Adam Mahdi

An ensemble Kalman filter for statistical estimation of physics constrained nonlinear regression models

A central issue in contemporary science is the development of nonlinear data driven statistical-dynamical models for time series of noisy partial observations from nature or a complex model. It has been established recently that ad-hoc quadratic multi-level regression models can have finite-time blow-up of statistical solutions and/or pathological behavior of their invariant measure. Recently, a new class of physics constrained nonlinear regression models were developed to ameliorate this pathological behavior. Here a new finite ensemble Kalman filtering algorithm is developed for estimating the state, the linear and nonlinear model coefficients, the model and the observation noise covariances from available partial noisy observations of the state. Several stringent tests and applications of the method are developed here. In the most complex application, the perfect model has 57 degrees of freedom involving a zonal (east-west) jet, two topographic Rossby waves, and 54 nonlinearly interacting Rossby waves; the perfect model has significant non-Gaussian statistics in the zonal jet with blocked and unblocked regimes and a non-Gaussian skewed distribution due to interaction with the other 56 modes. We only observe the zonal jet contaminated by noise and apply the ensemble filter algorithm for estimation. Numerically, we find that a three dimensional nonlinear stochastic model with one level of memory mimics the statistical effect of the other 56 modes on the zonal jet in an accurate fashion, including the skew non-Gaussian distribution and autocorrelation decay. On the other hand, a similar stochastic model with zero memory levels fails to capture the crucial non-Gaussian behavior of the zonal jet from the perfect 57-mode model.

Modeling the Afferent Dynamics of the Baroreflex Control System

In this study we develop a modeling framework for predicting baroreceptor firing rate as a function of blood pressure. We test models within this framework both quantitatively and qualitatively using data from rats. The models describe three components: arterial wall deformation, stimulation of mechanoreceptors located in the BR nerve-endings, and modulation of the action potential frequency. The three sub-systems are modeled individually following well-established biological principles. The first submodel, predicting arterial wall deformation, uses blood pressure as an input and outputs circumferential strain. The mechanoreceptor stimulation model, uses circumferential strain as an input, predicting receptor deformation as an output. Finally, the neural model takes receptor deformation as an input predicting the BR firing rate as an output. Our results show that nonlinear dependence of firing rate on pressure can be accounted for by taking into account the nonlinear elastic properties of the artery wall. This was observed when testing the models using multiple experiments with a single set of parameters. We find that to model the response to a square pressure stimulus, giving rise to post-excitatory depression, it is necessary to include an integrate-and-fire model, which allows the firing rate to cease when the stimulus falls below a given threshold. We show that our modeling framework in combination with sensitivity analysis and parameter estimation can be used to test and compare models. Finally, we demonstrate that our preferred model can exhibit all known dynamics and that it is advantageous to combine qualitative and quantitative analysis methods.

Modeling Cerebral Blood Flow Velocity During Orthostatic Stress

Cerebral autoregulation refers to the physiological process that maintains stable cerebral blood flow (CBF) during changes in arterial blood pressure (ABP). In this study, we propose a simple, nonlinear quantitative model with only four parameters that can predict CBF velocity as a function of ABP. The model was motivated by the viscoelastic-like behavior observed in the data collected during postural change from sitting to standing. Qualitative testing of the model involved analysis of dynamic responses to step-changes in pressure both within and outside the autoregulatory range, while quantitative testing was used to show that the model can fit dynamics observed in data measured from a healthy young and a healthy elderly subject. The latter involved analysis of structural and practical identifiability, sensitivity analysis, and parameter estimation. Results showed that the model is able to reproduce observed overshoot and adaptation and predict the different responses in the healthy young and the healthy elderly subject. For the healthy young subject, the overshoot was significantly more pronounced than for the elderly subject, but the recovery time was longer for the young subject. These differences resulted in different parameter values estimated using the two datasets.

Structural Identifiability of Viscoelastic Mechanical Systems

We solve the local and global structural identifiability problems for viscoelastic mechanical models represented by networks of springs and dashpots. We propose a very simple characterization of both local and global structural identifiability based on identifiability tables, with the purpose of providing a guideline for constructing arbitrarily complex, identifiable spring-dashpot networks. We illustrate how to use our results in a number of examples and point to some applications in cardiovascular modeling.

Bayesian Inference in Non-Markovian State-Space Models With Applications to Battery Fractional-Order Systems

Battery impedance spectroscopy models are given by fractional-order (FO) differential equations. In the discrete-time domain, they give rise to state-space models where the latent process is not Markovian. Parameter estimation for these models is, therefore, challenging, especially for noncommensurate FO models. In this paper, we propose a Bayesian approach to identify the parameters of generic FO systems. The computational challenge is tackled with particle Markov chain Monte Carlo methods, with an implementation specifically designed for the non-Markovian setting. Two examples are provided. In a first example, the approach is applied to identify a battery commensurate FO model with a single constant phase element (CPE) by using real data. We compare the proposed approach to an instrumental variable method. Then, we consider a noncommensurate FO model with more than one CPE and synthetic data sets, investigating how the proposed method enables the study of various effects on parameter identification, such as the data length, the magnitude of the input signal, the choice of prior, and the measurement noise.

At what data length do cerebral autoregulation measures stabilise

OBJECTIVEnCerebral autoregulation is commonly assessed through mathematical models that use non-invasive measurements of arterial blood pressure and cerebral blood flow velocity. There is no agreement in the literature as to what is the minimum length of data needed for the cerebral autoregulation coefficients to stabilise.nnnAPPROACHnWe introduce a simple empirical tool for studying the minimum length of time series needed to parameterise three popular cerebral autoregulation coefficients ARI, Mx and Phase (in the low frequency range [0.07-0.2] Hz), which can be easily applied in a more general context. We use our recently collected data, from which we select high quality (absence of non-physiological artefacts), baseline ABP-CBFV time series (16u2009min each). The data were beat-to-beat averaged and downsampled at 10 Hz.nnnMAIN RESULTnOn average, ARI exhibits greater variability than Mx and Phase, when calculated for short intervals; however, it stabilises fastest.nnnSIGNIFICANCEnOur results show that values of ARI, Mx and Phase calculated on intervals shorter than 3u2009min (1800 samples), 6u2009min (3600 samples) and 5u2009min (3000 samples), respectively, may be very sensitive to changes in the length of data interval.

Increased blood pressure variability upon standing up improves reproducibility of cerebral autoregulation indices

Dynamic cerebral autoregulation, that is the transient response of cerebral blood flow to changes in arterial blood pressure, is currently assessed using a variety of different time series methods and data collection protocols. In the continuing absence of a gold standard for the study of cerebral autoregulation it is unclear to what extent does the assessment depend on the choice of a computational method and protocol. We use continuous measurements of blood pressure and cerebral blood flow velocity in the middle cerebral artery from the cohorts of 18 normotensive subjects performing sit-to-stand manoeuvre. We estimate cerebral autoregulation using a wide variety of black-box approaches (including the following six autoregulation indices ARI, Mx, Sx, Dx, FIR and ARX) and compare them in the context of reproducibility and variability. For all autoregulation indices, considered here, the intra-class correlation was greater during the standing protocol, however, it was significantly greater (Fishers Z-test) for Mx (p < 0.03), Sx (p < 0.003) and Dx (p < 0.03). In the specific case of the sit-to-stand manoeuvre, measurements taken immediately after standing up greatly improve the reproducibility of the autoregulation coefficients. This is generally coupled with an increase of the within-group spread of the estimates.

Identifiability of Generalized Randles Circuit Models

The Randles circuit (including a parallel resistor and capacitor in series with another resistor) and its generalized topology have widely been employed in electrochemical energy storage systems, such as batteries, fuel cells, and supercapacitors, also in biomedical engineering, for example, to model the electrode–tissue interface in electroencephalography and baroreceptor dynamics. This paper studies identifiability of generalized Randles circuit models, that is, whether the model parameters can be estimated uniquely from the input–output data. It is shown that generalized Randles circuit models are structurally locally identifiable. The condition that makes the model structure globally identifiable is then discussed. Finally, the estimation accuracy with respect to noise-free, noisy, zero-mean, and nonzero-mean data is evaluated through extensive simulations. The existing tradeoff between the estimation of Warburg term and other parameters by using zero- and nonzero-mean data is fully discussed.

Reproducibility of dynamic cerebral autoregulation parameters: a multi-centre, multi-method study

OBJECTIVEnDifferent methods to calculate dynamic cerebral autoregulation (dCA) parameters are available. However, most of these methods demonstrate poor reproducibility that limit their reliability for clinical use. Inter-centre differences in study protocols, modelling approaches and default parameter settings have all led to a lack of standardisation and comparability between studies. We evaluated reproducibility of dCA parameters by assessing systematic errors in surrogate data resulting from different modelling techniques.nnnAPPROACHnFourteen centres analysed 22 datasets consisting of two repeated physiological blood pressure measurements with surrogate cerebral blood flow velocity signals, generated using Tiecks curves (autoregulation index, ARI 0-9) and added noise. For reproducibility, dCA methods were grouped in three broad categories: 1. Transfer function analysis (TFA)-like output; 2. ARI-like output; 3. Correlation coefficient-like output. For all methods, reproducibility was determined by one-way intraclass correlation coefficient analysis (ICC).nnnMAIN RESULTSnFor TFA-like methods the mean (SD; [range]) ICC gain was 0.71 (0.10; [0.49-0.86]) and 0.80 (0.17; [0.36-0.94]) for VLF and LF (pu2009u2009=u2009u20090.003) respectively. For phase, ICC values were 0.53 (0.21; [0.09-0.80]) for VLF, and 0.92 (0.13; [0.44-1.00]) for LF (pu2009u2009<u2009u20090.001). Finally, ICC for ARI-like methods was equal to 0.84 (0.19; [0.41-0.94]), and for correlation-like methods, ICC was 0.21 (0.21; [0.056-0.35]).nnnSIGNIFICANCEnWhen applied to realistic surrogate data, free from the additional exogenous influences of physiological variability on cerebral blood flow, most methods of dCA modelling showed ICC values considerably higher than what has been reported for physiological data. This finding suggests that the poor reproducibility reported by previous studies may be mainly due to the inherent physiological variability of cerebral blood flow regulatory mechanisms rather than related to (stationary) random noise and the signal analysis methods.

A model for generating synthetic arterial blood pressure

A new model capable of simulating many important aspects of human arterial blood pressure (ABP) is proposed. Both data-driven approach and physiological principles have been applied to describe the time series of diastolic, systolic, dicrotic notch and dicrotic peak pressure points. Major static and dynamic features of the model can be prescribed by the user, including heart rate, mean systolic and diastolic pressure, and the corresponding physiological control quantities, such as baroreflex sensitivity coefficient and Windkessel time constant. A realistic ABP generator can be used to compile a virtual database of signals reflecting individuals with different clinical conditions and signals containing common artefacts. The ABP model permits to create a platform to assess a wide range of biomedical signal processing approaches and be used in conjunction with, e.g. Kalman filters to improve the quality of ABP signals.