Computational modeling of bone fracture non-unions: four clinically relevant case studies
© Carlier et al. 2015
Received: 28 October 2015
Accepted: 11 November 2015
Published: 18 December 2015
The human skeleton has a remarkable regeneration capacity. Nevertheless, 5–10 % of the bone fractures fails to heal and develops into a non-union which is a challenging orthopedic complication requiring complex and expensive treatment. This review paper will discuss four different computational models, each capturing a particular clinical case of non-union: non-union induced by reaming of the marrow canal and periosteal stripping, non-union due to a large interfragmentary gap, non-union due to a genetic disorder [i.e. NF1 related congenital pseudoarthrosis of the tibia (CPT)] and non-union due to mechanical overload. Together, the four computational models are able to capture the etiology of a wide range of fracture non-union types and design novel treatment strategies thereof. Further research is required to corroborate the computational models in both animal and human settings and translate them from bench to bed side.
KeywordsComputational model Fracture healing Non-union
In case of injury, the majority of bone fractures can heal without the production of scar tissue. Unfortunately, fracture healing complications, such as delayed and non-unions, are associated with 5–10 % of the over 6 million fractures occurring annually in the USA [1, 2]. Fracture non-unions are challenging orthopedic complications requiring complex and expensive treatment including multiple surgical procedures and prolonged hospital stay [3–5]. As such, the resulting socio-economic burden is significant and rising according to the 2010 Global Burden of Disease study where musculoskeletal disorders accounted for 6–8 % of total disability-adjusted life years (DALYs) .
Although the field of orthopedics has experienced great advancements in the last decades, more knowledge of the complex physiological process of bone healing is a prerequisite for the prevention and effective treatment of complex fractures. (Patient-specific) Computational models have the potential to cope with this complexity. Moreover, computational models can help to integrate all the relevant, patient-specific information into a personalized diagnosis and optimal treatment.
This article will focus on the use of in silico models to improve our fundamental understanding of impaired bone regeneration and to design novel treatment strategies. It will first briefly summarize the biology of bone regeneration, including possible complications and treatment options. Subsequently, the added value of computational models will be illustrated with four different clinical cases of non-unions: non-union induced by reaming of the marrow canal and periosteal stripping, non-union due to a large interfragmentary gap, non-union due to a genetic disorder (i.e. NF1 related congenital pseudoarthrosis of the tibia (CPT)) and non-union due to mechanical overload. Finally, some prospects and conclusions will be formulated.
Biology of bone fracture healing
Normal and impaired bone regeneration
Although bone has a unique restorative capacity, the conditions for spontaneous bone healing are not always present leading to a delayed union or a non-union. The former is generally defined as the eventual bony union after an atypical long period of healing, in comparison to the normal healing period [7, 8]. The latter is characterized by an absence of healing during at least 6–12 months (in humans). The different types of fracture non-unions, i.e. hypertrophic, atrophic or oligotrophic, and synovial pseudarthrosis are classified based on their radiographic and histological appearance [7, 9].
Hypertrophic non-unions are mainly defined by an abnormal vascularity and abundant callus formation. In this type of non-union, the excessive motion at the fracture site prevents the bony bridging although the essential biological factors are present . Atrophic non-unions, however, are the result of inadequate biological conditions. They show little callus formation around the gap, which is mostly filled with fibrous tissue and little or no mineral deposition as evidenced by the blunted bony ends on radiographs . Oligotrophic non-unions combine the radiographic characteristics of atrophic non-unions, i.e. little to no callus formation, with the biological characteristics of hypertrophic non-unions, i.e. normal biological activity . In a synovial pseudarthrosis the fracture gap of the non-union is a fluid-filled cavity with a synovial-like membrane.
Current treatment strategies
Of the classical therapeutic methods currently available to treat non-unions, amputation is the treatment of last resort . Even when other treatment options are available, the choice is complex since at a more phenomenological level the outcome of bone healing depends on many mechanical and biological risk factors, such as excess motion, the interfragmentary gap size , the type of fracture [12–14], the particular bone , location of the trauma within the bone , loss of blood supply . Additional injuries such as severe periosteal and soft-tissue trauma [13, 14] as well as other preexisting patients risk factors including old age , cachexia and malnutrition , immune compromise , genetic disorders (e.g. neurofibromatosis type 1 ), osteoporosis , anticoagulants , smoking  and anti-inflammatory agents , may also compromise the optimal treatment.
Strategies for hypertrophic non-unions traditionally aim to restore the optimal mechanical environment for fracture healing [7, 24]. Enhanced stabilization and progressive compression such as in the Ilizarov method can allow to convert the abnormal fibrous, cartilaginous and adipose tissues between the mobile bony ends into bone without the need to remove them. In case of plate osteosynthesis a debridement is usually performed as the fracture is exposed anyway, whereas in intramedullary nailing the reaming of the canal contributes to the refreshment of the fracture zone. The key to success either with plates, intramedullary nails or external fixators is the reduction of any excessive motion allowing a sound biomechanical condition for bone healing [7, 24–28].
Successful healing of the more challenging (atrophic) non-unions will, however, not only require the removal of scar tissue and adequate stabilization of the fracture but also biological support (e.g. bone grafting, administration of growth factors) . Large quantities of bone are not only required for compromised bone healing processes, they are also indispensable for skeletal reconstructions of large bone defects created by trauma, infection, tumor resection and skeletal abnormalities . A widely used approach to stimulate or augment bone formation is distraction osteogenesis, a clinical procedure where bone regeneration is induced between two gradually distracted bony surfaces [30, 31]. This principle is applied in the bone transport technique, allowing huge defects to be replaced [32, 33]. However, due to the length of the treatment, as well as the technical demands and complications associated with distraction osteogenesis [30, 31], a number of bone grafting methods are more commonly performed in clinical practice when the defect size is not too extensive. Autologous bone grafting, i.e. the process by which bone is harvested from one anatomical site and transplanted to another site in the same patient, is still considered the “gold standard” since it effectively combines the required osteoinductive, osteogenic and osteoconductive properties . It has, however, several limitations which include donor site pain, increased blood loss, increased surgery times, increased risk for donor site infection and limited supply . An even more complex autologous bone transplant is the vascularized bone graft such as the fibula, which can be used to replace long-sized bone defects . Allogeneic bone grafts, where bone is harvested from human cadavers, sterilized and transplanted to the patient, lack donor site morbidity but are expensive and have an increased risk of viral transmission . Moreover, they have very limited biological activity as they are only osteoconductive without any osteogenic capacity and only sporadically a small osteoinductive capacity. As such they are inappropriate to treat atrophic non-unions. The use of synthetic calcium salt-based bone substitutes is sometimes suggested as an alternative to both autologous and allogeneic grafts since they are inexpensive and lack the risks of donor site morbidity and viral transmission . They are, however, only osteoconductive which limits their potential biological role in fracture healing [7, 38] and as such they cannot be recommended as a stand-alone treatment in hampered bone healing. The use of a “Masquelet-membrane” is to be considered as an enhanced bone grafting method which consists of two steps. In the first stage a polymethylmetacrylate cement spacer is placed in the defect which induces the formation of a vascularized membrane. In the second stage a non-vascularized graft will be inserted in the newly formed vascularized envelope which serves as a source of oxygen, nutrients and a cocktail of important growth factors [39, 40]. Besides bone grafting also bioactive molecules have been used to augment fracture healing. BMP-2 and BMP-7 have been shown to have significant osteogenic and angiogenic properties, which has led to their use in a variety of clinical conditions including non-unions, open fractures and joint fusions [29, 41]. An alternative approach is the local application of platelet-rich plasma, which is rich in many of the growth factors implicated in bone regeneration . Another promising strategy, which could potentially overcome the limitations of current bone regeneration therapies, is tissue engineering where an optimal bone healing microenvironment is created by seeding cells (osteogenesis) and growth factors (osteoinduction) on biocompatible scaffolds (osteoconduction) that will be implanted in a mechanically stabilized defect .
In silico modeling of bone fracture non-unions
Although the four cases-studies are based on the same general framework, there are some important differences which are detailed below and summarized in Fig. 3. Firstly, only case-study IV includes the role of mechanical influences on the healing processes for which the finite element analyses were carried out in MSC.Marc Mentat (Version 2005r2, Palo Alto, USA). The bioregulatory model of all case studies is solved in a custom finite volume code using Matlab (The MathWorks, Inc.). Secondly, case studies I and IV use a continuous description of bone regeneration (tissue level), including a continuous vascular density, whereas case studies II and III use a multi-scale description that combines a continuous tissue-level with a discrete cellular level (i.e. the vascular tree) and a detailed intracellular signaling network (i.e. Dll4-Notch signaling in the endothelial cells). Thirdly, case studies I and IV discriminate between a generic osteogenic (g b ) and chondrogenic (g c ) growth factor, whereas case studies II and III include one generic osteochondrogenic (g bc ) growth factor, whose influence of differentiation is either chondrogenic or osteogenic depending on the local oxygen tension. A complete description of the set of equations, the boundary and initial conditions, the parameter values, implementation details as well as some underlying assumptions and simplifications can be found in previous publications of the respective cases: case I , case II , case III , case IV .
Periosteal and endosteal injury
Non-union due to large interfragmentary gap
Non-union due to a genetic disorder
Clearly, this type of approach can give interesting insights into the varieties in phenotypes that can result from defects in a single gene. Even though at this moment it is not feasible to link patient-specific characteristics to a particular combination of model parameters, valuable information can be obtained at the population level. Currently we are testing treatments involving BMP administration in the entire simulated population in order to understand the cause of the highly variable outcome of this treatment reported in the literature [60–62]. As such, this case-study represents an important step towards real in silico clinical trials, where it is envisioned that computer models would be able to predict the efficacy of a “virtual” treatment on a group of “virtual” patients . Such in silico clinical trials could help to refine and reduce the size and duration of real clinical trials through better a design. Moreover, they have the potential to complement and replace real clinical trials, which would be particularly promising for orphan diseases (such as NF1-associated CPT), where only small cohorts of patients are available for clinical investigation, and for pediatric diseases where clinical trials are nearly impossible for ethical reasons.
Non-union due to mechanical overload
The above examples clearly illustrate that in silico techniques are able to investigate the etiology of a wide range of fracture non-union types and design novel treatment strategies thereof. More specifically, the first case-study demonstrated that the computational model can capture the essential aspects of an in vivo atrophic non-union and can help to explain and optimize experimental treatments, i.e. the location of the injection of a cell transplant. Similarly, the predictions of the second case-study showed that the effectiveness of a therapy, consisting of a single injection of osteochondrogenic growth factors, cells or a combination thereof in a large segmental bone defect, is strongly influenced by the (patient-specific) host environment and by the timing of injection. Moreover, case-study four evidenced that, depending on the dominating biology-mechanics interactions that are implemented in the computational model, different treatment strategies are required for the restoration of normal healing. As such, the results of these three case-studies clearly underline the need for patient-dependent modeling. However, at this moment it is not (yet) feasible to link patient-specific characteristics to a particular combination of model parameters (an in depth discussion of the key challenges associated with patient-specific modeling can be found in ). Nevertheless, case-study three nicely shows that the current approach can give interesting insights at the population level and represents an important first step towards in silico clinical trials.
The current computational framework has some limitations and the interpretation of the results should therefore be done carefully due to the following reasons. Firstly, the computational model only includes the repair phases of fracture healing, i.e. the soft and hard callus phase, and neglects the early inflammatory response and the bone remodeling phase. Other fracture healing models have accounted for the remodeling phase such as the work of Gómez-Benito et al. , Burke et al. , Byrne et al.  and Shefelbine et al. . Moreover, in the literature also detailed models exist of the bone remodeling process itself, including the studies of Ryser et al. [70, 71] and Buenzli et al. [72–74]. However, none of the state-of-the-art fracture healing models, to the author’s best knowledge, captures the inflammatory phase.
Secondly, the presented computational framework requires a simplified and fixed geometrical domain of a fracture callus (Fig. 3). As such, the implementation cannot account for tissue growth during callus formation, although this can be captured in the frameworks of Chen et al. , Simon et al.  and Gomez-Benito et al. . Moreover, the computational framework only allows 2D or 2D-axisymmetric calculations while others have simulated the regeneration processes in the inter-cortical region in 3D [68, 69, 77].
Thirdly, the computational model is based on experimental data from mouse models since these small animal models are increasingly used in bone healing studies due to their less expensive housing, shorter breeding cycles, well-defined genetic background and available (genetic) methods to study particular molecular mechanisms of action . However, rodents have a more primitive bone structure without a Haversian system and use resorption cavities for bone remodeling, which is different from large animals and adults . Moreover, in order to correctly mimic fracture healing in adults, animals of an age with completed bone growth should be used . Given that the computational model is corroborated with experimental data from mouse models, it is important to keep in mind the differences that exist between murine and human bone healing when extrapolating these findings to a clinical setting. Note that others have used ovine models to explore the predictive power of computational models of bone healing. Moore et al. report for example that the histological measures (amongst others Giemsa-Eosin staining and fluorochrome microscopy) match the predicted gradients in BMP, cells and tissue fractions over time in an ovine critical size defect model . Similarly, Chen et al. corroborate their predictions on the induction of non-unions in large gap sized and different mechanical conditions with experimental results obtained from an osteotomized ovine metatarsus .
Despite the above mentioned advances, several steps need to be taken in order to bring in silico models from bench to bed side . These steps include, amongst others, the establishment of patient-specific models as well as their corroboration in both small (e.g. mice) and large (e.g. sheep) animal models and a limited number of patient-specific study cases. Although this road to translation is challenging, we believe that it is important to focus future research efforts to overcome these challenges so that computational models of bone fracture healing are not only used as research tools in the experimental research phase but also aid in the advancement of individualized care and reduction of the associated health care costs.
In case of injury, the majority of bone fractures can heal without the production of scar tissue. Unfortunately, 5–10 % of the bone fractures fails to heal and develops into a non-union. This review illustrated the potential of computational models of fracture healing in contributing to a more profound understanding of the etiology and treatment of fracture non-unions. Four different cases of non-unions were discussed: non-union induced by periosteal and endosteal injury, non-union due to a large interfragmentary gap, non-union due to a genetic disorder (i.e. NF1 related congenital pseudoarthrosis of the tibia (CPT)) and non-union due to mechanical overload. Clearly, a treatment will be most beneficial if it tackles the underlying mechanism of action causing the hampered bone formation. The underlying mechanisms of action are, however, the result of complex non-linear biological and mechanical interactions occurring at various temporal and spatial scales. As such, a rigorous approach where in vivo and in silico methods work in tandem, are essential to deepen our fundamental understanding of (impaired) bone regeneration, to corroborate the existing computational models and to bring novel treatment strategies for challenging orthopedic cases from bench to bed side.
AC, JL, HVO and LG helped to draft the manuscript. AC wrote the manuscript. All authors read and approved the final manuscript.
LG and HVO acknowledge the European Research Council under the European Union's Seventh Framework Program ERC grant agreement numbers 279100 and 308223 respectively. This work has also been funded by BOF-KU Leuven GOA project 3M120209 and IWT 110702. AC is a post-doctoral fellow of the Research Foundation Flanders (FWO-Vlaanderen). This work is part of Prometheus, the Leuven R&D Division of Skeletal Tissue Engineering of the KU Leuven.
The authors declare that they have no competing interests.
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- Einhorn TA. The cell and molecular biology of fracture healing. Clin Orthop Relat Res. 1998;355:S7–21.View ArticleGoogle Scholar
- Bhandari M, Jain AK. Bone stimulators: beyond the black box. Indian J Orthop. 2009;43(2):109–10.View ArticleGoogle Scholar
- Ashman O, Phillips AM. Treatment of non-unions with bone defects: which option and why? Injury Int J Care Injured. 2013;44:S43–5.View ArticleGoogle Scholar
- Giannoudis PV, Atkins RM. Management of long-bone non-unions. Injury Int J Care Injured. 2007;38:S1–2.Google Scholar
- Giannoudis PV, Kontakis G. Treatment of long bone aseptic non-unions: monotherapy or polytherapy? Injury Int J Care Injured. 2009;40(10):1021–2.View ArticleGoogle Scholar
- Murray CJL, Vos T, Lozano R, Naghavi M, Flaxman AD, Michaud C, et al. Disability-adjusted life years (DALYs) for 291 diseases and injuries in 21 regions, 1990–2010: a systematic analysis for the Global Burden of Disease Study 2010. Lancet. 2012;380(9859):2197–223.View ArticleGoogle Scholar
- Roberts TT, Rosenbaum AJ. Bone grafts, bone substitutes and orthobiologics The bridge between basic science and clinical advancements in fracture healing. Organogenesis. 2012;8(4):114–24.View ArticleGoogle Scholar
- Marsh D. Concepts of fracture union, delayed union, and nonunion. Clin Orthop Relat Res. 1998;355(Suppl):S22–S30.Google Scholar
- Rodriguez-Merchan EC, Forriol F. Nonunion: general principles and experimental data. Clin Orthop Relat Res. 2004;419:4–12.View ArticleGoogle Scholar
- Bishop JA, Palanca AA, Bellino MJ, Lowenberg DW. Assessment of compromised fracture healing. J Am Acad Orthop Surg. 2012;20(5):273–82.View ArticleGoogle Scholar
- Cameron J, Milner D, Lee J, Cheng J, Fang N, Jasiuk I. Employing the biology of successful fracture repair to heal critical size bone defects. In: Heber-Katz E, Stocum DL, editors. New perspectives in regeneration. 367th ed. Berlin Hidelberg: Springer; 2013. p. 113–32.Google Scholar
- Fong K, Truong V, Foote CJ, Petrisor B, Williams D, Ristevski B, et al. (2013) Predictors of nonunion and reoperation in patients with fractures of the tibia: an observational study. BMC Musculoskel Dis.14.Google Scholar
- Krettek C, Schandelmaier P, Tscherne H. Nonreamed interlocking nailing of closed tibial fractures with severe soft-tissue injury. Clin Orthop Relat Res. 1995;315:34–47.Google Scholar
- Cross WW, Swiontkowski MF. Treatment principles in the management of open fractures. Indian J Orthop. 2008;42(4):377–86.View ArticleGoogle Scholar
- Dickson KF, Katzman S, Paiement G. The importance of the blood supply in the healing of tibial fractures. Contemp Orthop. 1995;30(6):489–93.Google Scholar
- Bak B, Andreassen TT. The effect of aging on fracture healing in the rat. Calcif Tissue Int. 1989;45(5):292–7.View ArticleGoogle Scholar
- Day SM, DeHeer DH. Reversal of the detrimental effects of chronic protein malnutrition on long bone fracture healing. J Orthop Trauma. 2001;15(1):47–53.View ArticleGoogle Scholar
- Claes L, Recknagel S, Ignatius A. Fracture healing under healthy and inflammatory conditions. Nat Rev Rheumatol. 2012;8(3):133–43.View ArticleGoogle Scholar
- Schindeler A, Little DG. Recent insights into bone development, homeostasis, and repair in type 1 neurofibromatosis (NFI). Bone. 2008;42(4):616–22.View ArticleGoogle Scholar
- Nikolaou VS, Efstathopoulos N, Kontakis G, Kanakaris NK, Giannoudis PV. The influence of osteoporosis in femoral fracture healing time. Injury. 2009;40(6):663–8.View ArticleGoogle Scholar
- Stinchfield FE, Sankaran B, Samilson R. The Effect of Anticoagulant Therapy on Bone Repair. J Bone Joint Surg Am. 1956;38(2):270–82.Google Scholar
- Scolaro JA, Schenker ML, Yannascoli S, Baldwin K, Mehta S, Ahn J. Cigarette Smoking Increases Complications Following Fracture. J Bone Joint Surg Am. 2014;96A(8):674–81.View ArticleGoogle Scholar
- Altman RD, Latta LL, Keer R, Renfree K, Hornicek FJ, Banovac K. Effect of nonsteroidal antiinflammatory drugs on fracture healing: a laboratory study in rats. J Orthop Trauma. 1995;9(5):392–400.View ArticleGoogle Scholar
- Dujardyn J, Lammens J. Treatment of delayed union or non-union of the tibial shaft with partial fibulectomy and an Ilizarov frame. Acta Orthop Belg. 2007;73(5):630–4.Google Scholar
- Miller DL, Goswami T. A review of locking compression plate biomechanics and their advantages as internal fixators in fracture healing. Clin Biomech (Bristol, Avon). 2007;22(10):1049–62.View ArticleGoogle Scholar
- Patel VR, Menon DK, Pool RD, Simonis RB. Nonunion of the humerus after failure of surgical treatment. Management using the Ilizarov circular fixator. J Bone Joint Surg Br. 2000;82(7):977–83.View ArticleGoogle Scholar
- Olson S, Hahn D. Surgical treatment of non-unions: a case for internal fixation. Injury. 2006;37(8):681–90.View ArticleGoogle Scholar
- Sledge SL, Johnson KD, Henley MB, Watson JT. Intramedullary nailing with reaming to treat non-union of the Tibia. J Bone Joint Surg Am. 1989;71A(7):1004–19.Google Scholar
- Dimitriou R, Jones E, McGonagle D, Giannoudis PV. Bone regeneration: current concepts and future directions. BMC Med. 2011;9:66.View ArticleGoogle Scholar
- Aronson J. Limb-lengthening, skeletal reconstruction, and bone transport with the Ilizarov method. J Bone Joint Surg Am. 1997;79(8):1243–58.Google Scholar
- Green SA, Jackson JM, Wall DM, Marinow H, Ishkanian J. Management of segmental defects by the Ilizarov intercalary bone transport method. Clin Orthop Relat Res. 1992;280:136–42.Google Scholar
- Lavini F, Dall’Oca C, Bartolozzi P. Bone transport and compression-distraction in the treatment of bone loss of the lower limbs. Injury Int J Care Injured. 2010;41(11):1191–5.View ArticleGoogle Scholar
- Mekhail AO, Abraham E, Gruber B, Gonzalez M. Bone transport in the management of posttraumatic bone defects in the lower extremity. J Trauma Injury Infect Critic Care. 2004;56(2):368–78.View ArticleGoogle Scholar
- Bauer TW, Muschler GF. Bone graft materials. An overview of the basic science. Clin Orthop Relat Res. 2000;371:10–27.View ArticleGoogle Scholar
- Pape HC, Evans A, Kobbe P. Autologous bone graft: properties and techniques. J Orthop Trauma. 2010;24:S36–40.View ArticleGoogle Scholar
- Gao YS, Ai ZS, Yu XW, Sheng JG, Jin DX, Chen SB, et al. Free vascularised fibular grafting combined with a locking plate for massive bone defects in the lower limbs: a retrospective analysis of fibular hypertrophy in 18 cases. Injury Int J Care Injured. 2012;43(7):1090–5.View ArticleGoogle Scholar
- Yuan H, Fernandes H, Habibovic P, de BJ, Barradas AM, de RA, et al. (2010) Osteoinductive ceramics as a synthetic alternative to autologous bone grafting. Proc Natl Acad Sci USA. 107(31):13614–9.Google Scholar
- Chai YC, Carlier A, Bolander J, Roberts SJ, Geris L, Schrooten J, et al. (2012) Current Views on Calcium Phosphate Osteogenicity and the Translation into Effective Bone Regeneration Strategies. Acta Biomater.Google Scholar
- Giannoudis PV, Faour O, Goff T, Kanakaris N, Dimitriou R. Masquelet technique for the treatment of bone defects: tips-tricks and future directions. Injury Int J Care Injured. 2011;42(6):591–8.View ArticleGoogle Scholar
- Masquelet AC, Begue T. The concept of induced membrane for reconstruction of long bone defects. Orthop Clin North Am. 2010;41(1):27–37.View ArticleGoogle Scholar
- Lissenberg-Thunnissen SN, de Gorter DJ, Sier CF, Schipper IB. Use and efficacy of bone morphogenetic proteins in fracture healing. Int Orthop. 2011;35(9):1271–80.View ArticleGoogle Scholar
- Alsousou J, Thompson M, Hulley P, Noble A, Willett K. The biology of platelet-rich plasma and its application in trauma and orthopaedic surgery: a review of the literature. J Bone Joint Surg Br. 2009;91(8):987–96.View ArticleGoogle Scholar
- Lammens J, Laumen A, Delport H, Vanlauwe J. The Pentaconcept in skeletal tissue engineering. A combined approach for the repair of bone defects. Acta Orthop Belg. 2012;78(5):569–73.Google Scholar
- Isaksson H. Recent advances in mechanobiological modeling of bone regeneration. Mech Res Commun. 2012;42:22–31.View ArticleGoogle Scholar
- Pivonka P, Dunstan CR. Role of mathematical modeling in bone fracture healing. Bone Key Rep. 2012; 1.Google Scholar
- Geris L, Gerisch A, Schugart RC. Mathematical modeling in wound healing, bone regeneration and tissue engineering. Acta Biotheor. 2010;58(4):355–67.View ArticleGoogle Scholar
- Geris L. Regenerative orthopaedics: in vitro, in vivo and in silico. Int Orthop (SICOT). 2014;38(9):1771–8.View ArticleGoogle Scholar
- Geris L, Reed AA, Vander SJ, Simpson AH, Van Oosterwyck H. Occurrence and treatment of bone atrophic non-unions investigated by an integrative approach. PLoS Comput Biol. 2010;6(9):e1000915.View ArticleGoogle Scholar
- Carlier A, van GN, Geris L, Carmeliet G, Van Oosterwyck H. (2014) Size does matter: an integrative in vivo-in silico approach for the treatment of critical size bone defects. PLoS Comput Biol. 10(11): e1003888.Google Scholar
- Carlier A, Brems H, Ashbourn JMA, Nica I, Legius E, Geris L. Capturing the wide variety of impaired fracture healing phenotypes in Neurofibromatosis Type 1 with eight key factors: a computational study. Scientific Reports (accepted) Google Scholar
- Geris L, Sloten JV, Van Oosterwyck H. Connecting biology and mechanics in fracture healing: an integrated mathematical modeling framework for the study of nonunions. Biomech Model Mechanobiol. 2010;9(6):713–24.View ArticleGoogle Scholar
- Reed AAC, Joyner CJ, Isefuku S, Brownlow HC, Simpson AHRW. Vascularity in a new model of atrophic nonunion. J Bone Joint Surg Br. 2003;85B(4):604–10.View ArticleGoogle Scholar
- Carlier A, van GN, Geris L, Carmeliet G, Van Oosterwyck H. Size does matter: an integrative in vivo-in silico approach for the treatment of critical size bone defects. PLoS Comput Biol. 2014;10(11):e1003888.Google Scholar
- Srouji S, Ben-David D, Kohler T, Muller R, Zussman E, Livne E. A model for tissue Engineering applications: femoral critical size defect in immunodeficient mice. Tissue Eng Part C Methods. 2011;17(5):597–606.View ArticleGoogle Scholar
- Zwingenberger S, Niederlohmann E, Vater C, Rammelt S, Matthys R, Bernhardt R, et al. Establishment of a femoral critical-size bone defect model in immunodeficient mice. J Surg Res. 2013;181(1):E7–14.View ArticleGoogle Scholar
- Manassero M, Viateau V, Matthys R, Deschepper M, Vallefuoco R, Bensidhoum M, et al. A novel murine femoral segmental critical-sized defect model stabilized by plate osteosynthesis for bone tissue engineering purposes. Tissue Eng Part C Methods. 2013;19(4):271–80.View ArticleGoogle Scholar
- Harris JS, Bemenderfer TB, Wessel AR, Kacena MA. A review of mouse critical size defect models in weight bearing bones. Bone. 2013;55(1):241–7.View ArticleGoogle Scholar
- Horn J, Steen H, Terjesen T. Epidemiology and treatment outcome of congenital pseudarthrosis of the tibia. J Child Orthop. 2013;7(2):157–66.View ArticleGoogle Scholar
- Pannier S. Congenital pseudarthrosis of the tibia. Orthop Traumatol Surg Res. 2011;97(7):750–61.View ArticleGoogle Scholar
- Lee FYI, Sinicropi SM, Lee FS, Vitale MG, Choi IH. Treatment of congenital pseudarthrosis of the tibia with recombinant human bone morphogenetic protein-7 (rhBMP-7)—A report of five cases. J Bone Joint Surg Am. 2006;88A(3):627–33.View ArticleGoogle Scholar
- Fabeck L, Ghafil D, Gerroudj M, Baillon R, Delince P. Bone morphogenetic protein 7 in the treatment of congenital pseudarthrosis of the tibia. J Bone Joint Surg Br. 2006;88B(1):116–8.View ArticleGoogle Scholar
- Anticevic D, Jelic M, Vukicevic S. Treatment of a congenital pseudarthrosis of the tibia by osteogenic protein-1 (bone morphogenetic protein-7): a case report. J Pediatric Orthop Part B. 2006;15(3):220–1.View ArticleGoogle Scholar
- Viceconti M, Morley-Fletcher E, Henney A, Contin M, El-Arifi K, M, et al. A Strategy for in silico Clinical Trials. 2015. Report No.: 611819.Google Scholar
- Geris L, Schugart R, Van Oosterwyck H. In silico design of treatment strategies in wound healing and bone fracture healing. Philos Transact A Math Phys Eng Sci. 2010;368(1920):2683–706.View ArticleGoogle Scholar
- Carlier A, Geris L, Lammens J, Van Oosterwyck H. Bringing computational models of bone regeneration to the clinic. WIREs Syst Biol Med. 2015;7:183–94. doi:10.1002/wsbm.1299.View ArticleGoogle Scholar
- Gomez-Benito MJ, Garcia-Aznar JM, Kuiper JH, Doblare M. Influence of fracture gap size on the pattern of long bone healing: a computational study. J Theor Biol. 2005;235(1):105–19.View ArticleMathSciNetGoogle Scholar
- Burke DP, Kelly DJ. Substrate stiffness and oxygen as regulators of stem cell differentiation during skeletal tissue regeneration: a mechanobiological model. PLoS One. 2012;7(7):e40737.View ArticleGoogle Scholar
- Byrne DP, Lacroix D, Prendergast PJ. Simulation of fracture healing in the tibia: mechanoregulation of cell activity using a lattice modeling approach. J Orthop Res. 2011;29(10):1496–503.View ArticleGoogle Scholar
- Shefelbine SJ, Augat P, Claes L, Simon U. Trabecular bone fracture healing simulation with finite element analysis and fuzzy logic. J Biomech. 2005;38(12):2440–50.View ArticleGoogle Scholar
- Ryser MD, Komarova SV, Nigam N. The cellular dynamics of bone remodeling: a mathematical model. Siam J Appl Math. 2010;70(6):1899–921.MATHView ArticleMathSciNetGoogle Scholar
- Ryser MD, Nigam N, Komarova SV. Mathematical modeling of spatio-temporal dynamics of a single bone multicellular unit. J Bone Miner Res. 2009;24(5):860–70.View ArticleGoogle Scholar
- Buenzli PR, Pivonka P, Gardiner BS, Smith DW. Modelling the anabolic response of bone using a cell population model. J Theor Biol. 2012;307:42–52.View ArticleMathSciNetGoogle Scholar
- Buenzli PR, Jeon J, Pivonka P, Smith DW, Cummings PT. Investigation of bone resorption within a cortical basic multicellular unit using a lattice-based computational model. Bone. 2012;50(1):378–89.View ArticleGoogle Scholar
- Buenzli PR, Pivonka P, Smith DW. Spatio-temporal structure of cell distribution in cortical Bone Multicellular Units: a mathematical model. Bone. 2011;48(4):918–26.View ArticleGoogle Scholar
- Chen G, Niemeyer F, Wehner T, Simon U, Schuetz MA, Pearcy MJ, et al. Simulation of the nutrient supply in fracture healing. J Biomech. 2009;42(15):2575–83.View ArticleGoogle Scholar
- Simon U, Augat P, Utz M, Claes L. A numerical model of the fracture healing process that describes tissue development and revascularisation. Comput Methods Biomech Biomed Engin. 2011;14(1):79–93.View ArticleGoogle Scholar
- Wehner T, Claes L, Niemeyer F, Nolte D, Simon U. Influence of the fixation stability on the healing time–a numerical study of a patient-specific fracture healing process. Clin Biomech (Bristol, Avon). 2010;25(6):606–12.View ArticleGoogle Scholar
- Histing T, Garcia P, Holstein JH, Klein M, Matthys R, Nuetzi R, et al. Small animal bone healing models: standards, tips, and pitfalls results of a consensus meeting. Bone. 2011;49(4):591–9.View ArticleGoogle Scholar
- Holstein JH, Garcia P, Histing T, Kristen A, Scheuer C, Menger MD, et al. Advances in the establishment of defined mouse models for the study of fracture healing and bone regeneration. J Orthop Trauma. 2009;23(5):S31–8.View ArticleGoogle Scholar
- Moore SR, Saidel GM, Knothe U, Knothe Tate ML. Mechanistic, mathematical model to predict the dynamics of tissue genesis in bone defects via mechanical feedback and mediation of biochemical factors. PLoS Comput Biol. 2014;10(6):e1003604.View ArticleGoogle Scholar