Heart failure (HF) is a broad spectrum of disease in which the heart is unable to supply blood at the rate required by the body. It is often stratified by ejection fraction (EF): heart failure with reduced ejection fraction (≤40%, HFrEF), where there is usually a decrease in stroke volume (SV) due to a failure of intrinsic inotropy or loss of functional heart muscle; and heart failure with preserved ejection fraction (≥50%, HFpEF), where there is often a decrease in SV because the end diastolic volume (EDV) has reduced due to loss of ventricular compliance.

HF is treatable but the prognosis is poor: in the UK, the 1-, 5- and 10-year mortality rates are 24.1, 54.5 and 75.5%, respectively, and these have only improved by around 7% each since the year 2000 (Taylor et al., 2019). Moreover, 79% of UK HF diagnoses are only made after an emergency hospital admission despite 41% of the patients visiting their GP in the preceding five years with at least one of the three major symptoms of HF: breathlessness, ankle swelling, and fatigue (Bottle et al., 2018). The survival rates following a diagnosis in primary care are, as expected, better than the average (Taylor et al., 2017), but there are clear missed opportunities for early diagnosis and intervention. There is also a significant disconnect between clinical guidelines and actual patient diagnoses: only 24% of patients were diagnosed according to the recommended pathway (Taylor et al., 2019). All of these factors necessitate an improved diagnosis pipeline.

One prominent issue is that the three main symptoms are not exclusive to HF. If one of these symptoms is identified, the current pathway recommends a blood test for brain natriuretic peptides (BNP); this is highly sensitive and can therefore effectively rule out those who do not have the disease, but BNP levels are also commonly elevated in other pathologies such as chronic kidney disease (Maisel et al., 2008; Tagore et al., 2008) and may be abnormally low in obese patients (Madamanchi et al., 2014). A positive BNP test should be followed by echocardiography, which is the gold-standard for diagnosis, but this is not available in primary care and, at least in the UK, a shortage of trained staff is a limiting factor with hospital wait times of up to 6 weeks (Cowie, 2017). Even for those patients who have a HF symptom recorded during a primary care consultation, 44% are not referred for a BNP test, for echocardiography, or to a specialist for further consultation, highlighting a lack of both confidence in current investigations and test availability (Bottle et al., 2018). There is an urgent need for a specific, low-cost, noninvasive method to improve HF screening in primary care and for treatment management at point-of-care.

Arterial pulse waves carry information about the performance of the heart and vessels, and several studies have identified statistically significant changes in the metrics derived from wave intensity (WI) in the presence of impaired heart performance (Curtis et al., 2007; Siniawski et al., 2009; Li and Guo, 2013; Takaya et al., 2013; Vriz et al., 2015) but only when averaged over groups. Furthermore, WI has generally been determined from measurements of blood velocity and pressure made throughout the cardiac cycle (Parker, 2009), but pressure waveforms with sufficient temporal resolution can only be obtained accurately by invasive methods or estimated inaccurately by noninvasive ones, which has inhibited the clinical realisability of WI-based diagnostic methods.

A new form of WI has recently been introduced that instead relies upon measurements of blood velocity and arterial diameter (Feng and Khir, 2010); this is significant because both can be noninvasively measured at the same arterial location and time, for example by ultrasound. Despite the intrinsic nonlinear relationship between arterial diameter and blood pressure arising from effects such as viscoelasticity and strain-stiffening, Reavette et al. (2020) found by numerical modelling that this noninvasive method gave results that were close to those of the invasive method.

As noted above, impaired SV is important in both HFrEF and HFpEF (A reduced SV is consistent with a preserved ejection fraction if EDV is reduced.) HF can co-exist with a normal resting SV if the heart compensates with increased inotropy; however, the hearts of such patients will struggle to fulfil physiological needs during exertion. At such times, SV can again become subnormal.

To elucidate whether diameter-based WI can detect impaired heart performance on an individual basis, we have generated an age-stratified virtual population of elderly subjects, modelling half as healthy controls and half as HF patients with an impaired SV. Simulations were performed using the PulseWaveSolver utility of Nektar++ (Cantwell et al., 2015), which solves the 1D equations of blood flow using a high-order discontinuous Galerkin method with a spectral/hp-element discretisation. This reduced-order modelling can accurately solve for the arterial area, velocity and pressure waveforms in complex arterial networks with reasonable computational cost (Alastruey et al., 2011; Sherwin et al., 2003) and has been validated against in-vitro (Alastruey et al., 2011; Matthys et al., 2007) and in-vivo (Mynard and Smolich, 2015; Olufsen et al., 2000; Pedregosa et al., 2011) data. We applied a support vector machine (SVM) classifier to metrics derived from wave intensity analysis (WIA) in the common carotid, brachial and radial arteries. These arteries are all accessible to ultrasound. Machine learning can identify and utilise complex relationships between metrics and has been successfully applied in numerous biomedical classification studies (Yamamoto et al., 2020; Zhao et al., 2020) and to in-silico datasets (Bikia et al., 2021; Jin et al., 2021). An SVM was chosen after it performed best in preliminary tests against other classification algorithms.

The 1D equations of conservation of mass and momentum

∂ A ∂ t + ∂ ∂ x ( A U ) = 0 , (1) ∂ U ∂ t + U ∂ U ∂ x + 1 ρ ∂ P ∂ x = f ρ A , (2) for cross-sectional area A, average cross-sectional velocity U and pressure P, frictional force per unit length f, and density of blood ρ = 1,060 kg m⁻³, together with the nonlinear tube law P = P d − β A 2 α ln [ 1 − α ⁡ ln (   A A d   ) ] + Γ A ∂ A ∂ t , (3) for reference diastolic area A_d and pressure P_d, stiffness and viscoelastic parameters β and Γ , respectively, and strain-stiffening modulus α , were solved in models of the largest 55 systemic arteries (Figure 1). The arteries were modelled as linearly tapering vessels, and terminal vessels were coupled to RCR Windkessel models (Alastruey et al., 2012). For further details of the numerical method and tube law, the reader is referred to Sherwin et al. (2003) and Reavette et al. (2020), respectively. The simulations and analysis were performed locally on an iMac (2017, 4.2 GHz Quad-Core Intel Core i7 processor, 32 GB RAM) using Nektar++ (Cantwell et al., 2015) and Python (Van Rossum and Drake, 2009), respectively.

Separating waves into their forwards and backwards components did not improve the model’s performance, nor did stratifying the data by age or sex; we therefore only present the results obtained from the unseparated waves for the entire dataset.

WI plots for the common carotid, brachial and radial arteries for three example subjects—one healthy control, one HF patient with an impaired LVET, and one HF patient with an impaired peak flow—are given in Figure 4.

The SWI and SWE are greatly reduced in all arteries for the patient with an impaired peak flow, but these same metrics are largely unchanged in the patient with impaired LVET. The DWI and DWE are reduced in all arteries for both HF patients, implying that these metrics have the potential to be the two biggest discriminators between the controls and patients. The SD-Delay is shorter for the patient with impaired LVET, as the diastolic relaxation occurs earlier due to the impairment. The R wave is approximately the same in the control and patient with an impaired LVET but is reduced in the patient with an impaired peak flow.

Overall, the SVM achieved high precision and recall when using the default decision threshold for all arteries and maintained a high precision with the 99% recall threshold for the radial. We conclude that arterial WIA has significant potential to improve the HF diagnostic pipeline.

These results were achieved without separating the WI into intensities of forwards- and backwards-travelling waves, which eliminates experimental errors introduced by determining the PWV from the lnDU -loop. Additionally, stratifying the data by sex and age group did not improve the classification. Both of these results led to a simpler analysis without sacrificing performance. We may have neglected modelling criteria that would cause greater differences between the WIs of different sexes and age groups; nevertheless, even without such stratification, this study has already supported the ability of WIA to screen for HF.

In this study, the SVM performed best on the radial data, and this may in part be caused by differences in the RCs between arteries. In general, there was more stratification for the RC in the radial than for the other arteries; the radial was modelled as a terminal vessel, and increasing the peripheral resistance to keep pressures in the physiological range increased the size of the R wave relative to the S wave. The trend towards larger reflections was not observed in the brachial, probably as a result of wave trapping: the geometry of the arterial system is optimised to minimise reflection of forwards-travelling waves at bifurcations—the well-matched condition—but this necessarily results in poor matching for backwards-travelling waves, with a greater proportion of these being re-reflected (Parker, 2009). Hence, reflections generated in the periphery will be less apparent in the brachial than in the radial. (The radial artery performs well in other applications where wave properties are important—see Zheng et al., 2012.) This does not, however, explain why the trend towards greater reflection in HF patients occurs in the carotid too, which itself is one bifurcation removed from terminal vessels (the internal and external carotids).

For the default decision threshold, some subjects that had a severely impaired SV were misclassified as healthy, which is a significant issue. Altering the threshold to increase recall prevented the most egregious misclassifications, and this caught all patients in the carotid but not in the brachial or radial. Although there was some crossover in the misclassified patients between arteries, it may be the case that WI data from the carotid, brachial and radial could be used together to improve the results obtained here. Another issue, albeit one less serious, is the misidentification of healthy subjects as having an impaired SV: as noted above, these errors, which are small in number, would be corrected at the echocardiogram stage. Due to the inner complexity of the SVM, it is difficult to suggest why patients were misclassified; these errors warrant further investigation since the method may miss subjects that require immediate treatment.

An SVM classifier worked well for all three arteries examined in this study and gave particularly good results for the radial artery data. It is a versatile algorithm capable of identifying complex nonlinear patterns between metrics when classifying. SVMs are particularly effective in high dimensional spaces, but this was not advantageous here given the relatively low number of features. They are also well suited to small- to medium-sized datasets. Both of these advantages in part explain why SVMs have been successfully used in other biomedical applications: fields such as genomics are often characterised by high-dimensional data (Amaratunga and Cabrera, 2018), and sample sizes in typical clinical trials tend to be relatively small. One notable drawback of an SVM is the lack of interpretability: knowledge of the features that drive classification is important for developing a stronger understanding of heart failure itself and helpful for clinical adoption. Another drawback is that SVMs do not ordinarily provide a probabilistic estimate when classifying each subject; classification is given only in a binary format. Platt scaling is a method that can transform the outputs into a probabilistic model (Platt, 1999), but this comes with its own disadvantages.

Clinically, this method would require spatiotemporally coincident measurements of diameter and velocity with sufficient temporal resolution to resolve the waveforms accurately. MRI has previously been used to calculate WIs with a temporal resolution of 10 ms (Bhuva et al., 2020; Biglino et al., 2012; Li et al., 2010; Neumann et al., 2018; Schäfer et al., 2018), but it is currently unavailable in primary care or on the ward and is likely to remain so. The present study used a frequency equivalent to 1,000 frames per second, which is achievable with ultrafast ultrasound devices; previous studies have indeed used ultrasound to determine diameter and velocity using a combination of M-mode and Doppler (Feng and Khir, 2010), but the optimal beam angle for Doppler is orthogonal to that for M-mode, which introduces inaccuracies. Ultrasound imaging velocimetry is, however, feasible, for it can accurately determine the velocity waveform even in the presence of the high gradients observed in early systole, and wall-tracking algorithms can determine the diameter waveform with the necessary spatiotemporal coincidence. Real-time implementation of an ultrafast ultrasound method is also possible: scans can be acquired in a few seconds, beamformed in real-time with GPU accelerated computing (Hyun et al., 2019), and clutter-filtered in real-time with randomised singular value decomposition (Song et al., 2017). Subsequent processing to determine the wave intensity metrics is not computationally intensive, and results could again be available within seconds. There are, however, key challenges that must be addressed to permit successful clinical adoption of such a method, the most important of which is proving that it gives reliable, reproducible and accurate results when applied to real human data.