Final Report Summary - OA AM (Articular contact mechanics with application to early diagnosis of osteoarthritis: asymptotic modelling of biomechanical contact phenomena under dynamic and impact )
Osteoarthritis (OA) is a degenerative disease of articular cartilage which involves its deterioration and affection of the underlying bone. Articular cartilage has no known function other than maintaining mechanical competence of joints, allowing bones to move against one another without friction. But there is no need to underline its significance to health of a human body, since almost all the load transmitted by a human joint goes through the articular cartilage, and it prevents biomechanical damage caused by severe loading. In OA, the cartilage thins, begins to fragment and wear away. The most common target joint affected by OA is the knee joint. Though the degeneration process at the OA knee joint is believed to result from a combination of mechanical loading and biological weakening of the cartilage matrix, considerable epidemiological evidence supports the concept that mechanical conditions producing increased load transfer across the joint and altered patterns of the contact pressure distribution can accelerate the initiation and progress of OA.
Thus, the ability to evaluate articular contact mechanics on a patient-specific basis could lead to improvements in the diagnosis and treatment of knee OA. For the aims of the first part of the research project, the contact pressure, determining the stress-strain field within the articular cartilage, is unique amongst other mechanical factors, because it is strongly affected by the changing cartilage geometry. Since there are no experimental methods available to assess articular contact mechanics noninvasively on a patient-specific basis, the construction and development of mathematical models for quantification of disease activity and progression in OA knee plays an important role in evaluation of mechanical factors affecting microstructural properties of articular cartilage and subchondral bone.
At present, there are no effective medical treatments to cure the OA degradation of articular cartilage. Treatment of articular cartilage degeneration is challenging because this tissue is incapable of quality repair or regeneration. Therefore, early detection of OA and the ability to monitor the disease progression are important for developing efficacious therapies. The second part of the research project is focused on the development of mathematical models for a breakthrough new minimally invasive methodology for in situ health monitoring of articular cartilage based on indentation type atomic force microscopy.
Today, mathematical modelling plays a crucial role in analysing biomechanical systems. However, many of the mathematical models developed for describing articular contact mechanics become so complex that only numerical solutions by computer seem to be feasible. This comes in apparent contradiction with the necessity to have a clear understanding of the underlying principles of OA, the causes of variation in articular cartilage properties, the effects of slight gradual changes in cartilage at the onset of OA upon the evolution of the contact distribution pattern.
Asymptotic modelling (AM) is a complex analytical mathematical technique of analysing and simplifying mathematical models. A transition from inorganic to organic material increases both the complexity of contact and fracture phenomena being investigated and the requirements to the feasibility of mathematical models obtained. The mathematical formulation of new contact problems in biomechanics has become so complicated that it is often impossible to solve them by many analytical methods. At the same time, numerical methods frequently become useless in capturing all the essential properties of the studied mathematical model of a complex biophysical reality. It should be emphasised that, whereas computer simulations produce numbers, graphs and pictures, the result of AM is the new mathematical problem that becomes amendable to mathematical analysis and numerical simulation. AM provides us with an understanding of the reality. In particular, AM being applied to the existing mathematical models of biomechanical contact aids in understanding of articular joint mechanics and etiology of the OA process.
The main research objectives of the project:
(1) development an advanced AM methodology for analytical evaluating the sensitivity of crucial parameters in articular joint contact mechanics of the contacting articular cartilage layers under dynamic and impact loading conditions due to small variations in thicknesses and biophysical microstructural properties, including microcracking and OA-related pathological changes in bone tissue;
(2) development of mathematical and computational models for nanoindentation testing of articular cartilage.
Main contributions of the research project:
- Asymptotic models for dynamic contact interaction of biphasic articular cartilages with non-axisymmetric variations in both the thicknesses and biomechanical properties with the effect of underlying bone tissue deformation taken into consideration were constructed. Based on these asymptotic models, a new methodology for modelling articular tibio-femoral contact was developed. In particular, it was established that deformation of a thin incompressible cartilage layer cannot be treated in the framework of Winkler foundation model, which is sometimes used for simplifying calculations in real-time computer simulations of human gait.
- Asymptotic model for contact interaction of biphasic articular cartilages under impact loading was introduced. In particular, the effect of cartilage viscoelastic properties on the accommodation mechanism in mechanical response of the articular cartilage layers subjected to repetitive mechanical loading in the knee joint during walking or running was modelled for the first time. It was established that the equivalent stiffness and damping coefficients for the HuntCrossley model, which widely used for modelling articular contact in multibody dynamic simulations of human movements, should depend on the gait frequency.
- Sensitivity analysis of articular contact mechanics due to variations in the knee joint system was performed. It was found that to minimise the influence of the cartilage thickness non-uniformity on the force-displacement relationship, the effective thicknesses of articular layers should be determined from a special optimisation criterion.
- Asymptotic models for time-dependent micro and nanoindentation of articular cartilage with the non-uniform microstructural properties not known a priori were developed. A simple asymptotic model to describe the mechanical behaviour of a relatively stiff infinite elastic fibre within an elastic matrix was constructed, and the fibre influence factor for the indentation stiffness was introduced. The dynamic indentation stiffness was studied with the aim of formulating new criteria for evaluation the quality of articular cartilage in order to be able to discriminate its degenerative state. In particular, evaluating the dynamic indentation stiffness at the turning point of the flat-ended indentation test, we introduced the so-called incomplete storage modulus. Implications to non-destructive indentation diagnostics of cartilage degeneration are drawn.
- Viscoelastic mathematical models were developed for the first time in order to evaluate the rebound indentation test which was very recently proposed for assessing the viability of articular cartilage.
Because of its important prevalence worldwide, OA is the leading cause of limitations in activities of daily living, and is second only to heart disease in causing work disability. Because of the longevity of working careers and the substantial prevalence of OA in middle-aged persons, OA causes a considerable socio-economic burden in lost time at work and early retirement. Symptomatic OA of the knee affects 10 % of adults over age 55, and a quarter of those affected are severely disabled. Generic persistent knee pain affects 25 % of adults over the age of 55. OA and knee pain in general, is associated with reduced activity levels, which then leads to further health problems. This means that health costs arising from OA may extend beyond the cost of directly treating the disease.
Because of the ageing of populations in the developed countries, the prevalence of OA is expected to increase. In Europe, 15.2 million of total prevalent cases of OA were diagnosed in 2007, and 15.8 million of new cases of OA are expected in 2012. Today in the European Union, a joint is replaced due to OA every 1.5 minutes. In 2060 it is projected that there will be one adult over the age of 65 to every two adults of working age the present ratio of one over 65 to every four of working age. We believe that implications drawn on the AM basis in the framework of the research project can be useful to aid understanding how to modify the complex process involved in OA pathogenesis as well as be employed for improving the arthroscopic indentation testing methodology for early detection of OA.
Thus, the ability to evaluate articular contact mechanics on a patient-specific basis could lead to improvements in the diagnosis and treatment of knee OA. For the aims of the first part of the research project, the contact pressure, determining the stress-strain field within the articular cartilage, is unique amongst other mechanical factors, because it is strongly affected by the changing cartilage geometry. Since there are no experimental methods available to assess articular contact mechanics noninvasively on a patient-specific basis, the construction and development of mathematical models for quantification of disease activity and progression in OA knee plays an important role in evaluation of mechanical factors affecting microstructural properties of articular cartilage and subchondral bone.
At present, there are no effective medical treatments to cure the OA degradation of articular cartilage. Treatment of articular cartilage degeneration is challenging because this tissue is incapable of quality repair or regeneration. Therefore, early detection of OA and the ability to monitor the disease progression are important for developing efficacious therapies. The second part of the research project is focused on the development of mathematical models for a breakthrough new minimally invasive methodology for in situ health monitoring of articular cartilage based on indentation type atomic force microscopy.
Today, mathematical modelling plays a crucial role in analysing biomechanical systems. However, many of the mathematical models developed for describing articular contact mechanics become so complex that only numerical solutions by computer seem to be feasible. This comes in apparent contradiction with the necessity to have a clear understanding of the underlying principles of OA, the causes of variation in articular cartilage properties, the effects of slight gradual changes in cartilage at the onset of OA upon the evolution of the contact distribution pattern.
Asymptotic modelling (AM) is a complex analytical mathematical technique of analysing and simplifying mathematical models. A transition from inorganic to organic material increases both the complexity of contact and fracture phenomena being investigated and the requirements to the feasibility of mathematical models obtained. The mathematical formulation of new contact problems in biomechanics has become so complicated that it is often impossible to solve them by many analytical methods. At the same time, numerical methods frequently become useless in capturing all the essential properties of the studied mathematical model of a complex biophysical reality. It should be emphasised that, whereas computer simulations produce numbers, graphs and pictures, the result of AM is the new mathematical problem that becomes amendable to mathematical analysis and numerical simulation. AM provides us with an understanding of the reality. In particular, AM being applied to the existing mathematical models of biomechanical contact aids in understanding of articular joint mechanics and etiology of the OA process.
The main research objectives of the project:
(1) development an advanced AM methodology for analytical evaluating the sensitivity of crucial parameters in articular joint contact mechanics of the contacting articular cartilage layers under dynamic and impact loading conditions due to small variations in thicknesses and biophysical microstructural properties, including microcracking and OA-related pathological changes in bone tissue;
(2) development of mathematical and computational models for nanoindentation testing of articular cartilage.
Main contributions of the research project:
- Asymptotic models for dynamic contact interaction of biphasic articular cartilages with non-axisymmetric variations in both the thicknesses and biomechanical properties with the effect of underlying bone tissue deformation taken into consideration were constructed. Based on these asymptotic models, a new methodology for modelling articular tibio-femoral contact was developed. In particular, it was established that deformation of a thin incompressible cartilage layer cannot be treated in the framework of Winkler foundation model, which is sometimes used for simplifying calculations in real-time computer simulations of human gait.
- Asymptotic model for contact interaction of biphasic articular cartilages under impact loading was introduced. In particular, the effect of cartilage viscoelastic properties on the accommodation mechanism in mechanical response of the articular cartilage layers subjected to repetitive mechanical loading in the knee joint during walking or running was modelled for the first time. It was established that the equivalent stiffness and damping coefficients for the HuntCrossley model, which widely used for modelling articular contact in multibody dynamic simulations of human movements, should depend on the gait frequency.
- Sensitivity analysis of articular contact mechanics due to variations in the knee joint system was performed. It was found that to minimise the influence of the cartilage thickness non-uniformity on the force-displacement relationship, the effective thicknesses of articular layers should be determined from a special optimisation criterion.
- Asymptotic models for time-dependent micro and nanoindentation of articular cartilage with the non-uniform microstructural properties not known a priori were developed. A simple asymptotic model to describe the mechanical behaviour of a relatively stiff infinite elastic fibre within an elastic matrix was constructed, and the fibre influence factor for the indentation stiffness was introduced. The dynamic indentation stiffness was studied with the aim of formulating new criteria for evaluation the quality of articular cartilage in order to be able to discriminate its degenerative state. In particular, evaluating the dynamic indentation stiffness at the turning point of the flat-ended indentation test, we introduced the so-called incomplete storage modulus. Implications to non-destructive indentation diagnostics of cartilage degeneration are drawn.
- Viscoelastic mathematical models were developed for the first time in order to evaluate the rebound indentation test which was very recently proposed for assessing the viability of articular cartilage.
Because of its important prevalence worldwide, OA is the leading cause of limitations in activities of daily living, and is second only to heart disease in causing work disability. Because of the longevity of working careers and the substantial prevalence of OA in middle-aged persons, OA causes a considerable socio-economic burden in lost time at work and early retirement. Symptomatic OA of the knee affects 10 % of adults over age 55, and a quarter of those affected are severely disabled. Generic persistent knee pain affects 25 % of adults over the age of 55. OA and knee pain in general, is associated with reduced activity levels, which then leads to further health problems. This means that health costs arising from OA may extend beyond the cost of directly treating the disease.
Because of the ageing of populations in the developed countries, the prevalence of OA is expected to increase. In Europe, 15.2 million of total prevalent cases of OA were diagnosed in 2007, and 15.8 million of new cases of OA are expected in 2012. Today in the European Union, a joint is replaced due to OA every 1.5 minutes. In 2060 it is projected that there will be one adult over the age of 65 to every two adults of working age the present ratio of one over 65 to every four of working age. We believe that implications drawn on the AM basis in the framework of the research project can be useful to aid understanding how to modify the complex process involved in OA pathogenesis as well as be employed for improving the arthroscopic indentation testing methodology for early detection of OA.