An implicit solver for 1D arterial network models

Development of non-invasive biomarkers for pre-eclampsia through data-driven cardiovascular network models

Computational models can be at the basis of new powerful technologies for studying and classifying disorders like pre-eclampsia, where it is difficult to distinguish pre-eclamptic patients from non-pre-eclamptic based on pressure when patients have a track record of hypertension. Computational models now enable a detailed analysis of how pregnancy affects the cardiovascular system. Therefore, new non-invasive biomarkers were developed that can aid the classification of pre-eclampsia through the integration of six different measured non-invasive cardiovascular signals. Datasets of 21 pregnant women (no early onset pre-eclampsia, n = 12; early onset pre-eclampsia, n = 9) were used to create personalised cardiovascular models through computational modelling resulting in predictions of blood pressure and flow waveforms in all major and minor vessels of the utero-ovarian system. The analysis performed revealed that the new predictors PPI (pressure pulsatility index) and RI (resistance index) calculated in arcuate and radial/spiral arteries are able to differentiate between the 2 groups of women (t-test scores of p < .001) better than PI (pulsatility index) and RI (Doppler calculated in the uterine artery) for both supervised and unsupervised classification. In conclusion, two novel high-performing biomarkers for the classification of pre-eclampsia have been identified based on blood velocity and pressure predictions in the smaller placental vasculatures where non-invasive measurements are not feasible.

The effect of prone positioning on maternal haemodynamics and fetal wellbeing in the third trimester-A primary cohort study with a scoping review

INTRODUCTION

Supine sleep position is associated with stillbirth, likely secondary to inferior vena cava compression, and a reduction in cardiac output (CO) and uteroplacental perfusion. Evidence for the effects of prone position in pregnancy is less clear. This study aimed to determine the effect maternal prone position on maternal haemodynamics and fetal heart rate, compared with left lateral position.

METHODS

Twenty-one women >28 weeks' gestation underwent non-invasive CO monitoring (Cheetah) every 5 minutes and continuous fetal heart rate monitoring (MONICA) in left lateral (20 minutes), prone (30 minutes), followed by left lateral (20 minutes). Anxiety and comfort were assessed by questionnaires. Regression analyses (adjusted for time) compared variables between positions. The information derived from the primary study was used in an existing mathematical model of maternal circulation in pregnancy, to determine whether occlusion of the inferior vena cava could account for the observed effects. In addition, a scoping review was performed to identify reported clinical, haemodynamic and fetal effects of maternal prone position; studies were included if they reported clinical outcomes or effects or maternal prone position in pregnancy. Study records were grouped by publication type for ease of data synthesis and critical analysis. Meta-analysis was performed where there were sufficient studies.

RESULTS

Maternal blood pressure (BP) and total vascular resistance (TVR) were increased in prone (sBP 109 vs 104 mmHg, p = 0.03; dBP 74 vs 67 mmHg, p = 0.003; TVR 1302 vs 1075 dyne.s-1cm-5, p = 0.03). CO was reduced in prone (5.7 vs 7.1 mL/minute, p = 0.003). Fetal heart rate, variability and decelerations were unaltered. However, fetal accelerations were less common in prone position (86% vs 95%, p = 0.03). Anxiety was reduced after the procedure, compared to beforehand (p = 0.002), despite a marginal decline in comfort (p = 0.04).The model predicted that if occlusion of the inferior vena cava occurred, the sBP, dBP and CO would generally decrease. However, the TVR remained relatively consistent, which implies that the MAP and CO decrease at a similar rate when occlusion occurs. The scoping review found that maternal and fetal outcomes from 47 included case reports of prone positioning during pregnancy were generally favourable. Meta-analysis of three prospective studies investigating maternal haemodynamic effects of prone position found an increase in sBP and maternal heart rate, but no effect on respiratory rate, oxygen saturation or baseline fetal heart rate (though there was significant heterogeneity between studies).

CONCLUSION

Prone position was associated with a reduction in CO but an uncertain effect on fetal wellbeing. The decline in CO may be due to caval compression, as supported by the computational model. Further work is needed to optimise the safety of prone positioning in pregnancy.

TRIAL REGISTRATION

This trial was registered at clinicaltrials.gov (NCT04586283).

First blood: An efficient, hybrid one- and zero-dimensional, modular hemodynamic solver

Low-dimensional (1D or 0D) models can describe the whole human blood circulation, for example, 1D distributed parameter model for the arterial network and 0D concentrated models for the heart or other organs. This paper presents a combined 1D-0D solver, called first_blood, that solves the governing equations of fluid dynamics to model low-dimensional hemodynamic effects. An extended method of characteristics is applied here to solve the momentum, and mass conservation equations and the viscoelastic wall model equation, mimicking the material properties of arterial walls. The heart and the peripheral lumped models are solved with a general zero-dimensional (0D) nonlinear solver. The model topology can be modular, that is, first_blood can solve any 1D-0D hemodynamic model. To demonstrate the applicability of first_blood, the human arterial system, the heart and the peripherals are modelled using the solver. The simulation time of a heartbeat takes around 2 s, that is, first_blood requires only twice the real-time for the simulation using an average PC, which highlights the computational efficiency. The source code is available on GitHub, that is, it is open source. The model parameters are based on the literature suggestions and on the validation of output data to obtain physiologically relevant results.

Modeling Reactive Hyperemia to Better Understand and Assess Microvascular Function: A Review of Techniques

Reactive hyperemia is a well-established technique for the non-invasive evaluation of the peripheral microcirculatory function, measured as the magnitude of limb re-perfusion after a brief period of ischemia. Despite widespread adoption by researchers and clinicians alike, many uncertainties remain surrounding interpretation, compounded by patient-specific confounding factors (such as blood pressure or the metabolic rate of the ischemic limb). Mathematical modeling can accelerate our understanding of the physiology underlying the reactive hyperemia response and guide in the estimation of quantities which are difficult to measure experimentally. In this work, we aim to provide a comprehensive guide for mathematical modeling techniques that can be used for describing the key phenomena involved in the reactive hyperemia response, alongside their limitations and advantages. The reported methodologies can be used for investigating specific reactive hyperemia aspects alone, or can be combined into a computational framework to be used in (pre-)clinical settings.

Automating fractional flow reserve (FFR) calculation from CT scans: A rapid workflow using unsupervised learning and computational fluid dynamics

Fractional flow reserve (FFR) provides the functional relevance of coronary atheroma. The FFR-guided strategy has been shown to reduce unnecessary stenting, improve overall health outcome, and to be cost-saving. The non-invasive, coronary computerised tomography (CT) angiography-derived FFR (cFFR) is an emerging method in reducing invasive catheter based measurements. This computational fluid dynamics-based method is laborious as it requires expertise in multidisciplinary analysis of combining image analysis and computational mechanics. In this work, we present a rapid method, powered by unsupervised learning, to automatically calculate cFFR from CT scans without manual intervention.

Personalising cardiovascular network models in pregnancy: A two-tiered parameter estimation approach

Uterine artery Doppler waveforms are often studied to determine whether a patient is at risk of developing pathologies such as pre-eclampsia. Many uterine waveform indices have been developed, which attempt to relate characteristics of the waveform with the physiological adaptation of the maternal cardiovascular system, and are often suggested to be an indicator of increased placenta resistance and arterial stiffness. Doppler waveforms of four patients, two of whom developed pre-eclampsia, are compared with a comprehensive closed-loop model of pregnancy. The closed-loop model has been previously validated but has been extended to include an improved parameter estimation technique that utilises systolic and diastolic blood pressure, cardiac output, heart rate, and pulse wave velocity measurements to adapt model resistances, compliances, blood volume, and the mean vessel areas in the main systemic arteries. The shape of the model-predicted uterine artery velocity waveforms showed good agreement with the characteristics observed in the patient Doppler waveforms. The personalised models obtained now allow a prediction of the uterine pressure waveforms in addition to the uterine velocity. This allows for a more detailed mechanistic analysis of the waveforms, eg, wave intensity analysis, to study existing clinical indices. The findings indicate that to accurately estimate arterial stiffness, both pulse pressure and pulse wave velocities are required. In addition, the results predict that patients who developed pre-eclampsia later in pregnancy have larger vessel areas in the main systemic arteries compared with the two patients who had normal pregnancy outcomes.

A physiologically realistic virtual patient database for the study of arterial haemodynamics

This study creates a physiologically realistic virtual patient database (VPD), representing the human arterial system, for the primary purpose of studying the effects of arterial disease on haemodynamics. A low dimensional representation of an anatomically detailed arterial network is outlined, and a physiologically realistic posterior distribution for its parameters constructed through the use of a Bayesian approach. This approach combines both physiological/geometrical constraints and the available measurements reported in the literature. A key contribution of this work is to present a framework for including all such available information for the creation of virtual patients (VPs). The Markov Chain Monte Carlo (MCMC) method is used to sample random VPs from this posterior distribution, and the pressure and flow-rate profiles associated with each VP computed through a physics based model of pulse wave propagation. This combination of the arterial network parameters (representing a virtual patient) and the haemodynamics waveforms of pressure and flow-rates at various locations (representing functional response and potential measurements that can be acquired in the virtual patient) makes up the VPD. While 75,000 VPs are sampled from the posterior distribution, 10,000 are discarded as the initial burn-in period of the MCMC sampler. A further 12,857 VPs are subsequently removed due to the presence of negative average flow-rate, reducing the VPD to 52,143. Due to undesirable behaviour observed in some VPs-asymmetric under- and over-damped pressure and flow-rate profiles in left and right sides of the arterial system-a filter is proposed to remove VPs showing such behaviour. Post application of the filter, the VPD has 28,868 subjects. It is shown that the methodology is appropriate by comparing the VPD statistics to those reported in literature across real populations. Generally, a good agreement between the two is found while respecting physiological/geometrical constraints.

A framework for incorporating 3D hyperelastic vascular wall models in 1D blood flow simulations

We present a novel framework for investigating the role of vascular structure on arterial haemodynamics in large vessels, with a special focus on the human common carotid artery (CCA). The analysis is carried out by adopting a three-dimensional (3D) derived, fibre-reinforced, hyperelastic structural model, which is coupled with an axisymmetric, reduced order model describing blood flow. The vessel transmural pressure and lumen area are related via a Holzapfel–Ogden type of law, and the residual stresses along the thickness and length of the vessel are also accounted for. After a structural characterization of the adopted hyperelastic model, we investigate the link underlying the vascular wall response and blood-flow dynamics by comparing the proposed framework results against a popular tube law. The comparison shows that the behaviour of the model can be captured by the simpler linear surrogate only if a representative value of compliance is applied. Sobol’s multi-variable sensitivity analysis is then carried out in order to identify the extent to which the structural parameters have an impact on the CCA haemodynamics. In this case, the local pulse wave velocity (PWV) is used as index for representing the arterial transmission capacity of blood pressure waveforms. The sensitivity analysis suggests that some geometrical factors, such as the stress-free inner radius and opening angle, play a major role on the system’s haemodynamics. Subsequently, we quantified the differences in haemodynamic variables obtained from different virtual CCAs, tube laws and flow conditions. Although each artery presents a distinct vascular response, the differences obtained across different flow regimes are not significant. As expected, the linear tube law is unable to accurately capture all the haemodynamic features characterizing the current model. The findings from the sensitivity analysis are further confirmed by investigating the axial stretching effect on the CCA fluid dynamics. This factor does not seem to alter the pressure and flow waveforms. On the contrary, it is shown that, for an axially stretched vessel, the vascular wall exhibits an attenuation in absolute distension and an increase in circumferential stress, corroborating the findings of previous studies. This analysis shows that the new model offers a good balance between computational complexity and physics captured, making it an ideal framework for studies aiming to investigate the profound link between vascular mechanobiology and blood flow.

Towards enabling a cardiovascular digital twin for human systemic circulation using inverse analysis

An exponential rise in patient data provides an excellent opportunity to improve the existing health care infrastructure. In the present work, a method to enable cardiovascular digital twin is proposed using inverse analysis. Conventionally, accurate analytical solutions for inverse analysis in linear problems have been proposed and used. However, these methods fail or are not efficient for nonlinear systems, such as blood flow in the cardiovascular system (systemic circulation) that involves high degree of nonlinearity. To address this, a methodology for inverse analysis using recurrent neural network for the cardiovascular system is proposed in this work, using a virtual patient database. Blood pressure waveforms in various vessels of the body are inversely calculated with the help of long short-term memory (LSTM) cells by inputting pressure waveforms from three non-invasively accessible blood vessels (carotid, femoral and brachial arteries). The inverse analysis system built this way is applied to the detection of abdominal aortic aneurysm (AAA) and its severity using neural networks.

Artificial intelligence approaches to predict coronary stenosis severity using non-invasive fractional flow reserve

Fractional flow reserve is the current reference standard in the assessment of the functional impact of a stenosis in coronary heart disease. In this study, three models of artificial intelligence of varying degrees of complexity were compared to fractional flow reserve measurements. The three models are the multivariate polynomial regression, which is a statistical method used primarily for correlation; the feed-forward neural network; and the long short-term memory, which is a type of recurrent neural network that is suited to modelling sequences. The models were initially trained using a virtual patient database that was generated from a validated one-dimensional physics-based model. The feed-forward neural network performed the best for all test cases considered, which were a single vessel case from a virtual patient database, a multi-vessel network from a virtual patient database, and 25 clinically invasive fractional flow reserve measurements from real patients. The feed-forward neural network model achieved around 99% diagnostic accuracy in both tests involving virtual patients, and a respectable 72% diagnostic accuracy when compared to the invasive fractional flow reserve measurements. The multivariate polynomial regression model performed well in the single vessel case, but struggled on network cases as the variation of input features was much larger. The long short-term memory performed well for the single vessel cases, but tended to have a bias towards a positive fractional flow reserve prediction for the virtual multi-vessel case, and for the patient cases. Overall, the feed-forward neural network shows promise in successfully predicting fractional flow reserve in real patients, and could be a viable option if trained using a large enough data set of real patients.

Determining the impacts of venoarterial extracorporeal membrane oxygenation on cerebral oxygenation using a one-dimensional blood flow simulator

Venoarterial extracorporeal membrane oxygenation (VA-ECMO) is a mechanical system that provides rapid and short-term support for patients with cardiac failure. In many patients, pulmonary function is also impaired, resulting in poorly-oxygenated cardiac outflow competing against well-oxygenated VA-ECMO outflow, a condition known as North-South syndrome. North-South syndrome is a primary concern because of its potential to cause cerebral hypoxia, which has a critical influence on neurological complications often seen in this patient population. In order to reduce ischemic neurological complications, it is important to understand how clinical decisions regarding VA-ECMO parameters influence blood oxygenation. Here, we studied the impacts of flow rate and cannulation site on oxygenation using a one-dimensional (1D) model to simulate blood flow. Our model was initially tested by comparing blood flow results to those observed from experimental work in VA-ECMO patients. The 1D model was combined with a two-phase flow model to simulate oxygenation. Additionally, the influence of various other clinician-tunable parameters on oxygenation in the common carotid arteries (CCAs) were tested, including, blood viscosity, cannula position within the insertion artery, heart rate, and systemic vascular resistance (SVR), as well as geometrical changes such as arterial radius and length. Our results indicated that blood oxygenation to the brain strongly depended on the cannula insertion site and the VA-ECMO flow rate with a weaker but potentially significant dependence on arterial radius. During femoral cannulation, VA-ECMO flow rates greater than ~4.9L/min were needed to perfuse the CCAs. However, axillary and central cannulation began to perfuse the CCAs at significantly lower flow (~1L/min). These results may help explain the incidence of cerebral hypoxia in this patient population and the common need to change cannulation strategies during treatment to address this clinical problem. While this work describes patient-averaged results, determining these relationships between VA-ECMO parameters and cerebral hypoxia is an important step towards future work to develop patient-specific models that clinicians can use to improve outcomes.

Computational instantaneous wave-free ratio (IFR) for patient-specific coronary artery stenoses using 1D network models

In this work, we estimate the diagnostic threshold of the instantaneous wave-free ratio (iFR) through the use of a one-dimensional haemodynamic framework. To this end, we first compared the computed fractional flow reserve (cFFR) predicted from a 1D computational framework with invasive clinical measurements. The framework shows excellent promise and utilises minimal patient data from a cohort of 52 patients with a total of 66 stenoses. The diagnostic accuracy of the cFFR model was 75.76%, with a sensitivity of 71.43%, a specificity of 77.78%, a positive predictive value of 60%, and a negative predictive value of 85.37%. The validated model was then used to estimate the diagnostic threshold of iFR. The model determined a quadratic relationship between cFFR and the ciFR. The iFR diagnostic threshold was determined to be 0.8910 from a receiver operating characteristic curve that is in the range of 0.89 to 0.9 that is normally reported in clinical studies.

Non-invasive coronary CT angiography-derived fractional flow reserve: A benchmark study comparing the diagnostic performance of four different computational methodologies

Non-invasive coronary computed tomography (CT) angiography-derived fractional flow reserve (cFFR) is an emergent approach to determine the functional relevance of obstructive coronary lesions. Its feasibility and diagnostic performance has been reported in several studies. It is unclear if differences in sensitivity and specificity between these studies are due to study design, population, or "computational methodology." We evaluate the diagnostic performance of four different computational workflows for the prediction of cFFR using a limited data set of 10 patients, three based on reduced-order modelling and one based on a 3D rigid-wall model. The results for three of these methodologies yield similar accuracy of 6.5% to 10.5% mean absolute difference between computed and measured FFR. The main aspects of modelling which affected cFFR estimation were choice of inlet and outlet boundary conditions and estimation of flow distribution in the coronary network. One of the reduced-order models showed the lowest overall deviation from the clinical FFR measurements, indicating that reduced-order models are capable of a similar level of accuracy to a 3D model. In addition, this reduced-order model did not include a lumped pressure-drop model for a stenosis, which implies that the additional effort of isolating a stenosis and inserting a pressure-drop element in the spatial mesh may not be required for FFR estimation. The present benchmark study is the first of this kind, in which we attempt to homogenize the data required to compute FFR using mathematical models. The clinical data utilised in the cFFR workflows are made publicly available online.

A novel porous media-based approach to outflow boundary resistances of 1D arterial blood flow models

In this paper we introduce a novel method for prescribing terminal boundary conditions in one-dimensional arterial flow networks. This is carried out by coupling the terminal arterial vessel with a poro-elastic tube, representing the flow resistance offered by microcirculation. The performance of the proposed porous media-based model has been investigated through several different numerical examples. First, we investigate model parameters that have a profound influence on the flow and pressure distributions of the system. The simulation results have been compared against the waveforms generated by three elements (RCR) Windkessel model. The proposed model is also integrated into a realistic arterial tree, and the results obtained have been compared against experimental data at different locations of the network. The accuracy and simplicity of the proposed model demonstrates that it can be an excellent alternative for the existing models.

A semi-active human digital twin model for detecting severity of carotid stenoses from head vibration-A coupled computational mechanics and computer vision method

In this work, we propose a methodology to detect the severity of carotid stenosis from a video of a human face with the help of a coupled blood flow and head vibration model. This semi-active digital twin model is an attempt to link noninvasive video of a patient face to the percentage of carotid occlusion. The pulsatile nature of blood flow through the carotid arteries induces a subtle head vibration. This vibration is a potential indicator of carotid stenosis severity, and it is exploited in the present study. A head vibration model has been proposed in the present work that is linked to the forces generated by blood flow with or without occlusion. The model is used to generate a large number of virtual head vibration data for different degrees of occlusion. In order to determine the in vivo head vibration, a computer vision algorithm is adopted to use human face videos. The in vivo vibrations are compared against the virtual vibration data generated from the coupled computational blood flow/vibration model. A comparison of the in vivo vibration is made against the virtual data to find the best fit between in vivo and virtual data. The preliminary results on healthy subjects and a patient clearly indicate that the model is accurate and it possesses the potential for detecting approximate severity of carotid artery stenoses.

A novel, FFT-based one-dimensional blood flow solution method for arterial network

In the present work, we propose an FFT-based method for solving blood flow equations in an arterial network with variable properties and geometrical changes. An essential advantage of this approach is in correctly accounting for the vessel skin friction through the use of Womersley solution. To incorporate nonlinear effects, a novel approximation method is proposed to enable calculation of nonlinear corrections. Unlike similar methods available in the literature, the set of algebraic equations required for every harmonic is constructed automatically. The result is a generalized, robust and fast method to accurately capture the increasing pulse wave velocity downstream as well as steepening of the pulse front. The proposed method is shown to be appropriate for incorporating correct convection and diffusion coefficients. We show that the proposed method is fast and accurate and it can be an effective tool for 1D modelling of blood flow in human arterial networks.

A data-driven model to study utero-ovarian blood flow physiology during pregnancy

In this paper, we describe a mathematical model of the cardiovascular system in human pregnancy. An automated, closed-loop 1D-0D modelling framework was developed, and we demonstrate its efficacy in (1) reproducing measured multi-variate cardiovascular variables (pulse pressure, total peripheral resistance and cardiac output) and (2) providing automated estimates of variables that have not been measured (uterine arterial and venous blood flow, pulse wave velocity, pulsatility index). This is the first model capable of estimating volumetric blood flow to the uterus via the utero-ovarian communicating arteries. It is also the first model capable of capturing wave propagation phenomena in the utero-ovarian circulation, which are important for the accurate estimation of arterial stiffness in contemporary obstetric practice. The model will provide a basis for future studies aiming to elucidate the physiological mechanisms underlying the dynamic properties (changing shapes) of vascular flow waveforms that are observed with advancing gestation. This in turn will facilitate the development of methods for the earlier detection of pathologies that have an influence on vascular structure and behaviour.

Influence of ageing on human body blood flow and heat transfer: A detailed computational modelling study

Ageing plays a fundamental role in arterial blood transport and heat transfer within a human body. The aim of this work is to provide a comprehensive methodology, based on biomechanical considerations, for modelling arterial flow and energy exchange mechanisms in the body accounting for age-induced changes. The study outlines a framework for age-related modifications within several interlinked subsystems, which include arterial stiffening, heart contractility variations, tissue volume and property changes, and thermoregulatory system deterioration. Some of the proposed age-dependent governing equations are directly extrapolated from experimental data sets. The computational framework is demonstrated through numerical experiments, which show the impact of such age-related changes on arterial blood pressure, local temperature distribution, and global body thermal response. The proposed numerical experiments show that the age-related changes in arterial convection do not significantly affect the tissue temperature distribution. Results also highlight age-related effects on the sweating mechanism, which lead to a significant reduction in heat dissipation and a subsequent rise in skin and core temperatures.

Numerical modelling of a peripheral arterial stenosis using dimensionally reduced models and kernel methods

In this work, we consider 2 kinds of model reduction techniques to simulate blood flow through the largest systemic arteries, where a stenosis is located in a peripheral artery, i.e., in an artery that is located far away from the heart. For our simulations, we place the stenosis in one of the tibial arteries belonging to the right lower leg (right posterior tibial artery). The model reduction techniques that are used are on the one hand dimensionally reduced models (1-D and 0-D models, the so-called mixed-dimension model) and on the other hand surrogate models produced by kernel methods. Both methods are combined in such a way that the mixed-dimension models yield training data for the surrogate model, where the surrogate model is parametrised by the degree of narrowing of the peripheral stenosis. By means of a well-trained surrogate model, we show that simulation data can be reproduced with a satisfactory accuracy and that parameter optimisation or state estimation problems can be solved in a very efficient way. Furthermore, it is demonstrated that a surrogate model enables us to present after a very short simulation time the impact of a varying degree of stenosis on blood flow, obtaining a speedup of several orders over the full model.