## 1 Introduction  
Bone modelling and remodelling are very important throughout the lifetime of a human being because both processes are related to the bone growth and healing processes. Dr. [PERSON] purported [PERSON]'s law, which states that the bone remodels itself to adapt to loads applied to the bone [1]. Because this is similar to the concept of topology optimisation in mechanical engineering, computer simulation studies dealing with the internal architecture of the long bone have been performed to understand the mechanism of the generation of trabecular bone architecture [2; 3; 4; 5]. However, abnormal bone formation not only in the trabecular bone region of the long bones but also on the outer surface of the bone is also critical because it is related to the mechanism of the development of orthopaedic disease and treatments. In spinal fusion surgery, the most popular surgical method for both the cervical and lumbar spine involves the placement of a spacer, which occupies the intervertebral disc space embedded between the upper and lower vertebral bodies of the operated motion segment. Although autologous bone grafting has been used in spinal fusion when the surgery is introduced, recently, the use of interbody cages filled with bone graft material has become more common, to avoid complications following autologous bone grafting [6; 7]. Even in multi-level fusion surgeries, cervical spinal fusion surgeries with a cage (expandable cage) showed similar results to surgeries with iliac crest autografts [8]. Because the purpose of spinal fusion surgery is to integrate two vertebral bodies of the operated motion segment into one solid structure, bone bridging through the prefilled bone graft (intragraft bone bridging) or around the interbody cage (extragraft bone bridging) is necessary to achieve a solid union of the vertebral bodies. A previously published clinical study showed that extragraft bone bridging is critical for achieving a solid union after anterior cervical fusion [6]. Therefore, investigation of bone modelling or remodelling after spinal fusion is crucial for securing a successful outcome from the surgery. Bone formation around a surgical implant after spinal surgery is observed not only after spinal fusion but also after other surgeries including motion-preserving spine surgeries using an artificial disc, as well as dynamic stabilisation [9; 10; 11; 12]. The concept of an artificial disc involves a ball-socket joint that is used to occupy the disc height while preserving spinal motions along the flexion-extension, lateral, and axial directions. Dynamic stabilisation involves a relatively flexible device, and its low stiffness is helpful for allowing segmental motions. However, many clinical studies have reported that spinal motion is not well preserved after motion-preserving spine surgeries. This is due to abnormally generated bone, which is called heterotopic ossification, that surrounds the embedded artificial disc, interrupting spinal motion as well as inducing the union of the superior and inferior vertebrae. While bone formation is essential for achieving successful spinal fusion surgery, it is a disadvantage and complication in motion-preserving spine surgeries [6; 9; 10; 11; 12]. Therefore, the accurate prediction of bone formation after spinal surgery is important for building preoperative plans and developing spinal surgical devices. Several biomechanical studies have been published in which spinal implants for treating spinal disorders or diseases were assessed using both experimental and computational methods [13; 14]. However, although bone formation surrounding the embedded implants is critical in the biomechanical assessment of the implant, only a few studies have investigated bone formation [10; 15; 16] and its biomechanical effects on the spinal motion segment.