**1. Introduction**

Adult acquired flatfoot deformity (AAFD) is a pathology that causes a progressive flattening of the foot arch, which has been traditionally related to a tibialis posterior tendon (TPT) dysfunction. However, some clinical studies found that a failure/rupture of the plantar fascia (PF) or the calcaneonavicular ligament (also spring ligament (SL)) could also generate the arch collapse and the forefoot abduction [1–5]. Treatment options depend on the injury stage. In the first stages, AAFD treatments are related to reinforcing the TPT [6]. Nevertheless, sometimes the foot deformation reappears over time, forcing surgeons to use more aggressive techniques, intervening directly over the foot's bone structure. If the foot deformity is still flexible (stages IIa and IIb), the most habitual procedure is medializing calcaneal osteotomy (MCO) [7,8], which allows both the progressive foot arch flattening and the foot pronation caused by the flatfoot deformity to be corrected [4,6,9]. This procedure provokes a supination momentum in the foot to compensate the pronation [1,10]. In this way, the foot's structural correction is achieved by MCO and its results are normally satisfactory. Nevertheless, some clinical studies have shown that this procedure generates long-term side-effects related to stress distribution changes in forefoot and metatarsals [11–13], which could increase the risk of bone fractures, as has been reported with Evans' osteotomy [14].

In a recent study published by our research group using a previous version of our foot model, we showed that MCO can reduce foot pronation on its own [9]. However, changes in the biomechanical stress caused in bones and the main soft tissues that support the arch remained unstudied. Even in the literature, these stress changes have not been sufficiently studied, because of the difficulty of measuring tissue stresses in cadavers.

Bayod,Larrainzar-Garijo,R.; Solórzano, B.D.; Cifuentes-De la Portilla, C. Biomechanical Effects of Medializing Calcaneal Osteotomy on Bones and the Tissues Related to Adult-Acquired Flatfoot Deformity: A Computational Study. *Mathematics* **2023**, *11*, 2243. https://doi.org/ 10.3390/math11102243

 J.;

Academic Editors: Fernando Simoes and Mauro Malvè

Received: 22 February 2023 Revised: 19 April 2023 Accepted: 29 April 2023 Published: 10 May 2023

**Citation:**

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Some cadaver-based models have been used to study the structural correction of the foot, evaluating changes in both the plantar footprint using force platforms, and foot arch falling using radiographic (Rx) images. For example, Patrick et al. [15] measured the subtalar joint pressure produced by MCO using a cadaveric model suffering with flatfoot. They introduced a pressure sensor in the posterior facet joint, obtaining some, but limited, information about the effects of MCO on hindfoot joint pressures. As can be noted, these kinds of studies require high economic investment in measurement equipment, as well as meticulous control over the tested tissues to guarantee their biomechanical characteristics [16].

An alternative now accepted by clinicians and biomechanical researchers for evaluating the complex biomechanics of the human foot is finite element modelling (FEM) [17–19]. There are many models that study foot biomechanics and the effects produced by some surgical techniques. However, none of them have been used to study the stress effects of MCO on foot tissues. This kind of model specifically evaluates foot structure deformation and plantar pressure measurement [13,19]. Thus, these models greatly simplify the tissue anatomy and do not take into account important aspects such as the biomechanical difference between cortical and trabecular bones (which is very important when tissue stresses are evaluated [20], nor the geometry of some soft tissues such as the plantar fascia, the spring ligament, ligaments, or tendons, which are habitually modelled as bar elements. Thus, previously reported models cannot measure and locate the stresses around the foot anatomy.

The objective of this research was to investigate the biomechanical effects in terms of stress concentrations and displacements that an MCO provokes in both foot bones and the main foot arch stabilizers (TPT, PF and SL), using an enhanced version of the model used in [9]. This analysis was performed by simulating different pathological scenarios related to AAFD development.

### **2. Materials and Methods**

This study was based on the foot model (segmentation and tissue properties) proposed by Cifuentes-De la Portilla et al [3], which has been used for the flatfoot evaluation of some other surgical procedures [21]. However, for this study, the entire model was reconstructed to simulate the MCO procedure, maintaining both tissue characteristics and loading conditions but including both the tibia and fibula bones to better represent the anatomical tendons' trajectories. The model used reconstructed a healthy human unloaded foot, based on CT images (radiographs of 0.6 mm/slide) acquired from the right foot of a 49-year-old man (weight = 75 Kg, height = 1.70 m).

### *2.1. FE Foot Model and Modifications*

Tissue segmentation and 3D reconstruction (bones, PF, TPT, Achilles' tendon, Peroneus Longus tendon (PLT) and Peroneus Brevis tendon (PBT)) were performed using MIMICS V. 10 (Materialize, Leuven, Belgium). The spring ligament (SL) and both plantar ligaments (short plantar ligament and long plantar ligament) were added following atlas images, following the surgeons' guidance due to the difficulty of segmenting these from the CT images. The tibia and fibula were reoriented with tools available in MIMICS from the scan position to correspond to the orientation during the stance phase of gait. The previous finite element model [9] was enhanced by adding the TPT, Achilles' tendon, PLT, and PBT.

To simulate the MCO, calcaneus bone was modified, performing a 45-degrees transversal cut and translating the segmen<sup>t</sup> medially by 10-mm (See Figure 1) [15,22]. This modification was performed following the guidance of a specialist in foot surgeries. Elements allowing internal fixation, such as plates, screws, and bone graft, were not simulated because a complete joint fusion was supposed. The complete FE model is shown in Figure 1. allowing internal fixation, such as plates, screws, and bone graft, were not simulated because a complete joint fusion was supposed. The complete FE model is shown in Figure 1.

**Figure 1.** Reconstruction and modifications in the model to simulate a medializing calcaneal osteotomy. The Achilles' tendon, tibialis posterior tendon, and both Peroneus tendons' geometries and the pieces of Tibia and Fibula bones were reoriented to obtain a vertical position.
