## 1 Introduction  
Knee osteoarthritis is a degenerative disease affecting the knee cartilage layers. Due to progressive knee cartilage wear, patients can experience pain, instability, loss of flexibility, joint stiffness and swelling. For patients with severe knee osteoarthritis, a total knee arthroplasty (TKA) surgery can help improve quality of life [1]. During a TKA procedure the femur and tibial bones are resected along the joint interface and replaced by metal components. Two-dimensional templating is the most frequently used method to preoperatively determine the required implant size. In 2D templating, anteroposterior and lateral radiographs are overlaid with 2D template shapes of the implants to determine the best fitting size. Despite its broad adoption, 2D templating has relatively low accuracy in determining the required size due to the nature of projection images [2, 3, 4, 5]. Three-dimensional templating based on computed tomography (CT) or magnetic resonance imaging (MRI) scans can overcome this limitation of 2D templating resulting in higher templating accuracy [6, 7, 8]. Three-dimensional templating allows the surgeon to position the implants on a 3D model of the joint to determine the optimal implant size and position. There are several benefits to preoperatively planning the implant size. First of all, determining the correct implant sizes is of importance from a clinical point of view. Femoral over sizing might lead to irritation of the soft-tissue structures surrounding the joint, increased patellofemoral pressure or reduced range of motion and, hence, should be avoided [9]. On the other hand, femoral undersizing can lead to femoral fracture due to anterior implant notching, flexion instability, or inferior patellar tracking due to a medialized implant placement [10]. Secondly, the extremely large and small implant sizes need to be specifically ordered as they are not consistently present in the hospital [11]. Furthermore, knowing the implant size up to a size smaller or larger reduces the sterilization cost and operation room setup time [12; 13]. Over recent decades, machine learning has been successfully applied to many medical applications, including implant size templating [14; 15; 16]. Several studies have investigated implant size prediction based on demographic data [11; 16; 17; 18; 19; 20; 21; 22]. The method by [PERSON] et al. used only shoe size and gender to obtain the highest reported accuracy of 63% and an accuracy of 99% for predicting up to one size larger or smaller [11]. Finally, [PERSON] et al. created a support vector machine based model that predicts implant size based on manually defined features deduced from 3D bone models, resulting in an absolute accuracy of 82.2%, significantly higher compared to models relying on demographic data [23]. Traditional methods for determining implant size are based on human predefined measurements. However, surgeons base the implant size on the size and morphology of the bone. Therefore, we hypothesize that shape based implant size prediction can yield higher accuracy compared to previously investigated methods. To the best of our knowledge, our method is the first to regress the implant size directly based on the 3D model of the bone. The computational method most similar to the one presented in this study, is an optimal-scoring-based method by [PERSON] et al. that induces sparsity [24]. This algorithm was evaluated on a classification task distinguishing male and female face silhouettes based on the point coordinates. The method by [PERSON] et al. obtains interpretability by introducing sparsity through the use of \(l_{1}\) regularization on the model coefficients corresponding to single coordinates of certain points. As a result, this method can only find directions of shape variations that are parallel to the coordinate system axes which is a major disadvantage. A second limitation is that proximity information is not incorporated into the model as neighbouring points in space should have similar contribution to the model.