Journal of Healthcare Engineering

Volume 2018, Article ID 7689589, 19 pages

https://doi.org/10.1155/2018/7689589

## Signal Space Separation Method for a Biomagnetic Sensor Array Arranged on a Flat Plane for Magnetocardiographic Applications: A Computer Simulation Study

^{1}Signal Analysis Inc., Hachioji, Tokyo, Japan^{2}Department of Advanced Technology in Medicine, Tokyo Medical and Dental University, 1-5-45 Yushima, Bunkyo-ku, Tokyo 113-8519, Japan

Correspondence should be addressed to Kensuke Sekihara; moc.ytfin@arahikes-k

Received 15 December 2017; Accepted 22 February 2018; Published 26 April 2018

Academic Editor: Vincenzo Positano

Copyright © 2018 Kensuke Sekihara. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Although the signal space separation (SSS) method can successfully suppress interference/artifacts overlapped onto magnetoencephalography (MEG) signals, the method is considered inapplicable to data from nonhelmet-type sensor arrays, such as the flat sensor arrays typically used in magnetocardiographic (MCG) applications. This paper shows that the SSS method is still effective for data measured from a (nonhelmet-type) array of sensors arranged on a flat plane. By using computer simulations, it is shown that the optimum location of the origin can be determined by assessing the dependence of signal and noise gains of the SSS extractor on the origin location. The optimum values of the parameters *L _{C}* and

*L*, which, respectively, indicate the truncation values of the multipole-order of the internal and external subspaces, are also determined by evaluating dependences of the signal, noise, and interference gains (i.e., the shield factor) on these parameters. The shield factor exceeds 10

_{D}^{4}for interferences originating from fairly distant sources. However, the shield factor drops to approximately 100 when calibration errors of 0.1% exist and to 30 when calibration errors of 1% exist. The shielding capability can be significantly improved using vector sensors, which measure the

*x*,

*y*, and

*z*components of the magnetic field. With 1% calibration errors, a vector sensor array still maintains a shield factor of approximately 500. It is found that the SSS application to data from flat sensor arrays causes a distortion in the signal magnetic field, but it is shown that the distortion can be corrected by using an SSS-modified sensor lead field in the voxel space analysis.

#### 1. Introduction

Development of a sensor system that can measure biomagnetic signals in room-temperature environments has gained great interest. One promising candidate among room-temperature sensors for biomagnetic systems is magnetoresistive (MR) sensors [1–4], which can lead to the development of novel low-initial-cost and maintenance-free biomagnetic systems. A potential near-future application of such systems is a low-cost magnetocardiography (MCG) system using MR sensors [5]. Such low-cost and maintenance-free MCG systems could replace the 12-lead electrocardiogram (ECG) now routinely used in daily clinical examinations.

However, to develop such low-cost systems, one major problem is the removal of ambient noise magnetic fields that exist in urban hospital environments. Biomagnetic signals are many orders of magnitude weaker than these ambient interference magnetic fields, called environmental noise. To reduce the influence of such environmental noise, biomagnetic measurements have traditionally relied on two kinds of hardware-based solutions: one is magnetically shielded rooms (MSRs) and the other is gradiometer sensors [6].

According to a purely technical point of view, use of an MSR would be advantageous even for MR sensor systems. However, the use of a high- or a medium-quality MSR may invalidate our goal, which is developing low-initial-cost biomagnetic systems, because MSRs are, in general, very costly and the use of even a medium-quality MSR causes a considerable increase of the total initial cost of the system. Gradiometers can significantly reduce the environmental noise, and the reduction ratio of a typical first-order gradiometer is believed to reach almost two orders of magnitude [6], although the noise reduction capability strongly depends on the precision in sensor manufacturing. However, for MR sensor systems, it is currently unknown whether gradiometers can be incorporated into the MR sensor hardware design. No gradiometer-type MR sensors have been developed so far.

Therefore, to attain the goal of developing low-cost biomagnetic systems, it may not be possible to rely on traditional hardware-based methods. This paper addresses a third option, developing software shielding methods, namely, efficient signal processing methods for environmental noise cancellation. A number of signal processing methods have been developed for this purpose, and arguments for and against those methods can be found in [7]. This paper focuses on a method called signal space separation (SSS), which was originally proposed for environmental noise cancellation for magnetoencephalography (MEG) SQUID sensor arrays [8–10].

The SSS method has an excellent characteristic that it imposes almost no prerequisites on the data or sources (i.e., it does not use any restrictive data or source models). One mild prerequisite of the SSS method, which can naturally be fulfilled, is that the region in which sensors are installed should be source-free (i.e., no current sources in that region). Under this assumption, the method decomposes the measured data into two components originated from the so-called “internal” and “external” regions. The internal region refers to a region that is closer to the origin than the sensors are, and the external region refers to the one that is farther from the origin than the sensors are.

The method can efficiently suppress interferences overlapped onto biomagnetic signals if clever choices of the origin location can make the internal and external regions match the signal and interference regions, respectively. It is not difficult to find such origin locations for helmet-type sensor arrays used in MEG. However, the SSS method has not been considered applicable to data from nonhelmet-type sensor arrays, such as sensor arrays arranged on a flat plane, which are usually used for MCG systems, because it does not seem possible to find an appropriate origin location for those nonhelmet-type arrays.

This paper presents a computer simulation-based investigation that explores the possibility of applying the SSS method to data measured from an array of sensors arranged on a flat plane. The goal of the investigation is to show that the SSS method is still effective for data from such flat sensor arrays. A series of computer simulations are performed assuming flat sensor arrays whose sensor arrangement is typical in MCG applications [11–15]. Using the results of these computer simulations, this paper seeks optimal values for numerical parameters used in the SSS method, showing that their choices are crucial for the effective use of the SSS method when applied to flat sensor arrays. It is found that the application of the SSS method to flat sensor data causes a distortion of signals but that this distortion can be corrected by using the SSS-modified lead field in the voxel space analysis.

This paper is organized as follows. In Section 2, the SSS method is described in detail, including the analysis of the problems caused when the SSS method is applied to flat sensor data. Section 3 presents computer simulation-based investigation, which shows the effectiveness of the SSS method for data from flat sensor arrays. This section includes empirical determinations of SSS parameters crucial to attain optimal performance of the SSS method. Section 4 summarizes the findings of the investigation.

#### 2. Signal Space Separation Method

##### 2.1. Data Model

Biomagnetic measurement is conducted using a sensor array, which simultaneously measures the signal with multiple sensors. Let us define the measurement of the *m*th sensor as *y _{m}*. The measurement from the whole sensor array is expressed as a column vector . Here, is the number of sensors, and the superscript indicates the matrix transpose. Throughout this paper, plain italics indicates scalars, lowercase boldface indicates vectors, and uppercase boldface indicates matrices.

The location in the three-dimensional space is represented by . The source magnitude at is denoted by a scalar . The source vector is denoted by *s*(**r**), and the source orientation is denoted by . We thus have the relationship: . Let us assume that a unit-magnitude source exists at . When this unit-magnitude source is directed in the *x*, *y*, and *z* directions, the outputs of the *m*th sensor are, respectively, denoted by and . Let us define an matrix whose *m*th row is equal to . This matrix , referred to as the lead field matrix, represents the sensitivity of the sensor array at **r**. When the unit-magnitude source at is oriented in the direction, the outputs of the sensor array are expressed as . This column vector , referred to as the lead field vector, represents the sensitivity of the sensor array in the direction of at the location .

The outputs of the sensor array **y** are expressed as the sum of a magnetic signal **b** and additive sensor noise, represented by a random vector

Here, the magnetic signal is expressed as

Here, , called the signal vector, represents the biomagnetic signal that is the target of the measurements, and , called the interference vector, represents the interference overlapped onto the signal . In this paper, the interference represents so-called environmental noise, and sources of environmental noise are assumed to be located much farther from the sensors than the sources of interest are. Sources of environmental noise are, in general, located from several meters (in cases of the noise sources such as electronic appliances in a laboratory) to several kilometers (in cases of urban environmental noise sources such as the subway noise) distant from the magnetically shielded room.

The sources that generate are confined to a region called the source space (e.g., the source space is the cardiac region for MCG and the brain region for MEG measurements). Let us assume that a total of discrete sources exist in the source space. Their locations are denoted by , their orientations by , and their magnitudes by . Then, the source distribution is expressed as
where indicates the Dirac delta function. The signal vector is expressed as
where represents the lead field vector of the *q*th source obtained such that .

##### 2.2. Derivation of SSS Basis Vectors

One fundamental assumption of the SSS method is that the sensors are installed in a source-free region, which is referred to as the sensor region. Then, the magnetic field at **r**, **B**(**r**), is expressed using the spherical polar coordinate by
where indicates the magnetic permeability of free space. In (5), and are the modified vector spherical harmonics [8, 16]. The index parameter is called the multipole-order or multipole parameter. In the right-hand side of (5), the first term represents the magnetic field generated from sources located closer to the origin than the sensors are.

The second term represents the magnetic field from sources located farther from the origin than the sensors are. The region closer to the origin than the sensors is referred to as the internal region, and the region farther from the origin than the sensors is referred to as the external region. Let us define the polar radial coordinate of the sensor nearest to the origin as and the radial coordinates of the sensor farthest from the origin as . The internal region is formally defined as the region with and the external region as the region with . The region with is called the intermediate region.

Let us derive the SSS basis vectors. To do so, the magnetic signal detected by the *j*th sensor is denoted by and the location and the normal vector of the *j*th sensor by and . Then, we have
where the notation “·” indicates taking the inner product between two vectors (note that when the area of the pickup coils is taken into consideration, the sensor signal is obtained as a surface integral of over the area of the *j*th pick-up coil). Here, and , respectively, represent magnetic components originating from the internal and external regions. These components are expressed as
where we set for simplicity. Let us define the internal and external components of the vector as and , which are expressed such that
where column vectors and are given by

Truncating the summation with respect to the multiple order to *L _{C}* for

**b**

_{int}and

*L*for

_{D}**b**

_{ext}, we finally obtain

Thus, defining we obtain where and . Here, is an matrix, and is an matrix, where

Note that the truncation values and correspond to the highest spatial frequencies possibly contained in and [9, 17], respectively. Therefore, setting these parameters at too low values may result in an insufficient representation of the signal vectors and . The effects of and for data from the 306-channel Elekta Neuromag have been investigated and values of and were found to be sufficient for such data sets in [8, 9].

##### 2.3. SSS Signal Extractors

Equation (12) is the basis for estimating the internal and external components and from given magnetic signal data **b**. That is, the least squares estimate is obtained as

Then, and are estimated as

We now derive SSS signal extractors and rewrite (15) and (16) using these extractors. To do so, let us define an operation to make a new column vector by using the *i*th to *j*th components of as (namely, ). From (14), we have

With a small positive constant , the relationship holds. Then, using the matrix inversion formula we get

Using (15) and (20), we obtain

Thus, the internal component can be extracted by multiplying with the magnetic-field data . That is, the matrix acts as a projector that passes the internal components and blocks the external ones (note that since and do not hold, is not actually a projector). Therefore, we call the SSS signal extractor in this paper.

In exactly the same manner, we can derive and the SSS external-signal extractor is derived as

This passes the external components but blocks the internal ones.

##### 2.4. Interference Suppression

A key condition for the success of the SSS interference suppression is that the origin is properly set such that the source space *Ω* is included within the internal region and the interference sources are located within the external region. A typical configuration between the helmet-type sensor array and the source space is depicted in Figure 1(a). As can be seen in this figure, an appropriate location of the origin can be found so that the internal region covers the whole source space and the external region covers all locations of interference sources. (Note that, in this paper, interference indicates only environmental noise, and its sources are assumed to be located much farther from the sensors than the signal sources.)