**1. Introduction**

Patello-femoral pain has a prevalence of more than 20% in the general population [1] and a high percentage of unsatisfied patients after total knee arthroplasty complain of anterior knee pain [2,3]. Even if satisfactory patellar tracking and kinematics seem to be evident on physical exam [4,5], it is still not entirely clear, what defines good patellar tracking and how healthy patellar kinematics can be quantified [6].

Although previous studies have investigated patellar kinematics [6,7], it remains unclear what ideal patellar kinematics encompass. What we know about patellar kinematics is that the patellar native facets or prosthetic button should be centered in the trochlear groove without subluxation or tilt throughout range of motion [6,7]. Even with descriptions of how the patella tracks from full extension to flexion, the issue of kinematic reference frame and rotations to describe motion has not been standardized.

The difficulty of converting between the multiple existing mathematical definitions of patellar kinematics or even understanding them properly is one of the major challenges for answering these questions. When we consider rotations, the way they are described has many implications. Researchers should be aware of these implications when they work with kinematic data concerning the patello-femoral joint. The choice of coordinate

**Citation:** Sauer, A.; Kebbach, M.; Maas, A.; Mihalko, W.M.; Grupp, T.M. The Influence of Mathematical Definitions on Patellar Kinematics Representations. *Materials* **2021**, *14*, 7644. https://doi.org/10.3390/ ma14247644

Academic Editor: Oskar Sachenkov

Received: 30 September 2021 Accepted: 9 December 2021 Published: 11 December 2021

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systems and the mathematical definition for describing patello-femoral kinematics can lead to substantial differences in the resulting curves and how one interprets whether normal kinematics have been established. Therefore, studies without a detailed description of the underlying definitions [8–12] are of limited value for researchers and clinicians.

There is a recommendation on definitions of joint coordinate systems for various joints from the International Society of Biomechanics. However, they do not provide a recommended definition for the knee joint, especially not for the patello-femoral joint [13–15]. However, since the publication of these recommendations, a lot of research has been conducted for the patello-femoral joint kinematics. To describe these kinematics, a definition using a floating axis for patellar spin [16], which has been recommended by Bull et al. [17], has been increasingly used in the past few years. This definition follows the same principle used by Grood and Suntay [18] for the tibio-femoral joint.

Nevertheless, there are still lots of studies that use more uncommon conventions for patello-femoral rotations which can lead to fundamental changes in the values for patellar flexion, spin and tilt [6,17]. Therefore, for interpretation of patello-femoral kinematics data a clear understanding of the underlying conventions is needed. If data with different underlying definitions should be compared, the ability to transform between several conventions is very helpful, but rarely described in literature.

The aim of this article is to give an overview of the different methods that can be used to describe patellar kinematics, with a particular focus on rotations. In contrast to previous publications, this article will consider the implications associated with each definition in greater detail. Additionally, the mathematics behind the patellar rotations will be given, including ways to convert data between some of the most common definitions. Therefore, the purpose is to enable researchers to choose the definition of patellar kinematics that is most suitable to them, as well as allow them to easily perform any conversions necessary to be able to compare the outcomes of different studies. As Supplementary Material, a Matlab template is provided to perform the most common conversions easily.
