Extracting new information from old waveforms: Symmetric projection attractor reconstruction: Where maths meets medicine.

NEW FINDINGS: What is the topic of this review? Symmetric Projection Attractor Reconstruction (SPAR) is a relatively new mathematical method that can extract additional information pertaining to the morphology and variability of physiological waveforms, such as arterial pulse pressure. Herein, we describe the potential utility of the method for more sensitive quantification of cardiovascular changes. What advances does it highlight? We use a simple example of a human tilt table to illustrate these concepts. SPAR can be used on any approximately periodic waveform and may add value to experimental and clinical settings, where such signals are collected routinely. ABSTRACT: Periodic physiological waveform data, such as blood pressure, pulse oximetry and ECG, are routinely sampled between 100 and 1000 Hz in preclinical research and in the clinical setting from a wide variety of implantable, bedside and wearable monitoring devices. Despite the underlying numerical waveform data being captured at such high fidelity, conventional analysis tends to reside in reporting only averages of minimum, maximum, amplitude and rate, as single point averages. Although these averages are undoubtedly of value, simplification of the data in this way means that most of the available numerical data are discarded. In turn, this may lead to subtle physiological changes being missed when investigating the cardiovascular system over time. We have developed a mathematical method (symmetric projection attractor reconstruction) that uses all the numerical data, replotting and revisualizing them in a manner that allows unique quantification of multiple changes in waveform morphology and variability. We propose that the additional quantification of these features will allow the complex behaviour of the cardiovascular system to be mapped more sensitively in different physiological and pathophysiological settings.

A mathematical analysis of rebound in a target-mediated drug disposition model: II. With feedback.

We consider the possibility of free receptor (antigen/cytokine) levels rebounding to higher than the baseline level after the application of an antibody drug using a target-mediated drug disposition model. It is assumed that the receptor synthesis rate experiences homeostatic feedback from the receptor levels. It is shown for a very fast feedback response, that the occurrence of rebound is determined by the ratio of the elimination rates, in a very similar way as for no feedback. However, for a slow feedback response, there will always be rebound. This result is illustrated with an example involving the drug efalizumab for patients with psoriasis. It is shown that slow feedback can be a plausible explanation for the observed rebound in this example.

Soluble expression, purification and functional characterisation of carboxypeptidase G2 and its individual domains.

Due to its applications in the treatment of cancer and autoimmune diseases, the 42 kDa zinc-dependent metalloenzyme carboxypeptidase G2 (CPG2) is of great therapeutic interest. An X-ray crystal structure of unliganded CPG2 reported in 1997 revealed the domain architecture and informed early rational drug design efforts, however further efforts at co-crystallization of CPG2 with ligands, substrates or inhibitors have not been reported. Thus key features of CPG2 such as the location of the active site, the presence of additional ligand-binding sites, stability, oligomeric state, and the molecular basis of activity remain largely unknown, with the current working understanding of CPG2 activity based primarily on computational modelling. To facilitate renewed efforts in CPG2 structural biology, we report the first high-yield (250 mg L(-1)) recombinant expression (and purification) of soluble and active CPG2 using the Escherichia coli expression system. We used this protocol to produce full-length enzyme, as well as protein fragments corresponding to the individual catalytic and dimerization domains, and the activity and stability of each construct was characterised. We adapted our protocol to allow for uniform incorporation of NMR labels ((13)C, (15)N and (2)H) and present preliminary solution-state NMR spectra of high quality. Taken together, our results offer a route for production and solution-state characterization that supports renewed effort in CPG2 structural biology as well as design of significantly truncated CPG2 proteins, which retain activity while yielding (potentially) improved immunogenicity.

A dynamical systems analysis of the data assimilation linked ecosystem carbon (DALEC) models.

Changes in our climate and environment make it ever more important to understand the processes involved in Earth systems, such as the carbon cycle. There are many models that attempt to describe and predict the behaviour of carbon stocks and stores but, despite their complexity, significant uncertainties remain. We consider the qualitative behaviour of one of the simplest carbon cycle models, the Data Assimilation Linked Ecosystem Carbon (DALEC) model, which is a simple vegetation model of processes involved in the carbon cycle of forests, and consider in detail the dynamical structure of the model. Our analysis shows that the dynamics of both evergreen and deciduous forests in DALEC are dependent on a few key parameters and it is possible to find a limit point where there is stable sustainable behaviour on one side but unsustainable conditions on the other side. The fact that typical parameter values reside close to this limit point highlights the difficulty of predicting even the correct trend without sufficient data and has implications for the use of data assimilation methods.

Cytochrome P450­catalyzed L-tryptophan nitration in thaxtomin phytotoxin biosynthesis.

Thaxtomin phytotoxins produced by plant-pathogenic Streptomyces species contain a nitro group that is essential for phytotoxicity. The N,N'-dimethyldiketopiperazine core of thaxtomins is assembled from L-phenylalanine and L-4-nitrotryptophan by a nonribosomal peptide synthetase, and nitric oxide synthase-generated NO is incorporated into the nitro group, but the biosynthesis of the nonproteinogenic amino acid L-4-nitrotryptophan is unclear. Here we report that TxtE, a unique cytochrome P450, catalyzes L-tryptophan nitration using NO and O(2).

A mathematical analysis of rebound in a target-mediated drug disposition model: I.without feedback.

We consider the possibility of free receptor (antigen/cytokine) levels rebounding to higher than the baseline level after one or more applications of an antibody drug using a target-mediated drug disposition model. Using geometry and dynamical systems analysis, we show that rebound will occur if and only if the elimination rate of the drug-receptor product is slower than the elimination rates of the drug and of the receptor. We also analyse the magnitude of rebound through approximations and simulations and demonstrate that it increases if the drug dose increases or if the difference between the elimination rate of the drug-receptor product and the minimum of the elimination rates of the drug and of the receptor increases.

Photoplethysmogram beat detection using Symmetric Projection Attractor Reconstruction.

Many methods have been proposed to detect beats in photoplethysmogram (PPG) signals. We present a novel method which uses the Symmetric Projection Attractor Reconstruction (SPAR) method to generate an attractor in a two dimensional phase space from the PPG signal. We can then define a line through the origin of this phase space to be a Poincaré section, as is commonly used in dynamical systems. Beats are detected when the attractor trajectory crosses the Poincaré section. By considering baseline drift, we define an optimal Poincaré section to use. The performance of this method was assessed using the WESAD dataset, achieving median F 1 scores of 74.3% in the Baseline phase, 63.0% during Stress, 93.6% during Amusement, and 97.7% during Meditation. Performance was better than an earlier version of the method, and comparable to one of the best algorithms identified in a recent benchmarking study of 15 beat detection algorithms. In addition, our method performed better than two others in the accuracy of the inter-beat intervals for two resting subjects.

ECG Feature Importance Rankings: Cardiologists vs. Algorithms.

Feature importance methods promise to provide a ranking of features according to importance for a given classification task. A wide range of methods exist but their rankings often disagree and they are inherently difficult to evaluate due to a lack of ground truth beyond synthetic datasets. In this work, we put feature importance methods to the test on real-world data in the domain of cardiology, where we try to distinguish three specific pathologies from healthy subjects based on ECG features comparing to features used in cardiologists' decision rules as ground truth. We found that the SHAP and LIME methods and Chi-squared test all worked well together with the native Random forest and Logistic regression feature rankings. Some methods gave inconsistent results, which included the Maximum Relevance Minimum Redundancy and Neighbourhood Component Analysis methods. The permutation-based methods generally performed quite poorly. A surprising result was found in the case of left bundle branch block, where T-wave morphology features were consistently identified as being important for diagnosis, but are not used by clinicians.

Advanced waveform analysis of the photoplethysmogram signal using complementary signal processing techniques for the extraction of biomarkers of cardiovascular function.

Introduction: Photoplethysmogram signals from wearable devices typically measure heart rate and blood oxygen saturation, but contain a wealth of additional information about the cardiovascular system. In this study, we compared two signal-processing techniques: fiducial point analysis and Symmetric Projection Attractor Reconstruction, on their ability to extract new cardiovascular information from a photoplethysmogram signal. The aim was to identify fiducial point analysis and Symmetric Projection Attractor Reconstruction indices that could classify photoplethysmogram signals, according to age, sex and physical activity. Methods: Three datasets were used: an in-silico dataset of simulated photoplethysmogram waves for healthy male participants (25-75 years old); an in-vivo dataset containing 10-min photoplethysmogram recordings from 57 healthy subjects at rest (18-39 or > 70 years old; 53% female); and an in-vivo dataset containing photoplethysmogram recordings collected for 4 weeks from a single subject, in daily life. The best-performing indices from the in-silico study (5/48 fiducial point analysis and 6/49 Symmetric Projection Attractor Reconstruction) were applied to the in-vivo datasets. Results: Key fiducial point analysis and Symmetric Projection Attractor Reconstruction indices, which showed the greatest differences between groups, were found to be consistent across datasets. These indices were related to systolic augmentation, diastolic peak positioning and prominence, and waveform variability. Both fiducial point analysis and Symmetric Projection Attractor Reconstruction techniques provided indices that supported the classification of age and physical activity, but not sex. Conclusions: Both fiducial point analysis and Symmetric Projection Attractor Reconstruction techniques demonstrated utility in identifying cardiovascular differences between individuals and within an individual over time. Future research should investigate the potential utility of these techniques for extracting information on fitness and disease, to support healthcare-decision making.

Novel breathing pattern analysis: Symmetric Projection Attractor Reconstruction improves identification of impending COPD re-exacerbations - a retrospective cohort analysis.

Respiratory waveforms can be reduced to simple metrics, such as rate, but this may miss information about waveform shape and whole breathing pattern. A novel analysis method quantifying the whole waveform shape identifies AECOPD earlier. https://bit.ly/3M6uIEB.

PTB-XL+, a comprehensive electrocardiographic feature dataset.

Machine learning (ML) methods for the analysis of electrocardiography (ECG) data are gaining importance, substantially supported by the release of large public datasets. However, these current datasets miss important derived descriptors such as ECG features that have been devised in the past hundred years and still form the basis of most automatic ECG analysis algorithms and are critical for cardiologists' decision processes. ECG features are available from sophisticated commercial software but are not accessible to the general public. To alleviate this issue, we add ECG features from two leading commercial algorithms and an open-source implementation supplemented by a set of automatic diagnostic statements from a commercial ECG analysis software in preprocessed format. This allows the comparison of ML models trained on clinically versus automatically generated label sets. We provide an extensive technical validation of features and diagnostic statements for ML applications. We believe this release crucially enhances the usability of the PTB-XL dataset as a reference dataset for ML methods in the context of ECG data.

Detecting beats in the photoplethysmogram: benchmarking open-source algorithms.

The photoplethysmogram (PPG) signal is widely used in pulse oximeters and smartwatches. A fundamental step in analysing the PPG is the detection of heartbeats. Several PPG beat detection algorithms have been proposed, although it is not clear which performs best.Objective:This study aimed to: (i) develop a framework with which to design and test PPG beat detectors; (ii) assess the performance of PPG beat detectors in different use cases; and (iii) investigate how their performance is affected by patient demographics and physiology.Approach:Fifteen beat detectors were assessed against electrocardiogram-derived heartbeats using data from eight datasets. Performance was assessed using theF1score, which combines sensitivity and positive predictive value.Main results:Eight beat detectors performed well in the absence of movement withF1scores of ≥90% on hospital data and wearable data collected at rest. Their performance was poorer during exercise withF1scores of 55%-91%; poorer in neonates than adults withF1scores of 84%-96% in neonates compared to 98%-99% in adults; and poorer in atrial fibrillation (AF) withF1scores of 92%-97% in AF compared to 99%-100% in normal sinus rhythm.Significance:Two PPG beat detectors denoted 'MSPTD' and 'qppg' performed best, with complementary performance characteristics. This evidence can be used to inform the choice of PPG beat detector algorithm. The algorithms, datasets, and assessment framework are freely available.

Detecting paroxysmal atrial fibrillation from normal sinus rhythm in equine athletes using Symmetric Projection Attractor Reconstruction and machine learning.

Background: Atrial fibrillation (AF) is a common cardiac arrhythmia in both human and equine populations. It is associated with adverse outcomes in humans and decreased athletic performance in both populations. Paroxysmal atrial fibrillation (PAF) presents with intermittent, self-terminating AF episodes, and is difficult to diagnose once sinus rhythm resumes. Objective: We aimed to detect PAF subjects from normal sinus rhythm equine electrocardiograms (ECGs) using the Symmetric Projection Attractor Reconstruction (SPAR) method to encapsulate the waveform morphology and variability as the basis of a machine learning classification. Methods: We obtained ECG signals from 139 active equine athletes (120 control, 19 with a PAF diagnosis). The SPAR method was applied to 9 short (20-second) ECG strips for each subject. An optimal SPAR feature set was determined by forward feature selection for input to a machine learning model ensemble of 3 different classifiers (k-nearest neighbors, linear support vector machine, and radial basis function kernel support vector machine). Imbalanced data were handled by upsampling the minority (PAF) class. A final subject classification was made by taking a majority vote over results from the 9 ECG strips. Results: Our final cross-validated classification for a subject gave an accuracy of 89.0%, sensitivity of 94.8%, specificity of 87.1%, and receiver operating characteristic area under the curve of 0.98, taking PAF as the positive class. Conclusion: The SPAR method and machine learning generated a final model with high sensitivity, suggesting that PAF can be discriminated from short equine ECG strips. This preliminary study indicated that SPAR analysis of human ECG could support patient monitoring, risk stratification, and clinical decision-making.

Ankle Doppler for Cuffless Ankle Brachial Index Estimation and Peripheral Artery Disease Diagnosis Independent of Diabetes.

Ankle brachial pressure index (ABPI) is the first-line test to diagnose peripheral artery disease (PAD). Its adoption in clinical practice is poor and its validity, particularly in diabetes, is limited. We hypothesised that ABPI can be accurately and precisely estimated based on cuffless Doppler waveforms. Retrospective analysis of standard ABPI and handheld Doppler waveform characteristics (n = 200). Prospective analysis of angle-corrected Doppler acceleration index (AccI, n = 148) and standard ABPI with testing of performance to diagnose PAD as assessed with imaging reference standards in consecutive patients. The highest AccI from handheld Doppler at ankle arteries was significantly logarithmically associated with the highest standard ABPI (E[y] = 0.32 ln [1.71 ∗ x + 1], p < 0.001, R2 = 0.68, n = 100 limbs). Estimated ABPI (eABPI) based on AccI closely resembled ABPI (r = 0.81, p < 0.001, average deviation −0.01 ± 0.13 [SD], n = 100 limbs). AccI from angle-corrected Doppler in patients without overt media sclerosis (ABPI ≤ 1.1) improved ABPI prediction (E[y] = 0.297 ∗ ln[0.039 ∗ x + 1], R2 = 0.92, p = 0.006, average deviation 0.00 ± 0.08, n = 100). In a population (n = 148 limbs) including diabetes (56%), chronic limb-threatening ischaemia (51%) and media sclerosis (32%), receiver operating characteristics analysis of (angle-corrected) eABPI performed significantly better than standard ABPI to diagnose PAD defined by ultrasound (ROC AUC = 0.99 ± 0.01, p < 0.001; sensitivity: 97%, specificity: 96%) at the ≤0.9 cut-off. This was confirmed with CT angiography (ROC AUC = 0.98, p < 0.001, sensitivity: 97%, specificity: 100%) and was independent of the presence of diabetes (p = 0.608). ABPI can be estimated based on ankle Doppler AccI without compression, and eABPI performs better than standard ABPI to diagnose PAD independent of diabetes. eABPI has the potential to be included as a standard component of lower extremity ultrasound.

Mathematical analysis of the pharmacokinetic-pharmacodynamic (PKPD) behaviour of monoclonal antibodies: predicting in vivo potency.

We consider the relationship between the target affinity of a monoclonal antibody and its in vivo potency. The dynamics of the system is described mathematically by a target-mediated drug disposition model. As a measure of potency, we consider the minimum level of the free receptor following a single bolus injection of the ligand into the plasma compartment. From the differential equations, we derive two expressions for this minimum level in terms of the parameters of the problem, one of which is valid over the full range of values of the equilibrium dissociation constant K(D) and the other which is valid only for a large drug dose or for a small value of K(D). Both of these formulae show that the potency achieved by increasing the association constant k(on) can be very different from the potency achieved by decreasing the dissociation constant k(off). In particular, there is a saturation effect when decreasing k(off) where the increase in potency that can be achieved is limited, whereas there is no such effect when increasing k(on). Thus, for certain monoclonal antibodies, an increase in potency may be better achieved by increasing k(on) than by decreasing k(off).

Symmetric Projection Attractor Reconstruction: Sex Differences in the ECG.

Background: The electrocardiogram (ECG) is a key tool in patient management. Automated ECG analysis supports clinical decision-making, but traditional fiducial point identification discards much of the time-series data that captures the morphology of the whole waveform. Our Symmetric Projection Attractor Reconstruction (SPAR) method uses all the available data to provide a new visualization and quantification of the morphology and variability of any approximately periodic signal. We therefore applied SPAR to ECG signals to ascertain whether this more detailed investigation of ECG morphology adds clinical value. Methods: Our aim was to demonstrate the accuracy of the SPAR method in discriminating between two biologically distinct groups. As sex has been shown to influence the waveform appearance, we investigated sex differences in normal sinus rhythm ECGs. We applied the SPAR method to 9,007 10 second 12-lead ECG recordings from Physionet, which comprised; Dataset 1: 104 subjects (40% female), Dataset 2: 8,903 subjects (54% female). Results: SPAR showed clear visual differences between female and male ECGs (Dataset 1). A stacked machine learning model achieved a cross-validation sex classification accuracy of 86.3% (Dataset 2) and an unseen test accuracy of 91.3% (Dataset 1). The mid-precordial leads performed best in classification individually, but the highest overall accuracy was achieved with all 12 leads. Classification accuracy was highest for young adults and declined with older age. Conclusions: SPAR allows quantification of the morphology of the ECG without the need to identify conventional fiducial points, whilst utilizing of all the data reduces inadvertent bias. By intuitively re-visualizing signal morphology as two-dimensional images, SPAR accurately discriminated ECG sex differences in a small dataset. We extended the approach to a machine learning classification of sex for a larger dataset, and showed that the SPAR method provided a means of visualizing the similarities of subjects given the same classification. This proof-of-concept study therefore provided an implementation of SPAR using existing data and showed that subtle differences in the ECG can be amplified by the attractor. SPAR's supplementary analysis of ECG morphology may enhance conventional automated analysis in clinically important datasets, and improve patient stratification and risk management.

Symmetric Projection Attractor Reconstruction analysis of murine electrocardiograms: Retrospective prediction of Scn5a+/- genetic mutation attributable to Brugada syndrome.

BACKGROUND: Life-threatening arrhythmias resulting from genetic mutations are often missed in current electrocardiogram (ECG) analysis. We combined a new method for ECG analysis that uses all the waveform data with machine learning to improve detection of such mutations from short ECG signals in a mouse model. OBJECTIVE: We sought to detect consequences of Na+ channel deficiencies known to compromise action potential conduction in comparisons of Scn5a+/- mutant and wild-type mice using short ECG signals, examining novel and standard features derived from lead I and II ECG recordings by machine learning algorithms. METHODS: Lead I and II ECG signals from anesthetized wild-type and Scn5a+/- mutant mice of length 130 seconds were analyzed by extracting various groups of features, which were used by machine learning to classify the mice as wild-type or mutant. The features used were standard ECG intervals and amplitudes, as well as features derived from attractors generated using the novel Symmetric Projection Attractor Reconstruction method, which reformulates the whole signal as a bounded, symmetric 2-dimensional attractor. All the features were also combined as a single feature group. RESULTS: Classification of genotype using the attractor features gave higher accuracy than using either the ECG intervals or the intervals and amplitudes. However, the highest accuracy (96%) was obtained using all the features. Accuracies for different subgroups of the data were obtained and compared. CONCLUSION: Detection of the Scn5a+/- mutation from short mouse ECG signals with high accuracy is possible using our Symmetric Projection Attractor Reconstruction method.

A New Model for the Dynamics of Hepatitis C Infection: Derivation, Analysis and Implications.

We review various existing models of hepatitis C virus (HCV) infection and show that there are inconsistencies between the models and known behaviour of the infection. A new model for HCV infection is proposed, based on various dynamical processes that occur during the infection that are described in the literature. This new model is analysed, and three steady state branches of solutions are found when there is no stem cell generation of hepatocytes. Unusually, the branch of infected solutions that connects the uninfected branch and the pure infection branch can be found analytically and always includes a limit point, subject to a few conditions on the parameters. When the action of stem cells is included, the bifurcation between the pure infection and infected branches unfolds, leaving a single branch of infected solutions. It is shown that this model can generate various viral load profiles that have been described in the literature, which is confirmed by fitting the model to four viral load datasets. Suggestions for possible changes in treatment are made based on the model.

A novel method to quantify arterial pulse waveform morphology: attractor reconstruction for physiologists and clinicians.

Current arterial pulse monitoring systems capture data at high frequencies (100-1000 Hz). However, they typically report averaged or low frequency summary data such as heart rate and systolic, mean and diastolic blood pressure. In doing so, a potential wealth of information contained in the high-fidelity waveform data is discarded, data which has long been known to contain useful information on cardiovascular performance. Here we summarise a new mathematical method, attractor reconstruction, which enables the quantification of arterial waveform shape and variability in real-time. The method can handle long streams of non-stationary data and does not require preprocessing of the raw physiological data by the end user. Whilst the detailed mathematical proofs have been described elsewhere (Aston et al 2008 Physiol. Meas. 39), the authors were motivated to write a summary of the method and its potential utility for biomedical researchers, physiologists and clinician readers. Here we illustrate how this new method may supplement and potentially enhance the sensitivity of detecting cardiovascular disturbances, to aid with biomedical research and clinical decision making.

Beyond HRV: attractor reconstruction using the entire cardiovascular waveform data for novel feature extraction.

Advances in monitoring technology allow blood pressure waveforms to be collected at sampling frequencies of 250-1000 Hz for long time periods. However, much of the raw data are under-analysed. Heart rate variability (HRV) methods, in which beat-to-beat interval lengths are extracted and analysed, have been extensively studied. However, this approach discards the majority of the raw data. OBJECTIVE: Our aim is to detect changes in the shape of the waveform in long streams of blood pressure data. APPROACH: Our approach involves extracting key features from large complex data sets by generating a reconstructed attractor in a three-dimensional phase space using delay coordinates from a window of the entire raw waveform data. The naturally occurring baseline variation is removed by projecting the attractor onto a plane from which new quantitative measures are obtained. The time window is moved through the data to give a collection of signals which relate to various aspects of the waveform shape. MAIN RESULTS: This approach enables visualisation and quantification of changes in the waveform shape and has been applied to blood pressure data collected from conscious unrestrained mice and to human blood pressure data. The interpretation of the attractor measures is aided by the analysis of simple artificial waveforms. SIGNIFICANCE: We have developed and analysed a new method for analysing blood pressure data that uses all of the waveform data and hence can detect changes in the waveform shape that HRV methods cannot, which is confirmed with an example, and hence our method goes 'beyond HRV'.