A Linear Transformation Approach for Estimating Pulse Arrival Time

We propose a new mathematical framework for estimating pulse arrival time PAT . Existing methods of estimating PAT rely on local characteristic points or global parametric models: local characteristic point methods detect points such as foot points, max points, or max slope points, while global parametric methods fit a parametric form to the anacrotic phase of pulse signals. Each approach has its strengths and weaknesses; we take advantage of the favorable properties of both approaches in our method. To be more precise, we transform continuous pulse signals into scalar timing codes through three consecutive transformations, the last of which is a linear transformation. By training the linear transformation method on a subset of data, the proposed method yields results that are robust to noise.We apply this method to real photoplethysmography PPG signals and analyze the agreement between our results and those obtained using a conventional approach.


Introduction
The importance of arterial stiffness as a cardiovascular disease index has been emphasized in recent years 1-6 , because arterial stiffness can be acquired using inexpensive and noninvasive methods such as pulse wave velocity PWV 7, 8 .PWV is considered to be a good indicator for assessing arterial stiffness because it shows a strong correlation with cardiovascular events and mortality 1, 9-15 .Furthermore, PWV can be used for the continuous assessment of cardiovascular homeostasis and regulation 16 .
One approach to assess PWV in vivo relies on tracking pressure pulses that arise from the onset of left ventricular ejection.This is the common method for acquiring PWV in

Representation of PPG Signals in Vector Space
Let Ω be the set of all continuous PPG signals measured from human arteries.We can define a sampling process ξ M as follows: where R M is an M-dimensional Euclidean space.The mapping ξ M reduces a continuous signal to an M-dimensional vector point.The vector point forms a lower-dimensional cluster in an M-dimensional space.Let us consider the cluster as being embedded by the manifold Υ.Our goal in this paper is to find a mapping τ between this manifold and PAT: Then, the parametric estimation of PAT is given by the composition of two mappings: PAT τ • ξ M .However, manifolds constructed through normal sampling at a constant frequency are highly curved.Training this type of manifold and mapping PAT using this manifold are challenging 22 .Let us consider simple translations of Gaussian peaks, as shown in Figure 1.The manifold constructed using simple translations is spirally curved.If slight variation is added to Gaussian peaks, it is not feasible to parameterize the manifold with well-defined functions.
Now, suppose that we find the sampling process ξ M such that the manifold Υ can be flat and isometric along PAT.For instance, if three vectors f, g, h ∈ Υ are collinear and have isometric timing codes t f , t g , t h , that is, t h −t g f t f −t h g t g −t f h 0, then we can always find the linear transformation ω such that t f ω T f, t g ω T g, and t h ω T h.This means that the special sampling process allows the mapping τ to be the simplest form by ω.However, we failed to find such a sampling process, regardless of the sampling frequencies applied to the continuous signals; convex combination of two different vectors f, g ∈ Υ cannot be used to represent the human artery PPG signal.
In this context, we propose adding another transformation between ξ M and τ.By considering the new mapping, we intend to keep the mapping, τ, the linear transformation.If we denote the novel mapping as ζ, we can estimate the PAT of the PPG signals as We refer to this as a linear transformation approach for estimating PAT.In following subsections, we describe the new transformation ζ in more detail.

Conjugate Transformations
In the previous subsection, we framed a set of three transformations to change continuous pulse functions into scalar timing codes.The first transformation, ξ M , is needed to reduce continuous pulse functions into M-dimensional vector points, while the third transformation τ is the linear transformation.The function of the second transformation, ζ, will be fully discussed in this subsection, in which we consider a well-known transformation called the convex conjugate or Legendre transformation.

Convex Conjugate
The convex conjugate is a transformation that maps a convex function onto another convex function 23, 24 .A convex function always has its conjugate function: the conjugate function is also a convex function.First, we outline why we need convex functions.A Gaussian function, which was exemplified as a pulse in the previous subsection, is the starting point for developing our idea.Gaussian functions have a single peak and are nonnegative over the entire region.Such a Gaussian function can be derived from a convex function by differentiating the convex function twice.We can therefore consider convex functions instead of Gaussian functions.A general type of pulse function that has mixed-signed values, unlike Gaussian functions, will be discussed later.
Let us consider two arbitrary convex functions.When their first derivatives become inverses of each other, two functions are referred to as "convex conjugate".If two functions f t and f t have such a relation, then where I is an identity function, that is, I • t t and I • t t.To find an explicit expression for f, we assume From 2.4 and 2.5 , we obtain Conversely, when we assume t d f/d t • t, we also obtain t df/dt • t.Thus, 2.4 gives the reciprocal expressions 2.5 and 2.6 , which are referred to as variable change.
When we assume a finite domain Ω on which a convex function is defined, the independent variable t on the domain can be changed into its conjugate variable t through convex conjugate.Then, the function form f can be changed into the form f by replacing t with t.To be precise, the explicit form of the convex conjugate from above relations is where the conjugate variable t is expressed as the gradient of f at t.By differentiating t t − f t with regard to t and equating this result to zero, we can confirm that t is expressed as the gradient of f.Conversely, the new convex function can be converted back to the original function in the same manner: In this case, the original variable t is expressed as the gradient of f at t. Variables t and t are basically conservative fields with regard to each other.Convex conjugation was originally derived from duality relationship between points and lines.The functional relationship specified by f t can be represented equally as well as a set of points t, or as a set of tangent lines specified by their gradients and intercept values.

Nonnegative Conjugate
Now, we introduce a new conjugate transformation termed nonnegative conjugate.This transformation is closely related to the former convex conjugate.If f is twice continuously differentiable and the domain is Ω, then we can characterize a convex function as follows: This is a link between convex conjugate and nonnegative conjugate based on the following definition.
Definition 2.1.Suppose that two convex functions f t and f t are in convex conjugate for t ∈ Ω and t ∈ Ω and their second derivatives f t and f t are denoted as I t and I t , respectively.Then I t and I t are said to be nonnegative conjugate of each other on domains Ω and Ω.
The two-dimensional conjugate transform that is analogous to this nonnegative conjugate has been applied to image morphing 25 .
Let us calculate the second derivatives directly.As mentioned in 2.5 and 2.6 , the first derivatives represent variable change between t and t.The second derivative of f t is given as Similarly, the second derivative of f t is given as From 2.10 and 2.11 , we obtain the following reciprocal relation between two second derivatives:

2.12
If we denote f t and f t as I t and I t , the expression can be rewritten as I t I t 1.Then, the nonnegative conjugate of I t can be expressed as

2.13
Like variable change in the convex conjugate transformation, the variable t of the nonnegative function I t on the domain Ω can be formally changed by using the nonnegative conjugate.This yields another nonnegative function I t on the domain Ω when the variable t is replaced with its conjugate variable t.Equation 2.13 has a very complex form, but the variable change between t and t has the following concise forms:

Nonnegative Conjugate of a Nonnegative Vector
In the previous section, we described a method of transformation based on the convex conjugate.However, although the nonnegative conjugate transforms a continuous function into another continuous function, the transformation ζ should map an M-dimensional vector onto another M-dimensional vector.Thus, we require a discrete version of the nonnegative conjugate.
Let us denote an M-dimensional column vector with nonnegative components as I, that is, I I 1 2.17 Then, from 2.13 , we obtain Finally, we can change it into the M-dimensional vector I by

2.19
Applying the same procedure to I, we can transform it back to the original vector I.However, this vector is not exactly same as the original vector.As the dimensionality of M increases, the error, I − I, converges to a zero vector.

Application to PPG Signals
All experiments and analyses were performed using ECG and PPG signals extracted from the publically available MIMIC database that contains data from intensive care unit patients admitted to Boston Beth Israel Hospital.ECG and PPG signals were measured to 500 and 125 samples per second, respectively.First, R peaks were detected from ECG based on the assumption that the R peak represents the onset time of left ventricular ejection.Therefore, the position of R peaks was used to segment single PPG pulses from the full PPG signals.
Raw PPG signals were low-pass filtered at 15 Hz, then single PPG pulses were separated by synchronized R peaks.The extracted single PPG pulses were resampled to 500 Hz to improve accuracy, and then FOOT and MAX points of single PPG pulses were detected by the traditional method that detects characteristic points 26 .Each single PPG pulse was divided into two parts by the FOOT point: the front part from the R peak to the FOOT point, and the rear part from the FOOT point to the next R peak.The time difference at each part was calculated in different ways than that used to estimate PAT.The time difference t a at the front part was derived by a simple translation to change the number of samples into time seconds , and the time difference t b at the rear part was calculated by the nonnegative conjugate transformation and linear projection, as shown in Figure 2.
Various single PPG pulses with different amplitudes, shapes, or pulse widths were represented as single points in an M-dimensional vector space after the nonnegative conjugate transformation.The points that corresponded to nonnegative conjugate vectors, I, were located on a same line in an M-dimensional vector space.This characteristic is referred to as collinearity.

Training a Projection Matrix W
To derive the matrix W that projects collinear points in an M-dimensional vector space into a one-dimensional time space, 10,000 different PPG pulses were extracted as the training set, and the linear projection matrix was trained according to the MAX point of the PPG pulse, because this is the most obvious characteristic point.Only rear parts of PPG pulses were used to train the linear projection matrix, which we derived by pseudoinverse operation between nonnegative conjugate transformed pulses and the known time information of MAX points as follows:

2.20
The matrix W was calculated from the training samples by using the pseudo-inverse relationship where , t b is the known time set from the R peaks to the MAX points, I is nonnegative conjugate transformed PPG pulse, and W is the derived linear projecting matrix.

PAT Estimation
Single PPG pulses extracted according to R peaks of ECG were divided into front and rear sections using the FOOT point, and time values were calculated for each section separately.First, the time of the front part, t a , was calculated using the number of samples between the R peak and FOOT point divided by the sampling rate.Second, the time of the rear part, t b , was acquired by linear operation between the linear projecting matrix W and the nonnegative conjugate transformed pulse I .Finally, the PAT was obtained by simple summation of t a and t b : 2.22

Results and Discussion
To assess the agreement between the traditional method and our novel PAT estimation method, we evaluated a subset of data from the MIMIC database.This database is part of the Physionet platform and contains data from over 72 intensive care unit patients at the Boston Beth Israel Hospital, but we selected only those records for which nonsaturated and nonmissing ECG and PPG signals were simultaneously measured and available 27, 28 .We adopted the agreement analysis proposed by Bland and Altman: given two different estimating methods, their agreement can be assessed by computing the standard deviation of two sets of estimates.The proposed strategy computes the differences between measurements provided by two methods and then computes their dispersion.Two methods have good agreement if dispersion is minimal 29 .
To acquire PAT estimates using our method, a linear projecting matrix was derived using a training process, and the derived linear projection matrix was applied to two test sets consisting of 2947 and 2890 PPG pulses, respectively.The mean difference and standard deviations of two sets of PAT estimates were calculated.Results of the agreement analysis are shown in Figure 3.The 95% limits of agreement were calculated as mean difference ± 1.96 * standard deviation at each set.
For the first test set, 98.9% of pulses were located between −34.3 ms and 28.3 ms as the limits of agreement, while in the second test set, 96.2% of the pulses were located between −33.4 ms and 25.3 ms.
The typical PAT estimation method, which detects characteristic points, is not accurate when applied to PPG pulse types with different morphologies.We therefore proposed a novel PAT estimation method that provides robust results by considering the morphological characteristics of PPG pulses according to the properties of blood vessels.We initially attempted to find a relation between a projecting factor and original PPG pulses, f, but were unsuccessful because of the broad dispersion of pulses in an M-dimensional vector space.We therefore decided to transform original PPG pulses into another form.We used Journal of Applied Mathematics the nonnegative conjugate transformation, because the nonnegative conjugate transformed signal, I, had the property of collinearity in an M-dimensional vector space and could be used to estimate the PAT by projection onto a one-dimensional time space.We derived a linear projection matrix through the training process for linear projection and applied this matrix to two different morphological PPG pulse sets.PAT values estimated from annotated MAX points and the proposed linear projecting method were in good agreement; over 95% of the data were included within the 95% limits of agreement.
Although our method provides results that appear to be highly accurate, it can show different results according to the linear projection matrix that is derived by different numbers of pulses and dimensionality.Therefore, an optimal combination that is applicable to a variety of morphological PPG pulse types should be determined.For instance, the size of the projecting matrix needs to be adjusted and the time delay caused by the nonnegative conjugate transformation needs to be addressed.Our novel approach still has some limitations in terms of its clinical application for real-time continuous monitoring of PAT as well as stiffness and blood pressure assessment using PPG; the linear projecting matrix needs to be optimized and the time delay caused by the nonnegative conjugate transformation needs to be addressed.Once these issues are addressed, however, our method has great potential in clinical practice to precisely assess cardiovascular risk associated with blood vessels.based on the nonnegative conjugate transformation.This easy, stable PAT estimation method relies on training of the linear model using various samples.Because our method extracts information from various pressure pulse, it can be applied to different morphological signals without special conditions.In conclusion, we developed a novel method that can be used to estimate PAT robustly for a variety of PPG signals with different morphological characteristics.

Figure 1 :
Figure 1:The estimation of PAT using a linear transformation approach.

Figure 2 :
Figure 2: Single PPG pulse processing to estimate PAT.

Figure 3 :
Figure 3: Agreement analysis between the two different PAT estimations.
Alternatively, the nonnegative conjugate can also be derived from the equidistribution principle.First, the conjugate variable t is introduced such that a nonnegative distribution I t becomes constant with 1 in the conjugate coordinate t: I t dt d t.The conjugate function I t also becomes constant with 1 in the original coordinate t by the same form: I t d t dt.As a result, we can obtain the reciprocal relation between I t and I t and biconjugacy from Note that equations in 2.15 are the same as 2.10 and 2.11 , respectively.