Comparison of constitutive models for meniscus and their effect on the knee joint biomechanics during gait

Abstract Mechanical behavior of meniscus can be modeled using constitutive material models of varying complexity, such as isotropic elastic or fibril reinforced poroelastic (FRPE). However, the FRPE material is complex to implement, computationally demanding in 3D geometries, and simulation is time-consuming. Hence, we aimed to quantify the most suitable and efficient constitutive model of meniscus for simulation of cartilage responses in the knee joint during walking. We showed that simpler constitutive material models can reproduce similar cartilage responses to a knee model with the FRPE meniscus, but only knee models that consider orthotropic elastic meniscus can also reproduce meniscus responses adequately.


Introduction
Meniscus is a crescent-shaped fibrocartilaginous, poroelastic tissue located between the tibial plateau and femoral condyles in the tibiofemoral joint.It is primarily composed of collagen, proteoglycans, and fluid.Due to the unique composition and collagen network structure, meniscus has optimal mechanical properties to resist compressive, tensile and shear forces (Aspden et al. 1985;Mow 1990;Masouros et al. 2008).Hence, it can distribute tibiofemoral joint loads into a larger area on joint surfaces and to provide support for joint stabilization during daily physical activities (Cameron and Macnab 1972;Walker and Erkman 1975;Shrive et al. 1978;Seedhom and Hargreaves 1979;Ahmed and Burke 1983;Proctor et al. 1989;Anderson et al. 1991;Zhu et al. 1994;Tissakht et al. 1996;Makris et al. 2011;Danso et al. 2015).The biomechanical contribution of meniscus can be impaired due to injuries, age-related degenerative changes and surgical treatments such as total and partial meniscectomy (Egner 1982;Drosos and Pozo 2004;Englund et al. 2009;Bedi et al. 2010;Seitz et al. 2013).These deficiencies of meniscus increase stress concentrations in articular cartilage that may lead to the development of osteoarthritis (OA) (Baratz et al. 1986; Lee et al. 2006;Neuman et al. 2008;Netravali et al. 2010;Mononen et al. 2015;Jiang et al. 2020).
Like cartilage, meniscus can be considered as a fibril reinforced poroelastic (FRPE) material due to its structural characteristics (Fithian et al. 1990;Danso et al. 2015;Kleinhans and Jackson 2018;Berni et al. 2021).The FRPE material model effectively considers the mechanical contribution of the fibrillar (collagen fibril network) and non-fibrillar components (proteoglycans and fluid), thus, it can be applied to simulate both instantaneous and static loading conditions of the knee with the same material parameters (Li et al. 1999;Li et al. 2000;Wilson et al. 2004;Julkunen et al. 2007;Shirazi et al. 2008;Kazemi et al. 2012;Mononen et al. 2012;R€ as€ anen et al. 2013;Ven€ al€ ainen et al. 2016;Ferroni et al. 2021;Norberg et al. 2021).However, implementation of the FRPE material model of meniscus into computational knee joint models is a demanding task due to the three-dimensional (3D) geometry and complex topology.Furthermore, it should be noted that the use of complex material models usually increases the time needed for implementation and simulation.This will limit their application when numerous knee joint models need to be constructed and analyzed, e.g. for large clinical cohorts.
Currently, due to its simplicity and close to structural correspondence, a transversely isotropic elastic (TIE) material has been often used to simulate the mechanical behavior of meniscus in biomechanical knee joint models (Haut Donahue et al. 2002;Klets et al. 2016;Mononen et al. 2012Mononen et al. , 2015;;Wilson et al. 2003;Wilson et al. 2019;Jiang et al. 2020).However, the TIE model presents some limitations: (i) the intrinsic orthotropic structure of the tissue is simplified (Haut Donahue et al. 2002;Mononen et al. 2012;Jiang et al. 2020) and (ii), the compression tension nonlinearity along the fiber direction and poroelastic properties are not considered.However, there exist no studies, where capabilities of different meniscus material models would have been studied under physiologically relevant loading conditions of the knee joint.This information would be important since the application of simpler meniscus material models would substantially accelerate the generation, implementation and simulation of biomechanical knee joint models.One option to overcome limitations present in TIE material may be found behind an orthotropic poroelastic material (OTPE), which can capture 3 D tissue responses (deformations) under various loading conditions.
This study aimed to evaluate the capabilities of different constitutive models of meniscus in the knee, namely, orthotropic poroelastic (OTPE), orthotropic elastic (OTE), and isotropic elastic (IE).Reaction forces through the knee joint and mechanical responses of cartilage were compared to the knee model with the FRPE meniscus.We hypothesized that a simpler meniscus material model, specifically the OTPE, can be used in the knee joint model to capture identical mechanical responses of cartilage with the knee model incorporating FRPE meniscus properties.

Approach, workflow, and analysis
The workflow of the study is presented in Figure 1.First, we obtained the FRPE material model parameters for meniscus from a previous study (Table 1).These FRPE material parameters are based on earlier experiments on human meniscus (Danso et al. 2014(Danso et al. , 2015;;Fithian et al. 1989Fithian et al. , 1990)).It should be considered that the used FRPE model for meniscus generates an approximation of average bulk tissue response, since the non-fibrillar matrix modulus (proteoglycan distribution), fluid distribution and collagen fibril orientation was kept constant through tissue depth and location.The details related to the FRPE material formulation are presented in the supplementary material (S1).
Then, in order to ensure feasible comparison between different meniscus models, material parameters for the simpler models (IE, OTE, OTPE) were optimized by matching their mechanical responses with those obtained from the FPRE material model.Thus, a cubic test model (1 mm Â 1 mm Â 1 mm) was constructed with the aforementioned FRPE material properties (Table 1) to represent a small fraction of meniscus.The mechanical response of the FRPE meniscus material served as a reference for the simpler material models.The cubic geometry was created and meshed in the Abaqus/CAE (v6.14-3,Dassault Systems, Providence, RI, USA) using 8-node porous continuum elements (type C3D8P).This model was constructed using a cartesian coordinate system (i.e.index notation correspondence: 1 ¼ x-direction ¼ radial direction; 2 ¼ y-direction ¼ axial direction; 3 ¼ z-direction ¼ circumferential direction).To mimic similar tissue deformations as under physiologically relevant loading such as walking (strain rate % 100%/s), the cubic test model with the FRPE properties was subjected to compression, tension, and shear by the application of a 10% ramp displacement within 0.1 s on the surface, while the bottom surface was fixed (Figure 1b).Free fluid flow (pore pressure ¼ 0) was allowed through the surfaces, which were not restricted by the displacement boundary conditions.Then, the external mechanical responses (reaction forces and displacements) of the FRPE meniscus test model were used as a reference to which the rest of the models were matched.This way the material properties for the simpler material models were obtained, except for the OTPE material.In the OTPE material, in order to have a correct contribution of fluid on the mechanical response, Poisson's ratios and shear moduli were obtained at equilibrium, whereas the rest of the material parameters were obtained similarly as for other simpler material models (Figure 1c, see further details related to this from section 2.2).In addition, see the supplementary materials (S2) for the mechanical responses of the cubic test models with the FRPE, OTPE, OTE and IE materials.
After the external mechanical responses of the IE, OTE and OTPE menisci were matched with those of the FRPE meniscus, the corresponding material properties of the IE, OTE and OTPE menisci were implemented in an existing knee joint model (Figure 1d, Orozco et al. 2018).Then, the stance phase of gait was simulated using different meniscus material formulations (see details from section 2.3).After the simulations, tibiofemoral and meniscotibial contact areas, contact pressures and reaction forces through the meniscus and tibial cartilage were calculated in medial and lateral compartments.To obtain a better overall picture from tissue responses, we evaluated the mechanical response of tibial cartilage (the maximum principal stress, the maximum principal strain, and the fluid pressure) by averaging the analyzed parameters over the tibiofemoral contact area (Figure 1e).All simulations were conducted in Abaqus/Standard (v6.14-3) using soils consolidation analysis (or static analysis in the elastic cubic models without fluid; see section 2.2).

Determination of OTPE, OTE and IE meniscus material model parameters
See the supplementary material (S3.1, S3.2 and S3.3) for details on how the material parameters for the OTPE, OTE and IE meniscus models were determined.

The knee joint model with different meniscus material models
A knee joint model geometry was created from magnetic resonance (MR) images of a healthy knee joint as described in detail in earlier studies (Halonen et al. 2016;Orozco Figure 1.General overview of the study workflow.a) Fibril-reinforced poroelastic (FRPE) material description for meniscus validated in previous studies was obtained from our previous study.b) The FRPE meniscus model properties were implemented into a cubic test geometry, where tensile, compressive, and shear tests were simulated to characterize material behavior (stress, strain, deformation) in all directions (primary collagen fibril orientation was along the z-axis).c) Simulated mechanical responses were used to adjust the material properties for simpler meniscus material models (isotropic elastic (IE), orthotropic elastic (OTE), and orthotropic poroelastic (OTPE)).d) Then, simpler meniscus material models with optimized material properties were implemented in the 3 D knee joint model.e) Finally, their effect on tibial cartilage and meniscus mechanical responses were compared during walking.
et al. 2018).Briefly, the knee joint of a volunteer was imaged with a magnetic resonance scanner using a 3 D fast spin-echo sequence (in-plane resolution ¼ 0.5 mm, slice thickness ¼ 0.5 mm, TR ¼ 1300 ms and TE ¼ 32.3 ms).The knee joint geometry (Figure 1e) was obtained by segmentation in Mimics (v.15.01, Materialise, Belgium).
The segmented geometry was imported into Abaqus/CAE.The femoral and tibial cartilage, the menisci, the patellar and quadriceps (PT and QT) tendons and all six main ligaments in the knee joint: anterior and posterior cruciate (ACL and PCL), medial and lateral collateral (MCL and LCL), and medial and lateral patellofemoral (MPFL and LPFL) ligaments were all modeled as solid geometries.The tibial and the femoral cartilage were meshed with hexahedral pore pressure elements (element type ¼ C3D8P), while menisci were meshed either with hexahedral pore pressure elements or without porosity (element type ¼ C3D8) depending on the constitutive model (see details for the mesh densities from the previous study (Halonen et al. 2016).Cartilage tissues were modeled as a fibril-reinforced poroviscoelastic (FRPVE) material (Wilson et al. 2004;Wilson et al. 2005).Ligaments, and patellar and quadriceps tendons were defined as the FRPE material.Menisci were modeled using the FRPE, OTPE, OTE and IE material formulations.To facilitate the collagen fibril orientation in the OTE and OTPE meniscus models, a cylindrical coordinate system was defined separately for lateral and medial menisci (Figure 2a).See Table 1 for cartilage and meniscus material parameters and the supplementary material (S1) for details on the fibril-reinforced material formulations.
Furthermore, to evaluate the effects of different model configurations (meniscus attachment point locations and mechanical properties of the knee ligaments) on the mechanical responses, we conducted additional simulations based on a parametric analysis.Moreover, as the knee geometry may also influence the model responses, an additional knee joint geometry was used to verify our conclusions (see the supplementary material (S5 and S6) for further details and results from these simulations).

Boundary conditions and gait cycle
The motions of the joint level models were controlled by the reference point located at the central point between lateral and medial epicondyles of femur (Mononen et al. 2013).As the subchondral bones were assumed as rigid, the translations of the femoral nodes at the cartilagebone interface were coupled with the translations of the reference point, whereas the tibial nodes at the cartilagebone interface were fixed in all directions.Frictionless surface-to-surface contact was defined for all contacting surfaces i.e. the contact surfaces between cartilages and menisci, the contact surfaces of solid ligaments and cartilage, and the contact surfaces of ACL and PCL.The master surfaces were determined as element-based surfaces, whereas the slave surfaces were defined as nodebased surfaces.The gait input (moment and forces, Figure 2b-d), the quadriceps muscle forces (divided into anterior-posterior and distal-proximal components) and other boundary conditions for the knee joint model were identical to the previous studies (Halonen et al. 2016;Orozco et al. 2018).See simulations and analyses from section 2.1.

Results
The simulated tensile (i.e.maximum principal) stress distributions on the tibial cartilage of the knee joint were similar with all meniscus material models during the entire stance phase of the gait (Figure 3, high-resulution image is provided in the supplementary material (S7)).However, when focusing on the tensile stresses in the meniscus, though the distributions were similar between the models, substantial differences in the peak values were observed during the entire stance phase of the gait.The peak values for tensile stresses were located constantly on the lateral meniscus.In the OTPE and OTE models, the peak values ($7MPa) were over two times higher compared to those of the FRPE model ($3 MPa).
The simulated reaction forces through the knee cartilage and meniscus surfaces were also very similar for each meniscus material model (Figures 4 and 5).The reaction forces through the tibiofemoral cartilage surfaces were practically identical with all meniscus material models (Figure 4).The largest difference in the simulated reaction forces ($100 N) through menisci was observed at the beginning of the stance phase in the lateral compartment between the models with FRPE and OTPE menisci (Figure 5).
The tibiofemoral contact areas and contact pressures were very similar between the meniscus material models (Figure 6).The simulated meniscotibial contact areas between the models with FRPE and orthotropic (OTE and OTPE) menisci were also similar.The knee model with IE menisci was not able to capture the same meniscotibial contact area, instead, it produced constantly smaller values when compared to the model with the FRPE meniscus (Figure 7a, c).Similar trend was seen in the simulated meniscotibial contact pressures.While the models with FRPE and orthotropic (OTE and OTPE) menisci produced similar meniscotibial contact pressures, the model with IE menisci produced constantly higher values (up to 75% and higher) when compared to the model with the FRPE meniscus (Figure 7b, d).
Maximum principal stress, maximum principal strain and pore pressures at the tibial cartilage surface between the different material models were almost identical during the entire stance phase (Figure 8).This was also the case in the additional knee joint geometry (see the supplementary material (S6)).

Discussion
In the present study, our main objective was to clarify what level of simplicity is enough to model meniscus in the FE knee joint model.We showed that the knee joint models with simpler material models of meniscus (OTE, OTPE and IE) can produce identical cartilage mechanical responses during physiologically relevant loading conditions with the knee model with the more complex FRPE meniscus.
In the knee joint models, almost identical reaction forces, contact areas and contact pressures through cartilage-cartilage and cartilage-meniscus contacts were observed between medial and lateral compartments with both orthotropic and FRPE meniscus material models during the entire gait.This suggests that when the material parameters of the simpler material models are properly calibrated using a similar workflow as in this study, the optimized simpler material models for meniscus may not significantly affect the contribution of the meniscus in transferring loads between the femur and tibia compared to the FRPE meniscus model.On the other hand, the knee model with IE meniscus material model was not able to capture simultaneously the reaction forces through meniscus nor the meniscotibial contact areas and pressures.This suggests that in order to obtain comparable contact areas, contact pressures and forces by the knee model with IE model, one single set of material parameters for IE meniscus does not suffice.
The simulated forces through the lateral and medial meniscus with IE, OTE and OTPE meniscus materials were similar to those of the model with FRPE meniscus material along with the stance phase of the gait cycle.However, in the lateral meniscus, there was a substantial difference in the simulated reaction forces of the models with IE and orthotropic menisci compared to those of the model with FRPE meniscus during the first $5% of the stance phase.This can be explained by the discrepancies in the implemented meniscus material models which may display different mechanical behavior under substantial strain and shear rates that are occurring simultaneously at the beginning of the loading response.
The magnitude and the distribution of the average maximum principal stresses, strains, and pore pressures were generally similar among the different material models throughout the gait cycle in both the medial and lateral compartments on the cartilage surface.This indicated that overall contribution of the medial and lateral meniscus to distribute joint loads are very similar in all meniscus models during walking activities.
However, it should be noted that when fluid flow in a cartilage or meniscus plays a crucial role in simulating mechanical responses that occur during slow loading rates (e.g.prolonged standing), the simpler material models (IE and OTE) no longer work as well as at fast loading rates where fluid does not have time to flow out of the tissue.
In earlier studies (Halonen et al. 2013;Haut Donahue et al. 2002;Klets et al. 2016;Mononen et al. 2012Mononen et al. , 2013Mononen et al. , 2015;;Vaziri et al. 2008;Wilson et al. 2003), menisci in the knee joint models were assumed as a transversely isotropic elastic (TIE) material.In those knee models, the TIE meniscus material model did not account for the time-dependent properties caused by fluid flow.Therefore, only the short-term loading of meniscus was considered, and the results mainly focused on cartilages.An important detail for the OTPE meniscus material used in the current study is that it can mimic both instantaneous and equilibrium deformation conditions.This is an important aspect, especially when modeling prolonged activities such as standing or kneeling with simpler constitutive models for meniscus.It is also worth mentioning that the optimized OTE material used here behaves practically as the TIE material (Figure 8).Therefore, current results can be considered as further confirmation for the applicability of the TIE material for meniscus, when simulating cartilage responses during physiologically relevant loading conditions within the knee joint.The option of using simpler material models for meniscus such TIE material enables easy implementation that is accessible to all, and faster simulation times compared to poroelastic alternatives such as OTPE or FRPE material models.Although the material properties in the commonly used TIE model (Mononen et al. 2012) differ significantly from the optimized OTE model, the support provided by the menisci appears to be similar based on similar simulated cartilage mechanical responses for the TIE and OTE models.This can be explained by the dependencies between Poisson's ratio and Young's modulus, which can produce similar mechanical response under several different combinations, especially during rapid loading when tissue volume is not assumed to change due to low fluid flow.In terms of clinical application, the models that are suitable should meet following criteria: speed and accuracy.Based on the results given by this study, the option utilizing the TIE material model offers mechanical responses for meniscus (forces through the meniscus) that meets both criteria.However, a particular attention should be given to how to select feasible Poisson's ratios into TIE models as they are highly dependent on loading conditions (instantaneous vs equilibrium).This study has a few limitations.First, the gait input data and the tissue geometries were taken from an earlier study, which consists of only one healthy male subject (Halonen et al. 2016).It is known that the capability of meniscus to transfer loads within the knee can vary highly in different subjects (Brial et al. 2019;Leatherman et al. 2014).Thus, we conducted additional simulations to show how different model configurations, i.e. changes in the meniscus attachment point locations, mechanical properties in knee ligaments, and knee geometry would affect the conclusions (see the supplementary material (S5 and S6)).These simulations resulted in similar conclusions as discussed.It should be also noted that in the current model presented here, the lateral meniscus transferred roughly 50-100% of the total joint force during the entire stance phase, while in the selected previous studies (Guess et al. 2015;Gilbert et al. 2014), medial meniscus was shown to be more heavily loaded.Hence the question about the influence of a constitutive model in the case of a heavily loaded medial meniscus remains still slightly open, though, the influence is likely not any different from the lateral joint compartment.Second, although only normal walking was simulated, this approach can be easily applied for other daily activities such as stair climbing, sit-to-stand and squatting.However, when the task is changed, one must be cautious if the task has a strongly varying timedependent component, i.e.only poroelastic formulations may be valid.Third, we did not have access to experimental data that would have contained both orthotropic and FRPE material properties of human meniscus.Thus, we selected the used approach in which the material properties of the orthotropic material were obtained by matching the force and displacement of these material models with the FRPE material model.Material parameters for the FRPE material was based on the experimental measurements for human and bovine tissues.The combination of human and bovine data can be justified by the lack of sufficient experimental data for human meniscus.Fourth, the mechanical properties for the FRPE model were based on the mechanical experimental measurements for certain meniscus locations.However, it is true that more realistic and inhomogeneous models considering location dependent changes in nonfibrillar matrix, depthwise collagen fibril orientation and fluid distribution might produce slightly different results, especially in certain local areas where the properties would differ from   the used ones.However, the same assumption about the orientation (and other properties) was used in both models, therefore, these assumptions should not affect the conclusions.Fifth, the current comparison of global and average stress values for each meniscus material model does not permit to obtain a conclusive assessment of menisci response.For instance, we did not analyze how the cartilage stresses under the meniscus changed due to different meniscus material models, since the focus on this study was to show that simpler material models of meniscus can reproduce identical articular cartilage responses at tibiofemoral contact region during walking compared with the knee model with the FRPE meniscus.Sixth, meniscal attachments were considered as linear springs, which is also a limitation.It has been shown that the mechanical response of meniscus attachment is hyperelastic (Abraham et al. 2011), which may have some influence on how meniscus behaves under loading.Thus, the validity of the applied material properties and model configuration should be compared against real experimental measurement, where simulated tissue deformations would be compared against MRI or x-ray imaging during performing a certain physical activity such as hopping or standing (Halonen et al. 2014;Sutter et al. 2014).
Complex material models are more capable to capture mechanical responses of soft tissues under various loading conditions compared to simpler models.However, it is not possible to ensure that mechanical parameters in complex models are subject-specific not to mention consideration of differences in mechanical properties in different meniscus regions (Danso et al. 2015;Fithian et al. 1990) when utilizing data from clinical setups.It has been shown experimentally how meniscus moves when the knee joint is compressed during knee flexion (Scholes et al. 2015).However, due to the fact that meniscus adapts its position due to changing tibiofemoral contact geometry, the motion of meniscus can be replicated with different constitutive models if the selected material properties reproduce similar responses (forces and deformations) under certain loading conditions.
In conclusion, the results showed that simpler meniscus material models, especially orthotropic ones, can be used in the knee joint models to simulate walking since they produce identical cartilage responses with the knee model incorporating more complex FRPE menisci.These simpler meniscus models are faster and easier to implement enabling their use in large cohort studies.

Figure 2 .
Figure 2. The main inputs of the knee joint model.a) The local cylindrical coordinate system was assigned on the lateral and medial menisci.Then, b) moments and rotations, c) quadriceps force components and d) translational forces were assigned to the knee joint model, as described in Orozco et al. (2018) and Halonen et al. (2016), to simulate the stance phase of the gait cycle.

Figure 3 .
Figure3.An overall view about (qualitative analysis) the maximum principal stress distributions on tibial cartilage and menisci with different material models of meniscus at different fractions of the stance phase of the gait cycle.The distributions are presented with the interval of 25% starting from 0% so that the time points of 25% and 75% represents loading response and terminal extension, respectively.

Figure 4 .
Figure 4.The simulated reaction force through the a) lateral and b) medial tibial cartilage surfaces as a function of the stance phase of the gait with different meniscus material models.

Figure 5 .
Figure 5.The simulated reaction force through the (a, b) lateral and (c, d) medial meniscus surfaces as a function of the stance phase of the gait with different meniscus material models.The force ratio in (b, d) indicates the ratio between total reaction forces through the tibiofemoral compartment (medial or lateral) and reaction forces through the meniscus.

Figure 6 .
Figure 6.The simulated tibiofemoral contact area and average contact pressure in (a, b) lateral and (c, d) medial compartments of the knee joint as a function of the stance phase of the gait with different meniscus material models.

Figure 7 .
Figure 7.The simulated meniscotibial contact area and average contact pressure in (a, b) lateral and (c, d) medial compartments of the knee joint as a function of the stance phase of the gait with different meniscus material models.

Figure 8 .
Figure 8.The simulated (a, b) lateral and medial average maximum principal stresses, (c, d) average maximum principal strains and (e, f) average pore pressures on the tibial cartilage as a function of the stance phase of the gait with meniscus material models.The commonly used (TIE) material model with parameters from Mononen et al. (2012) is presented only for a visual comparison.The average values were separately calculated over the tibiofemoral contact area on lateral and medial tibial cartilage for each time point.

Table 1 .
Material properties for the reference fibril-reinforced poroelastic (FRPE) meniscus material model and obtain properties for the other meniscus material models.¼ the fibril network modulus, E m ¼ the non-fibrillar matrix modulus, v m ¼ the Poisson's ratio of the non-fibrillar matrix, k 0 ¼ the initial permeability, M ¼ the strain-dependent permeability coefficient.OTPE and OTE material parameters: E 11 ¼ the radial elastic modulus, E 22 ¼ the axial elastic modulus, E 33 ¼ the circumferential elastic modulus, v 12 ¼ the Poisson's ratio between radial and axial planes, v 13 ¼ the Poisson's ratio between radial and circumferential planes, v 23 ¼ the Poisson's ratio between axial and circumferential planes, G 12 ¼ the shear modulus between radial and axial planes, G 13 ¼ the shear modulus between radial and circumferential planes, G 23 ¼ the shear modulus between axial and circumferential planes, and k ¼ the permeability.IE material parameters: E ¼ the elastic modulus, v ¼ the Poisson's ratio.