A one-dimensional finite element method for simulation-based medical planning for cardiovascular disease

We have previously described a new approach to planning treatments for cardiovascular disease, Simulation-Based Medical Planning, whereby a physician utilizes computational tools to construct and evaluate a combined anatomic/physiologic model to predict the outcome of alternative treatment plans for an individual patient. Current systems for Simulation-Based Medical Planning utilize finite element methods to solve the time-dependent, three-dimensional equations governing blood flow and provide detailed data on blood flow distribution, pressure gradients and locations of flow recirculation, low wall shear stress and high particle residence. However, these methods are computationally expensive and often require hours of time on parallel computers. This level of computation is necessary for obtaining detailed information about blood flow, but likely is unnecessary for obtaining information about mean flow rates and pressure losses. We describe, herein, a space-time finite element method for solving the one-dimensional equations of blood flow. This method is applied to compute flow rate and pressure in a single segment model, a bifurcation, an idealized model of the abdominal aorta, in three alternate treatment plans for a case of aorto-iliac occlusive disease and in a vascular bypass graft. All of these solutions were obtained in less than 5 min of computation time on a personal computer.     

On the paths of fluid particles in an axisymmetrical aneurysm

The aim of this study is the characterization of the pulsatile flow field by demonstration of the paths of single particles in a model of an axisymmetric aneurysm. The detailed analysis of the flow field can give additional information on the flow pattern and the time of transition of blood particles in the segment. The basis of the calculations is the system of the Navier-Stokes equations for incompressible Newtonian fluid flow. To solve these equations numerically the finite element method was used. The trajectory equations for a fluid particle were solved by use of a predictor-corrector procedure. The results of the computer simulation demonstrate the development, shift and disappearance of vortices in the excavation and give references to zones of stasis. This behavior can be an important factor in thrombogenesis.     

Numerical simulations of steady flow inside a three dimensional aortic bifurcation model

A finite element formulation of the Navier-Stokes equations for three dimensional flow is presented. The equations are solved using the finite element method. The model is constructed from a cast of a human aortic bifurcation. The numerical problems introduced by solving the equation system are discussed and special attention is paid to the selection of the linear equation solver. The simulations of the steady blood flow patterns in an aortic bifurcation is shown. The results of the numerical analysis are presented as three dimensional plots of velocity vectors, wall shear vectors, streamlines and pressure isobars. The flow simulations are done for Reynolds number 10. The flow patterns found in the bifurcation model are discussed in connection with proposed theories to explain the event of early atherosclerosis.     

基于卷积神经网络-长短期记忆神经网络模型利用光学体积描记术重建动脉血压波信号

目的 直接动脉血压（arterial blood pressure,ABP）连续监测是侵入式的,传统袖带式的间接血压测量法无法实现连续监测。既往利用光学体积描记术（photoplethysmography,PPG）实现了连续无创血压监测,但其为收缩压和舒张压的离散值,而非ABP波的连续值,本研究期望基于卷积神经网络-长短期记忆神经网络（CNN-LSTM）利用PPG信号波重建ABP波信号,实现连续无创血压监测。方法 构建CNN-LSTM混合神经网络模型,利用重症监护医学信息集（medical information mart for intensive care,MIMIC）中的PPG与ABP波同步记录信号数据,将PPG信号波经预处理降噪、归一化、滑窗分割后输入该模型,重建与之同步对应的ABP波信号。结果 使用窗口长度312的CNN-LSTM神经网络时,重建ABP值与实际ABP值间误差最小,平均绝对误差（mean absolute error,MAE）和均方根误差（root mean square error,RMSE）分别为2.79 mmHg和4.24 mmHg,余弦相似度最大,重建ABP值与实际ABP值一致性和相关性情况良好,符合美国医疗器械促进协会（Association for the Advancement of Medical Instrumentation,AAMI）标准。结论 CNN-LSTM混合神经网络可利用PPG信号波重建ABP波信号,实现连续无创血压监测。     

A Digital Model for the Venous Junctions

The venous network in the lower limbs is composed of a considerable number of confluent junctions. Each of these singularities introduces some blood flow disturbances. Each physiological junction is unique, in terms of its geometry as well as the blood flow rate. In order to account for this great variability, we developed a numerical model based on the use of the N3S code (a software package for solving Navier-Stokes equations). To test the validity of the model, one of the numerical simulations is compared with the data obtained in the corresponding experimental configuration. The velocity measurements were carried out with an ultrasonic pulsed Doppler velocimeter. We also measured pressure differences using differential sensors. The numerical computations were then used to obtain the values of the flow variables at any point, with various geometrical and flow configurations. As far as the velocity field is concerned, a very marked three-dimensional pattern with swirls was observed. The pressure evolution was also strongly disturbed, with a non-linear decrease. All these data indicate that confluence effects cannot be neglected when evaluating pressure decreases. With a tool of this kind, it is possible to accurately predict the disturbances associated with any geometrical configuration or any flow rate.     

Numerical 3D-simulation of pulsatile wall shear stress in an arterial T-bifurcation model

The structure of pulsatile blood flow and wall shear stress in a 90° T-bifurcation model is analysed numerically. The nonlinear Navier-Stokes equations for time-dependent incompressible Newtonian fluid flow are approximated using a newly developed pressure correction, finite element method. The wall shear stress is calculated from the finite element velocity field. The investigation shows viscous flow phenomena such as flow separation and stagnation and the distribution of high and low wall shear stress during the pulse cycle. Furthermore, the effect of a sharp corner the bifurcation edge on the wall shear stress is analysed. Detailed local flow investigation is required to examine fluid dynamic contribution to the development of arterial diseases such as atherosclerosis and thrombosis.     

A model for blood flow through a stenotic tube

This article deals with the introduction of the modified Casson's fluid model as the true representation for the blood for the steady laminar flow through a small diameter artery with axi-symmetric identical double stenoses in series. The governing equations are solved by using the finite element method. The results for the velocity profiles, the pressure and the wall shear stress distributions in addition to the location and length of the flow reversal zones have been brought out and discussed in reference to the severity of the disease. It has been observed that the non-Newtonian nature of the blood helps in reducing the magnitude of the peak wall shear stress at the throat and the length of the reversed flow regions in the post stenotic dilatation.     

Numerical simulation of blood flow in the left ventricle and aortic sinus using magnetic resonance imaging and computational fluid dynamics

Understanding cardiac blood flow patterns has many applications in analysing haemodynamics and for the clinical assessment of heart function. In this study, numerical simulations of blood flow in a patient-specific anatomical model of the left ventricle (LV) and the aortic sinus are presented. The realistic 3D geometry of both LV and aortic sinus is extracted from the processing of magnetic resonance imaging (MRI). Furthermore, motion of inner walls of LV and aortic sinus is obtained from cine-MR image analysis and is used as a constraint to a numerical computational fluid dynamics (CFD) model based on the moving boundary approach. Arbitrary Lagrangian–Eulerian finite element method formulation is used for the numerical solution of the transient dynamic equations of the fluid domain. Simulation results include detailed flow characteristics such as velocity, pressure and wall shear stress for the whole domain. The aortic outflow is compared with data obtained by phase-contrast MRI. Good agreement was found between simulation results and these measurements.     

Anisotropic adaptive finite element method for modelling blood flow

In this study, we present an adaptive anisotropic finite element method (FEM) and demonstrate how computational efficiency can be increased when applying the method to the simulation of blood flow in the cardiovascular system. We use the SUPG formulation for the transient 3D incompressible Navier–Stokes equations which are discretised by linear finite elements for both the pressure and the velocity field.

Given the pulsatile nature of the flow in blood vessels we have pursued adaptivity based on the average flow over a cardiac cycle. Error indicators are derived to define an anisotropic mesh metric field. Mesh modification algorithms are used to anisotropically adapt the mesh according to the desired size field. We demonstrate the efficiency of the method by first applying it to pulsatile flow in a straight cylindrical vessel and then to a porcine aorta with a stenosis bypassed by a graft. We demonstrate that the use of an anisotropic adaptive FEM can result in an order of magnitude reduction in computing time with no loss of accuracy compared to analyses obtained with uniform meshes.     

Automatic noninvasive measurement of systolic blood pressure using photoplethysmography

Background

Systolic blood flow has been simulated in the abdominal aorta and the superior mesenteric artery. The simulations were carried out using two different computational hemodynamic methods: the finite element method to solve the Navier Stokes equations and the lattice Boltzmann method.

Results

We have validated the lattice Boltzmann method for systolic flows by comparing the veloCity and pressure profiles of simulated blood flow between methods. We have also analyzed flow-specific characteristics such as the formation of a vortex at curvatures and traces of flow.

Conclusion

The lattice Boltzmann Method is as accurate as a Navier Stokes solver for computing complex blood flows. As such it is a good alternative for computational hemodynamics, certainly in situation where coupling to other models is required.     

Multiscale modelling as a tool to prescribe realistic boundary conditions for the study of surgical procedures

This work was motivated by the problems of analysing detailed 3D models of vascular districts with complex anatomy. It suggests an approach to prescribing realistic boundary conditions to use in order to obtain information on local as well as global haemodynamics. A method was developed which simultaneously solves Navier-Stokes equations for local information and a non-linear system of ordinary differential equations for global information. This is based on the principle that an anatomically detailed 3D model of a cardiovascular district can be achieved by using the finite element method. In turn the finite element method requires a specific boundary condition set. The approach outlined in this work is to include the system of ordinary differential equations in the boundary condition set. Such a multiscale approach was first applied to two controls: (i) a 3D model of a straight tube in a simple hydraulic network and (ii) a 3D model of a straight coronary vessel in a lumped-parameter model of the cardiovascular system. The results obtained are very close to the solutions available for the pipe geometry. This paper also presents preliminary results from the application of the methodology to a particular haemodynamic problem: namely the fluid dynamics of a systemic-to-pulmonary shunt in paediatric cardiac surgery.     

The Effect of Perfusion on Soft Tissue Mechanical Properties: A Computational Model

Some simple finite element models were constructed to investigate the magnitude and character of changes in mechanical properties of very soft tissues due to the loss of perfusion. Changes in the apparent stress-strain curve were used to characterise the effect of simulated blood perfusion pressure on the engineering stress-strain curve. The results indicated that the blood to tissue volume ratio and the perfusion pressure have the strongest effect on the effective stress-strain response of a representative tissue cell. Tissue viscoelasticity increased the sensitivity of the system to perfusion pressure changes.     

An integro-partial differential equation for modeling biofluids flow in fractured biomaterials

A novel mathematical model in the framework of a nonlinear integro-partial differential equation governing biofluids flow in fractured biomaterials is proposed, solved, verified, and evaluated. A semi-analytical solution is derived for the equation, verified by a mass-lumped Galerkin finite element method (FEM), and calibrated with two in vitro experimental datasets. The solution process uses separation of variables and results in explicit expression involving complete and incomplete beta functions. The proposed semi-analytical model shows reasonable agreements with the finite element simulator as well as with two in vitro experimental time series and can be successfully used to simulate biofluids (e.g. water, blood, oil, etc.) flow in natural and synthetic porous biomaterials.     

A Finite Element Model for Ice Ball Evolution in a Multi-probe Cryosurgery

The ice formation in a water body is examined for the computation of temperature field, phase change and a moving ice–water interface whose location is not known à priori. This is classically referred to as the Stefan problem [Rubinstein, L.I. (1971) The Stefan Problem (American Mathematical Society, Providence, Rhode Island 02904]. Based on the Duvaut [Duvaut, G. (1973) “Résolution d'un problème de Stefan” C.R. Acad. Sci. Paris 276, 1461–1463] transformation, the governing equations for heat conduction are formulated within a variational principle that is readily amenable to a standard finite element solution without remeshing. Numerical simulation results pertaining to the freezing of tumour tissue in a multi-cryoprobe cryosurgery are presented. These results lend both quantitative and graphical support to the current empirical standards of “effective therapy” in view of refining clinical applications.     

Accounting for spatial variation of trabecular anisotropy with subject-specific finite element modeling moderately improves predictions of local subchondral bone stiffness at the proximal tibia

IntroductionPreviously, a finite element (FE) model of the proximal tibia was developed and validated against experimentally measured local subchondral stiffness. This model indicated modest predictions of stiffness (R² = 0.77, normalized root mean squared error (RMSE%) = 16.6%). Trabecular bone though was modeled with isotropic material properties despite its orthotropic anisotropy. The objective of this study was to identify the anisotropic FE modeling approach which best predicted (with largest explained variance and least amount of error) local subchondral bone stiffness at the proximal tibia.MethodsLocal stiffness was measured at the subchondral surface of 13 medial/lateral tibial compartments using in situ macro indentation testing. An FE model of each specimen was generated assuming uniform anisotropy with 14 different combinations of cortical- and tibial-specific density-modulus relationships taken from the literature. Two FE models of each specimen were also generated which accounted for the spatial variation of trabecular bone anisotropy directly from clinical CT images using grey-level structure tensor and Cowin’s fabric-elasticity equations. Stiffness was calculated using FE and compared to measured stiffness in terms of R² and RMSE%.ResultsThe uniform anisotropic FE model explained 53–74% of the measured stiffness variance, with RMSE% ranging from 12.4 to 245.3%. The models which accounted for spatial variation of trabecular bone anisotropy predicted 76–79% of the variance in stiffness with RMSE% being 11.2–11.5%.ConclusionsOf the 16 evaluated finite element models in this study, the combination of Synder and Schneider (for cortical bone) and Cowin’s fabric-elasticity equations (for trabecular bone) best predicted local subchondral bone stiffness.     

Low Reynolds number steady state flow through a branching network of rigid vessels II. A finite element mixture model

This research aims at formulating and verifying a finite element mixture formulation for blood perfusion. The equations derived in a companion paper [3] are discretized according to the Galerkin method. A flow experiment in a rigid model of a vascular tree of about 500 vessels is performed in order to verify the finite element mixture formulation. Although the comparison of numerical results and experimental measurements is not conclusive as far as the validity of the theory is concerned, the results do suggest that the finite element model has predictive power in the case of low Reynolds number steady-state flow of a Newtonian fluid in a rigid vascular tree.     

Multiscale methodology for bone remodelling simulation using coupled finite element and neural network computation

The aim of this paper is to develop a multiscale hierarchical hybrid model based on finite element analysis and neural network computation to link mesoscopic scale (trabecular network level) and macroscopic (whole bone level) to simulate the process of bone remodelling. As whole bone simulation, including the 3D reconstruction of trabecular level bone, is time consuming, finite element calculation is only performed at the macroscopic level, whilst trained neural networks are employed as numerical substitutes for the finite element code needed for the mesoscale prediction. The bone mechanical properties are updated at the macroscopic scale depending on the morphological and mechanical adaptation at the mesoscopic scale computed by the trained neural network. The digital image-based modelling technique using μ-CT and voxel finite element analysis is used to capture volume elements representativeof 2 mm³ at the mesoscale level of the femoral head. The input data for the artificial neural network are a set of bone material parameters, boundary conditions and the applied stress. The output data are the updated bone properties and some trabecular bone factors. The current approach is the first model, to our knowledge, that incorporates both finite element analysis and neural network computation to rapidly simulate multilevel bone adaptation.     

Central Pressure Appraisal: Clinical Validation of a Subject-Specific Mathematical Model

IntroductionCurrent evidence suggests that aortic blood pressure has a superior prognostic value with respect to brachial pressure for cardiovascular events, but direct measurement is not feasible in daily clinical practice.AimThe aim of the present study is the clinical validation of a multiscale mathematical model for non-invasive appraisal of central blood pressure from subject-specific characteristics.MethodsA total of 51 young male were selected for the present study. Aortic systolic and diastolic pressure were estimated with a mathematical model and were compared to the most-used non-invasive validated technique (SphygmoCor device, AtCor Medical, Australia). SphygmoCor was calibrated through diastolic and systolic brachial pressure obtained with a sphygmomanometer, while model inputs consist of brachial pressure, height, weight, age, left-ventricular end-systolic and end-diastolic volumes, and data from a pulse wave velocity study.ResultsModel-estimated systolic and diastolic central blood pressures resulted to be significantly related to SphygmoCor-assessed central systolic (r = 0.65 p <0.0001) and diastolic (r = 0.84 p<0.0001) blood pressures. The model showed a significant overestimation of systolic pressure (+7.8 (-2.2;14) mmHg, p = 0.0003) and a significant underestimation of diastolic values (-3.2(-7.5;1.6), p = 0.004), which imply a significant overestimation of central pulse pressure. Interestingly, model prediction errors mirror the mean errors reported in large meta-analysis characterizing the use of the SphygmoCor when non-invasive calibration is performed.ConclusionIn conclusion, multi-scale mathematical model predictions result to be significantly related to SphygmoCor ones. Model-predicted systolic and diastolic aortic pressure resulted in difference of less than 10 mmHg in the 51% and 84% of the subjects, respectively, when compared with SphygmoCor-obtained pressures.