Noninvasive methods to study the knee joint has various applications, such as for better understanding of its biomechanics [6, 7, 36, 37], tissue health [12, 15, 21, 33], cartilage degeneration [2, 16, 18, 20], weight-bearing behavior of soft-tissues [3, 5, 15, 17, 19], and further using the information for optimization of implants design [4, 22] have long been an interest of clinical research community. A noninvasive method is proposed in the current study to estimate subject-specific material properties of whole tibiofemoral knee joint as Combined Compressive Stiffness (CCS) for use in FEM. In the CCS, the effect of all the contributing soft-tissues, non-Newtonian fluids, and their dynamic interaction during load was included. A simplified finite element modeling of the knee joint was conducted in the study to estimate CCS. Further, a generalized mathematical model of the stiffness versus strain relationship has been presented in the study. The stiffness calculated in the current study was the collective response of all tibiofemoral joint tissue and synovial fluid since the individual componential study may be insufficient to describe the complex physical interaction of these components in-situ conditions.
The subject-specific model in the current study is a simplified model, which is interested in Z-directional deformation in soft-tissue because of the 50%BW load. The femur and tibial bone have various regions with different material properties, such as a medullary cavity, compact bone, and epiphysis. Whereas uniform isotropic material properties were assigned in the current study to simplify the FEM model as bone components have non-significant contributions in Z-direction deformation during the applied load.
Furthermore, other subject-specific parameters such as muscle strength, synovial fluid as non-Newtonian fluid and its pressure, and anisotropic model of cartilage and bone might be included, making the model more realistic. However, the inclusion of all these subject-specific parameters will increase the complexity in FEM and further increase computational cost. Further, the problem with using these parameters is that most of this information cannot be accessed non-invasively, thus prohibits the use of such information in real clinical settings.
Experimental study
In the current study, subject-specific mBGFT and ΔmBGFT were computed from MRI images of the unloaded and loaded knee joint and used to calculate mean strain. Similarly, Chan et al. [18] computed articular cartilage strain during load, which overlooked other tissue effects in the in-situ environment. However, the current study considered the strain at the whole tibiofemoral joint, which includes the effects of each component of all WB-ST, synovial fluid, and interaction among them. The percentage of ΔmBGFT of unloaded mBGFT was observed in a range of 6–15%, which was similar to previously reported studies, 5.23% ± 6.20 using MRI and 4.57% ± 10.31 using X-ray [16]. In the current study, the intra-subject variation of percentage ΔmBGFT in subject-1, subject-2, and subject-3 was 1%, 4.8%, and 1.01%, respectively. However, the absolute deviation was 0.06 mm, 0.35 mm, and 0.08 mm, respectively, which might be due to variations in segmentation or change in bone alignment during experimental load conditions. The change in bone alignment can alter the distribution of load across the region [36], which is reflected as the variations in the observed value of compressive stiffness [27]. However, in the current study, FEM analysis incorporates the subject-specific change in force vector due to the change in alignment.
Simulation Study
The current study developed a method to estimate subject-specific stiffness for FEM and proposed a generalized mathematical model. The FEM was used as an efficient tool for evaluation of joint disorder [1], stress–strain distribution at articulating surface [1, 2], and estimation of body-weight for the onset of osteoarthritis (OA) [3] and optimization of implant selection [4]; however, previous studies [2, 3, 6, 7, 9, 37] has not used subject-specific material properties for FEM. Thus, a method to estimate subject-specific mechanical properties for FEM is proposed.
The average stiffness of the soft-tissues observed in the current study was 2.45 ± 0.13 MPa, for healthy subjects. Further, intra-subject variability in the estimated stiffness of two subjects was observed as 0.27 MPa and 0.12 MPa, respectively, within acceptable limits for FEM analysis because such small variation is trivial to yield any significant difference in results outcome.
The uniform mesh size may not be suited to the knee joint's complex geometry during FEM. Thus an adaptive convergence method was deployed for the refinement of mesh and to measure convergence error. The absolute convergence error was observed less than 1.02% for all the cases may depict further refinement of mesh have not significant change in the results.
The High computation time for the simulation studies restricts the use of this in individual data for clinical use, whereas the proposed generalized mathematical model is very handy and can even be used at a console and be useful in clinical practice.
Mathematical model
In the previous report, Butz et al. [38] estimated the subject-specific cartilage material properties using mathematical formula and stress–strain obtained from DENSE-FSE MR images. However, the mathematical equation used [38] did not consider the curvature shape of the tibiofemoral interaction region. In the ideal case of strain–stiffness–stress relationship (1), where contact region is a plain surface and area of contact remains constant under load conditions (Additional file 1: Fig. S2a), coefficient 'a' is denoted as stress and coefficient 'b' as '− 1'. However, in the curvature shape contact region as in the tibiofemoral joint, the contact area depends on the load and the stiffness of the tibiofemoral joint (Additional file 1: Fig. S2b). It has been observed that the inter-subject variation in estimated CCS was highly dependent on the scaling coefficient ‘a’. The scaling coefficient ‘a’ was derived from mean stress at the tibiofemoral joint, and it incorporates subject-specific features. Inter-subject variability in this scaling coefficient was observed because of inter-subject variability in tibiofemoral contact-area and anatomy. In contrast, intra-subject variability was observed because of variability in the orientation of bone and change in force vector during repeat scan.
Whereas the power coefficient ‘b’ was observed as a slightly lower negative value than ‘− 1’ because of the curvature shape of the tibiofemoral contact surface. Curvature shape changes the contact area with a change in stiffness; that is, more is the stiffness correspondingly lesser is the contact area. However, intra-subject variability in the power coefficient was observed (Table 2). This could be possible because of variability in the contact region due to orientation change during a repeat scan. The proposed generalized model (2) can estimate CCS using ΔmBGFT and mBGFT obtained by MRI images with the loading device. This study could also be extended to other imaging modalities such as standing X-ray [16] and standing MRI. However, for appropriate FEM analysis, it is recommended to drive the subject-specific mathematical equation to estimate the CCS of the tibiofemoral joint.
Clinical perspectives
Previous studies about biomechanical properties of individual tissues of the knee joint such as cartilage [24, 25, 27,28,29,30,31,32], and meniscus [32] may be useful for the development of tissue replacement biomaterial. However, understanding the knee dynamic by FEM modeling, CCS that is collective response complex interactions of all WB-ST and synovial fluid may be more appropriate.
The load distribution among the knee compartments depends on bone alignment, structure, and material properties. Therefore, a pre-surgical study might help clinicians to understand the joint’s loading pattern, thus minimizing post-surgery adversity [38]. In addition, a pre-surgical CCS evaluation may provide the best-suited customized material properties for an individual specific knee joint.
Further, load distribution studies are also important in partial knee replacement surgery. The pre-surgery evaluation of CCS may provide the whole joint’s stiffness; further, the partial volume material properties could be changed in the FEM, which may provide load distribution patterns for surgical planning.
The subject-specific model might help longitudinal studies of the subject, evaluating the effects of exercise, aging effect, or disease progression. From a clinical perspective, combined stiffness of the tibiofemoral joint may serve as an additional indicator of the joint's health.
Limitations
The proposed method could estimate the CCS of the tibiofemoral joint using FEM, which represented a simplified model of the complex knee joint. Further, stiffness of each knee joint’s component such as cartilage, meniscus, ligament, and synovial fluid, and their inter-and intra- component variability could only be accessed by an increase in the number of stiffness values for each compartment or component in FEM to develop a mathematical model; but, it increases the computation complexity; hence, it is not possible in routine clinical settings. Additionally, FEM is used largely for biomechanical studies. Nevertheless, simulation results may depart from realistic conditions, especially in complex structures such as knee joints. Furthermore, this study did not include subjects with osteoarthritis or patients with any other knee disease, which could show the variability between healthy and degraded soft-tissues. Thirdly, the effect of muscle tone on the bone gap is not considered in this study, which could be a further study with assessing the effect of muscle tone on knee joint health. Besides, the present model is derived from a small set of healthy clinical data; thus, the values of coefficients in the proposed generalized model for stress–strain of knee joint are subjected to further verification and experimentation with a larger dataset and across various pathological conditions.