## 1 Introduction  
Poor posture is often associated with a lack of postural control. It provides an unstable connection between the body's segments and unexpected strains on the supporting structures [1; 2; 3]. As such, it may represent a risk of back pain and disorders [4]. The World Health Organization outlines back pain as one of the most common health problems worldwide [5]. It is a leading cause of absenteeism at work [6] and significantly impacts personal, social, and government levels [7; 8; 9]. The spine is mobile in all three axes of the cartesian coordinate system as a translation or a rotation [10; 11; 12]. However, there is variability in mobility between spinal regions since movement between adjacent vertebrae is limited. Nevertheless, even small movements in one direction can provide an extensive range of motion to the global spine [11]. The cervical, thoracic and lumbar vertebrae are interconnected by fibrocartilaginous intervertebral discs [13]. Except for the atlas and axes, vertebrae share a typical structure, which includes the vertebral body and the arch. The vertebral body is the cylindrical part of the vertebra. The latter is the lateral and posterior aspects of the vertebra, consisting of paired pedicles and laminae, which join posteriorly, forming the spinous process [11; 12]. The apex of each vertebra is a cutaneous prominence visible along the back midline [14]. Pathological conditions or disorders may change the spine's biomechanics [12; 15], compromising the spinal alignment [16]. The quantitative evaluation of spine biomechanics is crucial to managing medical procedures. Radiological techniques are the most widely used for these purposes despite their association with ionising radiation [17; 18; 19]. Other conventional 3D imaging techniques, such as magnetic resonance imaging (MRI), require the patient to lie down, and many spinal deformations are lost [20]. On the other hand, non-invasive tools are not standard in clinics [21]. Some do not allow 3D analysis [22; 23], while others only assess partial segments [24; 25] or require contact points during evaluation [26]. Despite its ability to perform a biomechanical assessment of the spine, technologically advanced instruments, such as optical systems, have limitations, such as the background or lighting conditions [3; 27]. Our research team developed the Vertebral Metrics (VM) tool in previous research. It is a patented device (patents PT/103990, PCT/IB2009/005018, and US20110004125) that was designed to identify the spinous processes' apex position without ionising radiation [28]. As such, it combines a stereo vision system with software that recognises fluorescent marks in the skin [29; 30]. The device was developed, and the measurement resolution was determined [30]. Nevertheless, to assess the spine biomechanics, a model of the spine based on the apex position must be used. The previous model only provided the spine overview in the sagittal plane, so the information was insufficient and needed to be more comprehensive. Moreover, it only used one spring to simulate each intervertebral disk, which allows more degrees of freedom than the anatomical possibilities [29]. Therefore, the authors improved the concept and proposed a new mathematical model in the present paper. Modelling the spine is complex [31]. It may explain why the literature only describes a few studies, with most focusing only on the lumbar spine [32; 33]. The mathematical models can be continuous or discrete parameters. The first type is recommended as a first approximation because it does not consider mobility constraints [10; 32; 34]. The discrete type considers every single element. Generally, they represent the vertebrae by rigid bodies, whereas the ligamentous structures, muscles, or articulating facets are massless springs [34]. However, most discrete models use approximations of image techniques such as X-ray, magnetic resonance, or computer tomography [33; 35; 36].