In total knee arthroplasty (TKA), one common metric used to evaluate innovations in component designs, methods of component alignment, and surgical techniques aimed at decreasing the high rate of patient-reported dissatisfaction is tibiofemoral contact kinematics. Tibiofemoral contact kinematics are determined based on the movement of the contact locations in the medial and lateral compartments of the tibia during knee flexion. A tibial force sensor is a useful instrument to determine the contact locations, because it can simultaneously determine contact forces and contact locations. Previous reports of tibial force sensors have neither characterized nor corrected errors in the computed contact location (i.e., center of pressure) between the femoral and tibial components in TKA that, based on a static analysis, are caused by the curved articular surface of the tibial component. The objectives were to experimentally characterize these errors and to develop and validate an error correction algorithm. The errors were characterized by calculating the difference between the errors in the computed contact locations when forces were applied normal to the tibial articular surface and those when forces were applied normal to the tibial baseplate. The algorithm generated error correction functions to minimize these errors and was validated by determining how much the error correction functions reduced the errors in the computed contact location caused by the curved articular surface. The curved articular surface primarily caused bias (i.e., average or systematic error) which ranged from 1.0 to 2.7 mm in regions of high curvature. The error correction functions reduced the bias in these regions to negligible levels ranging from 0.0 to 0.6 mm (p < 0.001). Bias in the computed contact locations caused by the curved articular surface of the tibial component as small as 1 mm needs to be accounted for, because it might inflate the computed internal–external rotation and anterior–posterior translation of femur on the tibia leading to false identifications of clinically undesirable contact kinematics (e.g., internal rotation and anterior translation during flexion). Our novel error correction algorithm is an effective method to account for this bias to more accurately compute contact kinematics.

## Introduction

Clinicians along with engineers from both academia and industry are working to innovate component designs, methods of aligning the components, and surgical techniques to increase patient-reported satisfaction after total knee arthroplasty (TKA). Currently, 18–25% of patients are not satisfied following TKA [1–3]. With the projection that 3.5 × 10^{6} TKAs will be performed annually in the United States alone by 2030 [4], up to 875,000 patients will not be satisfied per year. Hence, it is important that all innovations are objectively evaluated so that iterations can be made efficiently to more rapidly increase patient satisfaction.

One metric that is often used to evaluate innovations is tibiofemoral contact kinematics [1–4]. Tibiofemoral contact kinematics are determined based on the movement of the contact locations in the medial and lateral compartments of the tibia [5]. Abnormal contact kinematics, which include external rotation of the tibia on the femur and anterior translation of the femur on the tibia during knee flexion, are associated with accelerated wear [6], limited flexion [7,8], and decreased function [9].

A useful instrument to determine tibial contact locations is a tibial force sensor, because it can simultaneously determine both contact locations and contact forces. When using a tibial force sensor, the contact locations are computed as the centers of pressure in the medial and lateral compartments of the tibia [10]. Thus, accurately computing contact locations using a tibial force sensor is important for objective evaluations of innovations based on tibiofemoral contact kinematics.

As with any measurement instrument, tibial force sensors must be calibrated to minimize the errors in the contact location computed using the sensor. Although the typical calibration procedure is performed by applying forces normal to the tibial baseplate [10–16], contact between the curved articular surfaces of the components occurs normal to the articular surfaces during use. Hence, the typical calibration procedure only accounts for one source of error, which is the error caused by imperfections in the design, manufacturing, and assembly of the sensor. A second source of error is the curved articular surface of a standard tibial component [17], which provides stability to the joint similar to that provided by the meniscus in the native knee [18,19]. Based on a static analysis, the curved articular surface will introduce bias (i.e., average or systematic error) in the computed contact location, which is directed toward the periphery of the tibial articular surface and is greater in regions of high curvature (see Appendix 1, which is available under the “Supplemental Materials” tab for this paper on the ASME Digital Collection). However, errors in the computed contact locations caused by the curved articular surface of the tibial component have not been accounted for previously [10–16]. It is important that these errors be accounted for to prevent incorrect interpretations of contact kinematics based on the computed contact locations.

Accordingly, the first objective of this study was to characterize the errors in the computed contact location caused by the curved articular surface of a standard tibial component. Because a large bias was found, the second objective was to develop and validate an error correction algorithm for the computed contact location to minimize the bias caused by the curved articular surface of a standard tibial component using a tibial force sensor developed by the authors [10].

## Methods

The tibial force sensor developed by the authors was used for these analyses, because its size, shape, and curvature match those of a standard tibial component [10]. This sensor is described in detail elsewhere [10], but briefly it may be broken down into five layers (Fig. 1). The first layer (most distal) consists of a modified tibial baseplate. The second layer consists of the printed circuit boards. The third layer consists of the medial and lateral transducer arrays. The fourth layer consists of the medial and lateral trays, which connect the medial and lateral transducer arrays to the medial and lateral articular surface inserts, respectively. The fifth layer consists of the medial and lateral articular surface inserts. Different configurations of the sensor, which allow the sensor to be used in different size knees, are possible by using different size and thickness tibial articular surface inserts with the corresponding conversion trays.

The contact location in a compartment is computed using the voltage output of each of the three transducers and nine inverse sensitivity coefficients [10]. During calibration of the sensor, the nine inverse sensitivity coefficients that minimized the differences between the computed and actual contact force and the computed and actual contact location, which are described by the medial–lateral (ML) and anterior-posterior (AP) coordinates, were determined (Fig. 2, Eqs. (1)–(3)). After calibration, the voltage outputs from the transducers were used to compute the ML and AP coordinates of the contact location.

where $Fcomputed,i$, $MLcomputed,i$, and $APcomputed,i$ are the computed contact force and computed ML coordinate and AP coordinate of the contact location, respectively; the index, *i*, indicates the compartment (i.e., either medial or lateral); ($ML1,i$, $AP1,i$), ($ML2,i$, $AP2,i$), and ($ML3,i$, $AP3,i$) are the ML and AP coordinates of the transducers; $c11,i$, …, $c33,i$ are the inverse sensitivity coefficients; and $V1,i$, $V2,i$, and $V3,i$ are the Wheatstone bridge voltage outputs.

To characterize the errors in the computed contact location caused by the curved articular surface of the tibial component and for later analysis to validate the error correction algorithm, contact locations were computed with forces applied in two orientations using two configurations of the tibial force sensor. These two configurations were the most-common configuration (i.e., the most-common size tibial baseplate with the most-common thickness tibial articular surface inserts) and the worst-case configuration (i.e., largest size tibial baseplate with the thickest tibial articular surface inserts). Forces were applied independently to each compartment both normal to the tibial baseplate and normal to the tibial articular surface at 19 locations, each of which represented a contact location, using the characterization and validation inserts for the most-common and worst-case configurations (Fig. 3). The set of applied forces ranged from 45 N to 450 N (set by the load capacity of the tibial force sensor) in 22.5 N increments. Force values from this set were randomly assigned to the contact locations in each compartment. Forces increasing to and decreasing from the randomly assigned value for a particular contact location were applied using a custom dead-weight loading fixture (Fig. 4). The loading increments were set by the weight of the cross member (22.5 N) and the weights of the available weight plates (22.5 N and 45 N) (e.g., loading sequence for a randomly selected value of 67.5 N was (1) 22.5 N, (2) 67.5 N, and (3) 22.5 N and loading sequence for a randomly selected value of 90 N was (1) 22.5 N, (2) 67.5 N, (3) 90 N, (4) 67.5 N, and (5) 22.5 N). The sagittal and coronal orientations of the applied force relative to the tibial articular surface of the tibial force sensor were set using a two-axis articulating fixture (Fig. 4). At each actual contact location for each loading orientation and increment of applied force, the contact location (expressed in terms of the ML and AP coordinates) was computed in each of the two compartments for each of the two configurations using Eqs. (2) and (3).

Using the computed contact locations above, three sets of errors for each of the two coordinates of computed contact location in each compartment were calculated for each of the two configurations (Table 1). The first set was the difference between the computed contact location when force was applied normal to the tibial baseplate (ML_{computed baseplate} and AP_{computed baseplate}) and the actual contact location (ML_{actual} and AP_{actual}) at each of the 19 locations for each increment of applied force. This first set of errors is due to imperfections in the design, manufacturing, and assembly of the sensor and served as a baseline in later analysis to determine how well the error correction algorithm corrected the errors in computed contact location caused by the curved articular surface. The second set of errors was the difference between the computed contact location when force was applied normal to the tibial articular surface (ML_{computed surface}, AP_{computed surface}) and the actual contact location at each of the 19 locations for each increment of applied force. This second set of errors included errors from both sources (i.e., imperfections and curvature) and was used in later analysis as an input to the error correction functions to validate the error correction algorithm. Termed precorrection errors, the third set of errors was the difference between the second set of errors and the first set of errors. This third set was used to characterize the errors caused by the curved articular surface. Per ASTM E177-13 and ISO 5725-1, each of the three sets of errors was described in terms of the bias (mean), precision (standard deviation), and root-mean-square error (RMSE) [20,21] for each of the two coordinates of contact location in each compartment.

Two possible methods were considered to minimize the errors in computed contact location caused by the curved articular surface of the tibial component. One method was to apply forces normal to the curved articular surface during the calibration procedure. Hence, the errors in computed contact location including both those caused by imperfections in the design, manufacturing, and assembly of the sensor and those caused by the curved articular surface would be minimized during the calibration procedure. However, because different configurations of the tibial force sensor are possible to allow for different size implants and different thickness tibial articular surface inserts (i.e., tibial liners), the calibration procedure would have to be repeated for each configuration used. The second method avoided repeated calibration procedures by taking advantage of the finding that the error in the computed contact location is related to the shape of the articular surface (see Appendix 1, which is available under the “Supplemental Materials” tab for this paper on the ASME Digital Collection). Using this relationship (Eq. (A7)), the error in the computed contact location caused by the curved articular surface at a particular location on the articular surface can be predicted. Therefore, the calibration procedure of the sensor would not need to account for the errors caused by the curved articular surface because these errors could be minimized by postprocessing the computed contact locations. This second method was selected for the present study because repeated calibration procedures are onerous.

As an overview of this second method, a three-step error correction algorithm was used to develop the error correction functions to postprocess computed contact locations to minimize the errors caused by the curved articular surface (Fig. 5). The first step was to determine error prediction functions that predict the error in each of the two coordinates of computed contact location in each compartment as a function of both tibial articular surface shape and applied force. The second step was to virtually generate computed contact locations by adding virtually generated errors determined using the error prediction functions to points on the articular surface of each configuration of the tibial force sensor; each point was considered an actual contact location. This second step replaced the repeated calibration procedures that would be required with the first method described above. The third step was to determine the error correction functions that minimized the error between the corrected contact locations (i.e., computed contact locations that have been corrected for errors caused by the curved articular surface) and the actual contact locations. The three steps are described in further detail below.

The first step in the error correction algorithm was to determine error prediction functions that predict the error in each of the two coordinates of the computed contact location in each compartment as a function of three factors that describe the articular surface shape and applied force (Fig. 5). These functions were determined using a four factor (i.e., three shape factors + applied force) full factorial face-centered response surface design (i.e., *α* = 1) [22]. The four factors were (1) the height above the mounting plane of the transducers (*h*), (2) the coronal orientation of the surface normal (*φ*), (3) the sagittal orientation of the surface normal (*θ*), and (4) the applied force (*F*). The minimum, average, and maximum values of each factor (Table 2) were determined based on the shape of the tibial articular surface and the load capacity of the tibial force sensor. Loads were applied to the tibial force sensor in each compartment at the centroid of the transducer array at 30 combinations of the four factors (16 corner points, 8 star points, and 6 center points) using the two-axis articulating fixture and 3D printed inserts (Figs. 4 and 6). The errors in the ML and AP coordinates of the computed contact locations were calculated as the differences between the contact locations computed using Eqs. (2) and (3) and the actual contact location (i.e., centroid of transducer array).

*F*). For the AP coordinate of the contact location, the three factors were the same except for the in-plane orientation of the surface normal which was instead the sagittal orientation ($\theta $). For each function (Eqs. (4) and (5)), eleven coefficients ($c0,\u2026,c10$ or $p0,\u2026,p10$) were determined that minimized the differences between the predicted ($eML$ and $eAP$) and computed errors using a standard least squares optimization (JMP Version 11.2.0; SAS Institute, Inc., Cary, NC).

^{2}In Eqs. (4) and (5), the index,

*i*, indicates the compartment (i.e., either medial or lateral).

*h*) and actual height of the point above the mounting plane of the transducers using a least squares optimization (

_{i}_{,}_{j}*cftool*, matlab R2014b; MathWorks, Natick, MA). The orientation of the surface normal relative to the normal of the tibial baseplate in both the coronal (

*φ*) and sagittal (

*θ*) planes was determined by taking the inverse tangent of the partial derivatives of the surface fit functions with respect to the ML and AP directions, respectively (Eqs. (7) and (8)). Virtually generated errors in the ML and AP coordinates of the contact locations were determined using the error prediction functions (Eqs. (4) and (5)) for each point in the point cloud. These errors were determined for each point for force values from 45 N to 450 N in 45 N increments. Finally, a virtually computed contact location ($MLvirtually\u2009computed,i,j$ and $APvirtually\u2009computed,i,j$) was generated for each point in each point cloud by adding the virtually generated errors in the ML and AP coordinates to the actual ML and AP coordinates of each point in the point cloud, respectively (Eqs. (9) and (10)). In Eqs. (6)–(10), the index,

*i*, indicates the compartment (i.e., either medial or lateral), and the index,

*j*, indicates the configuration (i.e., configuration 1,…, configuration

*j*,…, configuration

*n*).

^{3}Each point in the point clouds from step 2 was considered an actual contact location. The force values and the virtually computed contact locations from step 2 were used for this optimization. In Eqs. (11) and (12), the index,

*i*, indicates the compartment (i.e., either medial or lateral), and the index,

*j*, indicates the configuration (i.e., configuration 1,…, configuration

*j*,…, configuration

*n*).

Two more sets of errors were calculated to validate the error correction algorithm (Table 1). The error correction functions for the most-common configuration and the worst-case configuration were applied to the previously computed contact locations when forces were applied normal to the tibial articular surface. The fourth set of errors was the difference between the corrected ML and AP coordinates of the contact locations (ML_{corrected}, AP_{corrected}) and the actual contact locations. These errors in the corrected ML and AP coordinates of the contact locations in each compartment were described by the bias, precision, and RMSE [20,21] over all applied force values at all 19 locations for each configuration (i.e., a total of 237 errors were included for each coordinate of the contact location). Termed the postcorrection errors, the fifth set of errors was the difference between the fourth set of errors and the first set of errors calculated previously as the difference between the ML and AP coordinates of the computed contact location when force was applied normal to the tibial baseplate and those of the actual contact location. A further analysis was performed by dividing this fifth set of errors in the ML and AP coordinates of the computed contact location into two and three regions, respectively, in each compartment of the most-common configuration based on the coronal and sagittal orientations of the surface normals, respectively (Fig. 7). These errors were analyzed by region because both the height and the orientation of the surface normal vary by region due to the curvature of the articular surface.

To determine whether the postcorrection errors were less than the precorrection errors, a two-factor repeated measures analysis of variance (ANOVA) including interaction with a post hoc Tukey test was performed for each of the four coordinates of the computed contact location (JMP Version 11.2.0; SAS Institute, Inc., Cary, NC).^{4} The first factor was the correction status at two levels (precorrection and postcorrection). The second factor was the region of the articular surface at two levels for the ML coordinate (inner and middle) and at three levels for the AP coordinate (anterior, central, posterior). The level of significance was set at 0.05 for all analyses.

## Results

The curved articular surface (error set 3) introduced substantial error into the computed contact locations. The compartmental RMSEs for the AP and ML coordinates caused by the curved articular surface in each compartment over all regions ranged from 1.4 to 2.6 mm and from 1.9 to 2.5 mm, respectively (Table 3). The increase in compartmental error relative to the error when force was applied normal to the tibial baseplate ranged from 55% to 126%. In the anterior, posterior, and inner regions of the insert where the magnitudes of the orientation of the surface normals were greater than 5 deg (Fig. 7), the regional RMSE was due primarily to bias caused by the curved articular surface and ranged in magnitude from 1.0 to 2.7 mm (Fig. 8; Table 4).

The error correction functions significantly reduced the bias in the computed contact location caused by the curved articular surface. After applying the error correction functions, the RMSEs in the computed contact location including all regions of the articular surface in each compartment were reduced to between 1.0 and 1.2 mm for the most-common configuration and between 1.0 and 1.7 mm for the worst-case configuration (Table 5). Based on the Tukey tests on the interaction (i.e., correction status × region), the regional postcorrection bias (error set 5) in the computed AP coordinate of the contact location caused by the curved articular surface was reduced to a negligible level and was significantly less in both the anterior and posterior regions in both compartments (*p* < 0.001) than the precorrection bias (error set 3) (Figs. 8(c) and 8(d); Table 4). Likewise the postcorrection bias (error set 5) in the computed ML coordinate of the contact location caused by the curved articular surface was reduced to a negligible level and was significantly less in the inner region in both compartments (*p* < 0.001) than the precorrection bias (error set 3) (Figs. 8(a) and 8(b); Table 4).

## Discussion

Two sources of error in the computed contact location using a tibial force sensor are (1) imperfections in the design, manufacturing, and assembly of the sensor, and (2) the curved articular surface of the tibial component. Previous tibial force sensors have been developed to compute the contact location of the femoral component on the tibial component after TKA but have only minimized the first of these two sources of error through calibration by applying forces normal to the tibial baseplate. Hence, the objectives of this study were to (1) characterize errors caused by curved articular surface of the tibial component and (2) to develop and validate an error correction algorithm for the computed contact location that minimizes the errors caused by the curved articular surface. The first key finding was that the computed errors in the contact location caused by the curved articular surface (error set 3) were substantial and resulted primarily from bias in regions of high curvature. The second key finding was that the error correction functions significantly reduced the bias in computed contact locations caused by the curved articular surface to negligible levels.

There are several limitations that should be discussed prior to interpreting the results. First, the ranges of surface normal orientations and heights of the tibial articular surface of the sensor used in this study might be smaller than those of other insert designs. Hence, both our characterization of the errors caused by the curved articular surface of the tibial insert might be conservative, and this algorithm may need to be altered to handle larger ranges of surface normal orientation and height. Second, a different algorithm might be necessary for tibial force sensors that are capable of directly measuring the shear forces and/or moments which are caused by contact force applied to a curved articular surface. Third, although the errors caused by the curved articular surface and the validation results were determined for two of the possible configurations, these results should be representative of the other configurations because the error correction functions performed well in both the most-common configuration and the worst-case configuration. Fourth, the contribution of friction (a third source of error) was not considered in this error correction algorithm because it was accounted for by computing the contact location at a particular flexion angle as the average of the computed contact location at that flexion angle during knee flexion and that same flexion angle during knee extension (See Appendix 2, which is available under the “Supplemental Materials” tab for this paper on the ASME Digital Collection). Alternatively, values of the friction coefficients could have been used to correct friction effects in this error correction algorithm. However, our method for negating friction effects through averaging contact locations during flexion and extension is preferred because knowledge of coefficients of friction is not necessary. If future studies do not average contact locations during flexion and extension, then they should consider including a correction for friction in their error correction algorithm.

The first key finding was that the bias in the computed contact locations caused by the curved articular surface in regions of high curvature was substantial (error set 3, Table 4, and Fig. 8). This large regional bias was expected based on the results of the static analysis presented in Appendix 1 (Eq. (A7); Fig. A2), which is available under the “Supplemental Materials” tab for this paper on the ASME Digital Collection. The large regional bias is sufficient to artificially inflate computed tibiofemoral contact kinematics. To illustrate this inflation, consider a hypothetical case where 0.5 deg of external rotation (i.e., reverse rotation) of the tibia on the femur is computed due to the medial contact location remaining stationary and the lateral contact location translating anteriorly. If the bias were not accounted for in this calculation, then 3.1 deg of external rotation might have been computed due to the 1.8 mm bias error in the anterior region of the lateral compartment assuming an average width tibial component (Fig. 8; Table 4). Consider another hypothetical case where 0.5 mm of anterior translation of each contact location is computed. If the bias were not accounted for in this calculation, then between 2.3 and 2.8 mm of anterior translation might have been computed due to the 1.8–2.3 mm bias errors in the anterior regions of the lateral and medial compartments (Fig. 8; Table 4).

These inflated amounts of external rotation and anterior translation might lead to false identification of clinically undesirable contact kinematics. External rotation of the tibia on the femur between 3 deg and 6 deg [6,7,9] and changes in the anterior–posterior position of the femur on the tibia between 1 mm and 2 mm [6,8] are associated with accelerated wear [6], limited flexion [7,8], and decreased function [9]. Hence, as demonstrated by the two previous hypothetical cases, the errors in computed contact location caused by the curved articular surface of the tibial component should be corrected to prevent false identification of clinically undesirable contact kinematics.

In the preceding examples, it is important to note that the analysis relied on the regional bias (Table 4; Fig. 8) rather than the compartmental bias (Table 3). When comparing the compartmental bias to the regional bias in regions of high curvature for the AP coordinate, the magnitude of the compartmental bias is notably smaller. This is a result of the sign of the bias changing when the location of contact moves up the lip of the articular surface of the tibial component as it translates either anteriorly or posteriorly (Table 4, Fig. 8, and Fig. A2, which is available under the “Supplemental Materials” tab for this paper on the ASME Digital Collection). When the errors are summed over all points in a compartment, the bias for the AP tibial contact locations on the anterior and posterior lips negate each other, making the compartmental bias much lower. Therefore, the regional bias was also reported to give more realistic estimate of the possible errors in tibiofemoral contact kinematics due to bias.

In general, this first key finding highlights one of the challenges in designing a contact force sensor with a curved articular surface. These clinically important errors demonstrate that a method for minimizing errors in the computed contact location caused by a curved articular surface is necessary for future contact force sensors with such surfaces. This necessity likely applies to contact force sensors that are developed to compute the contact locations in other diarthrodial joints with curved articular surfaces such as the shoulder [23], spine [24,25], and hip [26].

The second key finding was that the error correction functions significantly reduced the bias errors in computed contact locations caused by the curved articular surface of the tibial component to negligible levels. The maximum bias postcorrection (0.5 mm) is about fivefold less in magnitude than the maximum bias precorrection (−2.4 mm) (Fig. 8, Table 4). Hence, this novel error correction algorithm successfully determined error correction functions that effectively reduced the errors caused by the curved articular surface.

For sensors similar to the one used in the present study, this error correction algorithm provides a template that can be implemented to minimize errors in the computed contact location caused by the curved articular surface of the tibial component and avoid repeated calibration procedures. For sensors with multi-axis transducers or other unique designs, this error correction algorithm should provide guidance for developing a sensor-specific error correction algorithm to account for errors caused by the curved articular surface without having to repeat the calibration procedure.

In conclusion, this study experimentally confirmed that the curved articular surface of the tibial component causes sufficiently large bias errors that might inflate the computed internal–external rotation and anterior translation of femur on the tibia leading to false identifications of clinically undesirable contact kinematics. Hence, tibial force sensors must minimize not just the errors caused by imperfections in the design, manufacturing, and assembly of the sensor but also the errors caused by the curved articular surface. The novel error correction algorithm developed and validated in the present study is an effective method to reduce the substantial bias errors caused by curved articular surface to negligible levels without repeated calibration procedures.

## Acknowledgment

We acknowledge the support of the National Science Foundation, Award No. CBET-1067527. We also acknowledge the support of Zimmer Biomet, Award No. CW88095 and in particular John Kyle Mueller who was our liaison at Zimmer Biomet.

## Nomenclature

- ANOVA =
analysis of variance

- AP =
anterior–posterior

- ASTM =
American Society of Testing and Materials

*c*=inverse sensitivity coefficient

*F*=force

*h*=height above the mounting plane of the transducers

- ISO =
International Standards Organization

- mm =
millimeter

- ML =
medial–lateral

- N =
Newton

- RMSE =
root-mean-square error

- TKA =
total knee arthroplasty

*V*=output voltage from Wheatstone bridge

- 3D =
three-dimensional

*α*=distance from center point to star point in the face-centered response surface design

*θ*=sagittal orientation of the surface normal

*μ*=coefficient of friction

*φ*=coronal orientation of the surface normal