This paper introduces a stochastic framework for a recently proposed discrete-time delay estimation method in Laguerre-domain, i.e. with the delay block input and output signals being represented by the corresponding Laguerre series. A novel Laguerre-domain disturbance model allowing the involved signals to be square-summable sequences is devised. The relation to two commonly used time-domain disturbance models is clarified. Furthermore, by forming the input signal in a certain way, the signal shape of an additive output disturbance can be estimated and utilized for noise reduction. It is demonstrated that a significant improvement in the delay estimation error is achieved when the noise sequence is correlated. The noise reduction approach is applicable to other Laguerre-domain problems than pure delay estimation.
Aim: To construct a Treatment Response Index from Multiple Sensors (TRIMS) for quantification of motor state in patients with Parkinson's disease (PD) during a single levodopa dose. Another aim was to compare TRIMS to sensor indexes derived from individual motor tasks.
Method: Nineteen PD patients performed three motor tests including leg agility, pronation-supination movement of hands, and walking in a clinic while wearing inertial measurement unit sensors on their wrists and ankles. They performed the tests repeatedly before and after taking 150% of their individual oral levodopa-carbidopa equivalent morning dose.Three neurologists blinded to treatment status, viewed patients' videos and rated their motor symptoms, dyskinesia, overall motor state based on selected items of Unified PD Rating Scale (UPDRS) part III, Dyskinesia scale, and Treatment Response Scale (TRS). To build TRIMS, out of initially 178 extracted features from upper- and lower-limbs data, 39 features were selected by stepwise regression method and were used as input to support vector machines to be mapped to mean reference TRS scores using 10-fold cross-validation method. Test-retest reliability, responsiveness to medication, and correlation to TRS as well as other UPDRS items were evaluated for TRIMS.
Results: The correlation of TRIMS with TRS was 0.93. TRIMS had good test-retest reliability (ICC = 0.83). Responsiveness of the TRIMS to medication was good compared to TRS indicating its power in capturing the treatment effects. TRIMS was highly correlated to dyskinesia (R = 0.85), bradykinesia (R = 0.84) and gait (R = 0.79) UPDRS items. Correlation of sensor index from the upper-limb to TRS was 0.89.
Conclusion: Using the fusion of upper- and lower-limbs sensor data to construct TRIMS provided accurate PD motor states estimation and responsive to treatment. In addition, quantification of upper-limb sensor data during walking test provided strong results.
A method for estimating the instantaneous heart rate (HR) using the morphological features of one electrocardiogram (ECG) cycle (beat) at a time is proposed. This work is not aimed at introducing an alternative way for HR estimation, but rather illustrates the utility of model-based ECG analysis in online individualized monitoring of the heart function. The HR estimation problem is reduced to fitting one parameter, whose value is related to the nine parameters of a realistic nonlinear model of the ECG and estimated from data by nonlinear least-squares optimization. The method feasibility is evaluated on synthetic ECG signals as well as signals acquired from MIT-BIH databases at Physionet website. Moreover, the performance of the method was tested under realistic free-moving conditions using a wearable ECG and HR monitor with encouraging results.
A novel model-based method for patient-specific detection of deformed electrocardiogram (ECG) beats is proposed and tested. Five parameters of a patient-specific nonlinear ECG model are estimated from data by nonlinear least-squares optimization. The normal variability of the model parameters is captured by estimated probability density functions. A binary classifier, based on stochastic anomaly detection methods, along with a pre-tuned classification threshold, is employed for detecting the abnormal ECG beats. We demonstrate the utility of the proposed approach by validating it on annotated arrhythmia data recorded under clinical conditions.
Modeling, control, and gas leakage detection in the coal injection process are discussed. It is shown that by use of model-based methods, the flow and pressure of the coal injection vessel are reliably controlled. With the new control law, the coal mass flow can be used as a control parameter for the blast furnace. High injection rates can be used and more coke substituted, This is expected to yield a cost reduction in the iron production. An experimental comparison of the conventional control unit with the one suggested in this article shows that an improvement of the process efficiency can be reached by other means than increasing the capacity of the plant
This article presents a way of modeling patient response to a pharmacotherapy by means of dynamic models with quantized output. The proposed modeling technique is exemplified by treatment of Parkinson's disease with Duodopa ®, where the drug is continuously administered via duodenal infusion. Titration of Duodopa ® is currently performed manually by a nurse judging the patient's motor symptoms on a quantized scale and adjusting the drug flow provided by a portable computer-controlled infusion pump. The optimal drug flow value is subject to significant inter-individual variation and the titration process might take up to two weeks for some patients. In order to expedite the titration procedure via automation, as well as to find optimal dosing strategies, a mathematical model of this system is sought. The proposed model is of Wiener type with a linear dynamic block, cascaded with a static nonlinearity in the form of a non-uniform quantizer where the quantizer levels are to be identified. An identification procedure based on the prediction error method and the Gauss-Newton algorithm is suggested. The datasets available from titration sessions are scarce so that finding a parsimonious model is essential. A few different model parameterizations and identification algorithms were initially evaluated. The results showed that models with four parameters giving accurate predictions can be identified for some of the available datasets.
The mathematical modeling of the human smooth pursuit system from eye-tracking data is considered. Recently developed algorithms for the estimation of Volterra-Laguerre (VL) models with explicit time delay are applied in continuous and discrete time formulations to experimental data collected from Parkinsonian patients in different medication states and healthy controls. The discrete VL model with an explicit time delay and the method for its estimation are first introduced in this paper. The estimated parameters of a second-order VL model are shown to capture the ocular dynamics both in health and disease. The possibility of including the estimated time delay, along with the VL kernel parameters, into the set of the model parameters is explored. The results obtained in continuous VL modeling are compared with those in discrete time to discern the effects due to the sampling enforced by the eye tracker used for data acquisition.
The problem of estimating nonlinear time-delay dynamics captured by continuous Volterra models from input-output data is treated. The delayed Volterra kernels are seen as impulse responses of linear time-invariant systems with time delay. Analytical expressions for the Laguerre series, where the Laguerre coefficients of the finite-dimensional part are admixed with the terms due to the delay, are provided. By utilizing the linearity of Volterra-Laguerre models in the unknown parameters, the model is estimated by a nonlinear least-squares method. An application of the proposed approach to the problem of Volterra modelling of the human smooth pursuit system from eye-tracking data is provided. The proposed approach demonstrates consistently accurate performance on both simulated and experimental data.
This paper deals with the identification of Volterra models that capture the dynamics of smooth pursuit eye movements recorded by an eye tracker. The framework is motivated by neurological applications but can also be useful in biometrics. In healthy subjects, ocular dynamics are predominantly linear, while neurological conditions inflict nonlinearity on smooth pursuit eye movements. Besides overparameterization, Volterra models may also exhibit functional dependence among the model coefficients. A combination of sparse estimation and Principal Component Analysis is shown to be instrumental in estimating parsimonious Volterra models from eye-tracking data. The efficacy of the approach is demonstrated on experimental data collected from Parkinsonian patients as well as healthy controls.
This paper solves the problem of estimating the time delay in a smooth nonlinear system modeled by a Volterra model (VM), along with the delay-free dynamics captured by the convolution operators. VMs with explicit time delay, whose convolution kernels are represented as series of Laguerre functions, are considered in continuous and discrete-time. Since a VM with a time delay is linear in the kernel parameters, the proposed modeling approach, on the one hand, allows for the use of efficient linear sparse estimation techniques and, on the other hand, extends the applicability of Volterra-Laguerre models (VLM) to a broader class of nonlinear systems. Hinging on recent progress in Laguerre domain modeling of linear time-delay systems, analytical expressions for the projections of kernels with a delayed argument on the Laguerre basis are obtained, featuring special kinds of polynomials that arise due to the presence of time delay. The mathematical modeling problem of the human smooth pursuit system from eye-tracking data illustrates the practical utility of the acquired theoretical results.
A closed-form Laguerre-domain representation of discrete linear time-invariant systems with constant input time delay is derived. It is shown to be useful in a I-2 -> I-2 system identification setup (with I-2 denoting square-summables signals) often arising in biomedical applications, where the experimental protocol does not allow for persistent excitation of the system dynamics. The utility of the proposed system representation is demonstrated on a problem of drug kinetics estimation from clinical data.
A parsimonious mathematical model of pulse modulated regulation of non-basal testosterone secretion in the male is developed. The model is of the third differential order, reflecting the three most significant hormones in the regulation loop, but is yet shown to be capable of sustaining periodic solutions with one or two pulses of gonadotropin-releasing hormone (GnRH) in each period. Lack of stable periodic solutions is otherwise a main shortcoming of existing low-order hormone regulation models. Existence and stability of periodic solutions are studied. The periodic mode with two GnRH pulses in the least period has not been described in medical literature, but is found to explain experimental data well.
It is shown that an impulsive system with a time-delay in the continuous part can be equivalently represented by discrete dynamics under less restrictive conditions on the time-delay value than considered previously.
Impulsive Goodwin's oscillator model is introduced to capture the dynamics of sustained periodic processes in endocrine systems controlled by episodic pulses of hormones. The model is hybrid and comprises a continuous subsystem describing the hormone concentrations operating under a discrete pulse-modulated feedback implemented by firing neurons. Time delays appear in mathematical models of endocrine systems due to the significant transport phenomena but also because of the time necessary to produce releasable hormone quantities. From a biological point of view, the neural control should be robust against the time delay to ensure the loop functionality over a wide range of inter-individual variability. The paper provides an overview of the currently available results and contributes a generalization of a Poincare mapping approach to study complex dynamics of impulsive Goodwin oscillator. Both pointwise and distributed time delays are considered in a general framework based on the Poincare mapping. Bifurcation analysis is utilized to illustrate the analytical results.
A popular biomathematics model of the Goodwin oscillator has been previously generalized to a more biologically plausible construct by introducing three time delays to portray the transport phenomena arising due to the spatial distribution of the model states. The present paper addresses a similar conversion of an impulsive version of the Goodwin oscillator that has found application in mathematical modeling, e.g. in endocrine systems with pulsatile hormone secretion. While the cascade structure of the linear continuous part pertinent to the Goodwin oscillator is preserved in the impulsive Goodwin oscillator, the static nonlinear feedback of the former is substituted with a pulse modulation mechanism thus resulting in hybrid dynamics of the closed-loop system. To facilitate the analysis of the mathematical model under investigation, a discrete mapping propagating the continuous state variables through the firing times of the impulsive feedback is derived. Due to the presence of multiple time delays in the considered model, previously developed mapping derivation approaches are not applicable here and a novel technique is proposed and applied. The mapping captures the dynamics of the original hybrid system and is instrumental in studying complex nonlinear phenomena arising in the impulsive Goodwin oscillator. A simulation example is presented to demonstrate the utility of the proposed approach in bifurcation analysis.
Objective: Deep Brain Stimulation (DBS) consists of delivering electrical stimuli to a brain target via an implanted lead to treat neurodegenerative conditions. Individualized stimulation is vital to ensure therapeutic results, since DBS may otherwise become ineffective or cause undesirable side effects. Since the DBS pulse generator is battery-driven, power consumption incurred by the stimulation is important. In this study, target coverage and power consumption are compared over a patient population for clinical and model-based patient-specific settings calculated by constrained optimization. Methods: Brain models for five patients undergoing bilateral DBS were built. Mathematical optimization of activated tissue volume was utilized to calculate stimuli amplitudes, with and without specifying the volumes, where stimulation was not allowed to avoid side effects. Power consumption was estimated using measured impedance values and battery life under both clinical and optimized settings. Results: It was observed that clinical settings are generally less aggressive than the ones suggested by unconstrained model-based optimization, especially under asymmetrical stimulation. The DBS settings satisfying the constraints were close to the clinical values. Conclusion: The use of mathematical models to suggest optimal patient-specific DBS settings that observe technological and safety constraints can save time in clinical practice. It appears though that the considered anatomy-related safety constraints depend on the patient and further research is needed in this regard. Power consumption is important to consider since it increases with the square of the stimuli amplitude and critically affects battery life. Significance: This work highlights the need of specifying the brain volumes to be avoided by stimulation while optimizing the DBS amplitude, in contrast to minimizing general stimuli overspill, and applies the technique to a cohort of patients. It also stresses the importance of taking power consumption into account.
Objective: Deep brain stimulation (DBS) consists of delivering electrical stimuli to a brain target via an implanted lead to treat neurological and psychiatric conditions. Individualized stimulation is vital to ensure therapeutic results, since DBS may otherwise become ineffective or cause undesirable side effects. Since the DBS pulse generator is battery-driven, power consumption incurred by the stimulation is important. In this study, target coverage and power consumption are compared over a patient population for clinical and model-based patient-specific settings calculated by constrained optimization.
Approach: Brain models for five patients undergoing bilateral DBS were built. Mathematical optimization of activated tissue volume was utilized to calculate stimuli amplitudes, with and without specifying the volumes, where stimulation was not allowed to avoid side effects. Power consumption was estimated using measured impedance values and battery life under both clinical and optimized settings.
Results: It was observed that clinical settings were generally less aggressive than the ones suggested by unconstrained model-based optimization, especially under asymmetrical stimulation. The DBS settings satisfying the constraints were close to the clinical values.
Significance: The use of mathematical models to suggest optimal patient-specific DBS settings that observe technological and safety constraints can save time in clinical practice. It appears though that the considered safety constraints based on brain anatomy depend on the patient and further research into it is needed. This work highlights the need of specifying the brain volumes to be avoided by stimulation while optimizing the DBS amplitude, in contrast to minimizing general stimuli overspill, and applies the technique to a cohort of patients. It also stresses the importance of considering power consumption in DBS optimization, since it increases with the square of the stimuli amplitude and also critically affects battery life through pulse frequency and duty cycle.
Deep Brain Stimulation (DBS) is a well-established treatment in neurodegenerative diseases, e.g. Parkinson's Disease. It consists of delivering electrical stimuli to a target in the brain via a chronically implanted lead. To expedite the tuning of DBS stimuli to best therapeutical effect, mathematical models have been developed during recent years. The electric field produced by the stimuli in the brain for a given lead position is evaluated by numerically solving a Partial Differential Equation with the medium conductivity as a parameter. The latter is patient- and target-specific but difficult to measure in vivo. Estimating brain tissue conductivity through medical imaging is feasible but time consuming due to registration, segmentation and post-processing. On the other hand, brain atlases are readily available and processed. This study analyzes how alternations in the conductivity due to inter-patient variability or lead position uncertainties affect both the stimulation shape and the activation of a given target. Results suggest that stimulation shapes are similar, with a Dice's Coefficient between 93.2 and 98.8%, with a higher similarity at lower depths. On the other hand, activation shows a significant variation of 17 percentage points, with most of it being at deeper positions as well. It is concluded that, as long as the lead is not too deep, atlases can be used for conductivity maps with acceptable accuracy instead of fully individualized though medical imaging models.
Deep Brain Stimulation (DBS) is an established therapy for treating e.g. Parkinson's disease, essential tremor, as well as epilepsy. In DBS, chronic pulsatile electrical stimulation is administered to a certain target area of the brain through a surgically implanted lead. The stimuli parameters have to be properly tuned in order to achieve therapeutical effect that in most cases is alleviation of motor symptoms. Tuning of DBS currently is a tedious task since it is performed manually by medical personnel in a trial-and-error manner. It can be dramatically improved and expedited by means of recently developed mathematical models together with control and estimation technology. This paper presents a control engineering perspective on DBS, viewing it as a control system for minimizing the severity of the symptoms through coordinated manipulation of the stimuli parameters. The DBS model structure comprises a stimuli model, an activation model, and a symptoms model. Each of those is individualized from patient data obtained through medical imaging, electrical measurements, and objective symptom quantification. The proposed approach is illustrated by simulation and clinical data from an individualized DBS model being developed by the authors.
Deep Brain Stimulation (DBS) is an established treatment, in e.g. Parkinson's Disease, whose underlying biological mechanisms are unknown. In DBS, electrical stimulation is delivered through electrodes surgically implanted into certain regions of the brain of the patient. Mathematical models aiming at a better understanding of DBS and optimization of its therapeutical effect through the simulation of the electrical field propagating in the brain tissue have been developed in the past decade. The contribution of the present study is twofold: First, an analytical approximation of the electric field produced by an emitting contact is suggested and compared to the numerical solution given by a Finite Element Method (FEM) solver. Second, the optimal stimulation settings are evaluated by fitting the field distribution to a target one to control the spread of the stimulation. Optimization results are compared to those of a geometric approach, maximizing the intersection between the target and the activated volume in the brain tissue and reducing the stimulated area beyond said target. Both methods exhibit similar performance with respect to the optimal stimuli, with the electric field control approach being faster and more versatile.
A simple non-intrusive approach to tremor quantification utilizing the repetitive nature of the phenomenon is proposed and implemented on a portable device equipped with a fused off-the-shelf sensor platform measuring 3D acceleration. The device can be automatically activated when picked up from a stationary position and acceleration measurements are performed for a certain time interval. This usage scenario naturally arises e.g. when a person lifts the cellular phone from a surface to the ear to make or answer a call. The relatively slow and damped voluntary movement is separated by filtering from the involuntary and repetitive tremor manifestations in the device position. Extreme points of the tremor signal are detected and the time stamps of the corresponding events are used to estimate of the momentary tremor amplitude and frequency. Kalman filtering of the estimates is applied further to obtain their smoothed versions.
The most prevalent steel-making process is the basic oxygen steel-making (BOS) process. Problems arise when the layer of foaming slag created on the surface of the molten metal exceeds the height of the vessel and overflows, causing metal loss, process disruption and environmental pollution. This phenomenon is commonly referred to as slopping. A method for automatic slopping detection is described in this contribution. The sound signal from a microphone located in the off-gas funnel is processed to obtain an estimate of the slag level in the converter. A model describing the relationship between off-gas flow rate, pressure and the slag level estimate is updated recursively in time. The output error is fed to a change detector yielding a warning system with three alarm levels indicating the persistence of slopping symptoms. The algorithm was tested on data from 100 heats at SSAB Oxelosund. Slopping was correctly detected in 80% of the blows.