On-line estimation and prediction of urea kinetics during hemodialysis: a simulation approach

A new method for the on-line estimation of urea kinetic parameters from blood urea concentration (BUN) continuously measured during a dialysis session is proposed. The method, based on the variable-volume double-pool model, is evaluated through a simulation approach in order to easily consider a large set of well-controlled test conditions. The model is characterized by six parameters, knowledge of which enables early prediction of the end dialysis urea concentration and the dose of dialysis. The sensitivity of the model predicted BUN with respect to the parameters was first analyzed to investigate which can be reliably estimated from blood urea measurements taken at a suitable rate. This analysis showed that the model predicted BUN is highly sensitive to the initial blood urea concentration and to the dialyzer clearance, normalized with respect to the total initial distribution volume, while it is scarcely influenced by the normalized ultrafiltration and urea generation rates. The new on-line estimation technique keeps these two last parameters constant and takes advantage of an original analytic solution of the second order urea kinetics. The results of the estimation process on realistic simulated data showed that the proposed method provides early and reliable estimates of the normalized clearance and of the end dialysis concentration. The transcellular mass transfer coefficient and the intra-extra cellular volume ratio can also be estimated, although with less accuracy. Moreover, it was shown that the use of the single-pool model, instead of the double-pool one, provides systematic errors on the estimates.

Influence of flow pattern on the parameter estimates of a simple breathing mechanics model

The first-order model of breathing mechanics is widely used in clinical practice to assess the viscoelastic properties of the respiratory system. Although simple, this model takes the predominant features of the pressure-flow relationship into account but gives highly systematic residuals between measured and model-predicted variables. To achieve a better fit of the entire data set, an approach hypothesizing deterministic time-variations of model parameters, summarized by information-weighted histograms was recently proposed by Bates and Lauzon. The present study uses flow and pressure data measured in intensive care patients to evaluate the real potential of this approach in clinical practice. Information-weighted histograms of the model parameters, estimated by an on-line identification algorithm, were first constructed by taking into account the parameter percentage standard deviations. Then, the influence of the respiratory flow pattern on the calculated histograms was evaluated by the Kolmogorov-Smirnov statistical test. The results show that the method gives good reproducibility under stable experimental conditions. In addition, for a given airflow waveform, an increase in respiratory frequency shifts the histograms representing time-varying viscous properties strongly versus lower values, whereas it shifts the histograms representing time-varying elastic properties slightly versus higher values. On the other hand, the same histograms were highly dependent on the airflow waveform, especially for the viscous properties. Even in a limited experimental work, in all the conditions considered, the method provides results which agree well with the physiological knowledge of nonlinear and multicompartment behavior of respiratory mechanics.

Sensitivity analysis for estimating urea kinetics parameters during hemodialysis

In this paper a time-varying volume, double-pool urea kinetics model is considered and a sensitivity analysis is carried out to determine those patient parameters that have greater influence on the time course of blood urea nitrogen concentration (BUN) during and between dialysis treatment. The model parameters include the urea generation rate, the initial distribution volume of the urea, the ratio between intracellular and extracellular volumes, and the mass transfer coefficient between the two pools. The analysis demonstrates that BUN is highly sensitive to the urea generation rate and total distribution volume whereas it is influenced by the remaining parameters to a much lesser extent. In addition, the location of the absolute maxima of BUN sensitivity functions suggests the rational placement of a reduced number of blood samples that may still allow sufficiently accurate estimates for the parameters of clinical interest, such as the urea generation rate, total distribution volume, and the ratio between intracellular and extracellular volumes. This conclusion has been confirmed by simulation studies where parameter estimation has been performed with a varying number of observation points.

Identification of the three-element windkessel model incorporating a pressure-dependent compliance

A new one-step computational procedure is presented for estimating the parameters of the nonlinear three-element windkessel model of the arterial system incorporating a pressure-dependent compliance. The data required are pulsatile aortic pressure and flow. The basic assumptions are a steady-state periodic regime and a purely elastic compliant element. By stating two conditions, zero mean flow and zero mean power in the compliant element, peripheral and characteristic resistances are determined through simple closed form formulas as functions of mean values of the square of aortic pressure, the square of aortic flow, and the product of aortic pressure with aortic flow. The pressure across as well as the flow through the compliant element can be then obtained so allowing the calculation of volume variation and compliance as functions of pressure. The feasibility of this method is studied by applying it to both simulated and experimental data relative to different circulatory conditions and comparing the results with those obtained by an iterative parameter optimization algorithm and with the actual values when available. The conclusion is that the proposed method appears to be effective in identifying the three-element windkessel even in the case of nonlinear compliance.

Two new algorithms for tracking arterial parameters in nonstationary noise conditions

Two new algorithms with reduced sensitivity to the changing environment are applied to tracking arterial circulation parameters. They are variants of the Least-Squares (LS) algorithm with Variable Forgetting factor (LSVF), and of the Constant Forgetting factor-Covariance Modification (CFCM) LS algorithm, devised to overcome their main practical deficiencies related to noise level sensitivity and the high number of design variables, respectively. To this end, adaptive mechanisms are incorporated to estimate observation noise variance in LSVF and the rate of change for the different parameters in CFCM. Specific computer simulation experiments are presented to compare their effectiveness with the original counterparts and to provide guidelines for their optimal tuning at different noise levels. Moreover, algorithm performance degradation, consequent on changes in the noise level compared to that assumed during the tuning phase, is analyzed. In particular, it is shown that, when the noise level changes with respect to the tuning value, the new LSVF algorithm is much more robust than the original one, whose performance degrades rapidly. The new CFCM algorithm is characterized by a reduced number of design variables with respect to its original counterpart. Nevertheless, it can be preferred only when low noise signals are used for estimation.

Comparative evaluation of different methods to estimate urea distribution volume and generation rate

Eight methods to estimate urea distribution volume and generation rate from blood urea samples measured in dialysis patients are reviewed. An analytical solution has been provided for a double-pool variable volume kinetic model to allow for faster and more accurate simulation and identification. The reliable parameter estimates provided by the double-pool kinetic model starting from seven samples, were assumed as references for the estimates obtained by the remaining methods. These include three kinetic models and four methods based on urea mass-balance. In particular, the estimation techniques differ in the number of compartments where urea is assumed distributed (double- and single-pool) or in the number of blood urea samples. Among the methods based on mass-balance, two techniques neglecting the weight loss or the urea generation during dialysis, were also analysed. The results obtained during hemofiltration sessions using three samples, usually available in clinical practice at the beginning and at the end of dialysis, demonstrate that a new method based on double-pool kinetics provides, on average, the most reliable estimates. Moreover, methods belonging to a single pool view and including both weight loss and urea generation during dialysis seem to underestimate by 1 divided by 2 liters the urea distribution volume. However, neglecting the weight loss or the urea generation can overcompensate this error, resulting in a significant overestimation of the distribution volume. Finally, it has been experimentally proved that the single-pool kinetic methods overestimate the urea production rate, while techniques based on mass balance provide more reliable values.

Comparison of algorithms for tracking short-term changes in arterial circulation parameters

Three recursive methods especially suited for identification of systems with rapidly changing parameters are applied to tracking of the viscoelastic properties of the systemic arterial bed. These methods include two least squares (LS) algorithms with constant or variable forgetting factor (RLS and LSVF) and a LS algorithm incorporating both a constant forgetting factor and covariance modification (CFCM). The methods are presented in a unified framework and their sensitivity with respect to the design variables is investigated using noisy data from computer simulations. All analysed methods have shown themselves to be able to satisfactory track rapid changes in peripheral resistance. The LSVF method, which offers slightly better performances than the classical RLS, may be preferred when calculation efficiency is the prime requirement. The CFCM algorithm, although maintaining reasonable simplicity, shows the best tracking ability also on varying of the noise sequence.

Real-time tracking of parameters of lung mechanics: emphasis on algorithm tuning

We consider the problem of tracking rapid changes in the viscous and elastic properties of the respiratory system by using mouth flow and transpulmonary pressure data measured during mechanical ventilation. A recursive least-squares algorithm with adjustable compensator is used for online estimation of an R-C model of the breathing mechanics. Specific simulation experiments are presented to provide guidelines to select suitable values for the key variable, which controls the compromise between tracking ability and noise sensitivity. The results obtained confirm the critical role of the optimum tuning in relation to the noise level. Experimental results obtained from data measured on mechanically-ventilated dogs, in which respiratory distress syndrome was intravenously induced by oleic acid, demonstrate that the tuned algorithm is able to track appropriately both the viscous and elastic properties of lung mechanics. Parameter estimates are consistent with those obtained by standard and robust offline algorithms and their time course is in qualitative agreement with known physiopathological behaviour.

Electrical analogs for monitoring vascular properties in artificial heart studies

The problem of choosing parametric descriptions of the systemic vascular bed suitable for monitoring beat-to-beat changes in peripheral vascular properties is considered. Three simple models with two, three, and five elements are compared, essentially exploiting the Akaike information criterion combined with reasonable requirements for estimate accuracy. Analysis of pressure data, which are either simulated by the five-element model or measured on a mock circulatory system during abrupt changes in peripheral resistance, suggests guidelines for model selection. In particular, the five-element model exhibits very close adherence to physical reality by allowing for reflection, while the classical windkessel model provides the most reproducible estimates.

Tracking time-varying properties of the systemic vascular bed

The problem of tracking changes in viscoelastic properties of the systemic arterial bed is considered and a recursive estimation procedure, belonging to the class of output-error algorithms with adjustable compensator, is developed and discussed. By means of computer simulations, suitable values are determined for the key design variable which controls the tradeoff between tracking ability and noise sensitivity of the algorithm. In this way, the algorithm allows on-line estimation of arterial compliance, peripheral resistance, and characteristic impedance on the basis of aortic pressure and flow signals. Furthermore, the results obtained from data numerically simulated, as well as measured on a mock circulatory system, demonstrate that the dominant arterial time-constant can be tracked by the algorithm using only measurements of the aortic pressure during diastole.

Sensitivity analysis of the systemic circulation with a view to computer simulation and parameter estimation

A sensitivity analysis study has been performed on a seven-parameter model of the systemic vascular bed in order to obtain structure reductions appropriate for simulation and estimation. This analysis considers separately the systolic and diastolic transfer functions between arterial and venous pressures in order to divide a non-linear problem in two distinct linear problems. The results obtained refer to nominal parameter values corresponding to normal circulatory conditions in man and supply guide-lines for an application-oriented selection of reduced models. Simple resistance-compliance models are preferred because the inertial effects appear to have only slight influence. In particular, the choice of a five-parameter model seems to be convenient for simulation purposes. An additional structure reduction is suggested to reach reliable results in parameter estimation problems. The resulting model is characterized by three elements: peripheral resistance, arterial compliance and venous compliance.

CADCS simulation of the closed-loop cardiovascular system

A pulsatile simulator of the closed-loop cardiovascular system, designed to solve simulation, identification and control problems in a research and education context, is presented. Its implementation makes use of a command-driven interactive program for simulation of non-linear ordinary differential equations. The flexibility of the simulator is demonstrated by the results presented which refer to a basal steady-state circulatory condition as well as a transient induced by an abrupt change in peripheral resistance.

Three-element model for total systemic circulation: emphasis on the accuracy of parameter estimates

In this study, the accuracy achievable in the parameter estimates of a three-element linear model for the systemic vascular bed is considered. The model neglects inertial effects and includes only three elements representing arterial compliance, peripheral resistance and venous compliance, in agreement with recent sensitivity investigations. Parameter estimation starts from arterial and right atrial pressure signals generated by a closed-loop simulator of the cardiovascular circulation and corrupted with normal noise to account for measurement errors. In this way, the influences of a wide variety of circulatory conditions were investigated. The results achieved give evidence that arterial compliance is generally well estimated, while venous compliance is more variable, particularly at high peripheral resistance when measured signals appear to be less sensitive to this parameter. However, presence of cardiac disease, such as heart failure and valvular stenosis has minimal influence on compliance estimates. These results suggest that this simple model can be conveniently applied even under noisy conditions.

Model selection for ventricular mechanics: a sensitivity analysis approach

Quantitative characterization of left ventricle pump properties has been recognized as being of great significance for both physiological and clinical purposes. Several descriptions have been proposed in the past to this end, where the ventricle is viewed as an isovolumic pressure generator coupled to an internal impedance, considered as either only viscous, only elastic or viscoelastic. Though these models have been used widely, the respective advantages and limits have not been fully elucidated. In this paper, six models for the left ventricular pumping function, of the viscoelastic type, are compared using both simulated and experimental data in a typical parameter estimation approach. Elastic and viscous parameters are estimated starting from ventricular pressure and aortic flow, together with the isovolumic pressure at the same preload. The basis for the comparison is the well-established criterion relating the fit obtained from collected data and the covariance matrix of the parameter estimates. The latter allows evaluation of the so-called indifference region in the parameter space, which is represented by an ellipse if both elastic and viscous elements are present. The properties of the indifference region are synthetically represented by two indices linked to the area and the eccentricity of the ellipse: the first represents the mean accuracy of the parameter estimate, the second gives information about the different sensitivities to variation of single parameters. This comparison, in both simulated and experimental cases, generally leads to preference for a model where elastance and viscosity vary with time in linear proportion to the isovolumically developed ventricular pressure. Appropriate description of the elastic effect reveals it to be very crucial while the viscous effect, though improving the fitting of data, is less critical.