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Multichannel Blind Deconvolution (MBD) is a powerful tool particularly for the identification and estimation of dynamical systems in which a sensor, for measuring the input, is difficult to place. This paper presents an MBD method, based on the Malliavin calculus MC (stochastic calculus of variations). The arterial network is modeled as a Finite Impulse Response (FIR) filter with unknown coefficients. The source signal central arterial pressure CAP is also unknown. Assuming that many coefficients of the FIR filter are time-varying, we have been able to get accurate estimation results for the source signal, even though the filter order is unknown. The time-varying filter coefficients have been estimated through the proposed Malliavin calculus-based method. We have been able to deconvolve the measurements and obtain both the source signal and the arterial path or filter. The presented examples prove the superiority of the proposed method, as compared to conventional methods.

In this paper, we present a new approach to monitor central arterial pressure using the Multichannel Blind Deconvolution (MBD) [

The MBD is the technique that allows the estimation of both an unknown input and unknown channel dynamics from only channel outputs. Although one cannot place a sensor [

The physiologic state of the cardiovascular (CV) system can be most accurately assessed by using the aortic blood pressure or CAP [

Because of the practical difficulty in measuring the arterial pressure waveform near the heart [

In this paper, we suggest characterizing the channels of the single-input, multioutput system model of the arterial tree by linear and time-variant FIR filters. If we make only one of the FIR filter parameters changing over time, then the problem is handled by the Ito calculus [

In this paper, we introduce a method that could estimate time-varying parameters/coefficients. It is based on the stochastic calculus of variations (Malliavin calculus) [

The proposed method is then applied to noninvasive monitoring of the cardiovascular system of the swine. The arterial network is modeled as a multichannel system where the CAP is the input and pressure profiles measured at different branches of the artery, for example, radial and femoral arteries, are the outputs. The proposed method would allow us to estimate both the waveform of the input pressure and the arterial channel dynamics from outputs obtained with noninvasive sensors placed at different branches of the arterial network. Numerical examples verify the major theoretical results and the feasibility of the method. In Section

The cardiovascular system is topologically analogous to a multichannel dynamic system. Pressure wave emanating from a common source, the heart, is broadcast and transmitted through the many vascular pathways. Therefore, noninvasive circulatory measurements taken at different locations (as shown in Figure

The

We estimate the FIR filter coefficients by the conventional methods in Section

Similarly,

The familiar scalar regression format as shown in (

Rearrange (

Once the coefficients of the FIR filter are estimated, we use inverse filtering to find an estimate for the source signal

The linear filter assumption is just an approximation to reality. Sometimes the media (arterial tree) is nonlinear, time varying, random, or all. Moreover and above the measured signals are usually noisy. The filter order is usually unknown. All the factors suggest that the FIR filter model is just an approximation. To compensate for these approximations, we suggest making some of the unknown filter coefficients varying with time, for example, (

We now recast the problem in the format that could be handled by the Malliavin calculus. Using (

To solve the estimation of the time-varying parameters problem, we imbed the sum of the processes (the observations) into another signal,

The augmented observed signal,

In this analysis, the presence of a deterministic component will make the exposition easier. In this case, It is

Assume that the unknown coefficients,

There are several commonly models that represent different physical situations. In our case, the signals,

The coefficients

Once the parameters

In order to further reduce the mathematical complexity, one could restrict the analysis to the estimation of only two coefficients

Our MBD technique is based on a set of assumptions as follows.

The common input is obtained at the output of two sensors. The channels relating the common input to each distinct output, as in Figure

The system is represented by FIR filters. The filters

The FIR filter parameters

An OU model is assumed to describe the signals/regressors

The time-varying FIR filter parameters

Equataion (

An absolute central aortic pressure waveform is determined by scaling the estimated input based on the measured waveforms. In certain embodiments, the reconstructed waveform is calibrated to absolute pressure based on the measured peripheral artery pressure waveforms. For example, the reconstructed waveform is scaled to have a mean value similar to the mean value of the measured waveforms. This scaling step is well justified, since the paths from the central aorta to peripheral arteries offer very little resistance to blood flow due to Poiseuille's law [

The reconstructed waveform

We now give a brief summary in an algorithmic form to describe the methodology as follows.

Insert data in [

Describe the form of the stochastic process OU using the formula in (

Estimate the parameters

Initiate an estimate for the unknown parameters

Use Monte Carlo simulation method to generate Wiener processes

Use (

Use (

Calculate the summed squared error of (

Change the estimate for the unknown parameters

Repeat steps (ii)–(vi) and stop when the error is minimum or the number of iterations is exceeded.

Calculate the matrix

Estimate the source signal by (

Calculate the estimated aortic pressure within scale factor by (

Segments of measured central arterial pressure (AP), femoral AP, and radial AP waveforms from one swine dataset.

The arterial pressure measurements at displayed points in the figure.

To test the proposed approach, first, we took real data from [

Multichannel blind deconvolution was experimentally evaluated with respect to measured data in which femoral artery pressure, radial artery pressure waveforms, and aortic pressure waveform were simultaneously measured (see Figure

The performance measure as the signal-to-noise ratio of the aortic pressure estimates (SNRE) using the conventional FIR-based method (FIR-2 model with two orders and FIR-4 model with 4 orders) and the proposed Malliavin-calculus-based method.

Methods | FIR-2 | FIR-4 | Malliavin calculus |
---|---|---|---|

SNRE | 15.125 db | 18.754 db | 25.111 db |

In the first example, it was assumed that the data were noise free but the order of each of the filters was unknown. In the second example, the data was assumed noisy and the order of each of the filters was unknown. The filter order was estimated using a corrected Akaike information criterion (AIC) [

In both cases, the proposed approach outperformed the conventional method. The pressure at the root of the aorta (central AP) was successfully estimated.

We have two measurements,

Figure

The estimated aorta AP using Malliavin calculus and measured aorta or central AP waveforms.

Typical estimate of

The performace measure SNRE of the estimated noise aorta AP at variant noise by using Malliavin and FIR_4 and FIR-2 methods.

Estimated aorta AP waveform using the conventional methods (FIR-2 model with two orders and FIR-4 model with four orders) and proposed method (Malliavin calculus).

In this example, white Gaussian noise was added to the original values of the measured pressures. It is assumed that the channels are represented by a second-order FIR filter and fourth order FIR filter (

The performance measure (SNRE) after added white Gaussian noise.

FIR-2 | FIR-4 | Malliavin calculus | ||
---|---|---|---|---|

Noise variance | SNR | SNRE | SNRE | SNRE |

1 | 38.199 db | 13.8976 db | 18.599 db | 24.927 db |

10 | 28.217 db | 13.479 db | 18.356 db | 23.443 db |

70 | 19.833 db | 12.748 db | 16.548 db | 18.808 db |

100 | 18.312 db | 12.441 db | 15.767 db | 17.664 db |

300 | 13.705 db | 10.851 db | 13.044 db | 13.768 db |

500 | 11.633 db | 9.7677 db | 11.466 db | 11.803 db |

700 | 10.309 db | 8.9428 db | 10.374 db | 10.523 db |

The multichannel blind deconvolution (MBD) technique was applied by using the conventional method (FIR-2 model with two orders and FIR-4 model with 4 orders) and the proposed method (Malliavin calculus). We compared between these methods by using the performance measure SNRE (

Figure

Estimated noisy aorta AP waveforms using the conventional methods FIR-4 model with 4 orders and FIR-2 Model with 2 Orders. (SNR = 19.833 db, SNRE = 16.548 db)

The estimated aorta AP using Malliavin calculus at high noise variance and measured aorta or central AP waveforms. (SNR = 19.833 db, SNRE = 18.808 db).

We have developed a new technique to estimate the aortic pressure waveform from multiple measured peripheral artery pressure waveforms. The technique is based on multichannel blind deconvolution in which two or more measured outputs (peripheral artery pressure waveforms) of a single input, multioutput system (arterial tree) are mathematically analyzed so as to reconstruct the common unobserved input (aortic pressure waveform). Each channel is modeled as an FIR filter. We assumed that all of the FIR filter parameters/coefficients are varying slowly with time. Their values were estimated using methods based on the Malliavin calculus. By this assumption, time-varying parameters, we were able to compensate for the wrong FIR filter order and the possible time variations/nonlinearities of the channels. The blind deconvolution problem was reformulated as a regression problem with unknown time-varying regression coefficients. We introduced a new method for the estimation of these slowly time-varying regression coefficients. The method relies heavily on the Malliavin calculus (stochastic calculus of variations) and the generalized Clark-Ocone formula to find a closed-form expression for the estimates of the unknown time-varying coefficients. Some approximations were needed to find mathematically tractable estimates. While the approach is quite general and can be applied to any signal model, we present only one signal model, the Ornstein Uhlenbeck process. Other models could have been used as well.

We tested the proposed technique in swine experiments, and our results showed superior performance for our proposed approach compared to conventional methods. Our way to reconstructed AP is simple and straightforward. Our method needs only the calculation of pressure wave components in the time domain and does not need calculations in the frequency domain and no need to large computer time. Because of this simplicity it is quite possible to implement this method in monitoring central pressure AP on-line. In the future, we suggest expanding this method by applying it to real data taken from human cardiovascular simulator. We suggest studying the case where the time-varying parameters/coefficients are rapidly changing over time. Another extension is the estimation of the flow at the root of the Aorta from peripheral measurements of the flow/pressure.

Let the augmented signal

Under the slowly varying-coefficient assumptions we have:

Let

Define

Notice that, for a random variable

Girsanov's theorem states that

The generalized version of the Clark-Ocone formula will now be used to find an estimate for the unknown time-varying coefficients

Let

In our case,

Equation (

Most of the time, it is difficult to find a closed-form expression for this equation. Instead we shall try to put it as a summation of three terms: (

Since

Notice that the drift and the diffusion coefficients of

Without an expression for

Akaike information criterion

Arterial pressure

Blood pressure

Clark-Ocone

Cardiovascular

Finite impulse response

Ordinary least square

Peripheral arterial pressure

Stochastic differential equation

Signal-to-noise ratio

Signal-to-noise ratio of the estimate

Total peripheral resistance.