Numerical optimization methods can be employed to tune the parameter values of the reduced-order models to match measured flow distribution or pressure values in the patient-specific model Using this approach to boundary condition specification, flow distribution and pressure wave propagation arise naturally through the coupling between the patient-specific three-dimensional model and the reduced-order model 44 , Figure 3 depicts an application of this approach to modeling blood flow in the pulmonary arteries of a year-old male with repaired tetralogy of Fallot and left pulmonary arterial stenosis.
Main pulmonary arterial flow was prescribed at the inlet of the computational model and morphometry-based impedance boundary conditions were prescribed at each outlet While, in this case, a one-dimensional nonlinear wave propagation model was used in the upstream domain, this example illustrates the importance of assigning realistic boundary conditions in patient-specific blood flow simulations.
A relatively new development in boundary condition specification has arisen due to difficulties in solving a class of problems where the patient-specific model is truncated in a region where flow is complex, recirculating or retrograde and a weakly enforced lumped-parameter or distributed reduced-order model is used to represent the downstream domain. Such numerical difficulties are inherent in problems related to modeling hemodynamics in the aorta or pulmonary arteries.
Formaggia et al. However, this approach requires an unconventional formulation of the Navier-Stokes equations and has not yet been proven to resolve boundary instability issues in complex hemodynamic simulations.
An alternate solution to this problem proposed by Kim et al. This approach, applied to the same boundaries where a lumped-parameter or distributed network boundary condition is assigned, has the remarkable effect of stabilizing many problems with complex flows near outlet boundaries. Blood can be accurately represented as an incompressible fluid whose constitutive behavior is usually approximated by a Newtonian model. Arterial blood flow has been traditionally represented using the Incompressible Navier-Stokes equations in a fixed Eulerian frame of reference.
However, blood velocity and pressure fields can be greatly influenced by the motion of external or internal vascular structures, such as the contracting cardiac muscle, moving heart valves, or deforming large arteries of the body.
In these situations, one must characterize the mechanical behavior of the moving vascular structure usually in a Lagrangian frame of reference , and its interactions with the blood flow. Modeling the interactions between an incompressible blood flow and a deforming vascular structure represents one of the major challenges in the field of cardiovascular mechanics. The evolution of this interface is determined by kinematic continuity of velocities and dynamic continuity of forces compatibility conditions between the blood flow and the vascular structure. The evolution of the ALE formulation is a boundary-fitting technique, where the fluid-solid interface is accurately captured due to continuous changes of the fluid grid.
However, in situations where the motion of the vascular structure is large, ALE formulations may result in time-consuming computations. Currently, modular also known as staggered and non-modular also known as monolithic pre-conditioners for the solution of the coupled algebraic system resulting from the space-time discretization of the FSI problem represent an active area of research.
Modular pre-conditioners allow for the use of independent, specialized fluid and structure solvers coupled via a relatively simple iterative scheme. However, these algorithms usually exhibit poor convergence behavior, especially in problems with large "added mass effect", where the density of the structure is comparable to the density of the fluid, as occurs in cardiovascular FSI problems.
Alternatively, non-modular pre-conditioners require a more elaborate coupling between the fluid and solid solvers, but they usually result in faster, more stable algorithms. Geometric Conservation Laws that govern the evolution of geometric parameters, including grid positions and velocities, have been developed to avoid overly-diffusive grid motion strategies. A Geometric Conservation Law requires that motion of grid is computed in a way such that the numerical scheme is able to reproduce exactly a constant solution.
The use of ALE formulations in cardiovascular applications was pioneered by Perktold and collaborators 54 , More recently, Gerbeau and colleagues 56 , 57 , 58 have performed pressure wave propagation simulations in cerebral aneurysm and carotid bifurcation models; Hughes and colleagues 59 , 60 have performed patient-specific FSI simulations using an isogeometric framework; Van de Vosse and colleagues 61 investigated a patient-specific abdominal aortic aneurysm model.
The Immersed Boundary Method, first introduced by Peskin 62 , 63 , is a non-boundary-fitting formulation that does not require any changes on the fluid computational grid. This formulation was first developed in the context of finite-difference approximations for the fluid domain, with a set of non-conforming, interconnected, elastic boundary points representing the structure. This structure interacts with the fluid via the introduction of body forces applied on the fluid domain at the position of the solid points.
The Immersed Boundary Method has used in cardiovascular applications by Lemmon and Yoganathan 64 to examine left ventricular dysfunction; Watton et al. On the microscales, the Immersed Boundary Method has also been applied to modeling the interactions of red blood cells and plasma in the mesocirculation The Fictitious Domain method was first developed by Glowinski et al.
Although closely related to the Immersed Boundary Method, the Fictitious domain method was developed in a finite-element context, introducing Lagrange multipliers to constrain the motion of the fluid and the solid at the interface.
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Later, Baaijens 69 developed an extension of the method suitable for slender structures. The method has been successfully applied to FSI simulations of the aortic valve 70 , Van de Vosse and colleagues 72 proposed a combination of ALE and Fictitious Domain methods and applied it to the simulation of valve dynamics in a simple left ventricular flow models. More recently, Van Loon et al.
In this formulation developed by Tezduyar et al. Here, the motion of the unknown boundary is defined in terms of other variables, such as velocities or displacements of the interface. This formulation has been applied to study FSI in a patient-specific model of a cerebral aneursym Figueroa, Taylor and colleagues 49 recently developed a Coupled-Momentum method to simulate blood flow and vessel deformation in arteries. This method embeds the elasto-dynamics equations into the variational form of the fluid via the definition of a fictitious body force that drives the motion of the vessel.
By using a thin-wall assumption, this body force is related to the traction at the fluid-solid interface. The formulation results in a robust, monolithic scheme that is highly efficient for large scale fluid-structure interaction and wave propagation problems provided that the assumptions of small deformation and thin walls are reasonable.
For example, Figure 5 shows results from a patient-specific model of blood flow from the aorta to the cerebral arteries for a patient with a right middle cerebral aneurysm. This computation included 3-element Windkessel outlet boundary conditions implemented using the Coupled Multidomain method 44 , constraints on the shape of outlet velocity profiles 51 and fluid-structure interactions including spatially varying tissue properties modeled using the Coupled Momentum method Figure 6 depicts the velocity magnitude and pressure fields at peak systole and mid-diastole for a patient-specific simulation of blood flow and vessel dynamics in the aorta of a 10 year old patient with aortic coarctation In this simulation, the Coupled Multidomain method 44 and Augmented Lagrangian methods 51 are used to couple the three-dimensional patient-specific model to a lumped-parameter heart model at the inlet, and 3-element Windkessel models at the outlets and to constrain the shape of velocity profiles at inlet and outlet boundaries.
The Coupled Momentum method 49 was used to solve the fluid-structure interaction problem on a finite element mesh with 2,, elements, , nodes and a time step size of 0. Velocity magnitude and pressure fields at peak systole and mid-diastole for a patient-specific simulation of blood flow and vessel dynamics in the aorta of a 10 year old patient with aortic coarctation. New imaging techniques enable the characterization of the motion and thickness of vascular structures; however, it remains a challenge to incorporate this in vivo information into FSI formulations.
A new class of "image-based FSI" problems will require the development of methods to assign tissue properties to the computer model from the medical image data, via the solution of an inverse problem.
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Atherosclerosis, the most prevalent of the acquired cardiovascular diseases, involves the accumulation of fatty material in the intima inner layer of arteries supplying the brain, heart, other vital organs, and lower extremities. However, while the biochemical stimuli for the development of atherosclerosis are diffuse throughout the body, the disease is very focal, localizing at branches and bends of the arterial tree.https://emurunamglob.ga
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There is substantial and mounting evidence for the role of hemodynamic factors in the localization of atherosclerosis 76 - Furthermore, atherosclerotic arteries continue to adapt in response to hemodynamic forces, in a mechanism known as compensatory remodeling Patient-specific models have been widely used to quantify hemodynamic conditions in arteries for the purpose of understanding localization of atherosclerosis. Due to the strong correlation between hemodynamics and atherosclerosis in the carotid bifurcation, it is not surprising that this vascular region has been widely studied.
In particular, the Steinman group at the University of Western Ontario and the Ethier group at the University of Toronto have made important contributions to the field 5 , 80 - The group of Xu at Imperial College in London have also made significant contributions towards understanding hemodynamic conditions in the carotid artery 84 , The infrarenal abdominal aorta is another site prone to atherosclerosis likely due to unique hemodynamic conditions including low wall shear stress, high oscillatory shear and generally low flow, particularly for sedentary individuals 41 , 87 - Tang et al.
When averaged over all subjects, wall shear stress significantly increased, whereas oscillatory shear index OSI decreased between rest and exercise at the supraceliac, infrarenal, and suprabifurcation levels, and significant differences in wall shear stress were found between anterior and posterior sections.
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Of note, in the atherosclerosis prone infrarenal aorta, wall shear stress increased six-fold between rest and exercise. Finally, with the advent of small animal imaging techniques, image-based modeling techniques have been applied to quantify hemodynamic conditions in rodent models often used for disease research. For example, using small animal imaging and computational methods Greve et al.
Importantly, wall shear stress in the aorta of a mouse is more than fold higher than that in humans.
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Feintuch et al. The recent review by Humphrey and Taylor 95 thoroughly discussed the role of computational mechanics in intracranial saccular and abdominal aortic aneurysm research. As noted in that review, image-based modeling techniques have been used in the last few years to conduct patient-specific investigations with the aim of determining whether biomechanical factors influence aneurysm initiation, growth and rupture 8 - 10 , 96 - In regards to hemodynamic forces, low wall shear stress, high wall shear stress and elevated dynamic pressure have been proposed to influence aneurysm growth.
For example, Acevedo-Bolton et al. Very recently, using a longitudinal study of 7 patients with inoperable cerebral aneurysms, Boussel et al. Undoubtedly, patient-specific computational methods will be used increasingly by investigators studying aneurysm growth and rupture.
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Interventional and surgical therapies used in the treatment of congenital and acquired cardiovascular diseases attempt to restore blood flow to compromised organs and tissues. Unfortunately, alternate treatments cannot be tested in the patients and physicians do not have the tools needed to evaluate the multiple options and design the optimal corrective procedures.
Instead, the current paradigm for medical planning for the treatment of cardiovascular disease relies exclusively on diagnostic imaging data and physical measurements to define the present state of the patient, empirical data to evaluate the efficacy of prior treatments for similar patients, and the judgment and experience of the physician to decide on a preferred treatment. However, diagnostic imaging, physical measurements and empirical data are insufficient to predict the outcome of a given treatment for an individual patient due to anatomic and physiologic variations and system complexity.
For cardiovascular treatment planning, these simulation systems require an integrated set of software tools to model the effect of alternate treatments on blood flow. The key elements of such a system include a human-computer interface, image segmentation, geometric solid modeling, operative planning, discretization methods, fluid-structure interaction methods, and scientific visualization techniques. Simulation-based medical planning systems must be efficient and minimize user intervention so that results can be obtain in hours instead of days to weeks.
These systems must also accurately describe the key hemodynamic variables, especially flow rate and pressure. In , Taylor and colleagues developed a prototype software system, Advanced Surgical Planning Interactive Research Environment ASPIRE , for patient-specific cardiovascular surgery planning using computational methods for modeling blood flow.
ASPIRE was demonstrated in the plenary research forum at the Society for Vascular Surgery SVS meeting and used successfully by four prominent vascular surgeons to evaluate alternate surgical plans for a case of lower extremity occlusive disease 2. Wilson et al. Modular software architecture was utilized to enable the use of best-in-class component technology and create a single application to go from medical imaging data to analysis results. Figure 7 provides an overview of the predictive medicine approach as applied to designing bypass surgery for patient with occlusive cardiovascular disease in aorta and iliac arteries In the last several years, a few groups have started to apply image-based modeling techniques to the assessment of surgical procedures for patients with congenital heart diseases.
Notable amongst these efforts is the work of Yoganathan and colleagues , and Migliavacca, Bove, de Leval and colleagues 42 , 43 , - to assess the hemodynamic efficiency of the total cavopulmonary connection. Marsden et al.
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