Introduction
Wound repair involves fibroblast proliferation, extracellular matrix production, and tissue remodeling (1). Depending on the location of a skin wound, the reconstructed tissue may experience undesirable deformities (2). In severe cases, hypertrophic tissues form, necessitating medical and surgical rehabilitation intervention to restore function and aesthetics (3). Scar contracture following a burn injury poses significant challenges that impair the patient’s functional abilities. The primary cause of contractures after burns is often inadequate medical attention, rehabilitation, and insufficient treatment. Z-plasty is a precise technique in plastic surgery that leverages the elasticity of the tissue surrounding a scar (4), determined through qualitative and quantitative assessments during a clinical examination. Z-plasty can be utilized in the following scenarios: i) aligning and altering the structure of tissues affected by stair-step deformity, where transferring tissue to a new position is advantageous, and ii) releasing contractions by lengthening the scar along its axis, thereby reducing tension on the scar. Classic Z-plasty surgery can be performed in two ways: by changing the size of the limb or adjusting the angle (3). Increasing the angle and enlarging the flaps can result in a tension increase of 7 to 10 times required for closure (5), making flap transfer challenging and leading to complications such as dog ears(3). Conversely, a slight angle may shorten the base of the flap, weakening vascular flow and leading to flap ischemia and necrosis. In Z-plasty, the triangular transposition flap is commonly employed, typically adjusted symmetrically to a 60-degree angle, necessitating a 75% increase in the length of the scar limb (2). Transposition flaps are frequently utilized in plastic surgery, especially in burn cases, and surgeons may have differing opinions on the ideal angle and rotation of the flap (3).
This study aimed to introduce a novel formula for determining the appropriate angle and rotation using mathematical relationships, potentially establishing a standardized procedure for repairing contractures and diminishing reliance on the surgeon’s experience. In conventional approaches, the design of a classical Z-plasty was employed, in which predetermined angles and limb lengths were utilized, and trigonometric relationships were applied to demonstrate the achievable gain. In contrast, the method was proposed is tailored to the patient’s clinical context and the surgeon’s specific objective regarding the extent of contracture release required to optimize the patient’s range of motion. By selecting the desired or appropriate gain and applying trigonometric calculations, a novel Z-plasty configuration with newly defined angles and limb lengths is established. Although the mathematical foundation resembles that of previous methods, the practical application is distinct, as it provides a customized design while maintaining the fundamental standards of a classical Z-plasty. This provides a more accurate, reproducible, and clinically applicable framework compared to earlier theoretical models.

Matereials and Method
As with all restorative surgical techniques, measurement accuracy and an appropriate model are essential in  Z-plasty (6). The fundamental basis of Z-plasty consists of two triangular flaps associated with transposition and rotation (7). In classical Z-plasty, incisions of equal length create two symmetric triangular flaps (8). A central limb can either be positioned over the scar or parallel to it, with two arms originating at different angles from this central limb; all three limbs are usually equal in length (9). After dissection, the two flaps are rotated in opposite directions to cover the remaining defect, if XY represents the scar.
The point where the two common limbs intersect is referred to as point O. The angle XOA, shown as β in the diagram, measures the degree of rotation of the common limb in a Z-plasty.
In figure 1, XY represents the scar. While lines AX and BY are drawn so that angles AXY and XYB equal θ°, resulting in flaps 1 and 2. Figure 2 illustrates the position of flaps 1 and 2 after Z-plasty. To apply trigonometric principles to our model, the dashed line XY, indicating the original scar line, is depicted in figure 3. Thus, the scar line XY represents the common limb of the Z-plasty before flap transposition, while line AB represents the position of the common limb after flaps 1 and 2 are transposed. The point where the two common limbs intersect is referred to as point O. The angle XOA, donated as β in the diagram, measures the degree of rotation of the common limb in Z-plasty. Next, consider the following:

Using trigonometric relationships (6,10), the desired angle and rotation was determined by calculating the appropriate increase in the length of contraction:

The angle in a Z-plasty can be calculated as follows:

The angle of rotation can be determined using the following relationship.

Results
As shown in the equations, the diagonal AB and the common limb (T) lengths are given. The size of the angles θ and β can be derived from equations 1 and 2. The biomechanics of skin tension may affect the mathematical predictions for estimating angles θ and β after completing the Z-plasty (5). It is known that the limbs of the Z can extend at various angles ranging from 30° to 90° (8). This study is mathematical; however, the skin doesn’t behave as a mathematical model, as Furnas et al (5) confirmed. It is essential to find a method to accurately measure the change in the direction of the common limb in the skin (11). Such a method would assist surgeons in understanding the relationship between the size of the defect, the angle, and the rotation of Z-plasty, both before and after the procedure, according to the length of the scar. A mathematical analysis of the degree of rotation for various angles of Z-plasty was conducted, with results tabulated in  table 1.
With careful planning and mastery in executing Z-plasty, impressive results can be achieved in alleviating contractures and limitations in the movement of limbs by reconstructing tissue and utilizing appropriately sized flaps (12). 

Table 1. The degree of angle Z and rotation for various length of contracture

Discussion
In 1837, Horners reported at the University of Pennsylvania on the ectropion of the left lower eyelid due to a burn scar (13). Denoiville performed the first Z-plasty on the right lower eyelid in 1863 (14). In 1904, Berger described Z-plasty as a technique involving the displacement of flaps to treat burn complications (15). In 1914, Morstein corrected a complication from finger flexion using Z-plasty (16). Limberg enhanced the flap design by introducing mathematical concepts (17), which were further developed by Davis (18). In 1963, McGregor performed Z-plasty on hand skin contractures and significantly popularized the technique (19). The method introduced in this study defines the optimal Z-angle and appropriate rotation, considering the desired increase in length as determined by the surgeon. Unlike previous methods, which relied on predetermined angles and limb lengths to demonstrate the expected gain in a classical Z-plasty, the author’s proposed approach is based on the patient’s clinical condition and the surgeon’s objective. By defining new angles and limb lengths through trigonometric calculations, the technique maintains the standards of the classical Z-plasty while offering a more tailored and clinically relevant design.
However, the geometrical principles were first elegantly described by Alexander Limberg in 1929 (17). The results of the present study align with these principles of Alexander Limberg, suggesting that the present study findings may provide guidance for plastic surgery using the Z-plasty technique (Chart 1). Schematic redrawing by the authors based on the original geometric description of Limberg (17). The plotted data correspond to the calculations performed in this study and illustrate the consistency between computational results and graphical representation.
This study is based primarily on mathematical modeling and geometrical analysis, which do not account for individual variations in tissue perfusion, elasticity, or intraoperative conditions. The surgeon’s clinical judgment plays a critical role in flap selection, design and intraoperative assessment of tissue perfusion and flexibility. These factors are indeed essential determinants of surgical outcomes. However, incorporating such qualitative and intraoperative variables into mathematical models requires further investigations and methodological development. This limitation has been acknowledged in the present study.

Conclusion
This paper proposes a new model based on trigonometric relationships to determine the angle size and Z-rotation. The primary advantage of the proposed model is that it simplifies the management of increased contraction length, ensuring that equations 1 and 2 always provide a unique solution. The concept behind this new model holds potential for future use. We implemented this model using MATLAB. 

Acknowledgement
This work was supported by the Islamic Azad University, Mashhad, Iran. The study was approved by the Ethics Committee of the Islamic Azad University, Mashhad, Iran, with ethical approval code: IR.IAU.MSHD.REC.1403.061.

Conflict of Interest
Authors declare no conflict of interest.