Finite element method in cardiac surgery
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《血管的通路杂志》
Department of Cardiac Surgery, Nottingham City Hospital, Hucknall Road, Nottingham, NG7 4HR, UK
Abstract
Finite element (FE) method is a form of mathematical modelling used to study the stress, strain and dynamics of a structure. The method has been used to analyse anatomical structures in the human body. In cardiac surgery it allows the quantifying of patient specific geometric features to both assess disease state and potentially aid in planning surgical intervention such as mitral valve repairs, prosthetic valve design or congenital reconstructive cardiac surgery. Important variables for a finite element model are model geometry, the type of mesh elements used, material properties and boundary condition. This review explains these variables, their limitations and describes their application in cardiac surgery.
Key Words: Finite element method; Cardiac surgery
1. Introduction
Finite Element (FE) method is a mathematical method that evolved in the late 1950s for structural analysis in the aerospace industry. Researchers have since adopted finite element method as a standard numerical technique for analysing complex structures and solving problems in solid and fluid mechanics. Recently there has been an increased use of FE method studies published in cardiac surgery. It is important for surgeons to have a level of understanding about the generation of FE models, their limitations and potential applications. This article aims to provide an over view of Finite Element Method and review studies published involving the technique in cardiac surgery.
2. Basic principles of finite element method
The main purpose in FE analysis is to define the relationship between force and deformation. It analyses the stress and strain of a given structure. In FE method the object under study is divided into small finite segments known as elements. The elements are assembled together and connected by nodes. The forces and deformation of each element will affect the behaviour of each adjacent element through the connecting nodes. The behaviour of a structure is represented by the displacement of these elements and their material properties.
The main parameters to be defined are stress and strain. Stress is the strength of the force from interactions such as stretching, squeezing or twisting. Stress is characterised as force per unit area. Strain is the resulting deformation. The stress/strain relationship defines how a tissue deforms under a given force [1]. To model a given structure using the FE analysis, three key inputs into the analysis must be specified without ambiguity. These are the model geometry, the material properties and the boundary conditions.
2.1. Model geometry
The geometry of a biological organ or structures of most FE models are derived from MRI, CT or Angiography imaging. These images are converted into the final FE geometry using special commercially available computer software. By using individual patient CT, MRI or angiogram images, patient specific FE geometry may be generated and for basic default geometry of a specific organ an average of many individual images, can be generated. Currently there are also commercially available geometry models of the heart and the great vessels (Fig. 1). As explained earlier the FE model is represented by elements. In FE modelling there are several types of element named, mainly according to the shape of the element. The element can be of shell, cube or plate type. For biological structures the shell type is normally used. The size and the number of the elements used to represent a structure will determine the accuracy of the output from the finite elements study. Basically small size and a large number of elements will lead to increased accuracy at the expense of increased computational time [1].
2.2. Material properties
The material properties of the structure under study are normally derived from biomechanical studies of the tissue being modelled. This property has to be an accurate description of the tissue for the FE analysis to be reliable. A biological structure will have both anisotropic and non-linear properties. Anisotropic tissue will deform differently depending on the direction of force. Non-linearity occurs when the rate of deformation changes as the force increases. Many researchers make the assumption that biological material is linear and isotropic due to the complexity and very high computational time that is required when modelling a non-linear and anisotropic object. Although this may affect the absolute value of the resulting stress of a structure studied, it however, normally will not affect the pattern of the stress across the whole structure.
In the FE model the material properties are represented by the Young modulus, which characterises the stiffness of the structure, and the Poisson ratio, which characterises the compressibility of the tissue (Table 1) [1]. Very stiff objects will have higher Young modulus and incompressible objects will have a higher Poisson ratio.
2.3. Boundary conditions
These are the forces or restrictions that apply to the surfaces of the geometry. These boundary conditions describe how the model is attached, and interacts with its surroundings. Examples would be the load applied to the aortic valve during diastole, the attachments of vessels or the restriction of radial movement of the aortic root under pressure from inside the aorta. The boundary conditions can be assumptions or can be accurate simulation of real life. An example would be the pressure recorded in the left ventricle during MRI acquisition of a heart, which will subsequently become the final geometry for the FE model. The simulation of a structure of an FE model depends heavily on the accurate boundary conditions. Spurious results may be generated if these boundary conditions are inaccurate and do not represent the surroundings of the real life structure being modelled.
3. Computational fluid dynamics and FE method
The addition of blood within the solid structure allows accurate analysis of the flow within the structure analysed. The simulation of solid and fluid structure requires special methods to allow coupling of these two elements together. One of the most often used methods is called the Arbitrary Lagrangian-Eulerian (ALE) method. This technique allows numerical simulation of multidimensional fluid dynamics and non-linear solid mechanics. With this technique clear delineation of free surfaces and fluid solid structure can be achieved [1].
4. FE model analysis
The output analysis of an FE model allows one to calculate the stress or strain of the model using commercially available software [2] (Video 1). Examples are LS DYNA or ABACUS. The computational time taken to analyse an FE model takes into account the size of the mesh of the geometry, the complexity of the boundary condition and the material properties. Highly complex mesh and non-linear structures can sometimes take days of computational times.
5. Model validation
Before an FE model can be applied to a clinical setting, the model needs to be validated. The validation process can be done by obtaining data from other different, well known clinical investigations like Doppler, catheterisation or MR studies or in vitro studies. Test results from multiple FE simulations can be compared with these clinical studies and, if comparable, the FE simulation can be assumed to be accurate.
6. FE models in cardiac surgery
The following are examples of FE simulation in cardiac surgery. They range from analysis of normal structures to simulation of diseases and planning of surgical procedures (Table 2).
6.1. Left ventricle
Cups et al. examined the effect of stress on the LV wall of patients with severe aortic regurgitation and in patients with normal aortic valves. In the study, the MRI images of these patients were used as the geometry input and the measured systolic and diastolic pressure specific to the patients were used as the boundary conditions. In their study the stress generated on the regional and global LV wall of patients with severe aortic regurgitation were significantly higher than in patients with normal valves. This illustrates the way FE method can study the stress of a structure non-invasively [3].
6.2. Mitral valve
Normal mitral valve function was examined by Kunzelman et al. looking at the deformation and stress under systemic loading conditions. The simulation demonstrated that annular contraction promotes valve closure whilst papillary muscle contraction tends to pull the leaflets apart. The combination of these two forces showed even stress distribution on valve leaflets [4].
Different leaflet and annular shape can be studied by changing the geometry as demonstrated by Salgo et al. In their study, increasing the annular height to commissural width ratio of 20% led to 37-fold reduction of the leaflet stress [5]. Similarly by changing the material properties of the valve leaflet simulation of stiff leaflet and the resulting stress can be achieved [6].
6.3. Aortic valve
Gnyasaneshwar et al. analysed the dynamics of the normal aortic valve with FE method. In this model the geometry of the valve was derived from MRI images and the material characteristics were assumed to be linear elastic, with the leaflet and the aortic sinus anisotropic. The study showed that the leaflets open initially with dilatation of the aortic root [7]. Using a combination of an FE method and fluid dynamics Nicosia et al. demonstrated the flow patterns of the blood across the valve and at the sinuses [8]. In studies of aortic valve insufficiency secondary to ageing or in Marfan syndrome, the resulting stress and strain from changes on the valve leaflets and the aortic root have been simulated by changing the geometry and the material properties of the models [9,10].
6.4. Congenital heart surgery
In a study of Total Cavopulmonary Connection the geometry of the investigated anastomosis was derived from a combination of angiography and MRI images. The boundary conditions at the inlet and outlet of the models were specified as the blood pressure and velocity of blood flows that were derived from Doppler measurements. The study showed the abrupt change in the geometry and velocity in the connection leading to energy loss with most of the superior vena cava flow going preferentially to the larger right lung. As the flow in this anastomosis is dependent on minimum energy loses and needs optimum flow distribution to both lungs, by changing the geometry, the FE method simulated a different kind of IVC anastomosis to identify the optimum flow pattern. The result of the study has led to alteration of the surgical technique [11].
6.5. Surgical procedures and valve designs
FE studies can analyse the effect of surgical procedures and valve designs. It has been used to analyse the effect of left ventricular reduction surgery and the resulting global and regional stress [12]. In prosthetic heart valve design FE method has allowed researchers to alter the shape of leaflets or use different material properties to study the net effect of the stress generated [13]. In mitral valve repair the FE method has been used to evaluate the effect of different repair techniques on leaflet stress distribution, essential for the longevity of the repair [14]. In aortic valve sparing operations FE methods were used to study the influence of different graft shape on the native aortic valve leaflet stress pattern with a view to optimise the surgical techniques [15].
7. Conclusion
We have shown multiple applications of FE simulation technology. These include the study of normal valve structure, the study of diseases mechanisms, the planning of medical treatments and design of artificial valves. The long-term future direction of the Finite Element method would be a patient-specific model using image-based geometric modelling techniques such as CT and MRI to construct the patient-specific anatomic models. The scope of Finite Element methods is very wide in cardiac surgery and once the models have evolved and are validated, this method may be used as routinely as echocardiogram or MRI scan.
Video 1. Example of stress analysis on the heart and the great vessels following blunt trauma to the chest.
References
Becker AA. An introductory guide to finite element analysis 2004;New York: ASME Press.
Richens D, Field M, Hashim S, Neale M, Oakley C. A finite element model of blunt traumatic aortic rupture. Eur J Cardiothorac Surg 2004; 25:1039–1047.
Cupps BP, Moustakidis P, Pomerantz BJ, Vedala G, Scheri RP, Kouchoukos NT, Davila-Roman VG, Pasque MK. Severe aortic insufficiency and normal systolic function: determining regional left ventricular wall stress by finite-element analysis. Ann Thorac Surg 2003; 76:3668–675.
Kunzelman KS, Cochran RP, Verrier ED, Eberhart RC. Anatomic basis for mitral valve modelling. J Heart Valve Dis 1994; 3:5491–496.
Salgo IS, Gorman 3rd JH, Gorman RC, Jackson BM, Bowen FW, Plappert T, St John Sutton MG, Edmunds Jr LH. Effect of annular shape on leaflet curvature in reducing mitral leaflet stress. Circulation 2002; 106:6711–717.
Kunzelman KS, Quick DW, Cochran RP. Altered collagen concentration in mitral valve leaflets: biochemical and finite element analysis. Ann Thorac Surg 1998; 66:6 SupplS198–205.
Gnyaneshwar R, Kumar RK, Balakrishnan KR. Dynamic analysis of the aortic valve using a finite element model. Ann Thorac Surg 2002; 73:41122–1129.
Nicosia MA, Cochran RP, Einstein DR, Rutland CJ, Kunzelman KS. A coupled fluid-structure finite element model of the aortic valve and root. J Heart Valve Dis 2003; 12:6781–789.
Grande-Allen KJ, Cochran RP, Reinhall PG, Kunzelman KS. Mechanisms of aortic valve incompetence: finite-element modeling of Marfan syndrome. J Thorac Cardiovasc Surg 2001; 122:5946–954.
Grande KJ, Cochran RP, Reinhall PG, Kunzelman KS. Mechanisms of aortic valve incompetence in aging: a finite element model. J Heart Valve Dis 1999; 8:2149–156.
de Leval MR, Dubini G, Migliavacca F, Jalali H, Camporini G, Redington A, Pietrabissa R. Use of computational fluid dynamics in the design of surgical procedures:application to the study of competitive flows in cavo-pulmonary connections. J Thorac Cardiovasc Surg 1996; 111:3502.
Guccione JM, Moonly SM, Wallace AW, Ratcliffe MB. Residual stress produced by ventricular volume reduction surgery has little effect on ventricular function and mechanics: a finite element model study. J Thorac Cardiovasc Surg 2001; 122:3592–599.
Lim KH, Candra J, Yeo JH, Duran CM. Flat or curved pericardial aortic valve cusps: a finite element study. J Heart Valve Dis 2004; 13:5792–797.
Kunzelman K, Reimink MS, Verrier ED, Cochran RP. Replacement of mitral valve posterior chordae tendineae with expanded polytetrafluoroethylene suture: a finite element study. J Card Surg 1996; 11:2136–145.
Grande-Allen KJ, Cochran RP, Reinhall PG, Kunzelman KS. Re-creation of sinuses is important for sparing the aortic valve: a finite element study. J Thorac Cardiovasc Surg 2000; 119:4 Pt 1753–763.(Shahrul Hashim and David )
Abstract
Finite element (FE) method is a form of mathematical modelling used to study the stress, strain and dynamics of a structure. The method has been used to analyse anatomical structures in the human body. In cardiac surgery it allows the quantifying of patient specific geometric features to both assess disease state and potentially aid in planning surgical intervention such as mitral valve repairs, prosthetic valve design or congenital reconstructive cardiac surgery. Important variables for a finite element model are model geometry, the type of mesh elements used, material properties and boundary condition. This review explains these variables, their limitations and describes their application in cardiac surgery.
Key Words: Finite element method; Cardiac surgery
1. Introduction
Finite Element (FE) method is a mathematical method that evolved in the late 1950s for structural analysis in the aerospace industry. Researchers have since adopted finite element method as a standard numerical technique for analysing complex structures and solving problems in solid and fluid mechanics. Recently there has been an increased use of FE method studies published in cardiac surgery. It is important for surgeons to have a level of understanding about the generation of FE models, their limitations and potential applications. This article aims to provide an over view of Finite Element Method and review studies published involving the technique in cardiac surgery.
2. Basic principles of finite element method
The main purpose in FE analysis is to define the relationship between force and deformation. It analyses the stress and strain of a given structure. In FE method the object under study is divided into small finite segments known as elements. The elements are assembled together and connected by nodes. The forces and deformation of each element will affect the behaviour of each adjacent element through the connecting nodes. The behaviour of a structure is represented by the displacement of these elements and their material properties.
The main parameters to be defined are stress and strain. Stress is the strength of the force from interactions such as stretching, squeezing or twisting. Stress is characterised as force per unit area. Strain is the resulting deformation. The stress/strain relationship defines how a tissue deforms under a given force [1]. To model a given structure using the FE analysis, three key inputs into the analysis must be specified without ambiguity. These are the model geometry, the material properties and the boundary conditions.
2.1. Model geometry
The geometry of a biological organ or structures of most FE models are derived from MRI, CT or Angiography imaging. These images are converted into the final FE geometry using special commercially available computer software. By using individual patient CT, MRI or angiogram images, patient specific FE geometry may be generated and for basic default geometry of a specific organ an average of many individual images, can be generated. Currently there are also commercially available geometry models of the heart and the great vessels (Fig. 1). As explained earlier the FE model is represented by elements. In FE modelling there are several types of element named, mainly according to the shape of the element. The element can be of shell, cube or plate type. For biological structures the shell type is normally used. The size and the number of the elements used to represent a structure will determine the accuracy of the output from the finite elements study. Basically small size and a large number of elements will lead to increased accuracy at the expense of increased computational time [1].
2.2. Material properties
The material properties of the structure under study are normally derived from biomechanical studies of the tissue being modelled. This property has to be an accurate description of the tissue for the FE analysis to be reliable. A biological structure will have both anisotropic and non-linear properties. Anisotropic tissue will deform differently depending on the direction of force. Non-linearity occurs when the rate of deformation changes as the force increases. Many researchers make the assumption that biological material is linear and isotropic due to the complexity and very high computational time that is required when modelling a non-linear and anisotropic object. Although this may affect the absolute value of the resulting stress of a structure studied, it however, normally will not affect the pattern of the stress across the whole structure.
In the FE model the material properties are represented by the Young modulus, which characterises the stiffness of the structure, and the Poisson ratio, which characterises the compressibility of the tissue (Table 1) [1]. Very stiff objects will have higher Young modulus and incompressible objects will have a higher Poisson ratio.
2.3. Boundary conditions
These are the forces or restrictions that apply to the surfaces of the geometry. These boundary conditions describe how the model is attached, and interacts with its surroundings. Examples would be the load applied to the aortic valve during diastole, the attachments of vessels or the restriction of radial movement of the aortic root under pressure from inside the aorta. The boundary conditions can be assumptions or can be accurate simulation of real life. An example would be the pressure recorded in the left ventricle during MRI acquisition of a heart, which will subsequently become the final geometry for the FE model. The simulation of a structure of an FE model depends heavily on the accurate boundary conditions. Spurious results may be generated if these boundary conditions are inaccurate and do not represent the surroundings of the real life structure being modelled.
3. Computational fluid dynamics and FE method
The addition of blood within the solid structure allows accurate analysis of the flow within the structure analysed. The simulation of solid and fluid structure requires special methods to allow coupling of these two elements together. One of the most often used methods is called the Arbitrary Lagrangian-Eulerian (ALE) method. This technique allows numerical simulation of multidimensional fluid dynamics and non-linear solid mechanics. With this technique clear delineation of free surfaces and fluid solid structure can be achieved [1].
4. FE model analysis
The output analysis of an FE model allows one to calculate the stress or strain of the model using commercially available software [2] (Video 1). Examples are LS DYNA or ABACUS. The computational time taken to analyse an FE model takes into account the size of the mesh of the geometry, the complexity of the boundary condition and the material properties. Highly complex mesh and non-linear structures can sometimes take days of computational times.
5. Model validation
Before an FE model can be applied to a clinical setting, the model needs to be validated. The validation process can be done by obtaining data from other different, well known clinical investigations like Doppler, catheterisation or MR studies or in vitro studies. Test results from multiple FE simulations can be compared with these clinical studies and, if comparable, the FE simulation can be assumed to be accurate.
6. FE models in cardiac surgery
The following are examples of FE simulation in cardiac surgery. They range from analysis of normal structures to simulation of diseases and planning of surgical procedures (Table 2).
6.1. Left ventricle
Cups et al. examined the effect of stress on the LV wall of patients with severe aortic regurgitation and in patients with normal aortic valves. In the study, the MRI images of these patients were used as the geometry input and the measured systolic and diastolic pressure specific to the patients were used as the boundary conditions. In their study the stress generated on the regional and global LV wall of patients with severe aortic regurgitation were significantly higher than in patients with normal valves. This illustrates the way FE method can study the stress of a structure non-invasively [3].
6.2. Mitral valve
Normal mitral valve function was examined by Kunzelman et al. looking at the deformation and stress under systemic loading conditions. The simulation demonstrated that annular contraction promotes valve closure whilst papillary muscle contraction tends to pull the leaflets apart. The combination of these two forces showed even stress distribution on valve leaflets [4].
Different leaflet and annular shape can be studied by changing the geometry as demonstrated by Salgo et al. In their study, increasing the annular height to commissural width ratio of 20% led to 37-fold reduction of the leaflet stress [5]. Similarly by changing the material properties of the valve leaflet simulation of stiff leaflet and the resulting stress can be achieved [6].
6.3. Aortic valve
Gnyasaneshwar et al. analysed the dynamics of the normal aortic valve with FE method. In this model the geometry of the valve was derived from MRI images and the material characteristics were assumed to be linear elastic, with the leaflet and the aortic sinus anisotropic. The study showed that the leaflets open initially with dilatation of the aortic root [7]. Using a combination of an FE method and fluid dynamics Nicosia et al. demonstrated the flow patterns of the blood across the valve and at the sinuses [8]. In studies of aortic valve insufficiency secondary to ageing or in Marfan syndrome, the resulting stress and strain from changes on the valve leaflets and the aortic root have been simulated by changing the geometry and the material properties of the models [9,10].
6.4. Congenital heart surgery
In a study of Total Cavopulmonary Connection the geometry of the investigated anastomosis was derived from a combination of angiography and MRI images. The boundary conditions at the inlet and outlet of the models were specified as the blood pressure and velocity of blood flows that were derived from Doppler measurements. The study showed the abrupt change in the geometry and velocity in the connection leading to energy loss with most of the superior vena cava flow going preferentially to the larger right lung. As the flow in this anastomosis is dependent on minimum energy loses and needs optimum flow distribution to both lungs, by changing the geometry, the FE method simulated a different kind of IVC anastomosis to identify the optimum flow pattern. The result of the study has led to alteration of the surgical technique [11].
6.5. Surgical procedures and valve designs
FE studies can analyse the effect of surgical procedures and valve designs. It has been used to analyse the effect of left ventricular reduction surgery and the resulting global and regional stress [12]. In prosthetic heart valve design FE method has allowed researchers to alter the shape of leaflets or use different material properties to study the net effect of the stress generated [13]. In mitral valve repair the FE method has been used to evaluate the effect of different repair techniques on leaflet stress distribution, essential for the longevity of the repair [14]. In aortic valve sparing operations FE methods were used to study the influence of different graft shape on the native aortic valve leaflet stress pattern with a view to optimise the surgical techniques [15].
7. Conclusion
We have shown multiple applications of FE simulation technology. These include the study of normal valve structure, the study of diseases mechanisms, the planning of medical treatments and design of artificial valves. The long-term future direction of the Finite Element method would be a patient-specific model using image-based geometric modelling techniques such as CT and MRI to construct the patient-specific anatomic models. The scope of Finite Element methods is very wide in cardiac surgery and once the models have evolved and are validated, this method may be used as routinely as echocardiogram or MRI scan.
Video 1. Example of stress analysis on the heart and the great vessels following blunt trauma to the chest.
References
Becker AA. An introductory guide to finite element analysis 2004;New York: ASME Press.
Richens D, Field M, Hashim S, Neale M, Oakley C. A finite element model of blunt traumatic aortic rupture. Eur J Cardiothorac Surg 2004; 25:1039–1047.
Cupps BP, Moustakidis P, Pomerantz BJ, Vedala G, Scheri RP, Kouchoukos NT, Davila-Roman VG, Pasque MK. Severe aortic insufficiency and normal systolic function: determining regional left ventricular wall stress by finite-element analysis. Ann Thorac Surg 2003; 76:3668–675.
Kunzelman KS, Cochran RP, Verrier ED, Eberhart RC. Anatomic basis for mitral valve modelling. J Heart Valve Dis 1994; 3:5491–496.
Salgo IS, Gorman 3rd JH, Gorman RC, Jackson BM, Bowen FW, Plappert T, St John Sutton MG, Edmunds Jr LH. Effect of annular shape on leaflet curvature in reducing mitral leaflet stress. Circulation 2002; 106:6711–717.
Kunzelman KS, Quick DW, Cochran RP. Altered collagen concentration in mitral valve leaflets: biochemical and finite element analysis. Ann Thorac Surg 1998; 66:6 SupplS198–205.
Gnyaneshwar R, Kumar RK, Balakrishnan KR. Dynamic analysis of the aortic valve using a finite element model. Ann Thorac Surg 2002; 73:41122–1129.
Nicosia MA, Cochran RP, Einstein DR, Rutland CJ, Kunzelman KS. A coupled fluid-structure finite element model of the aortic valve and root. J Heart Valve Dis 2003; 12:6781–789.
Grande-Allen KJ, Cochran RP, Reinhall PG, Kunzelman KS. Mechanisms of aortic valve incompetence: finite-element modeling of Marfan syndrome. J Thorac Cardiovasc Surg 2001; 122:5946–954.
Grande KJ, Cochran RP, Reinhall PG, Kunzelman KS. Mechanisms of aortic valve incompetence in aging: a finite element model. J Heart Valve Dis 1999; 8:2149–156.
de Leval MR, Dubini G, Migliavacca F, Jalali H, Camporini G, Redington A, Pietrabissa R. Use of computational fluid dynamics in the design of surgical procedures:application to the study of competitive flows in cavo-pulmonary connections. J Thorac Cardiovasc Surg 1996; 111:3502.
Guccione JM, Moonly SM, Wallace AW, Ratcliffe MB. Residual stress produced by ventricular volume reduction surgery has little effect on ventricular function and mechanics: a finite element model study. J Thorac Cardiovasc Surg 2001; 122:3592–599.
Lim KH, Candra J, Yeo JH, Duran CM. Flat or curved pericardial aortic valve cusps: a finite element study. J Heart Valve Dis 2004; 13:5792–797.
Kunzelman K, Reimink MS, Verrier ED, Cochran RP. Replacement of mitral valve posterior chordae tendineae with expanded polytetrafluoroethylene suture: a finite element study. J Card Surg 1996; 11:2136–145.
Grande-Allen KJ, Cochran RP, Reinhall PG, Kunzelman KS. Re-creation of sinuses is important for sparing the aortic valve: a finite element study. J Thorac Cardiovasc Surg 2000; 119:4 Pt 1753–763.(Shahrul Hashim and David )