Curve Approximation Models based on Statistical Distribution with Application to Photoplethysmography (PPG) Signal

Photoplethysmography (PPG) signal captures blood volume change in the arteries that carry blood. PPG signals are often used to check the cardiovascular health in patients. Health care automation have made more importance into study of PPG signal and its automatic prognosis and diagnosis. In this paper we aim to achieve motion artifact reduction using low rank minimization and PPG signal representation in mathematical form using statistical distribution models. The proposed approach has been tested for Gaussian, Bifurcated Gaussian, Exponentially broadened Gaussian and lognormal distributions. The accuracy of each distribution in modeling the PPG signal was also studied.


Introduction
Photoplethysmography (PPG) is a vital health care monitoring device used for monitoring the arterial blood oxygen saturation and blood volume change [1,2].Usually the volume of blood through the artery can vary depending on the arterial contraction and expansion. PPG is one of the vital parameter that gives the cardiovascular health of a person. The PPG signal is detected by either transmittance or reflectance method. The PPG signal consist of two or more peaks as shown in figure.1 The first peak is the main wave or main peak formed by blood volume transmitted from left ventricle of heart to finger. The rest of the peaks are called reflected waves or reflected peaks formed by the blood flows to aorta due to resistance difference in the lower extremities [3][4][5][6].Forward and reflected wave have equal importance for determining cardiovascular health. However this paper focuses on Arterial Stiffness disorder and it can be obtained by studying reflected wave of the PPG. In older person and those who are suffering from high arterial stiffness, reflected wave returns during early systole and it returns during late systole in younger person.
Thus the study of PPG signals becomes vital for detecting and diagnosing the arterial stiffness condition. This paper proposed an automated approach to mathematically model the PPG signal using statistical distribution function. The novelty in the proposed approach is that the mathematical representation of PPG signal makes it quantitatable, thereby computational analysis can be carried out. PPG wave decomposition separates the PPG into forward and reflected wave. The modeling of the PPG wave using statistical distribution models has been adopted previously with good performances [7].In most of the works, Gaussian forms a universal distribution function base. Rayleigh distribution provides another useful alternative to model PPG signal. The adaptive combination of Gaussian and Rayleigh can also be seen [8].A mixture of Gaussian have also been used for fitting PPG signal for carotid and radial artery pressure waveform [9,10].The lognormal basis distribution performed better than more generalized and global Gaussian functions [11].Also adaptive hybrid algorithms which work with combination of two or more distribution functions are more effective than a single distribution function [12,13].
In this work we introduced a method of motion artifact reduction using low rank minimization algorithm and a comparative analysis of four different distribution functions namely Gaussian, bifurcated Gaussian, Exponentially broadened Gaussian and lognormal distribution has been analyzed. The main contribution of the proposed approach is the error estimation for each of these four distributions. The errors computed help us to identify the distribution which succeeds in a hybrid adaptive environment which forms base of the forthcoming research.

PPG Datasets
A cohort of 120 subjects from the local area of southern Tamil Nadu was recruited. First group involved, Sixty-eight stable non-smoking subjects between the ages of 16 and 35 were included (36 men and 32 women) without any known hypertension, diabetes mellitus and no medications affecting the cardiovascular system. Second group involved a total of fifty-two subjects (32 males and 20 females) between the ages of 40 and 82 with hypertension and no cardiovascular medication. The signal sampling rate was taken as 125 samples per second. Subjects were required to be in a supine position for 10 minutes before fingertip signal acquisition. HRM-2115E, which functions in the transmission mode, is the optical sensor used here. For the analysis, the first 10 pulses were taken from each subject and the mean value of the parameters were taken into account. A written consent form was given to each participant to participate in the study, which was approved by the SRM Medical College and Research Center Ethical Committee, Chennai, Tamil Nadu.

Low Rank Minimization based Artifact removal algorithm
Using the optimized low rank minimization we can change the complex signal to minimum directions. The PPG signals are reduced to minimum dimensions because it has less number of rank. This method offers constraints on the envelope and any deformities of the signal. The method can be experimentally proven to reduce the dimension and make it simple without losing the information. The low rank optimization algorithm helps to reduce the PPG signal with lesser dimensions. The MA contaminated PPG signal can be written as where Y is the signal with noise Φ .The aim is to estimate Φ and X from Y and it is illustrated as an ill-posed problem. The ill posed problem states that the function that are available is less than functions not available. On of the advantage of the study is that the algorithm works good for estimation motion artifact free signal. ℑ() represents the algorithm in frequency domain where the convolution operation in time domain can be converted to multiplication operation, for the simplest way of separation of X and Y. This is more easier because the motion artifacts has higher frequencies.
The input PPG signal Y can be reduced to fewer dimension by reducing the number of rows ,hence this can reduce the rank of the matrix also.
The ill posed problem can be solved by reducing the redundancy that relates to the rank. The number of examinations is greater than or equal to the degree of freedom. The degree of freedom is given by $P+Q+t$ that denotes number of rows, column and rank of the matrix. The low rank matrix method used here is Augmented lagrangian method.
The optimization algorithm used here is SVD and it is expressed as = Σ (2) where Q is the nonsingular values and U, V are the singular matrix. When the signal consists of low rank and less observation, the signal is decomposed into N atoms and all the atoms form an atomic set B. The atomic set B formed by N atoms is = {± 1, ± 2, ± 3, … … . ± } (3) The SVD applied to atomic set B is given by = { : ‖ ‖ 2 = 1, ‖ ‖ 2 = 1}, ∈ ℜ , ∈ ℜ } (4) The convex optimization for motion artifact removal for the atomic set B is given as ‖ ‖ * = Φ (5) where ‖ ‖ * is the nuclear norm of the matrix. The motion artifact removal using independent component analysis is very familiar. But it work good when the signal has less frequency artifacts. This method also reduces the dimension of data for simpler analysis. Then for a sparse matrix B, the low rank constrained optimization problem can be modeled as: where ‖ ‖ represent the Frobenius norm and H is the motion artifact. The low rank optimization problem with SVD is given by U,Q and V are Singular Value Decomposition parameters. and are the regularisation parameters. The circular form of MA corrupted PPG is given as (9) This method varies from other methods that uses single rank matrix optimization. The iterative optimization step introduced in this work is given by Where +1 represent the recovered matrix after k+1 iterations. The low-rank matrix recovery is assisted with gradient priors to the preceding equation as +1 = − ∇ (11) where ∇ is the gradient prior and that helps to eliminate the noises in PPG signal without disturbing its fundamental and harmonics frequencies. Figure 2. Illustrates the motion artifact eliminated PPG after low rank matrix minimization.  Fig.2. Motion artifact eliminated PPG using low rank minimization

Statistical Modeling of Distribution
In this section we will see how any given PPG signal can be represented using statistical distribution function. The statistical modeling of PPG signals were applied to these database signals. The PPG signal is first sampled and these sample values are used to derive the distribution model function in an iterative manner. The iterative optimization converges when the error between the function values and sample values becomes close to zero. In this paper we modeled the PPG with four distribution function as shown in Table I. Table I Fig.3 to fig.6 represents the estimated distribution function and plots of residual errors. The residual error is calculated as the average difference between the sample points of the actual PPG and the estimated PPG through iterative optimization. Table II represents the residual errors of all distributions. The main goal of this method was to calculate the ideal mixture of different statistical distribution functions to model PPG signal. From fig.3 to fig.5 we can see that all the Gaussian and lognormal distributions fitted PPG perfectly. In the proposed method we found that the accuracy of lognormal distribution has improved to 0.3 percent with comparison with all other methods [10].

Results And Discussion
The position, width, height and area of all the distributions are given in distribution figures. These parameters are iteratively optimized according to the curve fitting with least square. As the baseline of signals is varying from peak to peak, we used linear baseline subtraction to correct the signal to baseline.
The accuracy of PPG waveform fitting using different distribution functions were calculated by comparing the curve fitted function f(n) with given original PPG F(n). The Mean Absolute Error (MAE) is given by 12) Where N is the total number of sample points for a complete PPG wave. The residual error is given as The value of residual error depends on the parameters computed during each iteration

Conclusion
In this paper, we have studied and compared how to model a PPG signal mathematically with best optimization-error trade off. This work will help us to computationally analyze the PPG signal for various medical conditions such as arterial stiffness. Three types of Gaussian functions are used for modeling and out of that bifurcated Gaussian gives less residual error when compared to fixed width Gaussian and exponentially broadened Gaussian. The better results obtained in bifurcated Gaussian is due to the variation in width of the wave with respect to the center position for each iteration as in PPG waveform. From the shape of PPG it is obvious that it has long tail features. Thus lognormal function works better than Gaussian function for the modeling of PPG pulse waveform as it has occurrences far from the head and center position. Also the proposed algorithm significantly reduces the cost of the PPG device in market expanding reach to more people. The drawback of the existing methodology which includes lack of adaptivity will be analyzed in future work along with use of multi-distribution function mode. In future work, the reflected wave parameters of the modelled PPG will also be studied for different age groups.