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Osteoporotic vertebral fractures represent major cause of disability, loss of quality of life and even mortality among the elderly population. Decisions on drug therapy are based on the assessment of risk factors for fracture, from bone mineral density measurements. The combination of biomechanical models with clinical studies could better estimate bone strength and support the specialists in their decision. A model to assess the probability of fracture, based on the Damage and Fracture Mechanics has been developed, evaluating the mechanical magnitudes involved in the fracture process from clinical bone mineral density measurements. The model is intended for simulating the degenerative process in the skeleton, with the consequent lost of bone mass and hence the decrease of its mechanical resistance which enables the fracture due to different traumatisms. Clinical studies were chosen, both in non-treatment conditions and receiving drug therapy, and fitted to specific patients according their actual bone mineral density measures. The predictive model is applied in a finite element simulation of the lumbar spine. The fracture zone would be determined according loading scenario (fall, impact, accidental loads, etc.), using the mechanical properties of bone obtained from the evolutionary model corresponding to the considered time. Bone mineral density evolution in untreated patients and in those under different treatments was analyzed. Evolutionary curves of fracture probability were obtained from the evolution of mechanical damage. The evolutionary curve of the untreated group of patients presented a marked increase of the fracture probability, while the curves of patients under drug treatment showed variable decreased risks, depending on the therapy type. The finite element model allowed obtaining detailed maps of damage and fracture probability, identifying high-risk local zones at vertebral body, which are the usual localization of osteoporotic vertebral fractures. The developed model is suitable for being used in individualized cases. The model might better identify at-risk individuals in early stages of osteoporosis and might be helpful for treatment decisions.

Currently, osteoporotic fractures represent major cause of disability, loss of quality of life and even death among the elderly population [

Vertebral fracture is the most common osteoporotic fracture, being more prevalent in women than in men [

Underdiagnosis of osteoporotic vertebral fracture is common for the lack of complementary tests in older women who visit the doctor complaining of back pain and sometimes because the symptomatology of the first fracture is not evident [

Different publications estimate that only 40% of fractures are diagnosed [

Decisions on drug therapy in osteoporotic patients without previous fractures are mainly based on the analysis of risk factors which predispose to fracture. A risk assessment tool called FRAX^{®} (Fracture Risk Assessment Tool) has been developed by the World Health Organization (WHO) for this purpose [^{®} tool. The probability of fracture is calculated on the basis of age, body mass index and several dichotomized variables (previous fracture, smoking, rheumatoid arthritis, etc.). Optionally, Bone Mineral Density (BMD) of the femoral neck can be included for risk calculation. Other studies have questioned the effectiveness of FRAX^{®} as a tool for predicting fracture risk [

Several previous surveys have assessed the risk of fracture using various methodologies mostly based on BMD measurements [

In the field of Finite Element (FE) simulation, different models were used for predicting bone strength or fracture risk at different ages. So, Lee [

Currently, the most popular clinical tool for fracture risk assessment is FRAX^{®}, which doesn’t consider bone strength as a relevant magnitude. All the aforementioned computational methods use clinical or mechanical magnitudes related to bone fracture in an independent way, without considering their mutual influence as it actually happens.

In view of the current models limitations, we have developed a model for predicting the risk of osteoporotic vertebral fractures based on the Damage Mechanics and Fracture Mechanics. The model is incorporated into a finite element code to simulate the damage evolution over time. Thus, we will estimate the probability of vertebral fracture in the mid and long terms. BMD measurements, from DXA scans, will be incorporated into the model in order to fit it to clinical conditions. The model is not intended for simulating the bone fracture, but to predict the degenerative process in the skeleton, with the consequent lost of bone mass and hence the decrease of its mechanical resistance which enables the fracture due to different traumatisms.

In order to develop a predictive model which takes into account the mechanical parameters involved in the fracture process, a correlation between these magnitudes and those ones measured in clinical terms is firstly required. To this effect, Carter and Hayes [

On the other hand, the relationship between the BMD value and the apparent density is adjusted, according to experimental results, as:

being λ a numerical parameter that depends on the sample data. For the present study a value of 0.36 was assigned for λ in order to fit the actual data presented in [

BMD is the current standard for diagnosis of osteoporosis. Over 100 worldwide published papers assessing lumbar spine BMD evolution, both in natural conditions and in patients under drug therapy were selected for analysis. Among them, two treatments have been selected for the comparative study: risedronate, a biphosphonate, in dosage of 5 mg per day [

Bisphosphonates settle in the bone tissue and its effect persists during some time after its administration, and hence it is completely accepted in the clinical practice the use of intermittent or discontinuous treatments. For the risedronate, several studies with a dosage of 5 mg/day [

With respect to the use of denosumab, its action principle deals with the blocking of natural bone resorption. Its effectiveness reducing the fracture risk has been demonstrated in a 5 years study [

Regarding the natural history of BMD, the average curve published by Mazess [

As the model evaluates the mechanical strength of bone from BMD, an analytical function expressing its evolution over time is needed. That function must be obtained from numerical data contained in different studies. Since the standard adjustment techniques do not provide enough accuracy and reliability to be applied to the predictive model, more complex adjustments have been set out to obtain continuous curves of regression. These make it possible to obtain continuous curves for the BMD evolution. The evolution trends of the different selected cases in the study can be extrapolated from those continuous curves, ensuring a consistent behavior. The following adjustments were proposed for the apparent density ρ:

• Polynomial:

• Exponential:

• Exponential asymptotic:

These adjustments (Equations (3), (4) and (5)) have been applied to the three examined curves (natural evolution and the two therapies). Several choices have been made during this adjustment: the lowest mean square error, the closest to unity correlation coefficient R^{2}, and a 10-year standardized follow-up period with the possibility of extrapolation to longer periods up to 15 years (

The Karganovin’s Damage Mechanics model [

being:

the equivalent strain, where

where E_{0} corresponds to the Young’s modulus of healthy bone and E is the actual value of Young’s modulus of the bone with cumulative mechanical damage. The

where _{cri}. Once the damage model is defined, a relationship between the damage level and the probability of fracture must be set. Obviously several factors are involved in the equation and may difficult the calculations: damage location, amount of

Fitting of BMD evolutionary curves: (a) Natural evolution (R^{2} = 0.9942); (b) Risedronate (5 mg/day) (R^{2} = 0.995); (c) Denosumab (60 mg/6 months) (R^{2} = 0.999)

damage, range and type of load cycles, and so on. However, since osteoporosis is a generalized disease affecting bone mass extensively, and it occurs in older people with little variation in their life habits, a simplified model could be used. In this work, a model of fracture probability based on the law of Paris [_{i} value to the D_{f} value was expressed as follows:

where a and b are material parameters associated with the law of Paris,

By normalizing the probability of fracture, assigning value 1 to critical damage (D = D_{cri}), and value 0 to no damage (D = 0), we finally obtain a measure of the probability of fracture as a function of both the number of cycles and the damage:

In Equation (11), the magnitude N_{max} represents the maximum number of load cycles necessary for a mechanical damage equivalent to the critical value, D_{cri}. A value 5 has been given to the β coefficient for cortical bone, according to [_{cri}, of 0.38 and a critical strain,

BMD evolutionary curves for a specific population are basic references and provide only general information. In order to apply the model to specific patients we must consider, in addition to the trend, the reference density value of the patient. According to the law of interpolated natural evolution, the density matching with the age of the patient is given by:

being

when a drug therapy is applied to the same patient, a similar correction is required because a new offset appears. So, the progression curve for this patient under a treatment would be:

In the Equations (12) to (14), the subscripts or superscripts N and T represent natural evolution or evolution with treatment, respectively. All these adjustments provide the estimated BMD value for any type of patient, at any age, and under any prescribed therapy. From this value, mechanical properties of bone can be calculated. It must be noticed that the considered curves represent the mean evolutionary curves for the population, and an individual patient could not follow the curve exactly but in an approximate way.

Densitometric data of the healthy lumbar spine have been taken as the starting point for this study, based on previous published works [

From all the previous calculations, an evolutionary algorithm has been implemented (

. BMD, apparent density and Young’s modulus for the data corresponding to the study of [48]

Zone | Standard BMD (mg/cm^{2}) | Apparent volumetric density (gr/cm^{3}) | Young modulus (MPa) |
---|---|---|---|

Vertebral wall | 1159 | 1.494 | 9593 |

Outer vertebral endplates | 1159 | 1.494 | 9593 |

Intermediate vertebral endplates | 610 | 1.186 | 4797 |

Centre of vertebral endplates | 221 | 0.822 | 1599 |

Apophyses | 321 | 0.941 | 2398 |

Evolutionary algorithm for the prediction of fracture probability

The vertebrae were meshed by means of tetrahedral elements with quadratic approximation in the I-deas pro- gram [

Complete FE model including vertebrae, discs and ligaments: (a) Frontal view; (b) Lateral view; (c) Dorsal view

internal to the most external (

Finally, the ligaments are modelled by means of tetrahedra and prisms with quadratic approximation; in addition, membrane elements have been used for capsular ligaments. The dimensions of those soft tissues correspond to average anatomical measurements [

The bone and ligament properties were taken from the bibliography. Concerning the bone, in [

The behaviour of the nucleus pulposus, like a non-compressible fluid, was simulated by means of the hyperelastic Mooney-Rivlin model (incompressible) incorporated in the Abaqus version 6.11 materials library [

As boundary conditions displacements in the wings of sacrum have been prevented. As load condition, a compression of 400 N was considered acting on the upper face of vertebra L4 (

The model has been used in predicting the evolution of vertebral fracture probability, by comparing the natural history and the expected evolution under different therapies.

Firstly, adjustment models were applied both to BMD physiological curve (weighted average from [

Model of the inter-vertebral disk and its layers of fibres

Zones of different elastic properties in the vertebral body

Boundary conditions: (a) Restrained displacements; (b) Load on L4

. Statistics for the FE model and mechanical properties of materials, according with [52]

Material | Young modulus (MPa) | Poisson coefficient | Element type | Number of elements |
---|---|---|---|---|

Outer vertebral endplates | 12000 | 0.3 | Tetrahedron | 3578 |

Intermediate vertebral endplates | 6000 | 0.3 | Tetrahedron | 2244 |

Centre of the vertebral endplates | 2000 | 0.3 | Tetrahedron | 831 |

Walls of the vertebral body | 12000 | 0.3 | Tetrahedron | 37205 |

Cancellous bone (inside vertebrae) | 100 | 0.2 | Tetrahedron | 44954 |

Posterior vertebra | 3000 | 0.3 | Tetrahedron | 47134 |

Cartilage | 50 | 0.4 | Wedge | 3086 |

Annulus fibrosus | 4.2 | 0.45 | Hexahedron | 8288 |

Nucleus pulposus (^{*}) | Incompressible material | Tetrahedron | 14410 | |

Annulus fiber layers 1 | 360 | 0.3 | Truss (^{**}) | 592 |

Annulus fiber layers 2 | 408 | 0.3 | Truss (^{**}) | 592 |

Annulus fiber layers 3 | 455 | 0.3 | Truss (^{**}) | 592 |

Annulus fiber layers 4 | 503 | 0.3 | Truss (^{**}) | 592 |

Annulus fiber layers 5 | 550 | 0.3 | Truss (^{**}) | 296 |

Ligament | Young modulus (MPa) | Transition strain (%) | Element type | Number of elements |

Anterior longitudinal ligament | 7.8 20.0 | 12.0 | Wedge (^{**}) | 9046 |

Posterior longitudinal ligament | 10.0 50.0 | 11.0 | Wedge (^{**}) | 3844 |

Ligamentum flavum | 15.0 19.0 | 6.2 | Tetrahedron (^{**}) | 3042 |

Intertransverse ligament | 10.0 59.0 | 18.0 | Tetrahedron (^{**}) | 6678 |

Capsular ligament | 7.5 33.0 | 25.0 | Membrane (^{**}) | 3220 |

Interspinous ligament | 8.0 15.0 | 20.0 | Tetrahedron (^{**}) | 2856 |

Supraspinous ligament | 10.0 12.0 | 14.0 | Tetrahedron (^{**}) | 2657 |

Iliolumbar ligament | 7.8 20.0 | 12.0 | Wedge (^{**}) | 816 |

(^{*}) C_{01} = 0.0343 MPa; C_{10} = 0.1369 MPa. An elastic analysis with Young modulus of 1.0 MPa and Poisson ratio of 0.49 was carried out with similar results and a volume change less than 0.6%. (^{**}) Only tension.

Comparisons among various treatments can be drawn regarding different simulations.

Equivalent comparisons can be made with different parameters. Thus,

As a final result, the evolutionary curves of fracture probability were obtained from the evolution of mechanical damage. The estimated probability, according to mechanical damage caused by strains, is calculated for the initial patient’s state (

Evolution of different magnitudes at L5 vertebra: (a) BMD; (b) Average Young’s modulus; (c) Average equivalent strain; (d) Average mechanical damage; (e) Evolution of the average fracture probability at L5 in different conditions; (f) Evolution of the average fracture probability increment at L5 in different conditions

In addition to previous results, programmed subroutines make it possible to obtain damage and fracture probability maps and to identify high-risk zones of the vertebral body (

The spinal localization of osteoporotic fractures is very common [

(a)

(b)

A novel method for estimating the risk in osteoporotic patients has been developed. Clinical data (DXA measures) and mechanical magnitudes related to bone strength were combined in this tool. The mechanical properties of bone are updated from BMD values obtained from clinical data of untreated patients and in those under different treatments. The model uses Damage and Fracture Mechanics concepts to evaluate the fracture probability in an evolutionary algorithm.

The model can be used in a personalized way from BMD measurements in each case. The model can contribute to the development of diagnostic tools for detection of early stages of osteoporosis. It may also be helpful for treatment decisions in selected patients. Many studies have been carried out, both in the clinical [

There are few studies in the bibliography, besides the tool FRAX^{®}, to determine the risk of vertebral fracture. Some authors use clinical data to assess the risk of new fracture after the diagnosis of first vertebral fracture [

Several predictive models can be found in the literature, but statistical models are currently the most reliable [

Concerning the finite element simulation, and based on previous micromechanical models [

From the mechanical point of view, the exposition of the bone to cyclic loads of high value in a damaged bone, once the degenerative process is started, decreases its strength over the time and produces a cumulative damage which can lead to a final fracture. It seems apparent that Damage Mechanics and Fracture Mechanics criteria should be incorporated in any model intending to obtain reliable results. In this regard, our model combines all these requirements, and might be useful as a basis for future more sophisticated models.

Moreover, this model enables to incorporate future developments with the same methodology. In the first term, a more accurate bone density distribution by individual elements in the mesh could be used. That requires a planned collection of BMD data, by means of DXA or CT scan images. More complex damage models can be added, including mechanical behavior of anisotropic or mixed models, based on both deformations and tensions. It would also be possible to include crack growth models fitting to the results of in vitro bone fracture. Finally, a parametric finite element model of the femoral head could be performed including both the loads produced on the bone and the shape and dimensions of a specific patient.

Despite DXA measurements just quantify bone mass and not bone quality, it is widely accepted as a macroscopic indicator of bone strength and stiffness and also that micro-fractures exert an important influence on the mechanical strength of the bone. Finally, clinical trials are needed to validate the proposed model in order to apply it to the clinical practice helping for treatment decisions.

A mechanical model based on Damage and Fracture Mechanics and DXA measurements, for predicting the probability of fracture in osteoporotic patients has been carried out. The model represents a first step towards the development of new tools for diagnosis and prevention of osteoporosis. The incorporation of clinical measurements and simulation results will be useful for an individualized monitoring and treatment in specific patients.

This work has been partially financed through the research project “Development and Clinical Application of a Mechanical Model for the Analysis of Spine Fractures due to Low Energy Trauma, based on Bone Mineral Density Measurements”, supported by Mutua Madrileña Foundation.

The authors declare that they have no conflict of interest.

AH and LG conceived the approach of this work. EL and LG conceived and developed the predictive model for fracture probability. AH, ALE and JM contribute with clinical measurements and experience with osteoporotic patients. EL, EI and SP conceived and developed the finite element model and carried out all the simulations. AH and LG coordinated the work between surgeons and engineers. All authors participated in the drawing up of the manuscript, and read and approved the final manuscript.