Analytical, stochastic and experimental solution of the osteosynthesis of the fifth metatarsal by headless screw

Citation: Vlˇcková, K.; Frydrýšek, K.; Bajtek, V.; Demel, J.; Pleva, L.; Havlíˇcek, M.; Pometlová, J.; Madeja, R.; Kratochvíl, J.; Krpec, P.; et al. Analytical, Stochastic and Experimental Solution of the Osteosynthesis of the Fifth Metatarsal by Headless Screw. Appl. Sci. 2022, 12, 9615. https://doi.org/10.3390/ app12199615 Academic Editor: Mark King Received: 9 June 2022 Accepted: 12 September 2022 Published: 25 September 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 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/). applied sciences Article Analytical, Stochastic and Experimental Solution of the Osteosynthesis of the Fifth Metatarsal by Headless Screw Kateˇrina Vlˇcková1,2,* , Karel Frydrýšek 1,2 , Vojtˇech Bajtek 1,2, JiˇríDemel 2,3, Leopold Pleva 2,3, Miroslav Havlíˇcek 4, Jana Pometlová2,3 , Roman Madeja 2,3, JiˇríKratochvíl5, Pavel Krpec 6, Paweł Osemlak 7, Kristina ˇ Cabanová8, Eva Olšovská8and Jana Vaculová9 1 Department of Applied Mechanics, Faculty of Mechanical Engineering, VSB—Technical University of Ostrava, 17. listopadu 2172/15, Poruba, 708 00 Ostrava, Czech Republic 2Institute of Emergency Medicine, Faculty of Medicine, University of Ostrava, Syllabova 19, Vítkovice, 703 00 Ostrava, Czech Republic 3Trauma Center, University Hospital Ostrava, 17. listopadu 1790, Poruba, 708 52 Ostrava, Czech Republic 4MEDIN, a.s., Vlachovicka 619, 592 31 NovéMˇesto na Moravˇe, Czech Republic 5Department of Machining Assembly and Engineering Metrology, Faculty of Mechanical Engineering, VSB—Technical University of Ostrava, 17. listopadu 2172/15, 708 00 Ostrava, Czech Republic 6V-NASS, a.s., Halasova 2938/1a, Vítkovice, 703 00 Ostrava, Czech Republic 7Pediatric University Hospital Named by Prof. Antoni G˛ebala in Lublin, ul. Prof. A. G˛ebali 6, Department of Pediatric Surgery and Traumatology, Medical University of Lublin, 20-093 Lublin, Poland 8Center of Advanced Innovation Technologies, VSB—Technical University of Ostrava, 17. listopadu 15/2172, Poruba, 708 00 Ostrava, Czech Republic 9Institute of Molecular and Clinical Pathology and Medical Genetics, University of Ostrava, Syllabova 19, Vítkovice, 703 00 Ostrava, Czech Republic *Correspondence: [email protected] Abstract: This paper evaluates the various approaches to strength and stiffness analysis of fracture osteosynthesis using a headless Herbert screw. The problem has been extensively addressed using several scientific approaches, namely the analytical approach, stochastic approach, experimental approach, and (marginally) using the finite elements method. The problem is illustrated on the use of a prototype headless screw Ti: 4.0/1.4 × 30/7 (manufacturer: Medin, Czech Republic) and the surgical treatment of the fifth metatarsal fracture. Mathematical equations for the analytical calculation of the maximum stresses in the screw were established for tensile/compression loading. This problem is also interesting because of its static indetermination in tension and compression; for this reason, it was necessary to use the deformation condition, i.e., the relationship between screw extension and bone contraction. The stochastic (probabilistic) approach, i.e., application of the Monte Carlo method, takes advantage of the mathematical equations derived during the analytical solution by respecting of the natural variabilities and uncertainties. The analytical and stochastic approaches were validated by measurements on porcine bones and by the finite element method. The data measured experimentally were also processed and used for deriving an equation, appropriately approximating the data. The main part of the measurement was to determine the axial force generated during osteosynthesis with a headless screw. The obtained compressional force was used to determine the maximal stress in the screw and bone. Finally, the methods were compared. In this paper, comprehensive and original approaches based on the authors’ experience with multiple methods are presented. Obtained results are necessary for headless screw designers during optimalization of the implants and are also useful for surgeons developing new surgical techniques. This biomechanical problem was solved in cooperation with the engineering industry and physicians to improve the quality of care for patients with trauma in orthopedics and surgery. Keywords: headless screw; fifth metatarsal fracture; osteosynthesis; tensile and compression loading; analytical solutions; probabilistic (stochastic) solutions; reliability; Monte Carlo method; experiments; finite element method; orthopedics; traumatology; biomechanics Appl. Sci. 2022,12, 9615. https://doi.org/10.3390/app12199615 https://www.mdpi.com/journal/applsci Appl. Sci. 2022,12, 9615 2 of 19 1. Introduction The presented paper focuses on an implant widely used in traumatology, namely a headless (so-called Herbert) screw. This screw (implant) is designed for surgical treatment of a fifth metatarsal bone using osteosynthesis. In general, osteosynthesis is a surgical method of treating fractures involving the fixation of bone fragments using internal (screws, splints, nails, wires, and combinations thereof) or external (external fixators) implants. The application of screws is important in the surgical treatment of complicated human or animal fractures and deformities in trauma or orthopedics. They serve both external and internal fixation in osteosynthesis, etc. Osteosynthesis in trauma surgery is a recognized and important medical method and an important topic of many studies [1–11]. Figure 1shows an X-ray snapshot of the fixation of a fifth metatarsal fracture with a headless screw. Metatarsal fractures are the most common fractures of the leg, the most common among them being the fifth metatarsal fracture. Metatarsal fractures are mostly caused by direct impact (car accident, fall, etc.) or during sports (typical injuries of football players, etc.). There are multiple types of metatarsal fractures, such as the Jones fractures, proximal diaphyseal, or fatigue fractures. Interested readers may find more information in [5,6]. Appl. Sci. 2022, 12, x FOR PEER REVIEW 2 of 20 Keywords: headless screw; fifth metatarsal fracture; osteosynthesis; tensile and compression loading; analytical solutions; probabilistic (stochastic) solutions; reliability; Monte Carlo method; experiments; finite element method; orthopedics; traumatology; biomechanics 1. Introduction The presented paper focuses on an implant widely used in traumatology, namely a headless (so-called Herbert) screw. This screw (implant) is designed for surgical treatment of a fifth metatarsal bone using osteosynthesis. In general, osteosynthesis is a surgical method of treating fractures involving the fixation of bone fragments using internal (screws, splints, nails, wires, and combinations thereof) or external (external fixators) implants. The application of screws is important in the surgical treatment of complicated human or animal fractures and deformities in trauma or orthopedics. They serve both external and internal fixation in osteosynthesis, etc. Osteosynthesis in trauma surgery is a recognized and important medical method and an important topic of many studies [1– 11]. Figure 1 shows an X-ray snapshot of the fixation of a fifth metatarsal fracture with a headless screw. Metatarsal fractures are the most common fractures of the leg, the most common among them being the fifth metatarsal fracture. Metatarsal fractures are mostly caused by direct impact (car accident, fall, etc.) or during sports (typical injuries of football players, etc.). There are multiple types of metatarsal fractures, such as the Jones fractures, proximal diaphyseal, or fatigue fractures. Interested readers may find more information in [5,6]. Figure 1. Application of a headless screw to the fifth metatarsal bone. Figure 2 indicates the basic anatomy of the fifth metatarsal bone and its surroundings. Figure 1. Application of a headless screw to the fifth metatarsal bone. Figure 2indicates the basic anatomy of the fifth metatarsal bone and its surroundings. In 1902, Sir Robert Jones described his own foot injury, which he suffered while dancing. It was a transverse fracture of the fifth metatarsal, without substantial dislocation in the junction of the diaphysis and metaphysis of that metatarsal; see Figure 2. This fracture is characterized by injury development caused by relatively low stress, minimal swelling, absence of hematoma, and prolonged healing. The classification of fractures in the proximal part of the fifth metatarsus is not uniform and completely clear. Some authors fundamentally differentiate the so-called true Jones fracture from stress fractures (fatigue fractures) of the proximal diaphysis of the fifth metatarsal. In 1960, Stewart reported that a large proportion of patients experienced prolonged healing or developed pseudoarthroses after the conservative treatment of the Jones fracture. Their study group, however, contained a large proportion of fatigue fractures of the body of the fifth metatarsal; see [5,6] or [11–13]. Appl. Sci. 2022,12, 9615 3 of 19 Appl. Sci. 2022, 12, x FOR PEER REVIEW 3 of 20 Figure 2. Anatomy of the 5th metatarsal bone: (1) head; (2) neck; (3) diaphysis; (4) base; (5) tuberosity. (A) peroneus tertius (also fibularis tertius) muscle; (B) peroneus brevis (also fibularis brevis) muscle; (C) plantar fascia. In 1902, Sir Robert Jones described his own foot injury, which he suffered while dancing. It was a transverse fracture of the fifth metatarsal, without substantial dislocation in the junction of the diaphysis and metaphysis of that metatarsal; see Figure 2. This fracture is characterized by injury development caused by relatively low stress, minimal swelling, absence of hematoma, and prolonged healing. The classification of fractures in the proximal part of the fifth metatarsus is not uniform and completely clear. Some authors fundamentally differentiate the so-called true Jones fracture from stress fractures (fatigue fractures) of the proximal diaphysis of the fifth metatarsal. In 1960, Stewart reported that a large proportion of patients experienced prolonged healing or developed pseudoarthroses after the conservative treatment of the Jones fracture. Their study group, however, contained a large proportion of fatigue fractures of the body of the fifth metatarsal; see [5,6] or [11–13]. Both conservative and surgical treatment of the fracture of the proximal part of the fifth metatarsal is possible. In conservative therapy, the application of a full, non-weightbearing cast or plastic fixation for 6–8 weeks is recommended. In surgical treatment, intramedullary screw fixation or osteosynthesis with a tension loop wiring is used. Spongioplasty or corticospongious graft application is often considered the method of choice for these fractures [5,6,11–13]. In 1962, Timothy J. Herbert started to use a compression screw he designed for the surgical treatment of scaphoid fractures. This screw was cannulated and threaded at both ends with threads of a non-uniform angle, which ensured compression between the fragments when the entire screw was inserted. The screw is not extracted after the fracture has healed. The interfragmentary compression with the Herbert screw is approximately 2.5 times lower than with the washer screw. Although Herbert designed the screw specifically to treat scaphoideal fractures, indications for its use in osteosynthesis were, over time, expanded to other fractures as well (olecranon, ankles, tibial heads, fifth metatarsus, etc.) [5,6,11–13]. Metatarsal fractures, see Figure 3, can be treated surgically or conservatively; this paper focuses on the headless screw technique, in which the fracture is immobilized by the screw thanks to the differential pitch of the threads at the threaded ends, which leads to pulling the bone fragments together and, subsequently, to healing. Figure 2. Anatomy of the 5th metatarsal bone: (1) head; (2) neck; (3) diaphysis; (4) base; (5) tuberosity. ( A ) peroneus tertius (also fibularis tertius) muscle; ( B ) peroneus brevis (also fibularis brevis) muscle; (C) plantar fascia. Both conservative and surgical treatment of the fracture of the proximal part of the fifth metatarsal is possible. In conservative therapy, the application of a full, non-weight-bearing cast or plastic fixation for 6–8 weeks is recommended. In surgical treatment, intramedullary screw fixation or osteosynthesis with a tension loop wiring is used. Spongioplasty or corticospongious graft application is often considered the method of choice for these fractures [5,6,11–13]. In 1962, Timothy J. Herbert started to use a compression screw he designed for the surgical treatment of scaphoid fractures. This screw was cannulated and threaded at both ends with threads of a non-uniform angle, which ensured compression between the fragments when the entire screw was inserted. The screw is not extracted after the fracture has healed. The interfragmentary compression with the Herbert screw is approximately 2.5 times lower than with the washer screw. Although Herbert designed the screw specifically to treat scaphoideal fractures, indications for its use in osteosynthesis were, over time, expanded to other fractures as well (olecranon, ankles, tibial heads, fifth metatarsus, etc.) [5,6,11–13]. Metatarsal fractures, see Figure 3, can be treated surgically or conservatively; this paper focuses on the headless screw technique, in which the fracture is immobilized by the screw thanks to the differential pitch of the threads at the threaded ends, which leads to pulling the bone fragments together and, subsequently, to healing. The application of the headless screw is shown graphically in Figure 4. The procedure of osteosynthetic treatment of the fifth metatarsal fracture is described in more detail in [ 3 ]. It is divided into several steps. A Jones fracture and a fatigue fracture are priori fractures without displacement, so in principle, it is not necessary to perform reposition. However, at the beginning of the surgery it is necessary to check whether displacement has occurred using an X-ray intensifier. Then, a (Kirschner) guide wire is drilled into the bone fragments and used to guide the implementation of the cannulated headless screw. The headless crew is introduced into the bone fragments to ensure that the bone fragments are connected in the correct position; see Figure 4. The correct length of the screw is detected using a drill. It is important that there is compression, that the screw is not too long, and that it is fully immersed in the bone. The resulting position, compression ratio, and length of the screw is checked with an X-ray intensifier in the operating room. The procedure is minimally invasive, and it is performed through a minimal incision of approximately 10 mm. After the procedure, a leg bandage is applied, and rigid fixation is not necessary. The load is recommended according to tolerance. Appl. Sci. 2022,12, 9615 4 of 19 Appl. Sci. 2022, 12, x FOR PEER REVIEW 4 of 20 Figure 3. New classification of the proximal 5th metatarsal fractures, according to K. T. Lee: (A1) acute complete fracture, (A2) chronic complete fracture, (B1) incomplete fracture below 1 mm, (B2) incomplete fracture of 1 mm or more. The application of the headless screw is shown graphically in Figure 4. The procedure of osteosynthetic treatment of the fifth metatarsal fracture is described in more detail in [3]. It is divided into several steps. A Jones fracture and a fatigue fracture are priori fractures without displacement, so in principle, it is not necessary to perform reposition. However, at the beginning of the surgery it is necessary to check whether displacement has occurred using an X-ray intensifier. Then, a (Kirschner) guide wire is drilled into the bone fragments and used to guide the implementation of the cannulated headless screw. The headless crew is introduced into the bone fragments to ensure that the bone fragments are connected in the correct position; see Figure 4. The correct length of the screw is detected using a drill. It is important that there is compression, that the screw is not too long, and that it is fully immersed in the bone. The resulting position, compression ratio, and length of the screw is checked with an X-ray intensifier in the operating room. The procedure is minimally invasive, and it is performed through a minimal incision of approximately 10 mm. After the procedure, a leg bandage is applied, and rigid fixation is not necessary. The load is recommended according to tolerance. Figure 3. New classification of the proximal 5th metatarsal fractures, according to K. T. Lee: ( A1 ) acute complete fracture, ( A2 ) chronic complete fracture, ( B1 ) incomplete fracture below 1 mm, ( B2 ) incomplete fracture of 1 mm or more. Appl. Sci. 2022, 12, x FOR PEER REVIEW 5 of 20 Figure 4. Procedure of the Herbert screw insertion. Frequent complications of surgical treatment of fractures of the fifth metatarsal are the failure of osteosynthesis, accompanied by necrosis of the surrounding tissue, nonunion, irritation of soft tissues, or chronic pain. Less commonly, osteosynthesis is accompanied by pseudoarthrosis, infectious complications, or thrombosis. This paper is particularly focusing on the headless cannulated Herbert screw Ti: 4.0/1.4 × 30/7 produced by the Czech company MEDIN a.s.; see [14]. This screw is depicted in Figure 5, and its basic dimensions are shown in Table 1. The diameters ØD1a [mm] and ØD1b [mm] are the median diameters of the resective threads. Figure 5. Description of the headless screw dimensions: (a) headless screw, (b) cross-section of the screw. Figure 4. Procedure of the Herbert screw insertion. Frequent complications of surgical treatment of fractures of the fifth metatarsal are the failure of osteosynthesis, accompanied by necrosis of the surrounding tissue, nonunion, irritation of soft tissues, or chronic pain. Less commonly, osteosynthesis is accompanied by pseudoarthrosis, infectious complications, or thrombosis. This paper is particularly focusing on the headless cannulated Herbert screw Ti: 4.0/ 1.4 ×30/7 produced by the Czech company MEDIN a.s.; see [ 14 ]. This screw is depicted in Appl. Sci. 2022,12, 9615 5 of 19 Figure 5, and its basic dimensions are shown in Table 1. The diameters ØD1a [mm] and ØD1b [mm] are the median diameters of the resective threads. Appl. Sci. 2022, 12, x FOR PEER REVIEW 5 of 20 Figure 4. Procedure of the Herbert screw insertion. Frequent complications of surgical treatment of fractures of the fifth metatarsal are the failure of osteosynthesis, accompanied by necrosis of the surrounding tissue, nonunion, irritation of soft tissues, or chronic pain. Less commonly, osteosynthesis is accompanied by pseudoarthrosis, infectious complications, or thrombosis. This paper is particularly focusing on the headless cannulated Herbert screw Ti: 4.0/1.4 × 30/7 produced by the Czech company MEDIN a.s.; see [14]. This screw is depicted in Figure 5, and its basic dimensions are shown in Table 1. The diameters ØD1a [mm] and ØD1b [mm] are the median diameters of the resective threads. Figure 5. Description of the headless screw dimensions: (a) headless screw, (b) cross-section of the screw. Figure 5. Description of the headless screw dimensions: ( a ) headless screw, ( b ) cross-section of the screw. Table 1. Definition of dimensions, see Figure 5. Length [mm] Diameter [mm] Thread Pitch [mm] L1 L1a L1b L1c D1a D1b D1c d1 PA PB 30 7 4 19 3.3 4.7 2.5 1.4 1.1 0.9 In this paper, strength and stiffness analysis of a headless screw is carried out using analytical, stochastic, and experimental approaches. The study focuses on osteosynthesis of the fifth metatarsal fractures, but the results can be used for other applications in trauma and orthopedics too. An original analytical design was devised, and an experiment was performed, in which the normal axial compressive forces acting on a headless screw during the osteosynthesis of porcine bones under laboratory conditions were measured and evaluated. The finite element method was also used to a minor degree. The information provided in this study fills the gaps in the knowledge of the biomechanical information of headless screws and serves as a basis for clinical applications for further improving patient care and new designs of headless screws. 2. Analytical Approach In the analytical approach, mathematical equations required for the analytical calculation of the maximum stress in the headless screw and fragments of the fractured fifth metatarsal are defined. Mathcad software was used for the actual analysis [ 15 ]. The acquired results originated from constant input values, and our analytical approach was, therefore, a deterministic approach. The follow-up stochastic (probabilistic) approach is discussed in Section 3. Only small deformations were assumed, inferring the validity of Hooke’s law. In this calculation, the material was considered homogeneous and isotropic, which is a standard and widely used simplification of the generally very complicated reality. These simplifications are acceptable and typical in common engineering tasks. The mutual closeness of fragments is a necessary condition for correct osteosynthesis. The drawing of fragments together is mediated by the different pitches of the ends of the headless screw. For this, the thread pitch PA [mm] must be greater than that of PB [mm] (i.e., PA >PB ); see Figure 5. If the bone fragments are close enough and the headless screw Appl. Sci. 2022,12, 9615 6 of 19 is implemented, the axial shifting (displacement) of the bone fragments ∆ depends on the number of rotations of the headless screw (n) [1] and is defined as ∆=n×(PA −PB)(1) The displacement causes compression and preload in the headless screw. The displacement ∆is shown in Figure 6as the required deformation condition (a special type of boundary condition in mechanics). Appl. Sci. 2022, 12, x FOR PEER REVIEW 6 of 20 Table 1. Definition of dimensions, see Figure 5. Length [mm] Diameter [mm] Thread Pitch [mm] L1 L1a L1b L1c D1a D1b D1c d1 PA PB 30 7 4 19 3.3 4.7 2.5 1.4 1.1 0.9 In this paper, strength and stiffness analysis of a headless screw is carried out using analytical, stochastic, and experimental approaches. The study focuses on osteosynthesis of the fifth metatarsal fractures, but the results can be used for other applications in trauma and orthopedics too. An original analytical design was devised, and an experiment was performed, in which the normal axial compressive forces acting on a headless screw during the osteosynthesis of porcine bones under laboratory conditions were measured and evaluated. The finite element method was also used to a minor degree. The information provided in this study fills the gaps in the knowledge of the biomechanical information of headless screws and serves as a basis for clinical applications for further improving patient care and new designs of headless screws. 2. Analytical Approach In the analytical approach, mathematical equations required for the analytical calculation of the maximum stress in the headless screw and fragments of the fractured fifth metatarsal are defined. Mathcad software was used for the actual analysis [15]. The acquired results originated from constant input values, and our analytical approach was, therefore, a deterministic approach. The follow-up stochastic (probabilistic) approach is discussed in Section 3. Only small deformations were assumed, inferring the validity of Hooke’s law. In this calculation, the material was considered homogeneous and isotropic, which is a standard and widely used simplification of the generally very complicated reality. These simplifications are acceptable and typical in common engineering tasks. The mutual closeness of fragments is a necessary condition for correct osteosynthesis. The drawing of fragments together is mediated by the different pitches of the ends of the headless screw. For this, the thread pitch PA [mm] must be greater than that of PB [mm] (i.e., PA>PB); see Figure 5. If the bone fragments are close enough and the headless screw is implemented, the axial shifting (displacement) of the bone fragments Δ depends on the number of rotations of the headless screw (n) [1] and is defined as Δ=n×(PA−PB) (1) The displacement causes compression and preload in the headless screw. The displacement Δ is shown in Figure 6 as the required deformation condition (a special type of boundary condition in mechanics). Figure 6. Deformation condition for solving the statically indeterminate issue (Δ = displacement caused by headless screw tightening, Δ1 = extension of the screw, Δ2,3 = shortening of bone fragments due to axial compression, and Δs = displacement of the inner sensor cylinder in the measuring). Figure 6. Deformation condition for solving the statically indeterminate issue ( ∆ = displacement caused by headless screw tightening, ∆ 1 = extension of the screw, ∆ 2,3 = shortening of bone fragments due to axial compression, and ∆s = displacement of the inner sensor cylinder in the measuring). Explanation of displacements in headless screw and bone fragments are shown in Figure 7. Appl. Sci. 2022, 12, x FOR PEER REVIEW 7 of 20 Explanation of displacements in headless screw and bone fragments are shown in Figure 7. Figure 7. Explanation of displacements Δ1, Δ2, and Δ3 (not in real scale). A sensor (i.e., axial force and displacement gauge) was also considered in the analytical solution to allow comparison with the experiments performed in Section 4. This task was solved as an interesting elasticity problem because it was a statically indeterminate task of the first degree in tension and compression. It was, therefore, necessary to determine the deformation condition. The deformation condition describes the relationship between the screw elongation (Δ1 [mm]) and bone fragment shortening (Δ2 [mm] and Δ3 [mm]); see Figures 6 and 7 and Equation (2). To derive Equation (2), the basic relationships of mechanics were used; namely, we used Hooke’s law, the relationship for axial tension/compression stress, and the relationship for relative longitudinal strain under tension/compression. Hence, Δ=Δ1+Δ2+Δ3+Δs= =F Lk1 𝐸𝑘∙ 𝐴𝑘1 +F 1 𝐸𝑘(L1b 𝐴𝑘3 +Lk2−L1b 𝐴𝑘2 )+F 1 𝐸1(L1a 𝐴𝑎+L1b 𝐴𝑏+L1c 𝐴𝑐)+F𝑎𝑆 (2) The internal normal force F is therefore described as: F= Δ Lk1 𝐸𝑘∙ 𝐴𝑘1+ 1 𝐸𝑘(L1b 𝐴𝑘3+Lk2−L1b 𝐴𝑘2 )+ 1 𝐸1(L1a 𝐴𝑎+L1b 𝐴𝑏+L1c 𝐴𝑐)+𝑎𝑆 (3) with the meaning of individual variables given in Table 2. Figure 7. Explanation of displacements ∆1, ∆2, and ∆3 (not in real scale). A sensor (i.e., axial force and displacement gauge) was also considered in the analytical solution to allow comparison with the experiments performed in Section 4. This task was solved as an interesting elasticity problem because it was a statically indeterminate task of the first degree in tension and compression. It was, therefore, necessary Appl. Sci. 2022,12, 9615 7 of 19 to determine the deformation condition. The deformation condition describes the relationship between the screw elongation ( ∆ 1 [mm]) and bone fragment shortening ( ∆ 2 [mm] and ∆ 3 [mm]); see Figures 6and 7and Equation (2). To derive Equation (2), the basic relationships of mechanics were used; namely, we used Hooke’s law, the relationship for axial tension/compression stress, and the relationship for relative longitudinal strain under tension/compression. Hence, ∆=∆1+∆2+∆3+∆s =FLk1 Ek·Ak1+F1 EkL1b Ak3+Lk2−L1b Ak2+F1 E1L1a Aa+L1b Ab+L1c Ac+FaS(2) The internal normal force F is therefore described as: F=∆ Lk1 Ek·Ak1+1 EkL1b Ak3+Lk2−L1b Ak2+1 E1L1a Aa+L1b Ab+L1c Ac+aS (3) with the meaning of individual variables given in Table 2. Table 2. Definition of deterministic input variables. Quantity Definition of Quantity Quantity Definition of Quantity Lk1, Lk2 Lengths of bone fragments ∆ Mutual displacement of the bone fragments (drawn to each other by the screw tightening) Ak1,Ak2,Ak3 Bone areas in the individual parts of the screw Ek Modulus of elasticity of the bone Aa,Ab,AcScrew areas in individual parts E1Modulus of elasticity of the screw L1a, L1b, L1cScrew lengths in individual parts aS Measurement—effect of the force sensor (correction to the real osteosynthesis not containing any sensor) This calculation and way of derivation is exposed in more detail in [ 16 ]. The abovedescribed analytical approach was also used to establish the equations of the stochastic approach (application of the Monte Carlo method), explained in the following section. The measurements are described in more detail in Section 4of this paper. The value of the tensile modulus of the bone was adopted from [17]. 3. Stochastic Approach The probabilistic calculation was performed in the Anthill software, in which the analysis was performed using the SBRA (simulation-based reliability assessment) method. The SBRA method applies the probability theory and statistics to stochastic inputs. The output variables affecting the safety of a structure or machined part are usually expressed as truncated histograms. Due to the random nature of the input variables, the resulting output variables also possess a random character. This allows a probabilistic assessment of the engineering problem [18,19]. The method is based on the direct Monte Carlo approach, which analyzes the reliability function using sequences of pseudo-random simulations (realizations). The simulation works with a large number of pseudo-random samples (in our case, 10 6 random simulations), and the input and output values are defined using bounded histograms [18–21]. In our calculation, input stochastic variables used screw and bone dimensions and their mechanical properties. Thanks to this, possible inaccuracies in production, different bone anatomy, and their mechanical properties could be included in the calculation. Output Appl. Sci. 2022,12, 9615 8 of 19 variables were axial normal force, maximal stress in the screw, and reliability function. Figure 8shows histogram of the axial normal force. Appl. Sci. 2022, 12, x FOR PEER REVIEW 8 of 20 Table 2. Definition of deterministic input variables. Quantity Definition of Quantity Quantity Definition of Quantity Lk1,Lk2 Lengths of bone fragments Δ Mutual displacement of the bone fragments (drawn to each other by the screw tightening) 𝐴𝑘1, 𝐴𝑘2,𝐴𝑘3 Bone areas in the individual parts of the screw 𝐸𝑘 Modulus of elasticity of the bone 𝐴𝑎,𝐴𝑏,𝐴𝑐 Screw areas in individual parts 𝐸1 Modulus of elasticity of the screw L1a,L1b,L1c Screw lengths in individual parts 𝑎𝑆 Measurement—effect of the force sensor (correction to the real osteosynthesis not containing any sensor) This calculation and way of derivation is exposed in more detail in [16]. The abovedescribed analytical approach was also used to establish the equations of the stochastic approach (application of the Monte Carlo method), explained in the following section. The measurements are described in more detail in Section 4 of this paper. The value of the tensile modulus of the bone was adopted from [17]. 3. Stochastic Approach The probabilistic calculation was performed in the Anthill software, in which the analysis was performed using the SBRA (simulation-based reliability assessment) method. The SBRA method applies the probability theory and statistics to stochastic inputs. The output variables affecting the safety of a structure or machined part are usually expressed as truncated histograms. Due to the random nature of the input variables, the resulting output variables also possess a random character. This allows a probabilistic assessment of the engineering problem [18,19]. The method is based on the direct Monte Carlo approach, which analyzes the reliability function using sequences of pseudo-random simulations (realizations). The simulation works with a large number of pseudo-random samples (in our case, 106 random simulations), and the input and output values are defined using bounded histograms [18–21]. In our calculation, input stochastic variables used screw and bone dimensions and their mechanical properties. Thanks to this, possible inaccuracies in production, different bone anatomy, and their mechanical properties could be included in the calculation. Output variables were axial normal force, maximal stress in the screw, and reliability function. Figure 8 shows histogram of the axial normal force. Figure 8. Histogram for axial normal force F in the headless screw (minimum = 62.7 N, median = 162.6 N, mean = 164.3 N, maximum = 357.9 N)—Anthill software. Figure 8. Histogram for axial normal force F in the headless screw (minimum = 62.7 N, median = 162.6 N, mean = 164.3 N, maximum = 357.9 N)—Anthill software. Reliability function Fs is defined as Fs =Re −σmax, (4) where Re is the yield stress of a screw and σmax is the maximal stress in the screw. The situation where Fs ≤ 0 expresses the probability of a condition where the thread cut in the bone fragment is sheared. The calculated reliability function Fs is shown in Figure 9. Thread shearing and thus, plastic deformation, occurred in the bone with a probability of only 0.0493 (i.e., of 4.93%). If thread is sheared during surgery, the headless screw can be replaced with a larger diameter headless screw, and the surgical procedure can subsequently be completed, which further reduces the risk to the patient. Appl. Sci. 2022, 12, x FOR PEER REVIEW 9 of 20 Reliability function Fs is defined as Fs=Re−σmax, (4) where Re is the yield stress of a screw and σmax is the maximal stress in the screw. The situation where Fs ≤ 0 expresses the probability of a condition where the thread cut in the bone fragment is sheared. The calculated reliability function Fs is shown in Figure 9. Thread shearing and thus, plastic deformation, occurred in the bone with a probability of only 0.0493 (i.e., of 4.93%). If thread is sheared during surgery, the headless screw can be replaced with a larger diameter headless screw, and the surgical procedure can subsequently be completed, which further reduces the risk to the patient. Figure 9. Distribution of the reliability function Fs—Anthill software. Figure 10 shows the 2D distribution of the reliability function Fs. Towards the center, the frequency of the reliability function values increased, with higher frequencies of occurrence shown in red. The green color indicates the limit line, which divides the reliability function into negative and positive parts. If the reliability function is negative (i.e., Fs < 0), the headless screw is in the region of permanent plastic deformation and thread shearing will occur. Conversely, if the reliability function is positive, it is in the region of elastic deformations, and no limit state causing thread shearing in the bone can occur. Figure 9. Distribution of the reliability function Fs—Anthill software. Figure 10 shows the 2D distribution of the reliability function Fs. Towards the center, the frequency of the reliability function values increased, with higher frequencies of occurrence shown in red. The green color indicates the limit line, which divides the reliability function into negative and positive parts. If the reliability function is negative (i.e., Fs < 0), the headless screw is in the region of permanent plastic deformation and thread shearing Appl. Sci. 2022,12, 9615 9 of 19 will occur. Conversely, if the reliability function is positive, it is in the region of elastic deformations, and no limit state causing thread shearing in the bone can occur. Appl. Sci. 2022, 12, x FOR PEER REVIEW 10 of 20 Figure 10. Two-dimensional distribution of the reliability function values (horizontal axis represents calculated stress and vertical axis is yield stress, steps: 1,000,000)—Anthill software. 4. Experimental Section The experiment aimed to obtain information about the dependency between the tightening of the headless screw and to determine the normal force acting on the headless, self-tapping cannulated screw (same as in the calculation, i.e., Ti: 4.0/1.4 × 30/7 mm by MEDIN a.s.; see Figure 5 and Table 1). The acquired results can then be compared with analytical, numerical, and stochastic calculations. The knowledge of axial forces acting during osteosynthesis is important when designing headless screws and their threads. In real-life fractures, bending stress is also present, but compression and tension are the dominant stresses on the bone fragments and screw during screw introduction (insertion). Moreover, this type of information is missing in literatures. The experiment was performed on a headless screw introduced into porcine leg bones purchased from a conventional butcher shop. The porcine bones are commonly used as a readily available substitute for human bones, including metatarsals [22]. The use of porcine bones was in accordance with ethical standards and did not require approval from an ethics committee. Tightening the screw inside the bone fragments elicited a compressive force measured by a holed strain gauge transducer. First, the porcine bone fragments were predrilled, then a tensometric sensor (LC 8150-375-500, Tensometric Messtechnik GmbH, Wuppertal, Germany) was placed between the bone fragments, and a headless screw was inserted. In accordance with the operating procedures, the headless screw was tightened to a tightening torque of 2.5 Nm. This value was experimentally verified by previous measurements to ensure that the screw threads were not stripped while driving the screw into the examined bone fragments. The force was measured in quarter-turn increments, i.e., 90°, until the headless screw was tightened to 3.5 turns; this also included the moment of the thread shear (i.e., the maximum moment) in the bone. (Thread shear will cause the Figure 10. Two-dimensional distribution of the reliability function values (horizontal axis represents calculated stress and vertical axis is yield stress, steps: 1,000,000)—Anthill software. 4. Experimental Section The experiment aimed to obtain information about the dependency between the tightening of the headless screw and to determine the normal force acting on the headless, self-tapping cannulated screw (same as in the calculation, i.e., Ti: 4.0/1.4 × 30/7 mm by MEDIN a.s.; see Figure 5and Table 1). The acquired results can then be compared with analytical, numerical, and stochastic calculations. The knowledge of axial forces acting during osteosynthesis is important when designing headless screws and their threads. In real-life fractures, bending stress is also present, but compression and tension are the dominant stresses on the bone fragments and screw during screw introduction (insertion). Moreover, this type of information is missing in literatures. The experiment was performed on a headless screw introduced into porcine leg bones purchased from a conventional butcher shop. The porcine bones are commonly used as a readily available substitute for human bones, including metatarsals [ 22 ]. The use of porcine bones was in accordance with ethical standards and did not require approval from an ethics committee. Tightening the screw inside the bone fragments elicited a compressive force measured by a holed strain gauge transducer. First, the porcine bone fragments were pre-drilled, then a tensometric sensor (LC 8150-375-500, Tensometric Messtechnik GmbH, Wuppertal, Germany) was placed between the bone fragments, and a headless screw was inserted. In accordance with the operating procedures, the headless screw was tightened to a tightening torque of 2.5 Nm. This value was experimentally verified by previous measurements to Appl. Sci. 2022,12, 9615 16 of 19 Appl. Sci. 2022, 12, x FOR PEER REVIEW 17 of 20 Figure 18. Elasticity modulus distribution in the 5th metatarsal bone—top: medial view; bottom: sagittal cutting with a headless screw. According to [29], simulation revealed that the critical equivalent von Mises stress can be detected at the notch of the screw; see Figures 19 and 20. Figure 19. Critical von Mises stress in a headless screw in the 5th metatarsal. Figure 18. Elasticity modulus distribution in the 5th metatarsal bone—top: medial view; bottom: sagittal cutting with a headless screw. According to [ 29 ], simulation revealed that the critical equivalent von Mises stress can be detected at the notch of the screw; see Figures 19 and 20. Appl. Sci. 2022, 12, x FOR PEER REVIEW 17 of 20 Figure 18. Elasticity modulus distribution in the 5th metatarsal bone—top: medial view; bottom: sagittal cutting with a headless screw. According to [29], simulation revealed that the critical equivalent von Mises stress can be detected at the notch of the screw; see Figures 19 and 20. Figure 19. Critical von Mises stress in a headless screw in the 5th metatarsal. Figure 19. Critical von Mises stress in a headless screw in the 5th metatarsal. Appl. Sci. 2022, 12, x FOR PEER REVIEW 18 of 20 Figure 20. Detail of critical stress in a headless screw in the 5th metatarsal. The calculation using the FEM method gives results similar to the solutions presented in this paper; nevertheless, it is not the main focus of this publication, just complementary information. A deeper look at the FEM solution, which is more complex and more difficult to relate to stochastic approaches, will be a subject of future publications. However, the author team feels that it is appropriate to mention the FEM method here, as well as the fact that FEM calculations have confirmed the facts obtained from other computational approaches and experiments performed in this study. 7. Conclusions The main objective of this study was to perform stress–strain computational analyses and an experimental analysis for the determination of the axial force generated during osteosynthesis with a headless Herbert screw (Ti: 4.0/1.4 × 30/7 mm, Medin a.s., Czech Republic). This screw is intended for the osteosynthesis of fragments of the fifth metatarsal fracture. Such implants are commonly used in medical practice. Due to the different pitches of the threads on both ends of the screw, the bone fragments are drawn to each other and compressed at the fracture site, which facilitates the process of fracture healing. The analytical (deterministic) calculation was defined as a tension–compression problem. This problem was statically indeterminate to the first degree, which required the use of an additional deformation boundary condition. The relationships derived from the analytical calculation were used for the stochastic (probabilistic) approach, which, through pseudo-random values generation from defined histograms, allows us to respect the real variability of the input and output data and to perform a probabilistic reliability assessment. Here, the Monte Carlo method was used for the solution and evaluation of 106 pseudo-random simulations. In addition, an experiment (10 measurements) aimed at determining the compressive normal forces acting on the Herbert screw during tightening was performed. This experiment revealed the average force required to shear the thread in the bone to be 204 N. This is an important finding for headless screw designers and for surgeons developing new surgical techniques. Next, a suitable function sufficiently approximating the measured values from the experiments was searched for. The excellent quality of the fit of the regression model to the experimental data was demonstrated using the coefficient of determination R2. Figure 20. Detail of critical stress in a headless screw in the 5th metatarsal. Appl. Sci. 2022,12, 9615 17 of 19 The calculation using the FEM method gives results similar to the solutions presented in this paper; nevertheless, it is not the main focus of this publication, just complementary information. A deeper look at the FEM solution, which is more complex and more difficult to relate to stochastic approaches, will be a subject of future publications. However, the author team feels that it is appropriate to mention the FEM method here, as well as the fact that FEM calculations have confirmed the facts obtained from other computational approaches and experiments performed in this study. 7. Conclusions The main objective of this study was to perform stress–strain computational analyses and an experimental analysis for the determination of the axial force generated during osteosynthesis with a headless Herbert screw (Ti: 4.0/1.4 × 30/7 mm, Medin a.s., Czech Republic). This screw is intended for the osteosynthesis of fragments of the fifth metatarsal fracture. Such implants are commonly used in medical practice. Due to the different pitches of the threads on both ends of the screw, the bone fragments are drawn to each other and compressed at the fracture site, which facilitates the process of fracture healing. The analytical (deterministic) calculation was defined as a tension–compression problem. This problem was statically indeterminate to the first degree, which required the use of an additional deformation boundary condition. The relationships derived from the analytical calculation were used for the stochastic (probabilistic) approach, which, through pseudo-random values generation from defined histograms, allows us to respect the real variability of the input and output data and to perform a probabilistic reliability assessment. Here, the Monte Carlo method was used for the solution and evaluation of 106pseudo-random simulations. In addition, an experiment (10 measurements) aimed at determining the compressive normal forces acting on the Herbert screw during tightening was performed. This experiment revealed the average force required to shear the thread in the bone to be 204 N. This is an important finding for headless screw designers and for surgeons developing new surgical techniques. Next, a suitable function sufficiently approximating the measured values from the experiments was searched for. The excellent quality of the fit of the regression model to the experimental data was demonstrated using the coefficient of determination R2. Finally, the results of all approaches were compared. As the approximation of the experiment using a cosine power function ( F=FE 2m[1−cos (πn nE) ]m ) is suitable and provided sufficient accuracy, it is obvious that the task can be solved analytically as a second-order differential equation, which subsequently leads to a significantly simpler and computationally less demanding solution than the use of the FEM approaches. Our model also enables a relatively simple application of the probabilistic approach, which is advantageous, as it allows respecting the real variability of input and output variables, which is typical of the real world. This paper also briefly mentions the numerical solution based on FEM. The results of all four approaches to the solution (i.e., deterministic, stochastic, experimental, and numerical) are in sufficient agreement and confirm the suitability of the examined screw for clinical use. This combined algorithm can be used as a basis for implant certification or recommendations for clinical testing of various types/applications of headless Herbert screws in medical or veterinary practice, as well as for the modification or change of screw design. The acquired data on axial forces in osteosynthesis of the fifth metatarsal were determined in an original and novel way in this study. From this perspective, the paper significantly contributes to the field of fracture biomechanics. The applications and conclusions obtained in this paper can be used in other fields as well—even in the construction industry in the analysis of joints in wooden structures [ 24 , 30 , 31 ]. Appl. Sci. 2022,12, 9615 18 of 19 Author Contributions: Supervision, K.F. and L.P.; Writing—original draft, K.V. and K.F.; Writing— review & editing, V.B., J.D., L.P., M.H., J.P., R.M., J.K., P.K., P.O., K. ˇ C., E.O. and J.V. All authors have read and agreed to the published version of the manuscript. Funding: This article was supported by a Czech project SP2022/26 and by projects CZ.02.1.01/0.0/ 17_049/0008407 and CZ.02.1.01/0.0/0.0/17_049/0008441 within the Operational Programme Research, Development, and Education financed by the European Union and from the state budget of the Czech Republic. Institutional Review Board Statement: Not relevant to this article. Informed Consent Statement: Not relevant to this article. Acknowledgments: This article was supported by a Czech project SP2022/26 and by projects CZ.02.1.01/0.0/17_049/0008407 and CZ.02.1.01/0.0/0.0/17_049/0008441 within the Operational Programme Research, Development, and Education financed by the European Union and from the state budget of the Czech Republic. Conflicts of Interest: The authors declare no conflict of interest. References 1. Bajtek, V.; Frydrýšek, K.; Pleva, L. Stochastic Strength Analysis of Compression Headless Screw. MM Sci. J. 2020 , 3837–3840. [CrossRef] 2. Frydrýšek, K.; Šír, M.; Pleva, L. Strength Analyses of Screws for Femoral Neck Fractures. J. Med. Biol. Eng. 2018 ,38, 816–834. [CrossRef] 3. Manak, P.; Drac, P. Osteosynthesis and Arthrodesis of the Skeleton of the Hand; Grada: Prague, Czech Republic, 2012; ISBN 978-80-2473873-4. 4. ˇ Cada, R.; Frydryšek, K.; Sejda, F.; Demel, J.; Pleva, L. Analysis of Locking Self-Taping Bone Screws for Angularly Stable Plates. J. Med. Biol. Eng. 2017,37, 612–625. 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