Fracture Gap Size Research Articles

Biomechanical analysis of anterior ring fixation of the ramus in type C pelvis fractures

This biomechanical study compared the stability of four different ramus fracture fixation methods for Type C pelvic ring injuries in the absence of posterior fixation. A 5-mm vertical osteotomy of the mid-superior and inferior pubic ramus was created in 12 synthetic pelvic models. Four surgical constructs were compared: (1) two-pin AIIS external fixation, (2) 3.5-mm reconstruction plating, (3) bicortical, fully threaded 3.5-mm, and (4) 6.5-mm pubic ramus screws. Specimens were tested in a simulated one-legged stance on a hemiarthroplasty implant in three stages: (1) no applied load, (2) application of the loading fixture preload to the sacrum (6N), and (3) following six cycles of a 250N load. Stability was assessed based on resultant displacement of the fracture sites at the superior ramus and the anterior sacroiliac joint. The bicortical, fully threaded 6.5-mm pubic ramus screw provided the most stable ramus fracture fixation (0.5 ± 0.4mm) displacement under load and was the only construct to finish testing without gross posterior pelvic disruption. Plate constructs finished the final loading stage with only a small increase (3.1 ± 2.3mm) in ramus fracture gap size, but had significant displacement at the SI joint (>20mm). 3.5-mm screw constructs had 1.6 ± 0.7mm of ramus displacement in the preload stage, but had complete posterior pelvic disruption (>20mm) that prevented further testing. External fixation was unstable at the ramus and sacroiliac sites in the initial setup. The bicortical, fully threaded 6.5-mm pubic ramus screw was the only anterior fixation construct tested that controlled motion at both the anterior and posterior pelvic rings in the absence of posterior fixation.

Comparison of 2-Dimensional and 3-Dimensional Metacarpal Fracture Plating Constructs Under Cyclic Loading

Metacarpal fractures are commonly treated by a variety of means including casting or open reduction internal fixation when unacceptable alignment is present following attempted closed reduction. Dorsal plating with either single-row 2-dimensional or double-row 3-dimensional plates has been proposed. This study's purpose was to determine if there are any differences in fixation construct stability under cyclic loading and subsequent load to failure between the lower profile 3-dimensional and the larger 2-dimensional plates in a metacarpal fracture gap sawbone model. Thirty metacarpal cortico-cancellous synthetic bones were cut with a 1.75-mm gap between the 2 fragments simulating mid-diaphyseal fracture comminution. Half of the metacarpals were plated with 2.0-mm locking 2-dimensional plates and half with 1.5-mm locking 3-dimensional plates. The plated metacarpals were mounted into a materials testing apparatus and cyclically loaded under cantilever bending for 2,000 cycles at 70 N, then 2,000 cycles at 120 N, and finally monotonically loaded to failure. Throughout testing, fracture gap sizes were measured, failure modes were recorded, and construct strength and stiffness values were calculated. All 3-dimensional constructs survived both cyclic loading conditions. Ten (67%) 2-dimensional constructs survived both loading conditions, whereas 5 (33%) failed the 120-N loading at 1377 ± 363 cycles. When loaded to failure, the 3-dimensional constructs failed at 265 N ± 21 N, whereas the 2-dimensional constructs surviving cyclic loading failed at 190 N ± 17 N. The shorter, thinner 3-dimensional metacarpal plates demonstrated increased resistance to failure in a cyclic loading model and increased load to failure compared with the relatively longer, thicker 2-dimensional metacarpal plates. The lower-profile 3-dimensional metacarpal plate fixation demonstrated greater stability for early postoperative resistance than the thicker 2-dimensional fixation, whereas the smaller size and lower profile of the 3-dimensional plates potentially reduces soft tissue stripping, implant prominence, and risk of extensor tendon irritation.

Does Lumbopelvic Fixation Add Stability? A Cadaveric Biomechanical Analysis of an Unstable Pelvic Fracture Model.

We sought to determine the role of lumbopelvic fixation (LPF) in the treatment of zone II sacral fractures with varying levels of sacral comminution combined with anterior pelvic ring (PR) instability. We also sought to determine the proximal extent of LPF necessary for adequate stabilization and the role of LPF in complex sacral fractures when only 1 transiliac-transsacral (TI-TS) screw is feasible. Fifteen L4 to pelvis fresh-frozen cadaveric specimens were tested intact in flexion-extension (FE) and axial rotation (AR) in a bilateral stance gliding hip model. Two comminution severities were simulated through the sacral foramen using an oscillating saw, with either a single vertical fracture (small gap, 1 mm) or 2 vertical fractures 10 mm apart with the intermediary bone removed (large gap). We assessed sacral fracture zone (SZ), PR, and total lumbopelvic (TL) stability during FE and AR. The following variables were tested: (1) presence of transverse cross-connector, (2) presence of anterior plate, (3) extent of LPF (L4 vs. L5), (4) fracture gap size (small vs. large), (5) number of TI-TS screws (1 vs. 2). The transverse cross-connector and anterior plate significantly increased PR stability during AR (P = 0.02 and P = 0.01, respectively). Increased sacral comminution significantly affected SZ stability during FE (P = 0.01). Two versus 1 TI-TS screw in a large-gap model significantly affected TL stability (P = 0.04) and trended toward increased SZ stabilization during FE (P = 0.08). Addition of LPF (L4 and L5) significantly improved SZ and TL stability during AR and FE (P < 0.05). LPF in combination with TI-TS screws resulted in the least amount of motion across all 3 zones (SZ, PR, and TL) compared with all other constructs in both small-gap and large-gap models. The role of LPF in the treatment of complex sacral fractures is supported, especially in the setting of sacral comminution. LPF with proximal fixation at L4 in a hybrid approach might be needed in highly comminuted cases and when only 1 TI-TS screw is feasible to obtain maximum biomechanical support across the fracture zone.

Influence of fracture geometry on bone healing under locking plate fixations: A comparison between oblique and transverse tibial fractures

Mechano-regulation plays a crucial role in bone healing and involves complex cellular events. In this study, we investigate the change of mechanical microenvironment of stem cells within early fracture callus as a result of the change of fracture obliquity, gap size and fixation configuration using mechanical testing in conjunction with computational modelling. The research outcomes show that angle of obliquity (θ) has significant effects on interfragmentary movement (IFM) which influences mechanical microenvironment of the callus cells. Axial IFM at near cortex of fracture decreases with θ, while shear IFM significantly increases with θ. While a large θ can increase shear IFM by four-fold compared to transverse fracture, it also result in the tension–stress effect at near cortex of fracture callus. In addition, mechanical stimuli for cell differentiation within the callus are found to be strongly negatively correlated to angle of obliquity and gap size. It is also shown that a relatively flexible fixation could enhance callus formation in presence of a large gap but could lead to excessive callus strain and interstitial fluid flow when a small transverse fracture gap is present. In conclusion, there appears to be an optimal fixation configuration for a given angle of obliquity and gap size.

The relationship between interfragmentary movement and cell differentiation in early fracture healing under locking plate fixation.

Interfragmentary movement (IFM) at the fracture site plays an important role in fracture healing, particularly during its early stage, via influencing the mechanical microenvironment of mesenchymal stem cells within the fracture callus. However, the effect of changes in IFM resulting from the changes in the configuration of locking plate fixation on cell differentiation has not yet been fully understood. In this study, mechanical experiments on surrogate tibia specimens, manufactured from specially formulated polyurethane, were conducted to investigate changes in IFM of fractures under various locking plate fixation configurations and loading magnitudes. The effect of the observed IFM on callus cell differentiation was then further studied using computational simulation. We found that during the early stage, cell differentiation in the fracture callus is highly influenced by fracture gap size and IFM, which in turn, is highly sensitive to locking plate fixation configuration. The computational model predicted that a small gap size (e.g. 1 mm) under a relatively flexible configuration of locking plate fixation (larger bone-plate distances and working lengths) could experience excessive strain and fluid flow within the fracture site, resulting in excessive fibrous tissue differentiation and delayed healing. By contrast, a relatively flexible configuration of locking plate fixation was predicted to improve cartilaginous callus formation and bone healing for a relatively larger gap size (e.g. 3 mm). If further confirmed by animal and human studies, the research outcome of this paper may have implications for orthopaedic surgeons in optimising the application of locking plate fixations for fractures in clinical practice.

Bone Mesenchymal Stem Cells with Growth Factors Successfully Treat Nonunions and Delayed Unions.

While the gold standard of treatment of nonunion is open autologous bone grafting, studies have shown that injecting bone marrow aspirate concentrates (BMAC) is effective in treating tibial nonunions with fracture gaps less than 5mm. We aim to demonstrate that combining BMAC with osteoinductive agents can effectively treat delayed or nonunion regardless of fracture gap size, nonunion site, or osteoinductive agent used. In this non-randomized retrospective-prospective cohort study, 49 patients with tibial nonunion met the inclusion criteria and underwent BMAC injection with demineralized bone matrix (DBM) and/or recombinant human bone morphogenic protein-2 (rhBMP-2). Radiologic healing of the fracture was the primary outcome. Patients were followed until radiographic union was achieved or another procedure was performed. Radiographic healing was defined as bridging of three out of four cortices on anteroposterior and lateral films. There was no difference in the healing rate (p = 0.81) between patients with fracture gaps less than and greater than 5mm. On multivariate analysis, the use of rhBMP-2 was associated with a lower healing rate compared to DBM (p = 0.036). Patients who underwent early intervention (within 6months of fixation) had higher union rates (p = 0.04). This study shows that percutaneous BMAC injection combined with either DBM and/or rhBMP-2 is a safe and effective treatment for delayed or nonunion regardless of the fracture gap size or fracture site. DBM may be superior to rhBMP-2 in this procedure.

Prediction of the Time Course of Callus Stiffness as a Function of Mechanical Parameters in Experimental Rat Fracture Healing Studies - A Numerical Study

Numerous experimental fracture healing studies are performed on rats, in which different experimental, mechanical parameters are applied, thereby prohibiting direct comparison between each other. Numerical fracture healing simulation models are able to predict courses of fracture healing and offer support for pre-planning animal experiments and for post-hoc comparison between outcomes of different in vivo studies. The aims of this study are to adapt a pre-existing fracture healing simulation algorithm for sheep and humans to the rat, to corroborate it using the data of numerous different rat experiments, and to provide healing predictions for future rat experiments. First, material properties of different tissue types involved were adjusted by comparing experimentally measured callus stiffness to respective simulated values obtained in three finite element (FE) models. This yielded values for Young’s moduli of cortical bone, woven bone, cartilage, and connective tissue of 15,750 MPa, 1,000 MPa, 5 MPa, and 1 MPa, respectively. Next, thresholds in the underlying mechanoregulatory tissue differentiation rules were calibrated by modifying model parameters so that predicted fracture callus stiffness matched experimental data from a study that used rigid and flexible fixators. This resulted in strain thresholds at higher magnitudes than in models for sheep and humans. The resulting numerical model was then used to simulate numerous fracture healing scenarios from literature, showing a considerable mismatch in only 6 of 21 cases. Based on this corroborated model, a fit curve function was derived which predicts the increase of callus stiffness dependent on bodyweight, fixation stiffness, and fracture gap size. By mathematically predicting the time course of the healing process prior to the animal studies, the data presented in this work provides support for planning new fracture healing experiments in rats. Furthermore, it allows one to transfer and compare new in vivo findings to previously performed studies with differing mechanical parameters.

Three key factors affecting treatment results of low-intensity pulsed ultrasound for delayed unions and nonunions: instability, gap size, and atrophic nonunion

BackgroundIf some predictable factors that affect the treatment results of low-intensity pulsed ultrasound (LIPUS) for delayed union or nonunion could be determined, these might provide us with suggestions for whether LIPUS should be used as an alternative treatment for surgery or an adjuvant therapy after surgery. Therefore, the objective of the present study was to determine what factors affected failure of fracture healing after LIPUS for delayed unions and nonunions. MethodsA one-year observational retrospective cohort study was conducted with a consecutive cohort of 101 delayed unions and 50 nonunions after long bone fractures that were treated with LIPUS between May 1998 and April 2007. The main outcome measure was radiographic determination of osseous bone union status within one year after start of LIPUS therapy. Statistical evaluation was used to recognize predictable factors that affect treatment results of LIPUS for delayed union and nonunion. ResultsDelayed union group (n=101): Seventy-five delayed unions (74.3%) united without an additional major surgical intervention. Failure of LIPUS therapy was associated with types of nonunion (atrophic/oligotrophic vs. hypertrophic, relative risk 23.72 [95% CI 1.20–11.5], p<0.01), instability at fracture site (unstable vs. stable, relative risk 3.03 [95% CI 1.67–5.49], p<0.001), and maximum fracture gap size not less than 9mm (relative risk 3.30 [95% CI 1.68–6.45]). Nonunion group (n=50): Thirty-four nonunions (68.0%) united without an additional major surgical intervention. Failure of LIPUS therapy was associated with method of fixation (intramedullary nail vs. others, relative risk 4.50 [95% CI 1.69–12.00], p<0.001), instability at fracture site (unstable vs. stable, relative risk 4.56 [95% CI 2.20–9.43], p<0.0001), and maximum fracture gap size not less than 8mm (relative risk 5.09 [95% CI 1.65–15.67]). ConclusionsLIPUS should be applied as an adjuvant therapy in combination with surgical intervention for an established atrophic nonunion with instability and/or with larger fracture gap.

Finite element analysis of tissue differentiation process of a tibia with various fracture configurations when a composite intramedullary rod was applied

Abstract Flexible implants for long bone fractures are receiving much attention in modern orthopedic surgery because of their excellent healing potential as biological fixations. In this study, we investigated the healing performance of composite intramedullary rods (IM rods) used for tibial diaphyseal fractures. A three dimensional tibia model was developed for various fracture gap sizes and oblique angles. To evaluate the types and time-dependent material properties of the healing tissues, a mechano-regulation algorithm with deviatoric strain was implemented in the finite element analysis. The results showed that the healing performance of a tibial fracture was affected by the modulus of the IM rod according to the fracture gap size and oblique angle. This study suggested the most appropriate IM rod modulus for bone fractures.

Biomechanical Evaluation of Fractured Tibia Externally Fixed With an LCP

A locking compression plate (LCP) can serve as an external fixation for fractured tibia. However, there is concern about the stability and endurance during partial weight bearing. This study was experimentally evaluated the effects of fracture gap sizes (1, 5, and 10 mm) on the stability and endurance of fractured tibia externally fixed with a 316L-stainless LCP. For stable fractured tibia (1-mm fracture gap), the large contact area of fracture surfaces resulted in nearly similar stiffness to that of intact tibia. The partial weight bearing is therefore possible. Whereas smaller contact area and no contact of fracture surfaces were observed for tibias with 5-mm and 10-mm fracture gaps, respectively. Their stiffnesses were significantly lower than those of intact tibia and tibia with 1-mm fracture gap. Thus, the partial weight bearing should be considered carefully in early phase of treatment. All LCP-tibial models were cyclically loaded beyond 500,000 cycles, that is, approximately 6 months of healing, without any failure of LCP. Thus, the failure of LCP is unlikely a critical issue for the present cases.

Characterization of cortical bone fracture with scanning confocal ultrasound and longitudinal acoustic velocity.

Non-invasive assessment of fracture, particularly in non-typical fracture, is a critical health problem. As a promising alternative to the x-ray, ultrasound has demonstrated potentials in early fracture diagnosis. A real-time scanning confocal ultrasound image was developed to evaluate bone defect and bone loss. The objectives of this study were to evaluate the cortical fracture gap size using quantitative ultrasound imaging and the longitudinal ultrasound velocity in bone to predict the fracture gap size. The total fractures were created by MTE compression at the middle diaphysis, with gap size varied from 1–5 mm (N=4). Fractures were tested with the ultrasound scanning with 0.5-mm resolution. The measured ultrasound signals were analyzed to calculate BUA (dB/MHz), ATT (dB), and transverse velocity (m/s). The longitudinal wave velocity was tested using three surface transducers. Strong correlations were observed between ultrasound and x-ray images in fracture size (R2=0.91). High correlation was found between gap size and the longitudinal velocity from 4000 to 3000-m/s for 0–5-mm gaps (R2=0.93). These results suggest that ultrasound is capable to predict bone fracture and provide useful information for longitudinal assessment of complications, such as non-union fracture, and for evaluating healing. [Work supported by NSBRI/NASA and NIH.]

Stiffness and Endurance of a Locking Compression Plate Fixed on Fractured Femur

The effects of locking screw position (long column fixation--long distance between the nearest screws to the fracture--and short column fixation--short distance between the nearest screws to the fracture) and fracture gap size (1-mm and 8-mm transverse fracture gap) on stiffness and fatigue of fractured femur fixed with a locking compression plate (LCP) were biomechanically evaluated. The stiffness of 1-mm fracture gap models and that of intact femoral model were in the range of 270-284 N/mm, while those of 8-mm fracture gap models were significantly lower (155-170 N/mm). After 1,000,000 cycles of loading, no fracture of LCP of 1-mm fracture gap models fixed in either long column or short column fashions occurred. On the other hand, the complete fractures of LCPs of 8-mm fracture gap models fixed in long column and short column fashions occurred after 51,500 and 42,000 cycles of loading, respectively. These results suggest that the full weight loading may be allowed for the patient with 1-mm transverse femoral fracture fixed with an LCP. On the other hand, the full load of walking should be avoided for the patient with 8-mm transverse femoral fracture fixed with an LCP before adequate healing.

Effect of fracture gap on stability of compression plate fixation: A finite element study

In compression plating, anatomical reduction and compression across the fracture site are the basic principles necessary to achieve primary bone healing. However, varying amounts of gap at the fracture site frequently occur due to technical pitfalls, such as overbending of the plate and inaccurate reduction, and due to the fracture configuration itself. Little is known as to how fracture gap affects stability of the bone-plate construct. We analyzed the effects of fracture gap size (1 and 4 mm) and bone defect (25%, 50%, 75%, 100%) on the biomechanical stability of the compression plate-bone construct through validated finite element analysis. The stiffnesses of eight different models were compared with the stiffness of an ideally compressed model (0 mm/0%). Stress concentration in form of peak von Mises stress (PVMS) was also evaluated. The decrease in stiffness depended mainly on the depth of bone defect. The decrease in stiffness was similar in models with the same defect and different gap size. Considerably more stress was concentrated around the central hole of the plate in gap models with the depth of bone defects of 75% and 100% than with smaller defects. We concluded that even a thin fracture gap (1 mm) with no contact between the fracture after plating decreases stiffness exponentially; contact at the fracture surfaces of > or =50% was necessary to avoid undue stress concentration in the plate.

Fracture and bone defect assessment using quantitative ultrasound wave propagation

Nondestructive evaluation of early fracture and monitoring its healing, particularly in non‐typical fracture, is critical yet challenge. Quantitative ultrasound test along the long bone is largely dependant on bone density, gap size, the level of maturity of the callus, and mineralized status. We propose that the ultrasonic velocity (UV) is dependent on fracture gap size and the degree of porosity. The objective of the study was to develop an analytical and a multiple ultrasound sensing system for non‐invasive fracture assessment. An analytical model was generated as a function between UV and the gap size in long bone. The ultrasound measurements were performed in the intact pig femur with controlled fracture sizes, e.g., 1mm, 3mm and 5mm. A serial ultrasound transducers were aligned 3 cm above bone with an angle approximately equal to the critical angle. The results indicated that the UV decreased with increase of fracture size in a linear manner (r=0.96), in which UV decreased 6.7%, 16.3% and 26.8% for 1, 3 and 5 mm gaps, respectively. The simulation result significantly matched the experiment measurement (r=0.98). The results demonstrated that quantitative ultrasound has potential to the diagnosis of fracture, monitoring fracture healing, and other bone disorders.

Torsional stiffness in healing fractures: influence of ossification

Background Different fracture fixation techniques and fracture environment influence bone formation in healing fractures. However, the influence on the development of biomechanical properties has not been clear described. We evaluated the influence of fracture fixation stability and fracture environment on mechanical properties in healing femoral fractures in rats.Methods Animals were treated surgically with external fixation: 1 group (27 rats) with 0-mm fracture gap size with bone ends touching, corresponding to an axial stiffness of 265 (SD 34) N/mm, and a second group (27 rats) with 2-mm fracture gap size corresponding to an axial stiffness of 30 (SD 2.1) N/mm. From each group, 6–7 animals were killed at 2, 4, 6 and 12 weeks. Torsional test revealed a delay in torsional stiffness in fractures in group 2 compared to group 1. In group 2, the torsional stiffness of the contralateral femora was found to be greater at 12 weeks than the torsional stiffness in group 1.Interpretation We found that during fracture healing, the development of torsional stiffness corresponds to the magnitude of endochondral ossification and late response of bone formation. A significantly increased torsional stiffness in the non-fractured leg of rats with delayed fracture healing was also found, possibly indicating a response to loading conditions or a systemic stimulation of bone mass.

The fracture gap size influences the local vascularization and tissue differentiation in callus healing.

Revascularization of a fracture depends on fracture stability and fracture gap conditions. The aim of the study was to determine quantitatively the revascularization and tissue differentiation in an animal model with different fracture gaps and controlled biomechanical conditions. The study was performed on ten sheep with an osteotomy on the right metatarsal. The fracture was stabilized by an external fixator that allowed adjustable axial interfragmentary movement. Two groups of five sheep each were adjusted to a medium sized gap (M, 2.1 mm) and a large gap (L, 5.7 mm) under comparable interfragmentary strain (30-32%). The animals were killed after 9 weeks, and the metatarsals were prepared for undecalcified histology and analysis of tissue differentiation and vessel distribution. Group M showed significantly more revascularization (M=1.62, L=0.85 vessels/mm2), more bone formation (M=37.2%, L=13.9%) and less fibrocartilage tissue (M=18.1%, L=39.1%) than group L. Larger vessels (>40 microm) were found mainly in the medullary channel, and smaller vessels (<20 microm) mainly in the peripheral callus. Histologically, group M showed partial bony bridging of the osteotomy gap, and the group L had delayed healing. A good reduction of a fracture with small interfragmentary gaps is important for its revascularization and healing.

Monitoring and healing analysis of 100 tibial shaft fractures.

We assessed the value of measuring biomechanical stiffness by assessing the fixator's external deformation as an objective means for monitoring fracture healing and determining the postoperative treatment regime, as compared to clinical and radiographic means of evaluation. One hundred patients with tibial shaft fractures managed by unilateral external fixation had their fracture stiffness monitored. Stiffness was measured and clinical and radiological examinations were performed every 3-4 weeks. The time required for healing as indicated by stiffness measurement was an average of 2.5 weeks earlier than by radiological assessment. Eighty-two patients healed within 19 weeks (12.1+/-3.3 weeks) and ten patients in the following 6 weeks (24+/-4.3 weeks). Eight patients did not show an increase in fracture stiffness and received intramedullary nailing at a second operation. The average healing time was 11.3+/-4 weeks for type A, 13.1+/-3.6 weeks for type B fractures, and 15.1+/-5.9 weeks for type C fractures. The healing time for closed fractures was 11.3+/-3.2 weeks and for open fractures 14+/-4.9 weeks. The measurement of fracture stiffness allows the detection of patients at risk for nonunions. The healing time increased with increasing fracture gap size and was less in patients with younger age, less complex fractures, and lesser degrees of soft tissue damage.

A Mathematical Framework to Study the Effects of Growth Factor Influences on Fracture Healing

During fracture healing, multipotential stem cells differentiate into specialized cells responsible for producing the different tissues involved in the bone regeneration process. This cell differentiation has been shown to be regulated by locally expressed growth factors. The details of their regulatory mechanisms need to be understood. In this work, we present a two-dimensional mathematical model of the bone healing process for moderate fracture gap sizes and fracture stability. The inflammatory and tissue regeneration stages of healing are simulated by modeling mesenchymal cell migration; mesenchymal cell, chondrocyte and osteoblast proliferation and differentiation, and extracellular matrix synthesis and degradation over time. The effects of two generic growth factors on cell differentiation are based on the experimentally studied chondrogenic and osteogenic effects of bone morphogenetic proteins-2 and 4 and transforming growth factor- β -1, respectively. The model successfully simulates the progression of healing and predicts that the rate of osteogenic growth factor production by osteoblasts and the duration of the initial release of growth factors upon injury are particularly important parameters for complete ossification and successful healing. This temporo-spatial model of fracture healing is the first model to consider the effects of growth factors. It will help us understand the regulatory mechanisms involved in bone regeneration and provides a mathematical framework with which to design experiments and understand pathological conditions.

An Efficient Boundary Integral Formulation for Flow Through Fractured Porous Media

In this paper we present a new model for flow in fractured porous media. We formulate our model in terms of a coupled system of boundary integral equations and present an efficient procedure for solving the equations using the boundary element method. In the new model, the flow in the matrix is governed by the usual Darcy law for porous media, with the fractures being treated as planar sources embedded in the matrix. The flow in an individual fracture is governed by a two-dimensional Darcy law (as in a Hele–Shaw cell), with an associated planar sink distribution. The essential feature of this approach is that the fractures are treated as special planes rather than narrow-gap voids. The error in the resulting system of equations is on the order of an intrinsic dimensionless parameter (the ratio of the fracture gap size to the scale of the volume under consideration). We also describe how we adapt the new model to compute effective grid block permeabilities. This was the principal motivation behind the development of the new model. Using effective grid block permeabilities to model flow in fractured oil and gas reservoirs is a much more efficient process than modeling the flow when every fracture is precisely represented. We present some numerical examples that illustrate the new flow model and how it is used to model flow in a reservoir.

EFFECTS OF AXIAL DYNAMIZATION ON BONE HEALING

The effects of early dynamization (physiologic axial compression) on canine fracture healing at six weeks were studied. Bilateral transverse mid-tibial osteotomies were created and initially stabilized with a 2-mm gap using relatively rigid external fixators. Seven days after osteotomy, the telescoping mechanism of one of the fixators on each dog was released (dynamized), resulting in physiologic loading of the osteotomy, while the contralateral fixator remained locked as a rigid control. The dynamized osteotomy closed, and increased functional weight bearing resulted from 3 weeks on. Radiographically the amount of periosteal callus increased over time, but no difference in callus size was seen between the dynamized fractures and the controls. Torsional mechanical testing found the dynamized osteotomies to be significantly stiffer, and they tended to tolerate more maximum torque than the controls. Microscopic evaluation found no difference in the volume of the periosteal and endosteal calluses or in the tissues constituting them. However, a significantly greater proportion of the dynamized osteotomy gap was filled with new bone. These results suggest that dynamization in this delayed union model improved fracture healing by reducing fracture gap size and increasing weight bearing, not by altering the pathway of fracture healing.