HHS Public Access Author manuscript Author Manuscript
Ser Biomech. Author manuscript; available in PMC 2016 April 14. Published in final edited form as: Ser Biomech. 2007 December ; 23(1): 96–103.
Mathematical modeling of fixation of a bone fragment in a new Double-needle external Fixator compared to hoffmann ii fixator M. Pitkin1, Y. Shukeylo2, and A. Gritsanov3 1Tufts
University, Boston, USA
2LETI
University, St. Petersburg, Russia
Author Manuscript
3Military
Medical Academy, St. Petersburg, Russia
Abstract
Author Manuscript
The outcome of rehabilitation after multiple bone fractures can be improved, and the reduction of the rate of amputation due to severe trauma can be achieved with the early use of external fixators [1, 2]. Effectiveness of the fixator depends on the stability of the bone fragments during the evacuation of a patient to specialized facilities [3, 4]. The paper is devoted to the mathematical modeling of the stability of a bone fragment in an external fixator. The vertical displacement of the end of a bone fragment loaded with a standardized force and moment is suggested to be the measure of stability. The finite element analysis (FEA) model developed has been applied to the new Double-Needle Ilizarov External Fixator (DNIF) [5], which does not penetrate the medullary canal, and to the Hoffmann II1 external fixator. Vertical displacement in DNIF (4.78·10−5m) obtained via FEA was approximately of one order smaller than in the corresponding Hoffmann II fixators (4.196·10−4m). The initial hypothesis has been confirmed that the stability of fixation with the DNIF was greater when compared to Hofmann II.
Keywords bone fracture; external fixator; bone displacement; FEA modeling
1. Introduction
Author Manuscript
Limb amputations constitute one of the most devastating consequences of wars, natural disasters and terrorist attacks [6–8]. Among victims affected by explosive injuries in liberated Kuwait between March and December of 1991, 26.9% required amputations [9]. During mine clearance [10], 59% of the injuries resulted in multiple fractures, with an overall amputation rate of 30%. Early surgical management and evacuation of battle casualties have greatly reduced mortality and subsequent disabilities from the war. Russian military orthopaedic surgeons reported an increased rate of limb salvage during the war in Afghanistan, 1979–1989, due in large part to the wide use of the Ilizarov method and apparatus for the fixation of multiple open fractures [3]. Reports on limb salvation with this technique were also provided by other surgeons [11–16]. 1Stryker Trauma AG (SA), Selzach, Switzerland, www.osteosynthesis.stryker.com
Pitkin et al.
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The application of a traditional Ilizarov apparatus or Hoffman II external fixator (Fig.1) requires drilling bone fragments through the medullary canal, creating the potential for secondary complications [17]. To avoid drilling, a “needle” Ilizarov apparatus was introduced by Russian military surgeons [2, 3] during the war in Afghanistan, 1979–1989. The apparatus allowed for easy external fixation of multiple fractures during medical evacuation from the battlefield to the hospital. However, the insufficient stability of fixation prevented this apparatus from wide implementation into practice.
Author Manuscript
To increase the stability of fixation while refraining from drilling through the medullary canal, the first prototype of the new double-needle Ilizarov Fixator (DNIF)2 was manufactured with paired needles (Fig.2). The number of base rings, ranging from two to four, depends on the specific site of application and on a patient’s conditions. Two or three double-needle clamps are secured on each ring, and their position relative to the ring can be adjusted in accordance with the needed direction of the corresponding pair of needles. A unicortical penetration of a needle is performed manually. After initial contact of the needle with the bone, the desired depth of the needle’s penetration can be regulated. The current paper presents a model the comparison of the stability of fixation in the external apparatuses. The model uses the method of finite element analysis (FEA). The results have suggested that the stability of fixation in the new DNIF apparatus is greater compared to that of the Hoffman II external fixator.
2. Methods 2.1. Finite Element Analysis of the stability of bone fixation in DNIF and Hoffmann II apparatuses
Author Manuscript
2.1. 1. Materials modeling assumptions—The material for all components of both fixators is assumed to be elastic, isotropic, continuous, and uniform. The material’s characteristics were taken from the specification of stainless steel X18H10T [18]: modulus of normal elasticity – 190 GPa; density – 7800 kg/m3; ultimate stress – 600 MPa; yield strength – 200 MPa. Material’s characteristics for the cancellous bone were as follows: modulus of normal elasticity – 480 MPa; density – 500 kg/m3; ultimate strength – 10 MPa [19]. Material’s characteristics for the compact bone were: modulus of normal elasticity – 16.2 MPa; density – 2000 kg/m3; ultimate strength – 170 MPa [20]. To simplify calculations, the Poisson ratio for all components was taken to be 0.3.
Author Manuscript
2.1.2. Modeling of the tibia—A three-dimensional model of the human tibia was created with computerized tomography (Siemens Somatom Emotion™) in a conventional manner [21]. CT image data of a cadaver tibia were transferred to a computer, and the cross-section of bone for each slice was generated. The model constructed was used as input data for the engineering program SolidWorks 20063. A gap of 50 mm was generated in the middle of the virtual model to simulate a bone fracture. Both distal and proximal fragments of the bone were fixed in the models of the new DNIF and Hoffmann II apparatuses.
2Poly-Orth International, Sharon, MA 3SolidWorks Corporation, Concord, MA
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2.1.3. Modeling of the DNIF—A CAD model of the DNIF is shown in Fig.3. The apparatus has two base rings 1 connected with three telescopic bars 4. Each of the rings has three double-needle clamps 2, whose position relative to a ring (40° and 90°) was chosen in accordance with the principles established for the classic Ilizarov method [22]. The needles 3 have diameter of 4 mm, and are 150 mm in length. The length of the conical end penetrating the bone cortical layer is 7 mm. The rings 1 have an inner diameter of 160 mm, and the outer diameter of 190 mm; their thickness is 5 mm. The inner diameter of the telescopic bar 4 is 6 mm. The dimensions of the clamps 2 are 50×51×20 mm and their parallelepiped shape was a simplification of the real shape seen in Fig.2. The distance between the rings is 300 mm.
Author Manuscript
2.1.4. Modeling of Hoffmann II apparatus—Since the Hoffmann II design allows for many different configurations of the components, we selected a configuration close to that of the new DNIF [23]. A CAD model of fixation of the fractured bone fragments in the Hoffmann II external apparatus is shown in Fig.4. Four independent parallelepiped supports are connected with two bars 1 with a diameter of 5 mm, and two mounting arcs 2 with a radius of 70 mm. Two clamps 3 have two pins 4 each. The diameter of the pins is 5 mm, and the distance between the pins installed one clamp is 30 mm. The distance between inner pins belonging to the separate clamps, is 210 mm.
Author Manuscript
2.1.5. Modeling of the external load—We assume that the shank and the foot are suspended in the external fixator with the rings (DNIF) or the parallepiped blocs (Hoffmann II) as the base cantilever supports, the bones have a pin-joint connection and the position of the leg in horizontal. In addition to the load from the bone fragments, the apparatuses’ elements are loaded by the foot and the thigh (Fig.5). We assume also the mass of the patient to be 80 kg and height to be 180 cm. That yields the mass of the foot to be 1.1. kg and of the thigh to be 11.5 kg [24]. The estimate the loading effect from the foot yields the force P = 10N; and the moment M0 = 0.51N·m. The effect of the loading from the thigh is represented by the force P1 = 51N (Fig.6). We assume a rigid connection of the elements of the apparatuses and their combined deformation as a whole. Stress-strain calculations in the elements of the apparatuses were performed with the program CosmosWorks 20054
3. Results
Author Manuscript
Vertical displacement of the points of a cross-section in the fracture zone was determined with FEA modeling (Fig.7). The results of calculating the vertical displacement of the bone fragments and the greatest equivalent stress (Von Mises) in the needles of the DNIF are presented in Table 1. Both characteristics were calculated for two configurations with respect to the number of needles used for fixation: 11 needles (as shown in Fig.3) and 12 needles, and for two levels of the needles’ penetration of the bone: 10 mm and 3.5 mm. In Hoffmann II apparatus (Fig.9), the vertical displacement obtained with the FEA yields u = 4.196·10−4 m (see Table 1). The FEA model is responsive not only to the bone’s displacement, but also to the deformation of the Hoffmann II apparatus’s frame. 4Structural Research & Analysis Corp., Santa Monica, CA
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Displacement of the cross-section of the tibia in the fracture zone as the function of the force acting from the foot through the external fixators is plotted in Fig.8, where (1) represents the Hoffmann II external fixator; (2) – DNIF with 10 mm penetration by the needles; (3) – DNIF with 3.5 mm penetration by the needles. The stress-strained state of the system “DNIF-bone fragments” with 11 and 12 needles didn’t differ significantly. The greatest equivalent stress in the needles occurs in the outer row for both fragments for both configurations. In a configuration with 10 mm penetration of the needles, their greatest equivalent stress is 9.083 MPa, and doesn’t exceed 4.5% of the dangerous yield strength of the material (200 MPa). That percentage is higher (12%) in a configuration with the 3.5 mm penetration, where the greatest equivalent stress yields 25.57 MPa. The greatest equivalent stress in the bone was about 2 MPa, or approximately 20% of the dangerous ultimate stress for the cancellous bone.
Author Manuscript
In the Hofmann II apparatus, the greatest equivalent stress of 135.8 MPa occurs in bar 1 (Fig. 3). That is about 67% of the dangerous ultimate stress of the metal (200 MPa). The greatest equivalent stress in the bone was about 2 MPa.
4. Discussion
Author Manuscript
In the new double-needle fixator (DNIF), clusters of paired needles replace wires or pins penetrating the bone in Hoffman II fixator. Stability of fixation in both external apparatuses was evaluated using the vertical displacement of the end of a fractured bone fragment. The mathematical modeling allowed for the calculation of the vertical displacement of a bone using finite element analysis. Results of calculations suggest that the stability of fixation in the new device is greater even without the bicortical drilling of a bone fragment which is currently accepted for early treatment of multiple fractures [4]. In modeling the Hoffmann II external fixator we considered a double-bar configuration, which is known to be more stable compared to a single-bar construct [4]. Thus, the conclusion about greater stability of the new fixator was reached using conservative approach.
Author Manuscript
The greater stability of a new double-needle fixator compared to Hoffmann II external fixator can be explained by the circumferential placement of the needles in the DNIF. Another potential advantage of the new fixator is that it is applied to the bone periosteum without opening of the medullary canal. It will potentially combine the advantages of the existing, stable half-pin external fixators and the pinless AO external fixator, which doesn’t penetrate the medullar canal but does not demonstrate sufficient stability of fixation [25, 26]. Further studies are required to verify the stability of fixation and the ease of application with the new double-needle external fixator.
5. Conclusions A mathematical modeling of fixation of a bone fragment in two external apparatuses has been developed. We have considered the vertical displacement of the bone fragment as an indicator of the stability of its fixation in the apparatus. Results of calculations suggest that
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the stress-strained state of the system “Hoffmann II – bone fragments” is similar to the system “DNIF-bone fragments.” The displacement of a bone fragment in Hoffmann II apparatus doesn’t exceed 1 mm, which is of one order greater than in the new DNIF apparatus. Further study is required to verify this theoretical result.
Acknowledgments The funding for the study was provided in part by NIH Grant R43HD04793. The authors thank ARETE Company, St. Petersburg, Russia, and Controlled Atmosphere Welding Technologies, North Attleboro, MA, USA, for prototyping the DNIF.
References Author Manuscript Author Manuscript Author Manuscript
1. Dougherty PJ, Vickaryous B, Conley E, Hickerson K. A comparison of two military temporary femoral external fixators. Clin Orthop Relat Res. 2003; (412):176–83. [PubMed: 12838069] 2. Nechaev, EA.; Gritsanov, AI.; Fomin, NF.; Minnulin, IP. Mine Blast Trauma. Experience from the war in Afghanistan. St. Petersburg, Russia/Varnamo, Sweden English version: Russian Ministry of Public Health and Medical Industry; 1995. p. 463 3. Gritsanov AI, Averkiev VA, Vardak MM, Rakhman M. Transosseous osteosynthesis in the system for treating victims with penetrating gunshot wounds of the large joints of the upper extremity. Voen Med Zh. 1991; (8):29–32. [PubMed: 1755221] 4. Dougherty PJ, Silverton C, Yeni Y, Tashman S, Weir R. Conversion from temporary external fixation to definitive fixation: shaft fractures. J Am Acad Orthop Surg. 2006; 14(10 Suppl):S124–7. [PubMed: 17003183] 5. Pitkin M, Gritsanov AI. Double-needle external fixator. 2006 US Patent Pending. 6. Alm A, Johansson H, Liljedahl SO, Sjodahl R. Train catastrophy outside Mjolby. Lakartidningen. 1975; 72(26):2767–72. [PubMed: 1142914] 7. Lerner A, Nierenberg G, Stein H. Ilizarov external fixation in the management of bilateral, highly complex blast injuries of lower extremities: a report of two cases. J Orthop Trauma. 1998; 12(6): 442–5. [PubMed: 9715456] 8. Islinger RB, Kuklo TR, Mchale KA. A review of orthopedic injuries in three recent U.S. military conflicts. Mil Med. 2000; 165(6):463–5. [PubMed: 10870364] 9. Bajec J, Gang RK, Lari AR. Post Gulf war explosive injuries in liberated Kuwait. Injury. 1993; 24(8):517–20. [PubMed: 8244542] 10. Brown R, Chaloner E, Mannion S, Cheatle T. 10-year experience of injuries sustained during clearance of anti-personnel mines. Lancet. 2001; 358(9298):2048–9. [PubMed: 11755615] 11. Coupland RM, Howell PR. An experience of war surgery and wounds presenting after 3 days on the border of Afghanistan. Injury. 1988; 19(4):259–62. [PubMed: 3229841] 12. Bhatnagar MK, Smith GS. Trauma in the Afghan guerrilla war: effects of lack of access to care. Surgery. 1989; 105(6):699–705. [PubMed: 2727898] 13. Green SA. Ilizarov external fixation. Technical and anatomic considerations. Bull Hosp Jt Dis Orthop Inst. 1988; 48(1):28–35. [PubMed: 2840145] 14. Dipasquale D, Ochsner MG, Kelly AM, Maloney DM. The Ilizarov method for complex fracture nonunions. J Trauma. 1994; 37(4):629–34. [PubMed: 7932895] 15. Owen MA. Use of the Ilizarov method to manage a septic tibial fracture nonunion with a large cortical defect. J Small Anim Pract. 2000; 41(3):124–7. [PubMed: 10759382] 16. Dougherty PJ. Wartime amputations. Mil Med. 1993; 158(12):755–763. [PubMed: 8108012] 17. Dougherty, PJ. Extremity Fractures. In: Bellamy, RF., editor. In: Emergency War Surgery. Borden Institute, Walter Reed Army Medical Center; Washington, DC: 2004. Series: The Textbooks of Military Medicine 18. Physical-Characteristics. The reference book. Moscow: Energoizdat; 1991. p. 1232 Ser Biomech. Author manuscript; available in PMC 2016 April 14.
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19. Ding M. Age variations in the properties of human tibial trabecular bone and cartilage. Acta Orthop Scand Suppl. 2000; 292:1–45. [PubMed: 10951715] 20. Vitins V, Dobelis M, Middleton J, Limbert G, Knets I. Flexural and creep properties of human jaw compact bone for FEA studies. Comput Methods Biomech Biomed Engin. 2003; 6(5–6):299–303. [PubMed: 14675950] 21. Kang YK, Park HC, Youm Y, Lee IK, Ahn MH, Ihn JC. Three dimensional shape reconstruction and finite element analysis of femur before and after the cementless type of total hip replacement. J Biomed Eng. 1993; 15(6):497–504. [PubMed: 8277755] 22. Ilizarov GA. Basic principles of transosseous compression and distraction osteosynthesis. Ortop Travmatol Protez. 1971; 32(11):7–15. [PubMed: 5141248] 23. Yeni, YN.; Basho, R.; Wybo, C.; Dougherty, PJ.; Gritsanov, AI.; Pitkin, M. Double-needle Ilizarov External Fixator vs Hoffman II. Fourth Conference “US-Russian Program in Prosthetics and Rehabilitation”; New England Sinai Hospital, Stoughton, MA. June 18; 2007. p. 18-20. 24. Mccronville, J.; Churchill, T.; Kaleps, I.; Clauser, C.; Cuzzi, J. Report AFAMRL-TR-80-119. Anthropology research Project, Inc; Yellow Springs, OH: 1980. Anthropometric relationships of body and body segments moments of inrtia. 25. Alonso JE, Regazzoni P. The use of the Ilizarov concept with the AO/ASIF tubular fixateur in the treatment of segmental defects. Orthop Clin North Am. 1990; 21(4):655–65. [PubMed: 2216400] 26. Cheung KY, Wong MS, Choi SH. Hybrid configuration of the AO/ASIF pinless external fixator in the treatment of compound fractures of distal tibial shafts. Kong J Orthop Surg. 2001; 5(1):58–63.
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Fig.1.
Hoffmann II external fixator
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Fig 2.
The DNIF. A three-part fractured bone model is fixed with one redundant ring and several redundant needles to demonstrate that the apparatus is modular and extendable.
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Fig.3.
Model of fixation of the fractured bone fragments in the DNIF: 1 - base ring; 2 – doubleneedle clamp; 3 – a needle; 4 – telescopic bar.
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Fig. 4.
Model of fixation of the fractured bone fragments in the Hoffmann II external apparatus: 1 – connecting bar; 2 – clamp: 3 – mounting arc: 4 – pin.
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Fig. 5.
External forces applied to the fractured bone fragments from the foot (P) and the thigh (P1). a=5 cm – distance from the foot’s center of mass to the center of the ankle joint.
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Fig.6.
Model of loading of the fractured bone fragments in the Hoffmann external apparatus
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Fig.7.
FEA SolidWorks Model of the DNIF for calculation of the vertical displacement in the fracture zone (configuration with 11 needles).
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Fig.8.
Displacement of the cross-section of tibia in the fracture zone as the function of the force acting from the foot through the external fixators: (1) – Hoffmann II external fixator; (2)– DNIF with 10 mm penetration of the needles; (3) – DNIF with 3.5 mm penetration of the needles.
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Fig 9.
SolidWorks model of the Hoffmann II external fixator for calculation of the vertical displacement of a bone fragment in the fracture zone.
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Table 1
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Displacement of bone fragments and stresses in Hoffmann II fixator and DNIF Type of Fixator
Vertical displacement
Greatest equivalent stress (Von Mises)
Hoffmann II
4.196·10−4m
135.8 MPa
DNIF - 11 needles with 10 mm penetration
4.72·10−5m
9,083 MPa
DNIF - 12 needles with 10 mm penetration
4.78·10−5m
9,072 MPa
DNIF - 11 needles with 3.5 mm penetration
9.8·10−5m
20.16 MPa
DNIF - 12 needles with 3.5 mm penetration
13.9·10−5m
25.57 MPa
Author Manuscript Author Manuscript Author Manuscript Ser Biomech. Author manuscript; available in PMC 2016 April 14.
Ser Biomech. Author manuscript; available in PMC 2016 April 14. Published in final edited form as: Ser Biomech. 2007 December ; 23(1): 96–103.
Mathematical modeling of fixation of a bone fragment in a new Double-needle external Fixator compared to hoffmann ii fixator M. Pitkin1, Y. Shukeylo2, and A. Gritsanov3 1Tufts
University, Boston, USA
2LETI
University, St. Petersburg, Russia
Author Manuscript
3Military
Medical Academy, St. Petersburg, Russia
Abstract
Author Manuscript
The outcome of rehabilitation after multiple bone fractures can be improved, and the reduction of the rate of amputation due to severe trauma can be achieved with the early use of external fixators [1, 2]. Effectiveness of the fixator depends on the stability of the bone fragments during the evacuation of a patient to specialized facilities [3, 4]. The paper is devoted to the mathematical modeling of the stability of a bone fragment in an external fixator. The vertical displacement of the end of a bone fragment loaded with a standardized force and moment is suggested to be the measure of stability. The finite element analysis (FEA) model developed has been applied to the new Double-Needle Ilizarov External Fixator (DNIF) [5], which does not penetrate the medullary canal, and to the Hoffmann II1 external fixator. Vertical displacement in DNIF (4.78·10−5m) obtained via FEA was approximately of one order smaller than in the corresponding Hoffmann II fixators (4.196·10−4m). The initial hypothesis has been confirmed that the stability of fixation with the DNIF was greater when compared to Hofmann II.
Keywords bone fracture; external fixator; bone displacement; FEA modeling
1. Introduction
Author Manuscript
Limb amputations constitute one of the most devastating consequences of wars, natural disasters and terrorist attacks [6–8]. Among victims affected by explosive injuries in liberated Kuwait between March and December of 1991, 26.9% required amputations [9]. During mine clearance [10], 59% of the injuries resulted in multiple fractures, with an overall amputation rate of 30%. Early surgical management and evacuation of battle casualties have greatly reduced mortality and subsequent disabilities from the war. Russian military orthopaedic surgeons reported an increased rate of limb salvage during the war in Afghanistan, 1979–1989, due in large part to the wide use of the Ilizarov method and apparatus for the fixation of multiple open fractures [3]. Reports on limb salvation with this technique were also provided by other surgeons [11–16]. 1Stryker Trauma AG (SA), Selzach, Switzerland, www.osteosynthesis.stryker.com
Pitkin et al.
Page 2
Author Manuscript
The application of a traditional Ilizarov apparatus or Hoffman II external fixator (Fig.1) requires drilling bone fragments through the medullary canal, creating the potential for secondary complications [17]. To avoid drilling, a “needle” Ilizarov apparatus was introduced by Russian military surgeons [2, 3] during the war in Afghanistan, 1979–1989. The apparatus allowed for easy external fixation of multiple fractures during medical evacuation from the battlefield to the hospital. However, the insufficient stability of fixation prevented this apparatus from wide implementation into practice.
Author Manuscript
To increase the stability of fixation while refraining from drilling through the medullary canal, the first prototype of the new double-needle Ilizarov Fixator (DNIF)2 was manufactured with paired needles (Fig.2). The number of base rings, ranging from two to four, depends on the specific site of application and on a patient’s conditions. Two or three double-needle clamps are secured on each ring, and their position relative to the ring can be adjusted in accordance with the needed direction of the corresponding pair of needles. A unicortical penetration of a needle is performed manually. After initial contact of the needle with the bone, the desired depth of the needle’s penetration can be regulated. The current paper presents a model the comparison of the stability of fixation in the external apparatuses. The model uses the method of finite element analysis (FEA). The results have suggested that the stability of fixation in the new DNIF apparatus is greater compared to that of the Hoffman II external fixator.
2. Methods 2.1. Finite Element Analysis of the stability of bone fixation in DNIF and Hoffmann II apparatuses
Author Manuscript
2.1. 1. Materials modeling assumptions—The material for all components of both fixators is assumed to be elastic, isotropic, continuous, and uniform. The material’s characteristics were taken from the specification of stainless steel X18H10T [18]: modulus of normal elasticity – 190 GPa; density – 7800 kg/m3; ultimate stress – 600 MPa; yield strength – 200 MPa. Material’s characteristics for the cancellous bone were as follows: modulus of normal elasticity – 480 MPa; density – 500 kg/m3; ultimate strength – 10 MPa [19]. Material’s characteristics for the compact bone were: modulus of normal elasticity – 16.2 MPa; density – 2000 kg/m3; ultimate strength – 170 MPa [20]. To simplify calculations, the Poisson ratio for all components was taken to be 0.3.
Author Manuscript
2.1.2. Modeling of the tibia—A three-dimensional model of the human tibia was created with computerized tomography (Siemens Somatom Emotion™) in a conventional manner [21]. CT image data of a cadaver tibia were transferred to a computer, and the cross-section of bone for each slice was generated. The model constructed was used as input data for the engineering program SolidWorks 20063. A gap of 50 mm was generated in the middle of the virtual model to simulate a bone fracture. Both distal and proximal fragments of the bone were fixed in the models of the new DNIF and Hoffmann II apparatuses.
2Poly-Orth International, Sharon, MA 3SolidWorks Corporation, Concord, MA
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Author Manuscript
2.1.3. Modeling of the DNIF—A CAD model of the DNIF is shown in Fig.3. The apparatus has two base rings 1 connected with three telescopic bars 4. Each of the rings has three double-needle clamps 2, whose position relative to a ring (40° and 90°) was chosen in accordance with the principles established for the classic Ilizarov method [22]. The needles 3 have diameter of 4 mm, and are 150 mm in length. The length of the conical end penetrating the bone cortical layer is 7 mm. The rings 1 have an inner diameter of 160 mm, and the outer diameter of 190 mm; their thickness is 5 mm. The inner diameter of the telescopic bar 4 is 6 mm. The dimensions of the clamps 2 are 50×51×20 mm and their parallelepiped shape was a simplification of the real shape seen in Fig.2. The distance between the rings is 300 mm.
Author Manuscript
2.1.4. Modeling of Hoffmann II apparatus—Since the Hoffmann II design allows for many different configurations of the components, we selected a configuration close to that of the new DNIF [23]. A CAD model of fixation of the fractured bone fragments in the Hoffmann II external apparatus is shown in Fig.4. Four independent parallelepiped supports are connected with two bars 1 with a diameter of 5 mm, and two mounting arcs 2 with a radius of 70 mm. Two clamps 3 have two pins 4 each. The diameter of the pins is 5 mm, and the distance between the pins installed one clamp is 30 mm. The distance between inner pins belonging to the separate clamps, is 210 mm.
Author Manuscript
2.1.5. Modeling of the external load—We assume that the shank and the foot are suspended in the external fixator with the rings (DNIF) or the parallepiped blocs (Hoffmann II) as the base cantilever supports, the bones have a pin-joint connection and the position of the leg in horizontal. In addition to the load from the bone fragments, the apparatuses’ elements are loaded by the foot and the thigh (Fig.5). We assume also the mass of the patient to be 80 kg and height to be 180 cm. That yields the mass of the foot to be 1.1. kg and of the thigh to be 11.5 kg [24]. The estimate the loading effect from the foot yields the force P = 10N; and the moment M0 = 0.51N·m. The effect of the loading from the thigh is represented by the force P1 = 51N (Fig.6). We assume a rigid connection of the elements of the apparatuses and their combined deformation as a whole. Stress-strain calculations in the elements of the apparatuses were performed with the program CosmosWorks 20054
3. Results
Author Manuscript
Vertical displacement of the points of a cross-section in the fracture zone was determined with FEA modeling (Fig.7). The results of calculating the vertical displacement of the bone fragments and the greatest equivalent stress (Von Mises) in the needles of the DNIF are presented in Table 1. Both characteristics were calculated for two configurations with respect to the number of needles used for fixation: 11 needles (as shown in Fig.3) and 12 needles, and for two levels of the needles’ penetration of the bone: 10 mm and 3.5 mm. In Hoffmann II apparatus (Fig.9), the vertical displacement obtained with the FEA yields u = 4.196·10−4 m (see Table 1). The FEA model is responsive not only to the bone’s displacement, but also to the deformation of the Hoffmann II apparatus’s frame. 4Structural Research & Analysis Corp., Santa Monica, CA
Ser Biomech. Author manuscript; available in PMC 2016 April 14.
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Author Manuscript
Displacement of the cross-section of the tibia in the fracture zone as the function of the force acting from the foot through the external fixators is plotted in Fig.8, where (1) represents the Hoffmann II external fixator; (2) – DNIF with 10 mm penetration by the needles; (3) – DNIF with 3.5 mm penetration by the needles. The stress-strained state of the system “DNIF-bone fragments” with 11 and 12 needles didn’t differ significantly. The greatest equivalent stress in the needles occurs in the outer row for both fragments for both configurations. In a configuration with 10 mm penetration of the needles, their greatest equivalent stress is 9.083 MPa, and doesn’t exceed 4.5% of the dangerous yield strength of the material (200 MPa). That percentage is higher (12%) in a configuration with the 3.5 mm penetration, where the greatest equivalent stress yields 25.57 MPa. The greatest equivalent stress in the bone was about 2 MPa, or approximately 20% of the dangerous ultimate stress for the cancellous bone.
Author Manuscript
In the Hofmann II apparatus, the greatest equivalent stress of 135.8 MPa occurs in bar 1 (Fig. 3). That is about 67% of the dangerous ultimate stress of the metal (200 MPa). The greatest equivalent stress in the bone was about 2 MPa.
4. Discussion
Author Manuscript
In the new double-needle fixator (DNIF), clusters of paired needles replace wires or pins penetrating the bone in Hoffman II fixator. Stability of fixation in both external apparatuses was evaluated using the vertical displacement of the end of a fractured bone fragment. The mathematical modeling allowed for the calculation of the vertical displacement of a bone using finite element analysis. Results of calculations suggest that the stability of fixation in the new device is greater even without the bicortical drilling of a bone fragment which is currently accepted for early treatment of multiple fractures [4]. In modeling the Hoffmann II external fixator we considered a double-bar configuration, which is known to be more stable compared to a single-bar construct [4]. Thus, the conclusion about greater stability of the new fixator was reached using conservative approach.
Author Manuscript
The greater stability of a new double-needle fixator compared to Hoffmann II external fixator can be explained by the circumferential placement of the needles in the DNIF. Another potential advantage of the new fixator is that it is applied to the bone periosteum without opening of the medullary canal. It will potentially combine the advantages of the existing, stable half-pin external fixators and the pinless AO external fixator, which doesn’t penetrate the medullar canal but does not demonstrate sufficient stability of fixation [25, 26]. Further studies are required to verify the stability of fixation and the ease of application with the new double-needle external fixator.
5. Conclusions A mathematical modeling of fixation of a bone fragment in two external apparatuses has been developed. We have considered the vertical displacement of the bone fragment as an indicator of the stability of its fixation in the apparatus. Results of calculations suggest that
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the stress-strained state of the system “Hoffmann II – bone fragments” is similar to the system “DNIF-bone fragments.” The displacement of a bone fragment in Hoffmann II apparatus doesn’t exceed 1 mm, which is of one order greater than in the new DNIF apparatus. Further study is required to verify this theoretical result.
Acknowledgments The funding for the study was provided in part by NIH Grant R43HD04793. The authors thank ARETE Company, St. Petersburg, Russia, and Controlled Atmosphere Welding Technologies, North Attleboro, MA, USA, for prototyping the DNIF.
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Fig.1.
Hoffmann II external fixator
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Fig 2.
The DNIF. A three-part fractured bone model is fixed with one redundant ring and several redundant needles to demonstrate that the apparatus is modular and extendable.
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Fig.3.
Model of fixation of the fractured bone fragments in the DNIF: 1 - base ring; 2 – doubleneedle clamp; 3 – a needle; 4 – telescopic bar.
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Fig. 4.
Model of fixation of the fractured bone fragments in the Hoffmann II external apparatus: 1 – connecting bar; 2 – clamp: 3 – mounting arc: 4 – pin.
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Fig. 5.
External forces applied to the fractured bone fragments from the foot (P) and the thigh (P1). a=5 cm – distance from the foot’s center of mass to the center of the ankle joint.
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Fig.6.
Model of loading of the fractured bone fragments in the Hoffmann external apparatus
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Fig.7.
FEA SolidWorks Model of the DNIF for calculation of the vertical displacement in the fracture zone (configuration with 11 needles).
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Fig.8.
Displacement of the cross-section of tibia in the fracture zone as the function of the force acting from the foot through the external fixators: (1) – Hoffmann II external fixator; (2)– DNIF with 10 mm penetration of the needles; (3) – DNIF with 3.5 mm penetration of the needles.
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Fig 9.
SolidWorks model of the Hoffmann II external fixator for calculation of the vertical displacement of a bone fragment in the fracture zone.
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Table 1
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Displacement of bone fragments and stresses in Hoffmann II fixator and DNIF Type of Fixator
Vertical displacement
Greatest equivalent stress (Von Mises)
Hoffmann II
4.196·10−4m
135.8 MPa
DNIF - 11 needles with 10 mm penetration
4.72·10−5m
9,083 MPa
DNIF - 12 needles with 10 mm penetration
4.78·10−5m
9,072 MPa
DNIF - 11 needles with 3.5 mm penetration
9.8·10−5m
20.16 MPa
DNIF - 12 needles with 3.5 mm penetration
13.9·10−5m
25.57 MPa
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![Universal joint slippage as a cause of Hoffmann half-frame external fixator failure [corrected].](/assets/img/hold.png)
