication February 2013.Supplemental digital content is available for this article. Direct URL citations appear in the printed text and are provided in the HTML and PDF versions of this article on the journal’s Web site ( ).AbstractABSTRACT: The mechanical properties of the interface between the human body and the ground play an important role in attenuating the foot strike impact during locomotion. Understanding the properties of such an impact-attenuating system and the mathematical models governing its behavior has major implications in determining the load on the musculoskeletal system during locomotion. This interface consists of the plantar soft tissue and the sole complex of the shoe that together act like a sophisticated suspension system with generic viscoelastic properties. The interface has generally been modeled as a system of spring and damper, in which the reaction force deformation is expressed by a mathematical equation that represents the reaction force as a nonlinear function of the deformation and deformation rate of the interface. This overview intends to provide an insight into the different mathematical models that have been used to describe such relationship and into further understanding the role of the reaction model parameters in determining the behavior of the interface under compression. Various models included within this review ranged from the models representing the plantar soft tissue behavior during barefoot walking to those that consider the sole complex force–deformation behavior during shod foot running. The barefoot models are categorized under in vitro/in situ and in vivo, whereas the models representing the sole complex behavior are investigated before discussing the shod foot models. The mathematical models varied from those in which the reaction force was a nonlinear function of interface deformation to those that considered the deformation rate of the interface as a contributing factor to the interface reaction force. Ultimately, the implication of the reaction models in determining the load on the musculoskeletal system is discussed.In general terms, the impact profile can be indicated as the pattern of vertical ground reaction force during the initial contact between the human body and the ground. This profile normally associated with the impact peak (the maximum force reached during this period) and with the loading rate (the rate of force increase) and has been linked to various musculoskeletal injuries. In particular, the loading rate has been found to be significantly different for the runner groups with stress fracture injuries as compared with the controls (56).The transient peak within the force curve depends on the deceleration of different body segments, including the wobbling mass of the soft tissue and the rigid mass of the bones and also on the deceleration of the bony segments in immediate contact with the ground surface (37). Hence, the impact profile can potentially be influenced or attenuated by several parameters, including active strategies like muscle activity during contact (38,50,51), referred to here as “active mechanisms,” and also by the mechanical behavior of the interface between the ground and the bone of a segment in immediate contact with the ground, referred to here as “passive mechanisms” (27).The impact attenuation involves spreading the force over a longer period to create the impulse required to bring the body to rest through both of these active and passive mechanisms (5,6). The active mechanism of impact attenuation acts through an eccentric contraction of muscles that allow lower extremity joints to undergo hip and knee flexion along with ankle dorsiflexion (12,26,33). The active mechanism also includes muscle activity before the foot makes contact with the ground, which can affect the deceleration of wobbling mass during the impact (38,49).The passive mechanism normally includes all the elements in the interface between the skeleton and the ground, including the shoe sole, the heel pad, and the cartilage at the interconnecting articular tissue that together act to attenuate the impact during the foot–ground contact. With a different stiffness and energy return ability for each of the constituting elements in the sole complex (14) and soft tissue (7,19,46), the behavior of the interface is affected by the way these different constituting elements interact (21,22).The passive and the active impact attenuation mechanisms are interlinked (10,11). Despite the earlier studies revealing that the passive mechanisms have a dominant role as compared with the active mechanisms (27), more recent studies revealed a close interconnection between the two (32,38,49,50,53,55).Although the passive mechanism acts through the compression of the interface between the ground and the body, the combined mechanical properties of the interface elements affect the loading of the musculoskeletal system during locomotion (1,10,13,36,38,50). Hence, understanding the mechanical properties of passive impact-attenuating system and its behavior can have major implications in determining the external and internal load on the musculoskeletal system.The Importance of the Interface Reaction ModelsIn the majority of experimental studies in which the role of interface on the loading of the musculoskeletal system is established (1,10,13,38,50), shore hardness and density of the sole complex were used as a quantitative measure of the interface. Similarly, in experimental studies, the plantar soft tissue behavior (2,3,23), a descriptive graphical representation of the behavior of the interface, is represented. Although these general descriptive assessments of the sole complex provided generic information of the interface behavior under load, they do not provide a quantitative measure of the complex force–deformation relationship during locomotion. In addition, the lack of a quantitative detailed measure of the behavior of each constituting element in the interface (i.e., the sole complex and plantar soft tissue) would make the assessment of the interaction between the two more complicated because of the interplay between the two (1,21,22).To reflect such complex and interactive behavior, an interface reaction model in which the force–deformation behavior of the interface is specifically demonstrated has been used (7–9,15–19,24,29,30–32,34,37,39,40,44,45,48,53–55,57). These models, in addition to the generic application in musculoskeletal simulations (9,16–18,32) and vibration modeling (24,30–31,37,39,40,53–55), has been also used in quantifying the plantar soft tissue behavior during loading (15,19,29,42,45,57,46). By providing a detailed understanding of the behavior of the interface during locomotion, the interface reaction models provide a deeper understanding of the complex interaction between the active and the passive mechanisms. This, in addition to the provision of a systematic method of analyzing the effect of interface on the impact profile (27,32,38,50,53,55), provides an insight into the way the muscle activity hence joint loading is regulated (9,16–18,32,37,39,55). Thus, understanding the mathematical models that determine the behavior of the interface under loading provides an in-depth understanding of the load on the musculoskeletal system, which can have major implications in assessing the risk of musculoskeletal injuries.Purpose and Structure of the ReviewThis article intends to provide an overview of the mathematical models that have been used to describe the viscoelastic behavior of the interface between the skeleton and the ground, specifically in relation to how the stiffness and damping characteristics of the interface affect the relationship between the reaction force and the deformation pattern. It should be emphasized that the term reaction force within this review refers to the force that results from the compression of the viscoelastic interface. The pattern in which this force develops (as a result of deformation or compression between the skeleton and the ground during loading) can be described by the interface reaction model.The effectiveness of the ground/skeleton interface as the mechanism of impact attenuation is dependent on both the mechanical properties of the plantar soft tissue (7,19,46) and the shoe sole complex (14). As a result, the mathematical model describing the behavior of the plantar soft tissue and shoe sole complex is presented and discussed before presenting the models describing the shod foot behavior.Barefoot ModelsIn a barefoot condition, the attenuation of impact depends on the mechanical properties of the region of the foot that first contacts the ground during a foot strike. In normal circumstances during barefoot walking, the heel is the first point of contact; hence, the mechanical properties of the heel pad have been studied in detail with a wide range of mathematical models designed to investigate its behavior (1–4,7,15,19,42,46).Heel pad function and behavior.The area underneath the calcaneus, known as the the heel pad, consists of a fatty tissue cushion along with a structure constituting of two distinct regions. This includes a layer containing stiffer microchamber and a deeper layer of relatively compliant macrochambers (19). The properties of the fat pad under loading are influenced by the thickness and integrity of the fat pad structure, which is in turn determined by the characteristics of septa that separate the fat into compartments or cells (20).The septa that are reinforced internally with elastic transverse and diagonal fibers connect the thicker walls (20). During loading, the fat chambers undergo deformation, and the septa act as a support structure to prevent bulging. These structures are fixed to the bone from the top (superior aspect) and to the bottom (the inferior aspect) to a thick fibrous subdermal layer known as internal heel cup (20).On the basis of the histologic observations, it was hypothesized that the microchambers as the smaller structures are more effective in resisting deformation than the macrochambers (20). However, differentiating the contribution of different layers and chambers in impact attenuation is not straightforward. Hence, most models developed for the bare foot have considered cushioning characteristics of the heel pad in bulk.The studies on the mechanical properties of the heel pad include in vivo and in vitro/in situ tests. In the in vivo situation, the heel area of the foot is affected by an instrumented pendulum while the subject kneels on a support surface with the knees against the wall and the forefoot clamped (1). In in vitro studies, the isolated heel pad (4,29) is loaded using a dynamic loading machine to determine the force and deformation during a loading cycle. In some in situ studies, the heel pad while intact with the underlying bone was loaded by either an instrumented pendulum (2,42) or in a universal testing machine (4,23).In vitro/in situ testsLedoux and Belevins (29) proposed the following equation to reflect the stress–strain behavior of the plantar soft tissue in the foot:Equation (Uncited)A skinless 2 × 2-cm specimen from a freshly frozen cadaver samples was tested with a displacement control triangular wave. They showed that when the specimens from different plantar sites were loaded up to 20% of body weight, there was a significantly higher modulus of elasticity and lower energy loss for the subcalcaneal heel pad when compared with the other areas. Despite this, there was no significant difference in the values of A and B reported for different sites. Using the same equation, Pai and Ledoux (45) found no significant differences in either A or B values for different sites or between the diabetic and the nondiabetic specimens. The lack of significant difference in the values between the diabetic and the nondiabetic soft tissue was attributed to the changes in the structural level that may not have reflected themselves adequately at the material level (43,45).In addition, the in vitro tests of isolated heel pad showed an energy loss of 30%, and the in vivo results showed a strong damping of approximately 95% energy dissipation and double compliance as compared with in vitro tests (2). To account for this discrepancy, it was recommended that the investigation of the human heel pad would produce more reliable results if the heel pad were to be tested in situ under the conditions proposed by Aerts et al. (2). In this condition, a cylindrical mass instrumented with an accelerometer was used as a pendulum to impact the heel region of the amputated intact foot, where the heel bone was fixed to a wall (2).Several studies assessed the behavior of the heel pad in in situ conditions (2,42,23). Noe et al. (42) modeled the behavior of heel pad and the polymeric impact absorber during the heel strike as a system of spring and damper. The spring stiffness and the viscosity were selected so that the acceleration of foot specimen equals the acceleration measured during the heel strike. Despite presenting the two sets of spring damper elements placed in series, no mathematical model was provided to represent the behavior of the system. However, in another study, Ker (23) proposed a nonlinear equation in which force was related to stiffness. The constant within this equation was determined using a nonlinear regression between the force and the stiffness. Although the stiffness value is variable depending on the loading cycle stage, it was only reported when the force is equal to body weight, hence making the comparison of the reaction forces at different stages of loading cycle impossible. In general, both the in vitro and in situ tests have not been commonly used in the assessment of plantar soft tissue in situ; hence, most of the modeling studies have used experimental data from in vivo indentation techniques to reflect the mathematical model of the plantar soft tissue under loading.In vivo testsIn an attempt to quantify the heel pad characteristics, a function represented by the following equation was fitted to the force deformation data:Equation (Uncited)where a and b are constants that were extracted by fitting this function to the data. The parameters for the two groups of participants including those with heel pain were compared with the group without heel pain (46). Although there was a significant difference in the b value between the groups (0.85 for the group with heel pain vs 0.91 for the control group), no significant difference was found for the a value between the groups (0.12 for the group with heel pain and 0.13 for the control group).Although both parameters (a as the factor and b as the exponent) determine the value of the curve in practice, the value of exponent b can be considered as a more influential in determining the shape of the force–deformation curve. It is evident that the higher the factor a, the higher the stiffness (as the load required to deform the heel pad to a certain deformation); whereas the higher the exponent factor b, the less the rate of changes in the stiffness up to a point where they intersect (at 5.8 mm of deformation).It was shown by Rome et al. (46) that the stiffness (as the load required for equal deformation) was less for the group with plantar heel pain as compared with the normal control up to the critical deformation, which was indicated by having a lower factor (a) value. Although the observed differences between the two groups were directly related to the curve parameters, the maximum load (i.e., 20 N) under which the foot was loaded was much lower when compared with the load on the heel pad during the stance phase of walking.To quantify the mechanical properties of the heel pad, Challis et al. (7) modeled the heel pad as a system of elastic element that resembles a soft tissue behavior using a force deformation function identical with equation 1. The coefficients were estimated using a nonlinear least square method to fit the function represented by equation 1 to the force–deformation data gathered through indentation using a triangular deformation control with a constant deformation rate of 1 mm·s−1. A curve-fitting technique was used to find the stiffness at a force corresponding to 5% of body weight of subjects. A significant difference in stiffness between the cyclist and the runner groups was found and associated with the activity (7). Although the changes in the mechanical properties of soft tissue as a result of activity reveal themselves in the load deformation behavior of the interface because the values for a and b were not reported, the effect of reaction model parameters cannot be determined.To provide a model in which changes in the mechanical properties of the heel pad can be readily reflected on the model parameter, another model was proposed in other studies (8,57) to quantify the heel pad elastic properties,Equation (Uncited)where E is the Young modulus of elasticity, α is the radius of indenter, ν is the Poison ratio, h is the tissue thickness, and κ is a function of Poisson ratio and the ratio of radius of indenter to the tissue thickness. By assuming a Poisson ratio of 0.45, Zheng et al. (57) evaluated the Young modulus of elasticity based on the slope of a linear regression line to the force deformation.Interestingly, the method provided a range of different modulus of elasticity for the different regions of the plantar surface of the foot ranging from approximately 40 to 50 kPa for healthy adults, and the value increased by approximately 160% on average when similar sites in a diabetic group were tested (57). Although differences were reported between the pathological soft tissue and that of a healthy adult, there was no indication on how function κ changed with the tissue thickness, making it impossible to reconstruct the force deformation behavior across all loading limits.Furthermore, limiting the maximum deformation to 10% of the initial thickness may have affected the parameters extracted from the model. Because heel pad follows a curvilinear trend in the force–deformation relationship similar to any other viscoelastic material, it is likely that the stiffness reported would vary if a higher deformation is achieved (44).The same formula (equation 3) was used by Chao et al. (8), where the Young modulus of elasticity of two different age groups, including the young (22–35 yr) and the older (55–74 yr) adults, were compared. To control the loading and indentation pattern, Chao et al. (8) introduced a noncontact optical coherence tomography combined with an air-jet indentation system, where four cycles of loading and unloading at a deformation rate of 0.4 mm·s−1 were carried out for each indentation trial loading cycle. A significant increase in the Young modulus of the second metatarsal region was observed in the older group as compared with the younger adults.Despite the success of Chao et al. (8) in controlling for the loading rate, which was an improvement to what was reported by Zhen et al. (57), the maximum indentation force (1.4 N) was much lower than what is exerted to the same area during walking, which raises the same issue in relation to extrapolating the stiffness results to the higher loads.Loading and Unloading CurvesIn all of the above-reviewed methods (7,8,57), the stiffness coefficient was calculated by fitting the proposed model to the force deformation graph during loading and unloading. However, no separate reaction model parameters were considered for the loading and unloading phases. To consider the differences in the force deformation behavior between loading and unloading stages, Hsu et al. (19) proposed a stress–strain equation (equation 4) for which the constants were different for the loading and unloading:Equation (Uncited)where σ represents stress and ε is the strain defined as the ratio of heel pad deformation under loading to the unloaded thickness. σmax represents the maximum stress constant, and [Latin Small Letter Open E]max represents the maximum strain constant. α is the curvature constant that represents a different value for loading and unloading curve. Hsu et al. (19) performed an in vivo test of the heel pad of a specimen, using ultrasound indentation where the heel-pad thickness was measured in a loading and unloading cycle to the maximum pressure of 74 kPa, with increments of 12.4 kPa.A significant difference in the curvature constant for the unloading curve between the diabetic and the healthy specimens was found (11.8 ± 5.1 vs 8.5 ± 2.6), whereas no significant difference in α for loading between the groups existed. Although the higher value of α corresponded to a higher curvature, it was evident that this can distinguish between the loading and the unloading curvature, with the unloading curvature always higher than loading. This is the result of the fact that the energy stored during loading as the area below the loading curve is higher than the energy return represented by the area below the unloading curve.Role of ViscosityIn addition to elastic forces that conserve energy, a nonconservative force acts as another component of reaction force that needs to be overcome when the plantar soft tissue is deformed, resulting in hysteresis. The viscosity component plays an important role in determining the changes in the mechanical properties of soft tissue. For example, when an energy dissipation ratio was calculated as the ratio of the area surrounded by the loading and the unloading curve divided by the area below the loading curve, a significantly different energy dissipation ratio between the diabetic groups (36% ± 8.7%) and healthy adult groups (27.9% ± 6.2%) were reported. It was also reported that the difference between unloading and loading curvature constants was significantly higher for the diabetic when compared with healthy adults (19), which was assumed to be related to the breakdown of collagen fibrils.To identify a direct association between energy dissipation ratio and reaction model parameters, the viscous term was added to the reaction force model as an additional term to the elasticity (15). Gefen et al. (15) considered the effect of viscosity by proposing a viscoelastic model based on the mechanical behavior of a Voight–Kelvin model comprising a linear elastic and a nonlinear damper model:Equation (Uncited)where σ represents stress and ε is the strain, E is the elastic parameter, and η represents the viscous parameter of the soft tissue. In their study, the changes in the thickness of the heel pad of participants walking on an instrumented platform were measured using digital fluoroscopy synchronized with dynamic plantar pressure measurement. Then the stress and strain values were calculated based on the changes in the thickness of the plantar tissue and the corresponding pressure. Eventually, a curve represented by equation 5 was fitted to the stress–strain data gathered during the experiment. E = 175 kPa and η = 22 kPa were reported, where a constant strain rate of 2.5 ± 0.5·s−1 was assumed. It seems that this assumption was necessary to allow the elastic and viscous parameters to be extracted through a curve-fitting approach, although the measured strain rate showed a constant change with higher strain rate during loading as compared with unloading. Despite the advantages of the model introduced by Gefen et al. (15), the assumption of linear elasticity may be scrutinized as an oversimplification of the elastic behavior of soft tissue.Barefoot Model OverviewIn vivo versus in vitro/in situ.Some studies report the differences between the mechanical properties of an isolated and intact fat pad and investigated the cause of these differences (2–4). Despite the evidence that the changes in mechanical properties and, hence, the reaction model parameters happen at the structural level (19,45), the mathematical model that represents the behavior of plantar soft tissue in in situ conditions have been scarce.In most in vivo studies (7,8,19,46), the maximum load applied to the plantar soft tissue was much lower than the actual load applied to the heel pad during locomotion. Because of the nonlinear (curvilinear) force–deformation relationship inherent in viscoelastic behavior, the extrapolation of the mechanical properties can potentially lead to large errors (44).The determination of the mechanical properties of the plantar soft tissue during walking seems to reveal a more realistic behavior of soft tissue under load (15,52). However, the increased level of exposure to radiation could be debilitating. The ultrasound indentation techniques replicating the observed load pattern when combined with an appropriate reaction model may provide more insight into the complex behavior of plantar soft tissue, although a two-dimensional measurement of soft tissue deformation may not reveal the three-dimensional nature of soft tissue deformation.Loading frequency and rest time.It has been reported that the loading frequency affects the stiffness of plantar soft tissue (29,43,45). Despite this, a reaction model that includes loading frequency as a direct input into the model does not exist. When considering the effect of rest time, Ker (23) reported a 3.7% increase in energy loss when the rest time between the loading cycles was increased from 1 to 10 s. More recently, Pai and Ledoux (45), using a similar method to the one proposed by Ledoux and Blevins (29), indicated no significant difference between the time dependent mechanical properties of soft tissue between the diabetic and the nondiabetic patients or between the different plantar regions of the foot when isolated plantar tissue is tested in vitro.The Need for New Barefoot ModelsThere has been a paucity of studies in which the complete foot rather than the isolated heel pad is tested, and there is a lack of knowledge on how the reaction model parameters are influenced by the diseases and loading pattern, that is, walking pace, effect of rest time, and frequency on the reaction model parameters.There is also a need for a more detailed model that can possibly consider the effect of different layers of soft tissue and relate the microproperties to the differences observed in the load deformation pattern. For example, Hsu et al. (19) measured the stiffness of micro- and macrochambers separately using ultrasound indentation at constant loading and unloading speed, and an increased macrochambers but decreased microchambers stiffness for the heel pad tissue was reported for diabetics and was attributed to diminished cushioning ability of diabetic heel region.Despite this, the findings were based on a linear stress–strain relationship that can be considered as an oversimplification of the soft tissue force deformation behavior. There is a need for more detailed models. The models presenting the behavior of the entire soft tissue when combined with appropriate nonlinear models can provide more insight into the mechanical properties of the soft tissue during loading. Also, such models need to be linked to the histological studies at the microlevel to shed more light on the interconnection between the positioning and the structure of the fat globules in the collagen septa in both the micro- and the macrochambers and the role they play with regard to the behavior of the fat pad under loading. In addition, there is also a need for models that can identify the effect of skin tissue on the loading and impact attenuation properties of the whole plantar soft tissue structure.Such models can have important implications in developing a more detailed understanding of the differences between healthy/unhealthy as well as the age groups under each condition. Previously, it was estimated that the stiffness coefficient of the soft tissue of the first metatarsal region was 28% lower as compared to the second metatarsal region (8), and there is a need for methods that can provide a different set of reaction model parameters for different sites of the plantar region of the foot under the loading pattern specific to that region. Furthermore, such a model should consider the effect of rest time in between the loading cycles.Sole Complex ModelsThe sole complex consists of several different layers made from different materials. This complex that is positioned between the ground and the heel tissue consists of the outsole, the midsole, and the insole made from rubber or polymer foam material. Depending on the structure of the foam as being open or closed cell, the behavior of the foam can vary in compression (25). The open cell foam (like polyurethane [PU]) allows a free passage of air through the cells, whereas in a closed cell foam (like ethylene vinyl acetate [EVA]), the cell acts like an air chamber (14,48), and the microscopic characteristics affect the behavior of the material under loading (25).Like the heel pad, the measurement of the impact attenuation properties of the shoe sole has been performed on the insole, midsole, and outsole. Because some of the polymers used as midsole are made of closed cell polymer foams like EVA, Verdejo and Mills (48) proposed the following equation to define the behavior of the midsole under loading:Equation (Uncited)where σ0 is a constant dependent on the polymer material or initial compressive collapse stress, p0 is the gas pressure in the undeformed foam cell, and R represents the relative foam density. The ratio of [Latin Small Letter Open E] / (1 − [Latin Small Letter Open E] − R) was defined as the gas volumetric strain, and σ0 and p0 were found by fitting a linear regression fit to the graph of stress against the gas volumetric strain. By testing the foams after approximately 1300 km of running, the authors found that the softening of the foam was due to the decrease in the so-called initial compressive collapse stress rather than the changes in the gas pressure inside the foam cells.Although the model was used to provide useful information on the stress–strain relationship in closed cell foams, it could not be used for open cell foams, which are also commonly used in footwear (e.g., PU). A method that can evaluate the performance of both closed and open cell foams was proposed by Garcia et al. (14). For determining the effect of loading frequency and thickness on insole, Garcia et al. (14) proposed a linear relationship between the stress and the strain.By observing the force deformation behavior of the foams under harmonic loading, and through the use of Fourier conversion, stiffness (the ratio between the stress and the strain) was defined as a harmonic function in which the amplitude represented rigidity, and the phase shift between the load and the displacement was defined as the loss tangent and considered as the ratio of lost energy to stored energy. Using this method, it was concluded that the rigidity increases with an increase in the loading frequency and thickness of the tested insole materials, including a PVC foam of open structure as well as a microair elastomer and cellular urethane. However, there was no consistent trend for the loss tangent observed for either increasing thickness or loading frequency between the groups.Although both approaches proposed by Verdejo and Mills (48) and Garcia et al. (14) enables considering the behavior of material during loading in conditions corresponding to loading values and frequencies that happen in a realistic scenario, the assumption of linear relationship between the load and the displacement may be considered as an oversimplification of the nonlinear behavior of a viscoelastic material. When evaluating the impact absorbency of an outsole made of rubber, Silva et al. (47) observed an exponential increase and a linear decrease in the energy absorbed with an increase in the thickness and an increase in shore A hardness, respectively. Despite the finding, no model was proposed to relate the force deformation behavior under loading (47).In general, although the models developed for testing the material commonly used as insole, midsole, or outsole provide an understanding of the behavior of the sole components during loading, it does not provide an understanding of how the entire sole complex behaves as a whole. Several factors like the sole unit geometry, thickness, and various constituting components and the material used within each of these components (47) contribute to the impact absorption characteristics of sole complex of the footwear.To assess the behavior of the sole complex as a whole, Ly et al. (31) proposed a model (equation 7) in which the material constants of the midsole were determined:Equation (Uncited)where R0 is the radius of the indenter and b is a constant representing the nonlinear model. An oscillating sinusoidal force of amplitude 1200 N at 1.5 Hz to the rear area of the sole was applied. The parameters K and b were determined by fitting a power law function to the entire data during complete cycle. The value for parameter c was determined by calculating the energy dissipation during a loading/unloading cycle for different range of midsole shore hardness. The reconstructed reaction force over a range of deformation and deformation rates is presented in Appendix 1 (Supplemental Digital Content 1, Figure 6, ). Despite the ability of the method to identify the effect of the shore hardness on the sole reaction model, the assumption of damping term being a linear function of deformation rate was not fully justified (31).To evaluate a sole-specific reaction model parameters, Naemi and Chockalingam (34) assumed a nonlinear vicoelastic model represented by nonlinear stiffness and a nonlinear damping terms (equation 10) that was introduced first by Cole et al. (9) to model the behavior of shoe/ground during running, where the reaction force is a nonlinear function of deformation and deformation rate. Based on the fact that the elastic and the viscous reaction forces act in favor during loading and against each other during unloading, the viscous and elastic components were extracted (34). Using a curve-fitting technique, a parametric curve represented by reaction force model was fitted to each of the components force–deformation data to extract the parameters. Furthermore, by integrating the parametric curve for loading and unloading, a parametric energy return efficiency function was calculated, which directly showed the effect of the reaction model parameters on the energy return efficiency.Despite the importance of the sole complex models in determining the behavior of the sole complex under loading, the models on their own cannot predict the interaction between the body and the ground during running and walking. For this reason, the shod foot models in which the combined effects of plantar soft tissue and sole complex are taken into account have been specifically developed to assess the effect on locomotion.Shod Foot ModelsAlthough a direct measurement of force during locomotion provides a more realistic data on the effect of certain type of footwear on the ground reaction force (10,36), a detailed understanding of the force–deformation behavior of the shod foot cannot be determined. Furthermore, the interaction between the active and the passive mechanism (explained earlier) will make the comparison of the shod foot behavior difficult to control. To control for these differences, several methods have been used. These include assessing the ground reaction force after a drop jump when the subjects’ knee joint were fixed (22) or using the human pendulum approach when a subject is laid on the back and suspended while the knee movement is fixed (28).In the later approach, a vertically mounted force plate was used to measure the ground reaction forces, and the amount of swing was adjusted to simulate the foot touchdown speed. Although these two approaches can assimilate conditions in relation to the way the active mechanism of the leg is controlled (the knee fixed or fully extended), the difference may arise as a result of the dissimilarities in trunk movement. These variations can influence the center of mass deceleration pattern under weight-bearing (vertical) and non–weight- bearing (supine) conditions. In another study to control the variables, an accelerometer-instrumented pendulum was used to apply impact forces to the heel region while the subject knelt on a support surface with knees against the wall and the forefoot clamped (1).In practice, the impact-attenuating properties of the passive mechanism during shod locomotion depends on the behavior of the heel pad and the ways its natural impact-attenuating mechanisms interact with shoe sole complex (1,10,21,22). To investigate this interaction, the behavior of the heel pad when confined with insoles was studied (21,22). It was concluded that for subjects with a low degree of heel pad impact absorbency, the impact absorption during the heel strike can be increased with an appropriate shoe. Although this emphasizes the effect of the shoe in modifying the behavior of soft tissue in shod condition, it clearly highlighted the need to investigate the properties of the shod foot as a system comprising interacting impact-attenuating mechanisms, including the heel pad and the shoe sole.To relate the impact-attenuating properties of the shod foot to the stiffness and damping characteristics in more simplistic models, the reaction force was assumed to be equal to the sum of force from linear spring and damper:Equation (Uncited)By comparing the acceleration of the foot, calculated based on the model with the experimental measures, Kim et al. (24) calculated a spring stiffness of 9300 N·m−1 while the damping factor was found to be 5.4 N·s·m−1. When a similar linear model was used for running by Nigg and Anton (35), a spring stiffness was found to be 125–250 kN·m−1 and was chosen in such a way that the deformation of foot mass under a load of 2500 N was 2 to 1 cm with viscosity was reported between 2500 and 17,500 (N·s·m−1).The discrepancy of the values reported may be a result of the fact that in the model proposed by Nigg and Anton (35), kN·m−1 a spring representing the ground stiffness and a damper representing the combined viscosity of the sole and the heel pad in series were positioned in parallel to a stiffer spring, representing the stiffness of the sole and heel pad. However, the model proposed by Kim et al. (24) did not include ground stiffness, and the spring and the damper representing the shod foot were in parallel. This also indicates that the range for the constants for a model is influenced by the way the different components of the model are configured and the way the reaction model parameters are determined.Furthermore, considering the nonlinear behavior of a viscoelastic material including the plantar soft tissue and the sole of the shoe, a linear system is an oversimplification of the viscoelastic characteristics of the shod foot. Gilchrist and Winter (18) introduced a nonlinear model for the ground reaction force in which the damping terms was a nonlinear function of the spring deformation. The model proposed changes in the damping over the initial heel contact, and the damping was assumed to stay constant over the stance phase. A stiffness of 40 kN·m−1 and a maximum damping factor of 300 N·s·m−1 were reported. The model proved to be useful for gait simulation (17) but required knowing how the damping factor changes and how they interact at different deformations.To reflect the nonlinear and interacting fashion in which the deformation and deformation rate determine the reaction force in the shod foot, Gerristen et al. (16) proposed the following function:Equation (Uncited)Although it is quoted that using the nonlinear model provides a better continuity of the numerical solutions during the simulation, there was no rationale on the choice of the function. The parameters of a = 0.25 × 109 N·m−3 and b = 1.0 s·m−1 were proposed for achieving realistic ground reaction forces during simulation (i.e., 1066 N). The reconstructed reaction force over a range of deformation and deformation rates is presented in Appendix 1 (Supplemental Digital Content 1, Figure 1, ). It was also reported that the higher stiffness resulted in a higher impact force and a lower deformation. Although an increase in the damping coefficient b from 0 to 1.5 (s·m−1) seems to decrease the impact peak of the simulated ground reaction force, a gradual increase was observed when the value for b increased from 1.5 to 2.5 (s·m−1).Despite the ability of the model to predict the nonlinear behavior of the system and to show the effect of viscosity on the impact profile of the simulated ground reaction force, the choice of 3 as the exponent would restrict the system to behave in a predetermined fashion. To reflect the more realistic behavior of the interface, a more versatile reaction model was proposed by Cole et al. (9). This consisted of a nonlinear system of spring and damper in which the stiffness term is considered as a power function of deformation, whereas the damping term was assumed to be a power function of the deformation rate and deformation.In the proposed model, the shod foot was considered as a mechanical system represented by a nonlinear spring and damper, and the reaction force was formulated according to the following equation (10):Equation (Uncited)where the constants a, b, c, d, and e are representing the specific parameters. The constant parameters are determined using a trial and error procedure in which the force–deformation curve predicted from the simulated pendulum impact tests are compared with the force deformation curve of the pendulum test data provided by Aerts and de Clercq (1). The model proposed a set of parameters for a typical shod foot and used in a musculoskeletal simulation by which Cole et al. (9) characterized the effect of muscle activity on joint loading and ground reaction force. The reconstructed reaction force over a range of deformation and deformation rates is presented in Appendix 1 (Supplemental Digital Content 1, Figure 2, ).The same model (equation 10) was used to investigate the effect of shoe hardness on the impact forces in heel toe running (53). Wright et al. (53) proposed a different set of reaction model parameters for the hard and soft shoe. Using the same method, two sets of parameters were obtained for the hard and soft shoe corresponding to the shore hardness of 65 and 40, respectively. The reconstructed reaction force over a range of deformation and deformation rates is presented in Appendix 1 (Supplemental Digital Content 1, Figure 3, ). No significant difference in the simulated ground reaction impact peak between the hard and the soft shoe was observed when a whole body musculoskeletal simulation was performed. Despite this, the muscle forces for the two different conditions were found to be significantly different and lead to a different joint loading as a result of using a different shoe sole hardness.Different sets of parameters for the soft and hard shoe were proposed by Liu and Nigg (30) to investigate the effect of mass distribution on the impact profile during running. By using a mass spring damper model that represented the rigid body mass and wobbling mass of the segments, it was shown that the impact profile of the ground reaction force is influenced by the shoe hardness. The reconstructed reaction force over a range of deformation and deformation rates is presented in Appendix 1 (Supplemental Digital Content 1, Figure 4, ).Zadpoor et al. (54) proposed a different value for a as compared with what was recommended by Liu and Nigg (30). Appendix 1 (Supplemental Digital Content 1, Figure 5, ) shows the reconstructed reaction force over a range of deformation and deformation rates based on the model parameters recommended by Zadpoor et al. (54).Zadpoor et al. (54) also showed that the difference in this value reflect itself on the ground reaction force impact profile.Implications in Vibration ModelsThe ground reaction force that is predicted with the vibrating models has been shown to be strongly dependent on the sole reaction model parameters when passive multibody models were simulated (30,31,54). These observations, although in line with experimental studies in which the foot was affected while the leg was fixed (1,27), are contradictory to the findings of other experimental studies in which the data were collected during running (11,13), which showed low dependency of the ground reaction force on the interface stiffness (40). To reflect such differences that arise as a result of muscle activity (38), more recently the active simulation of the multibody models in which a nonlinear controller was introduced to mimic the functionality of the central nervous system and muscle activity was investigated (39,40,55). Such studies show the role of active mechanism in tuning the mechanical properties of the soft-tissue, revealing that the footwear model influences the muscle activity in a way so that the ground reaction force is kept constant (39,41,55). Although these studies indicate that the sole reaction model parameters do not influence the ground reaction impulse, the specific role of sole reaction model parameters in modifying muscle activity warrants further investigations.Implications in Musculoskeletal SimulationsThe effect of the sole reaction model parameters on muscle activity during the impact phase has been investigated in several studies (9,16,53). More recently, Miller and Hamill (32) reported a modified muscle activity as a result of the effect of the shoe. The ground reaction impact peak when using a harder shoe increased by only 2% body weight and decreased by 7% of body weight for the male and female simulated subjects, respectively, when compared with the use of a softer shoe. When the effect of shoe cushioning on the joint loading during running was investigated, Miller and Hamill (32) reported an increased loading rate of 25% for male and 23% for the female simulated subjects, respectively, when a hard shoe is used as opposed to a soft shoe. More importantly, when using a hard shoe, there was an increase of 20% and 5% in the compressive and shear forces applied to the tibia for the male and female simulated model as compared with the corresponding values when using a soft shoe.In general, in line with the previous experimental studies (11,13), the musculoskeletal simulation studies combined with the sole reaction models for the soft and hard shoe (32,53) reveal an increase in the internal loading as a result of changes in the shoe hardness. Although these changes may not be reflected in the impact profile of the ground reaction force, the changes in the reaction model parameters influence the muscle activity, hence the joint loading.The Future and the Need for Subject/Shoe-Specific ModelsDespite the valuable information that can be gained by using the interface models in simulation studies to answer generic questions a case-specific reaction model that can be used for specific subject/shoe scenarios does not exist.Furthermore, the behavior of the model that is based on the data gathered in mechanical test rigs may not adequately reflect the specific force–deformation pattern that happens during locomotion; hence, there is a need to develop a testing procedure that can determine the reaction model during locomotion in realistic loading conditions.Furthermore, there is a paucity of subject-specific models that take into account the unique mechanical properties of the plantar soft tissue under the unique loading pattern specific to the individual. This, when combined with the shoe-specific model unique to an individual, can be a useful tool in investigating the effect of individualized interface (the sole of the shoe and the plantar soft tissue) on the ground reaction force profile, the vibration of soft tissue, and the muscle activity and joint loading.ConclusionsThe reaction model parameters relate the reaction force to the deformation pattern of the interface. These parameters determine the deformation behavior of the interface when the interface is loaded. As the interface between the ground and the body includes viscoelastic materials such as sole unit and plantar fat pad, the mathematical models that represents the force-deformation pattern generally consisted of sum of a stiffness component, in which the force is a nonlinear function of deformation—and a damping component, for which the force is correlated to the deformation rate. 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[CrossRef] [Medline Link] [Context Link] GROUND REACTION MODEL PARAMETERS; SOLE REACTION MODEL; FORCE–DEFORMATION PATTERN; SHOD FOOT MODEL; BAREFOOT MODEL; PLANTAR SOFT|00005768-201308000-00013#xpointer(id(R2-13))|11065405||ovftdb|SL00004539199618841711065405P140[Medline Link]|00005768-201308000-00013#xpointer(id(R3-13))|11065213||ovftdb|SL00004617199528129911065213P141[CrossRef]|00005768-201308000-00013#xpointer(id(R3-13))|11065405||ovftdb|SL00004617199528129911065405P141[Medline Link]|00005768-201308000-00013#xpointer(id(R5-13))|11065213||ovftdb|SL00004617199124109511065213P143[CrossRef]|00005768-201308000-00013#xpointer(id(R5-13))|11065405||ovftdb|SL00004617199124109511065405P143[Medline Link]|00005768-201308000-00013#xpointer(id(R6-13))|11065213||ovftdb|SL0000461719922522311065213P144[CrossRef]|00005768-201308000-00013#xpointer(id(R6-13))|11065405||ovftdb|SL0000461719922522311065405P144[Medline 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Link]|00005768-201308000-00013#xpointer(id(R31-13))|11065213||ovftdb|SL0000461720104331011065213P169[CrossRef]|00005768-201308000-00013#xpointer(id(R31-13))|11065405||ovftdb|SL0000461720104331011065405P169[Medline Link]|00005768-201308000-00013#xpointer(id(R32-13))|11065213||ovftdb|SL0013099920091248111065213P170[CrossRef]|00005768-201308000-00013#xpointer(id(R32-13))|11065405||ovftdb|SL0013099920091248111065405P170[Medline Link]|00005768-201308000-00013#xpointer(id(R35-13))|11065213||ovftdb|00005768-199501000-00017SL000057681995279211065213P173[CrossRef]|00005768-201308000-00013#xpointer(id(R35-13))|11065404||ovftdb|00005768-199501000-00017SL000057681995279211065404P173[Full Text]|00005768-201308000-00013#xpointer(id(R35-13))|11065405||ovftdb|00005768-199501000-00017SL000057681995279211065405P173[Medline Link]|00005768-201308000-00013#xpointer(id(R37-13))|11065213||ovftdb|SL0000461719993284911065213P175[CrossRef]|00005768-201308000-00013#xpointer(id(R37-13))|11065405||ovftdb|SL0000461719993284911065405P175[Medline Link]|00005768-201308000-00013#xpointer(id(R38-13))|11065213||ovftdb|SL0000461720033656911065213P176[CrossRef]|00005768-201308000-00013#xpointer(id(R38-13))|11065405||ovftdb|SL0000461720033656911065405P176[Medline Link]|00005768-201308000-00013#xpointer(id(R39-13))|11065213||ovftdb|SL0000461720114498411065213P177[CrossRef]|00005768-201308000-00013#xpointer(id(R39-13))|11065405||ovftdb|SL0000461720114498411065405P177[Medline Link]|00005768-201308000-00013#xpointer(id(R40-13))|11065213||ovftdb|SL000018602011225112111065213P178[CrossRef]|00005768-201308000-00013#xpointer(id(R40-13))|11065405||ovftdb|SL000018602011225112111065405P178[Medline Link]|00005768-201308000-00013#xpointer(id(R41-13))|11065213||ovftdb|SL0000420620125979711065213P179[CrossRef]|00005768-201308000-00013#xpointer(id(R41-13))|11065405||ovftdb|SL0000420620125979711065405P179[Medline Link]|00005768-201308000-00013#xpointer(id(R42-13))|11065213||ovftdb|SL000053351993152311065213P180[CrossRef]|00005768-201308000-00013#xpointer(id(R42-13))|11065405||ovftdb|SL000053351993152311065405P180[Medline Link]|00005768-201308000-00013#xpointer(id(R43-13))|11065213||ovftdb|SL00004617201043175411065213P181[CrossRef]|00005768-201308000-00013#xpointer(id(R43-13))|11065405||ovftdb|SL00004617201043175411065405P181[Medline Link]|00005768-201308000-00013#xpointer(id(R44-13))|11065213||ovftdb|SL00005328201013207100111065213P182[CrossRef]|00005768-201308000-00013#xpointer(id(R44-13))|11065405||ovftdb|SL00005328201013207100111065405P182[Medline Link]|00005768-201308000-00013#xpointer(id(R45-13))|11065213||ovftdb|00000570-201105000-00012SL00000570201139151711065213P183[CrossRef]|00005768-201308000-00013#xpointer(id(R45-13))|11065404||ovftdb|00000570-201105000-00012SL00000570201139151711065404P183[Full Text]|00005768-201308000-00013#xpointer(id(R45-13))|11065405||ovftdb|00000570-201105000-00012SL00000570201139151711065405P183[Medline Link]|00005768-201308000-00013#xpointer(id(R46-13))|11065213||ovftdb|SL0000904320011690111065213P184[CrossRef]|00005768-201308000-00013#xpointer(id(R46-13))|11065405||ovftdb|SL0000904320011690111065405P184[Medline Link]|00005768-201308000-00013#xpointer(id(R48-13))|11065213||ovftdb|SL0001050720032356711065213P186[CrossRef]|00005768-201308000-00013#xpointer(id(R48-13))|11065405||ovftdb|SL0001050720032356711065405P186[Medline Link]|00005768-201308000-00013#xpointer(id(R49-13))|11065213||ovftdb|SL00004617200336176111065213P187[CrossRef]|00005768-201308000-00013#xpointer(id(R49-13))|11065405||ovftdb|SL00004617200336176111065405P187[Medline Link]|00005768-201308000-00013#xpointer(id(R50-13))|11065213||ovftdb|00005768-200209000-00021SL00005768200234152911065213P188[CrossRef]|00005768-201308000-00013#xpointer(id(R50-13))|11065404||ovftdb|00005768-200209000-00021SL00005768200234152911065404P188[Full Text]|00005768-201308000-00013#xpointer(id(R50-13))|11065405||ovftdb|00005768-200209000-00021SL00005768200234152911065405P188[Medline Link]|00005768-201308000-00013#xpointer(id(R51-13))|11065405||ovftdb|SL00004560200191130711065405P189[Medline Link]|00005768-201308000-00013#xpointer(id(R52-13))|11065213||ovftdb|00009043-200905000-00012SL0000904320092439711065213P190[CrossRef]|00005768-201308000-00013#xpointer(id(R52-13))|11065404||ovftdb|00009043-200905000-00012SL0000904320092439711065404P190[Full Text]|00005768-201308000-00013#xpointer(id(R52-13))|11065405||ovftdb|00009043-200905000-00012SL0000904320092439711065405P190[Medline Link]|00005768-201308000-00013#xpointer(id(R53-13))|11065213||ovftdb|SL0000904319981352111065213P191[CrossRef]|00005768-201308000-00013#xpointer(id(R53-13))|11065405||ovftdb|SL0000904319981352111065405P191[Medline Link]|00005768-201308000-00013#xpointer(id(R54-13))|11065213||ovftdb|SL00004617200740201211065213P192[CrossRef]|00005768-201308000-00013#xpointer(id(R54-13))|11065405||ovftdb|SL00004617200740201211065405P192[Medline Link]|00005768-201308000-00013#xpointer(id(R55-13))|11065213||ovftdb|SL0000461720104318611065213P193[CrossRef]|00005768-201308000-00013#xpointer(id(R55-13))|11065405||ovftdb|SL0000461720104318611065405P193[Medline Link]|00005768-201308000-00013#xpointer(id(R56-13))|11065213||ovftdb|00009043-201101000-00003SL000090432011262311065213P194[CrossRef]|00005768-201308000-00013#xpointer(id(R56-13))|11065404||ovftdb|00009043-201101000-00003SL000090432011262311065404P194[Full Text]|00005768-201308000-00013#xpointer(id(R56-13))|11065405||ovftdb|00009043-201101000-00003SL000090432011262311065405P194[Medline Link]|00005768-201308000-00013#xpointer(id(R57-13))|11065213||ovftdb|SL0000793520002645111065213P195[CrossRef]|00005768-201308000-00013#xpointer(id(R57-13))|11065405||ovftdb|SL0000793520002645111065405P195[Medline Link]10773376Mathematical Models to Assess Foot–Ground Interaction: An OverviewNAEMI, ROOZBEH; CHOCKALINGAM, NACHIAPPANApplied Sciences845