Calibration and Thermal Analysis of a
Conformal SERF Magnetometer Array
for MEG Applications
A thesis
submitted by
Elizabeth V. Alexander
In partial fulfillment of the
requirements for the degree of
Master of Science
in
Mechanical Engineering
TUFTS UNIVERSITY
May 2013
ADVISER:
Jason Rife, Department of Mechanical Engineering, Tufts University
COMMITTEE MEMBERS:
Joe Kinast, Draper Laboratory
Mike Zimmerman, Department of Mechanical Engineering, Tufts University
c
2013
Elizabeth Alexander, All rights reserved
The author hereby grants to Tufts University and The Charles Stark Draper Laboratory,
Inc. permission to reproduce and to distribute publicly paper and electronic copies of this
thesis document in whole or in part.
Abstract
In magnetoencephalography (MEG), a measurement of the magnetic field produced
by the brain, it is critical for the distance between the magnetometers and the brain
to be small and unchanging. Limitations of SQUIDs, the magnetometers used in
most MEG systems, make this requirement difficult to meet. SERF magnetometers
have been proposed as an alternative technology for use in more portable and flexible
MEG systems.
This research considers two of the challenges associated with an array of SERF
magnetometers attached to a conformal cap. First, for meaningful measurements
in MEG, it is important for the locations and orientations of magnetometers to be
known. This information can be estimated using the magnetometers’ measurements
of a known magnetic field produced by a configuration of magnetic dipoles, but the
estimation algorithm is extremely sensitive to the dipole configuration. This thesis introduces a graphical tool to assess the quality of a configuration of magnetic
dipoles. Second, the core of SERF magnetometers has a high operating temperature. This work finds a thermal insulation layout that allows for a small overall
magnetometer package size while maintaining proper operation temperature and
preventing thermal injury to a patient.
ii
Acknowledgements
Thank you to Jason Rife and Joe Kinast for making this research possible by coordinating with Draper Laboratory and for their constant support over the past two
years. In particular, thank you to Joe Kinast for always patiently explaining new
concepts and encouraging me when lab work was at its most challenging. Thank
you to Jason Rife for continually pushing me to make progress on my research and
for all his help improving this document. I consider myself lucky to have had the
opportunity to work with two such kind, knowledgeable individuals.
For providing valuable feedback on my research and presentations, I would like
to thank Professor Zimmerman, Daniel Jones, my parents, and the ASAR Lab
group. Thank you to all of my friends and family for keeping my spirits up and for
the constant encouragement. To my friends and colleagues at Tufts and Draper,
thank you for your support and friendship; getting to know you all has enriched my
graduate school experience.
iii
Contents
1
Chapter 1: Introduction
1
1.1
Background and Motivation . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.1
Current MEG Technology . . . . . . . . . . . . . . . . . . . .
1
1.1.2
Emerging Magnetometer Technology . . . . . . . . . . . . . .
5
1.1.3
Magnetometer Array on a Conformal Cap . . . . . . . . . . .
7
1.2
2
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.3
Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.4
Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
Chapter 2: Calibration
13
2.1
Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.2
Mathematical Model for Calibration System . . . . . . . . . . . . . .
14
2.3
Procedure for Evaluating Calibration Geometries . . . . . . . . . . .
19
2.3.1
Calibration Estimation Algorithm . . . . . . . . . . . . . . .
19
2.3.2
GRaphical Estimation-quality Assessment Tool (GREAT) . .
23
Application of GREAT . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.4.1
Magnetic Dipole Configuration Exploration . . . . . . . . . .
27
2.4.2
Recommended Geometry . . . . . . . . . . . . . . . . . . . .
33
2.5
Discussion and Future Work . . . . . . . . . . . . . . . . . . . . . . .
34
2.6
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
2.4
3
Challenges Implementing a Conformal Cap SERF Array for MEG
Chapter 3: Thermal Design
36
3.1
Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
3.1.1
Requirements for Preventing Thermal Injury . . . . . . . . .
37
3.1.2
Magnetometer Operating Temperature Requirement . . . . .
38
3.1.3
Conduction PDE . . . . . . . . . . . . . . . . . . . . . . . . .
38
Finite Element Analysis . . . . . . . . . . . . . . . . . . . . . . . . .
40
3.2.1
40
3.2
Modeling Assumptions . . . . . . . . . . . . . . . . . . . . . .
iv
4
3.2.2
Components, Materials, and Thermal Properties of Model . .
41
3.2.3
Inputs and Outputs of the Finite Element Model . . . . . . .
44
3.2.4
FEA Temperature Results . . . . . . . . . . . . . . . . . . . .
44
3.3
Design Space Metamodel
. . . . . . . . . . . . . . . . . . . . . . . .
45
3.4
Thermal Insulation Optimization . . . . . . . . . . . . . . . . . . . .
47
3.4.1
Verification of Results . . . . . . . . . . . . . . . . . . . . . .
50
3.5
Discussion and Future Work . . . . . . . . . . . . . . . . . . . . . . .
51
3.6
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
Chapter 4: Conclusion
53
4.1
Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
4.2
Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
A Appendix A: FEA Results
55
B Appendix B: Magnetic Field Produced by Current-Carrying Coil
57
C Appendix C: Head Dimensions
58
D Appendix D: Magnetometer and Dipole States Used for Figures Created
Using GREAT
59
E Appendix E: Additional Finite Element Model Details
62
References
63
v
List of Figures
1
SERF Magnetometer Components . . . . . . . . . . . . . . . . . . .
7
2
Illustration of Conformal Cap Concept . . . . . . . . . . . . . . . . .
8
3
Illustration of Calibration Components: Conformal Cap, Magnetometers, and Magnetic Dipoles
. . . . . . . . . . . . . . . . . . . . . . .
14
4
Head and Conformal Cap Models . . . . . . . . . . . . . . . . . . . .
17
5
Conformal Cap Levels . . . . . . . . . . . . . . . . . . . . . . . . . .
17
6
Block Diagram of Calibration Algorithm . . . . . . . . . . . . . . . .
19
7
Coil Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
8
Magnetic Field Magnitude vs. Displacement . . . . . . . . . . . . . .
22
9
Magnetic Field Magnitude Multiplied by the Cube of the Displacement 22
10
Location Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
11
Calculation of Cost Function Change . . . . . . . . . . . . . . . . . .
26
12
Example GREAT Plot . . . . . . . . . . . . . . . . . . . . . . . . . .
27
13
Coil and Head Axes . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
14
Effect of Rotating Magnetic Dipole About X-Axis
. . . . . . . . . .
29
15
Effect of Rotating Magnetic Dipole About Z-Axis . . . . . . . . . . .
30
16
Coil Position with Respect to Head Axes . . . . . . . . . . . . . . . .
31
17
Effect of Magnetic Dipole Position . . . . . . . . . . . . . . . . . . .
31
18
Quantity of Coils Shown on Head Axes . . . . . . . . . . . . . . . . .
32
19
Effect of Quantity of Magnetic Dipoles . . . . . . . . . . . . . . . . .
32
20
Dipole Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
21
Effect of Many Magnetic Dipoles in Varied Positions and Orientations 33
22
Single Magnetometer from Conformal Cap, Quartered . . . . . . . .
23
Alkali Vapor Chamber (White), Other Device Components (Light
36
Gray), and Skin (Dark Gray) . . . . . . . . . . . . . . . . . . . . . .
37
24
Surface to Which Load is Applied . . . . . . . . . . . . . . . . . . . .
39
25
Surfaces to Which Boundary Conditions Apply . . . . . . . . . . . .
39
26
Top View of Magnetometer, Shaded Portion Modeled in ABAQUS .
41
vi
27
SERF Magnetometer Components . . . . . . . . . . . . . . . . . . .
42
28
Thermal Insulation Thicknesses . . . . . . . . . . . . . . . . . . . . .
43
29
FEA Temperature Results . . . . . . . . . . . . . . . . . . . . . . . .
45
30
Overall Sensor Dimensions . . . . . . . . . . . . . . . . . . . . . . . .
49
31
FEA Temperature Results for Optimal Insulation Thicknesses . . . .
51
32
Head Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
List of Tables
1
Data for Estimator Self-Consistency Check . . . . . . . . . . . . . .
23
2
Thermal Conductivities and Component Thicknesses . . . . . . . . .
43
3
Cases for Which Magnetometer Operating Temperature Exceeded
150◦ C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
4
Magnetometer Operating Temperature Fit Equation Coefficients . .
47
5
FEA Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
6
FEA Data (Continued) . . . . . . . . . . . . . . . . . . . . . . . . . .
56
7
Head Size Data Corresponding to Figure 32
. . . . . . . . . . . . .
58
8
Magnetometer Locations for All Plots in Section 2.4.1 . . . . . . . .
59
9
Figure 14 Dipole States . . . . . . . . . . . . . . . . . . . . . . . . .
59
10
Figure 15 Dipole States . . . . . . . . . . . . . . . . . . . . . . . . .
60
11
Figure 17 Dipole States . . . . . . . . . . . . . . . . . . . . . . . . .
60
12
Figure 19 Dipole States . . . . . . . . . . . . . . . . . . . . . . . . .
61
13
Figure 21 Dipole States . . . . . . . . . . . . . . . . . . . . . . . . .
61
vii
1 Chapter 1: Introduction
Magnetoencephalography (MEG), a brain imaging technique, is extremely useful
in basic brain research and before surgery for localization of brain structures. A
typical MEG system uses superconducting quantum interference devices (SQUIDs),
magnetometers with operating temperatures so low that they must be insulated
with significant vacuum space and cooled with liquid helium. These large systems
require a large magnetically shielded room (MSR). The size of the MSR as well as
the frequent need to refill the liquid helium contribute to the extremely high cost of
the MEG system. The vacuum space used for thermal insulation causes the system
to be rigid and sized poorly for most patients. As such, the magnetometers may not
be close enough to the brain, and the head may move during MEG, leading to poor
measurements. Additional detail on the operation of MEG systems is provided in
Section 1.1.
1.1 Background and Motivation
1.1.1
Current MEG Technology
Brain imaging techniques are used extensively for many procedures including basic
research, clinical medicine [1], presurgical brain mapping, and studying epilepsy [2].
With them, attaining high simultaneous spatial and temporal accuracy is challenging [1]. Some of these techniques, such as functional magnetic resonance imaging (fMRI) and positron emission tomography, have excellent spatial accuracy [3],
whereas others, such as electroencephalography (EEG) and magnetoencephalography (MEG), have superior temporal resolution [3]. MEG, a non-invasive brain
imaging technique, is superior to most of its brain imaging counterparts by being
non-invasive [4] and temporally accurate [1]. While it may not be as sensitive to
deep brain sources as is EEG [2], MEG outshines EEG by not requiring skin contact
with the patient [3] and by measuring signals that are insensitive to the distorting
effects of the hair, scalp, and skull [1].
MEG signals are directly related to neuronal currents, which is what causes
1
MEG to have excellent temporal resolution; unfortunately, localization of the origins
of these signals, a very important part of MEG, is very complicated [3, 5]. The
determination of the quantity and locations of the current sources in the brain that
explain the recorded MEG signals is a challenge termed the inverse problem [6].
This inverse problem has no unique solution [3]; that is to say there are an infinite
number of current source arrangements that could produce a set of recorded MEG
signals [3, 6]. For example, a signal which appears to come from a single location
may actually originate from a few activated areas of the brain. [6].
Much effort has gone into developing methods of source localization [7], resulting
in the creation of numerous inverse modeling algorithms [1,8]. Typically, fewer than
10 current dipoles are used to approximate the brain current sources [5]. Better
source localization has been obtained by combining multiple brain imaging techniques, such as fMRI, MEG, and EEG [5] and MEG and EEG [2,6]. Frequently, the
sensitivity of MEG measurements to tissue conductivity is not considered [9], but
sometimes incorporating conductivity anisotropies and inhomogeneities helps [6].
Currently, the ability of sources to be localized is limited by the inverse problem
and the overly-large distance between sources and sensors [6].
Biologically generated magnetic fields are very weak [10]. MEG signals, or the
biologically generated magnetic field emanating from the brain, are the result of
thousands of neurons acting in synchrony [1], and are usually less than 10−12 T [3].
Earth’s steady magnetic field is approximately 0.5×10−4 T, and the fluctuating urban
magnetic field is around 10−7 T [3], so external stray fields must be minimized [10].
As such, most MEG systems operate in magnetically-shielded rooms (MSRs) [11–13]
that are box-shaped and around 2.5m on a side [4], with passive and active shielding
and consisting of multiple layers of different materials [11–13]. The shielding factor
for MSRs, or the extent to which they suppress external magnetic fields, depends
on the frequency [12, 13]. A quality system may having a shielding factor in the
millions for 1-10Hz [11, 13].
MSRs are a large part of the size and the cost of an MEG system [4]. Operating
in a remote location [14] and using gradiometers (to reduce the ambient magnetic
2
field) instead of magnetometers [1, 4] can make unshielded operation more feasible
[4, 14]. Unshielded MEG has been demonstrated in the past, but using a pair of
two-dimensional gradiometers [4], which is only a small fraction of the number of
magnetometers used in a standard MEG.
In current whole-head MEG setups, the magnetometers are housed in rigid helmets sized for the majority of adults [15]. Although this allows most heads to fit
within them, the helmets are not optimized for extreme head sizes [16], such as the
small heads of children [16, 17]. There are several consequences of performing MEG
with standard helmets on smaller heads: the distance between the sensors and the
cortical sources is larger [18], there is additional room for heads to move around
within the helmet [19], and when the crown-neck distance is also small, the head
cannot be fully inserted into the helmet [16]. All of these consequences make MEG
and its inverse problem more challenging.
Magnetic field strength decreases with the cube of the distance from the source
[16, 19]. When magnetometers are too far from the brain, they cannot effectively
pick up the signals produced by the brain [16, 17]. Furthermore, when the head is
closer to one part of the helmet (and a subset of sensors) than to another part, there
will be different sensitivities and signal-to-noise ratios (SNRs) of different parts of
the brain [19,20]. It has been shown that the detection of frontal activity in adults is
strongly affected by the position of the head in the front-back direction with respect
to the helmet [20] and that minimizing the distance between the source and sensors
by appropriate positioning of the head drastically improves source measurement [15].
This leads to the natural conclusion that the magnetometers must be placed as close
as possible to the most important regions of the brain [16, 19]. Unfortunately this
is not always possible: the distance between sensors and sources is greater with
smaller heads [18]. Additionally, a short crown-neck distance, frequently exhibited
by children, prevents the head from being fully inserted into the helmet [16], and
slouching during the MEG [20] can lead to an even greater distance between the
head and the sensors at the top of the helmet [16, 20]. Although a child-sized MEG
helmet may not be able to fit as many magnetometers [16], MEG using smaller
3
helmets has been performed on neonates [3], and it is widely believed that smaller
helmets can improve data quality [15, 17].
Inaccuracies in brain activity localization can be caused by head movement during MEG [20]. Sometimes MEG studies use averaging to separate the signal from
the area being measured from the signal within other areas or background noise [3].
In this case, head movement could be especially detrimental [20]. Ideally, no head
movement would occur during MEG [17,18]. Anesthesia is often used with pediatric
patients to meet this goal [16, 18]. Unfortunately, it is often necessary for a patient
to remain awake and perform tasks during MEG, so anesthesia is frequently not
an option. In this case, with current MEG setups, the head cannot remain completely still, so the feasible goal is for the head to remain as still as possible [16].
Movement can be mitigated with the use of bite bars [6, 7, 19] and padding [16, 20],
but they require much setup time [19], may adversely affect performance on tasks
performed by the subject during MEG [20], and are unable to completely prevent
head movement [19].
Another method for dealing with head movement is to monitor the head position
and use that information in the data analysis [20]. Information from magnetic coils
can be used to track head position [3, 19, 21], but it requires post-processing of
data [19,20]. Coils are used for head localization [15,18] but are often only monitored
before and after data collection [18] or at a few distinct points in time during MEG
[6]. Continuous localization can be done using currents in coils that have frequencies
outside the frequency range of the brain activity of interest [17, 20]. Even when
head position is monitored continuously, errors can arise from the placement and
calibration of the fiducial coils: the stylus tip can be placed imprecisely, the subject
can move during the lengthy digitization procedure, and electromagnetic digitizers
have precision of about 1mm [21]. These factors combine to yield sensor position
measurement errors on the order of several millimeters. [21]. It is clear that better
source estimates could be made if head movement were to be compensated for
[20], but with currently-available instrumentation and technology, maintaining head
position requires excessive complexity [19] and time [16].
4
To summarize, MEG has many advantages relative to other brain imaging modalities, but it currently also has many practical problems and high costs [3]. Superconducting Quantum Interference Devices (SQUIDs), the primary type of magnetometer used in measuring biomagnetic fields [22], operate at 4 Kelvin and must be
contained within a large dewar of liquid helium [3]. The best MEG measurements
require a shielded room, which must be quite large [4] in order to house the dewar.
Larger MSRs typically cost more [4] and the liquid helium dewars must be refilled
one or two times per week [3], causing MEG to have both high startup and operating costs. The helmet used in standard MEG setups, rigid in order to house the
SQUIDs with ample vacuum space for insulation, prevents the sensors from being in
close proximity to the head, and allows head movement, both of which lead to poor
MEG measurements. Poor measurements make the inverse problem more difficult
to solve.
1.1.2
Emerging Magnetometer Technology
Atomic magnetometers have a much shorter history measuring biomagnetic signals,
but they make many improvements in MEG possible. They have been used in
the past for measuring the cardiomagnetic field, one of the strongest biomagnetic
signals [10]. Their operation is nearly cost-free, they require little maintenance,
and they are less expensive than SQUIDs [10]. The spin-exchange relaxation-free
(SERF) magnetometer is a type of optical atomic magnetometer pioneered by the
Romalis Group at Princeton University [22]. SERF magnetometers require a very
low bias magnetic field (≤nT [22]) in order to achieve their very high sensitivity [22]
and have a bandwidth that would work well for detecting biomagnetic signals [22].
The application of SERF magnetometers to MEG has an inherent advantage:
the sensors do not require cryogenic cooling the way SQUIDs do; instead, they must
be heated [22], which can be achieved with a single heater attached to each sensor,
requiring little room. Although SERF magnetometers have been operated without
shields in a laboratory environment [14], they need a very low bias magnetic field
in order to achieve high sensitivity [22]. Since the thermal regulation equipment is
5
small, the size of the magnetic shields used could be significantly reduced, thereby
reducing the cost of the MEG system [23].
A SERF magnetometer has been successfully used for MEG by multiple research
groups [23,24]. It has been suggested that four large SERF magnetometers would be
sufficient for full head coverage [23], measuring the magnetic field at many different
parts of each cell [14] causing each cell to act as many magnetometers, however,
this arrangement of sensors might still allow for head movement and sub-optimal
proximity to the head, two of the major drawbacks of the current MEG setup.
Furthermore, the larger the sensor, the higher the power required for operation [25].
Since the invention of the SERF magnetometer, various organizations have
sought to improve its design. Twinleaf sells a SERF magnetometer called a SERF-2
Compact vector magnetometer. It has a field sensitivity of less than 5 fT Hz−1/2
and dimensions of 1.9cm × 2.7cm × 6cm [26] Other SERF magnetometers have
been designed and fabricated with an emphasis on reducing their dimensions and
power consumption [27–30]. A process was developed to manufacture the cesium
and buffer gas-filled glass cells crucial to a SERF magnetometer with increasingly
smaller dimensions and many more at a time. [29]
SERF magnetometers are primarily made up of seven components as shown in
Figure 1. A glass cell contains an alkali vapor (labeled “atoms” in the figure). Surrounding the glass cell is thermal insulation, and surrounding the thermal insulation
is some packaging. To measure the magnetometer’s operating temperature, there is
typically a thermocouple in contact with an exterior face of the glass cell. There are
typically one or two heaters found on the top and bottom of the glass cell. Chapter
3 will consider the case with a single heater on top of the glass cell. Finally, there
will be some fiber optics and other cables that travel from the interior to the exterior
of the magnetometer.
6
Figure 1: SERF Magnetometer Components
Much effort has gone into developing chip-scale atomic magnetometers and methods for manufacturing them [27–30]. Chip-scale magnetometers have the advantage
of lower-power operation and the potential for low cost manufacture of large quantities of them [25]. This technology makes flexible and conformal geometries of
hundreds of SERF magnetometers feasible [25], which could allow for dense packing
of sensors and high spatial resolution measurements [25].
1.1.3
Magnetometer Array on a Conformal Cap
A conformal cap containing hundreds of SERF magnetometers would have many
advantages over existing MEG technology, with few drawbacks. Figure 2 shows an
illustration of this concept as well as a mockup to demonstrate that it is possible
to fit 300 magnetometers on a conventional swim cap, which is a type of conformal
cap. Elasticized caps are certainly feasible for MEG, as they have been used in
the past to hold the magnetic coils used for head-tracking [16]. A conformal cap
would be able to stretch to fit snugly on most head sizes and shapes, bringing the
magnetometers as close to the head as possible without shaving the head or becoming an invasive procedure, and preventing all but the tiniest of movements of the
head with respect to the magnetometers. These two features alone allow for vastly
better sensitivity than does traditional MEG technology. Without the need for bite
bars or foam padding to immobilize the head, setup time could be significantly reduced, and patient comfort could be dramatically improved. In addition, the cap
and magnetometers would have little inertia, allowing them to move where the head
moves with little delay, allowing longer filtering time to smooth measurements with7
out concerns about blurring. Unrestricted movement has the potential to not only
increase subject comfort and decrease the patient’s feeling of confinement, but also
may allow for greater flexibility and freedom in which tasks are performed by the
subject during MEG. Low mechanical complexity, in addition to low setup time and
effort, high patient mobility, and minimal distance of sensors from the head, makes a
conformal cap containing hundreds of magnetometers a promising option for MEG.
Figure 2: Illustration of Conformal Cap Concept
1.2 Challenges Implementing a Conformal Cap SERF Array for MEG Applications
In conventional MEG setups using SQUIDs, many of the challenges have been solved,
from calibration, to thermal requirements, to methods of solving the inverse problem. A conformal cap with attached SERF magnetometers is a new technology
with new technical challenges. In magnetoencephalography, in order for the sensor
readings to be meaningful, it is important for the positions of all magnetometers to
be known [21]. When measuring magnetic field vectors rather than just the magnetic field magnitude, the magnetometers’ orientations must also be known. This
thesis considers the case in which a few hundred spin-exchange relaxation-free magnetometers are attached to a tight-fitting cap, made of an elastomer, that stretches
to fit snugly on most heads. This cap, which can be likened to a swim cap, will be
called a “conformal cap” in the remainder of this document. When this conformal
cap is stretched over a patient’s head, the magnetometers move with respect to each
other, causing the relative location and orientation of each magnetometer to change.
In this document, determining the positions and orientations of the magnetometers
8
will be called “calibration.” Calibration can be carried out in a number of ways;
this work explores one method.
There are a number of ways to determine the positions and orientations of the
magnetometers: fix them in a rigid structure so that their locations and orientations with respect to each other never change, monitor the positions and orientations
continuously as the sensors are placed on the head, or solve for the positions and
orientations after the sensors have been placed on the head. A rigid structure is
what has traditionally been used in magnetoencephalography. The SQUIDs were
fixed within a helmet and dewar of liquid helium, expected never to move with respect to each other, meaning that they could be calibrated once (or infrequently)
and only their position with respect to the head would need to be solved for during each MEG session. A conformal cap arrangement of magnetometers gains the
advantages of increased proximity and reduced head movement, but makes a rigid
array infeasible. Alternatively, during each MEG session, the magnetometers could
be placed in specific locations on a conformal cap, their locations and orientations
monitored continuously during placement. Unfortunately, the operation could require excessive setup time and would require precise, expensive equipment and many
additional sensors that would also need to be calibrated. Solving for the positions
and orientations of the magnetometers after they have been placed on the subject’s
head allows for a number of calibration solutions. One possibility is to use an external camera, as has been used in SQUID MEG systems in the past [21]. Another
possibility is to expose the magnetometers to a known magnetic field and use no sensors other than the magnetometers themselves to perform calibration. This method
involves reduced mechanical and electrical complexity, lessened cost, and possibly
even decreased setup time.
This thesis seeks to address two major technical challenges associated with this
proposed MEG setup: calibration and thermal insulation of the magnetometers.
1. Calibration Dipole Quality: When a conformal cap is stretched over a
patient’s head, the positions and orientations of magnetometers change with
9
respect to each other and the head. The accuracy with which these positions
and orientations are known can significantly affect the localization of the MEG
sources [21]. An important technical challenge is to solve for the positions and
orientations, with respect to the head, of all of the magnetometers attached
to the conformal cap. This information can be estimated, using the magnetometers’ readings of a known magnetic field produced by a set of magnetic
dipoles. Unfortunately, the ability of this estimation algorithm to converge
quickly and accurately is highly influenced by this dipole configuration. Performing a thorough analysis of a dipole configuration can take a significant
amount of time, so it would be useful to have a tool to speed up this analysis.
2. Thermal Insulation: SERF magnetometers have an operating temperature high enough to destroy human skin [22, 31]: 120-150◦ C. Attaching SERF
magnetometers to a conformal cap to be worn on the head would bring the
magnetometers into close proximity with human skin, making thermal injury
a possibility. In addition, this high operating temperature may be difficult
to achieve with low power heaters. As such, another significant challenge of
this setup would be to thermally insulate the magnetometers sufficiently to
keep their interiors at an adequate operating temperature while keeping the
side of them that comes closest to the head at a temperature low enough to
prevent skin damage. A second part to the challenge is making sure that the
thermal insulation does not cause the magnetometer to become so large that
a sufficient quantity of them would not be able to fit on a conformal cap and
that it does not cause too much of a standoff between source and sensor.
1.3 Thesis Contributions
The goal of this thesis is to address two of the challenges associated with applying
SERF magnetometers to MEG. The thesis contributions are summarized as follows:
1. Development of a Tool to Assess the Quality of Dipole Geometry
for Conformal Cap Calibration: A potentially low cost solution for deter10
mining the locations and orientations of SERF magnetometers on a conformal
cap is to measure their response to a known magnetic field. However, it is not
immediately clear what magnetic dipole geometry is most useful. This thesis
develops a tool to compare prospective dipole geometries and applies that tool
to identify a promising candidate geometry.
2. Design of Compact Thermal Packaging: To achieve high sensor density,
magnetometers must be clustered close together on the conformal cap, which
requires them to have compact packaging. Compact packaging is difficult because reduced insulation increases the risk of burns and makes it harder for the
magnetometers to reach their operating temperature. This thesis conducted
a computational analysis to show that it is possible to meet thermal requirements using a device with 1.3cm per side cube overall package dimensions.
1.4 Thesis Overview
The remainder of this thesis is divided into three chapters.
Chapter 2 introduces a graphical tool for assessing the quality of a magnetic
dipole configuration to be used for calibrating SERF magnetometers on a conformal
cap. It begins by defining the requirements for this graphical tool followed by a
description of a calibration estimation algorithm. It continues by describing how
the graphical tool works, then it provides examples, followed by the best dipole
configuration yet found. Finally, the options for future work related to calibration
are discussed.
Chapter 3 suggests a thermal insulation scheme that allows for all temperature
requirements to be met and for the magnetometer to have small overall dimensions.
The chapter begins by defining the thermal and size requirements for the magnetometer. It continues with a description of the finite element model that was created
for the magnetometer followed by a description of the limited quanity of thermal
insulation layouts that were analyzed in the finite element software. Next, it details the metamodel that was constructed from the finite element results. Then it
11
describes how this metamodel was used to optimize the thermal insulation scheme.
Finally, suggestions future work for thermal analysis are presented.
Chapter 4 summarizes the thesis contributions and discusses their impact.
12
2 Chapter 2: Calibration
In magnetoencephalography, the positions and orientations of the magnetometers
must be known in order for measurements to be meaningful. As discussed in Chapter
1, a simple solution to this calibration problem is to use the measurements from the
magnetometers when they are exposed to a known magnetic field. This calibration
method has the advantage of requiring no sensors other than the magnetometers
attached to the conformal cap. The known magnetic field to which the magnetometers are exposed can be generated by a set of magnetic dipoles with known strength,
position, and orientation. Some dipole configurations will allow the magnetometer positions and orientations to be solved for quickly and accurately, and other
configurations will not.
This chapter introduces a graphical tool for comparing the quality of magnetic
dipole configurations. First the requirements for the calibration algorithm are defined, followed by a description of the chosen calibration algorithm. Next, a tool for
comparing dipole configurations is described, and the effect of the positions, orientations, and quantities of dipoles is explored. Finally, this tool is used to show how
one dipole configuration that allows for quick and accurate calibration compares to
other lower-quality dipole configurations.
2.1 Problem Statement
For magnetometer measurements in MEG to be meaningful, the magnetometer positions and orientations must be known. The magnetometers are situated on a
conformal cap, and their locations and orientations relative to each other change
when the cap is placed on a patient’s head. The calibration problem discussed in
this chapter uses magnetometer measurements of a known magnetic field produced
by magnetic dipoles to estimate the positions and orientations of those magnetometers. Unfortunately, not all magnetic dipole configurations will allow for quick or
accurate estimation of the positions and orientations of the magnetometers. Figure
3 shows the magnetometers on a conformal cap and an example magnetic dipole
13
configuration surrounding the head.
Figure 3: Illustration of Calibration Components: Conformal Cap, Magnetometers,
and Magnetic Dipoles
Assessing the quality of a dipole configuration is difficult to quantify. The estimation problem will be very sensitive to different dipole configurations, as it is
underconstrained for arbitrary head geometry. The estimator may not converge to
a solution for some dipole configurations or for certain initial conditions. If it is
able to converge to a solution, it may not do so quickly or to the correct solution.
For a particular dipole configuration, it is possible to run thousands of cases with
different initial conditions and noise to assess its quality, but that would take a very
long time. As such, it would be beneficial to have a tool that is capable of quickly
assessing the quality of a dipole configuration.
The primary goal of this chapter is to find a way to assess whether a particular
dipole configuration will allow for quick and accurate estimation. The inputs to this
problem are the magnetic dipole parameters: the quantity of dipoles, their positions,
their orientations, and their strength. The output is whether that particular dipole
configuration can quickly and reliably estimate the positions and orientations of the
magnetometers.
2.2 Mathematical Model for Calibration System
This section defines the calibration problem of using measurements of a magnetic
field to determine the locations and orientations of all SERF magnetomters in an
14
array attached to a conformal cap. The calibration method makes use of an array
of N SERF magnetometers attached to a conformal cap and M magnetic dipoles
that surround the head. Figure 3 shows an example of such a setup.
The conformal cap has N magnetometers attached to it, and each of these sensors
has a state xn , where n = 1 to N . If the head’s coordinate system is comprised of
three orthogonal axes, x, y, and z, with the head centered at the origin, each sensor’s
state vector is composed of its position along each axis (x, y, z) and rotation about
each axis (θx , θy , θz ). Note that in this section, T indicates the transpose.
xn = [ x y z θx θy θz ]T
(1)
This work assumes that once the conformal cap has been stretched and placed over
the head, the magnetometer array states x do not change.
x = [ x1 x2 x3 . . . xN ]T
(2)
It is assumed that each magnetometer is capable of simultaneously measuring all
three components of a magnetic field1 . As such, a general magnetometer measurement vector, Bvm , will be composed of three magnetic field measurements relative
to that particular sensor’s coordinate frame (Bsx , Bsy , and Bsz ) and noise (ε).
Bvm = [ Bsx Bsy Bsz ]T + ε
(3)
For this work, the measurement will consist only of the magnitude of the magnetic
field. This scalar measurement will be called Bsm .
Bsm = kBvm k + ε =
q
2 + B2 + B2 + ε
Bsx
sy
sz
(4)
The goal of the calibration algorithm is to solve for all of the magnetometer states,
1
In reality, most SERF magnetometers measure a single vector component of the total field, but
can in principle be operated to measure all three vector components.
15
x, given the magnetic field measurement for all magnetometers,
y = [ Bsm,1 Bsm,2 Bsm,3 . . . Bsm,N ]T
(5)
For calibration, the magnetometers measure the magnetic field produced by one
or more magnetic dipoles. Suppose that M dipoles are used in the calibration
process. Each dipole’s position and orientation in the head’s coordinate system
defines its state xm , where m = 1 to M .
xm = [ x y z θx θy θz ]T
(6)
Fm , the magnetic field vector at magnetometer n produced by dipole m, is dependent
on the relative states of the magnetometer and dipole, xn − xm , and the dipole’s
strength Am . This work considers magnetic dipoles in the form of current-carrying
coils. The equations for the magnetic field produced by a current-carrying coil are
lengthy and can be found in Appendix B. B(xn ) is the true magnetic field, which is
the superposition of the magnetic fields produced by all of the dipoles at the location
of magnetometer n.
B(xn ) =
M
X
Fm (xn , xm , Am )
(7)
m=1
An estimate of magnetometer states x̂ can be obtained by evaluating Equation
7 at the estimate and matching the result to the sensor measurement vector y.
y = [ kB(xˆ1 )k kB(xˆ2 )k kB(xˆ3 )k . . . kB(xˆn )k ]T
(8)
The problem of estimating x from y is underconstrained. Ideally, the dipole
states and strengths would not be necessary to fully constrain the problem. In
this case, the number of unknown variables is 6N + 7M : six for the state of each
magnetometer and dipole, and one for each dipole’s strength. One equation can be
written for each magnetometer for a total of N equations (see Equation 8, where y
16
is the magnetic field measurement vector). As 6N + 7M is greater than N , there
are too many unknowns to generate a unique solution for x.
To reduce the number of unknowns, several assumptions are made. First, assuming that the dipole states and strengths are known reduces the number of unknowns
by 7M . The following assumptions further reduce the number of unknowns to only
3 unknowns.
• The head can be modeled as an ellipsoid with dimensions ha , hb , and hc (as
shown in Figure 4).
• After being stretched over the head, the conformal cap can be modeled as half
of an ellipsoid with dimensions ca , cb , and cc (as shown in Figure 4).
• The conformal cap stretches to be half the size of the head, such that ca = ha ,
cb = hb , and cc = hc
Figure 4: Head and Conformal Cap Models
• The layout of magnetometers for the unstretched conformal cap is known. The
cap is divided into p levels (labeled L1 , L2 , ... , Lp in Figure 5). Magnetometers
are found only on these levels, not on any other part of the cap.
Figure 5: Conformal Cap Levels
17
• The quantity of magnetometers in each level is known. Each level is the shape
of an ellipse. On any given level, the magnetometers are equally spaced about
the perimeter of the ellipse. This relationship is defined in Equation 9, where ∆
is the distance between sensors along the elliptical contour of the level, S is the
perimeter of the elliptical contour, and Ns is the quantity of magnetometers
on level Ls .
∆ = S/Ns
(9)
• When the cap is stretched, the levels are a distance d1 in the Z direction
from each other, and the top level is the same distance (d2 ) in the Z direction
from the top of the cap that the bottom level is from the bottom of the cap.
Equation 10 gives the spacing of the levels.
cb = (p − 1)d1 + 2d2
(10)
• When the cap is stretched, each cap level is still in the shape of an ellipse, and
the magnetometers are still equally spaced about the perimeter of each level.
This means that Equation 9 is still valid, but will have different values of ∆
and S.
• For each magnetometer, the face that is in contact with the conformal cap
remains tangent to the surface of the cap.
These assumptions make it so that if ha , hb , and hc are known, the magnetometer
states x are also known. This means that there are only 3 unknowns. As long as at
least 3 magnetometers are available for calibration, enough equations can be written
for the calibration problem to be solved. As the MEG system is likely to have a
few hundred magnetometers, significantly more equations can be written than the
quantity of unknowns, which will allow for some redundancy when dealing with
noisy measurements.
18
2.3 Procedure for Evaluating Calibration Geometries
This section provides detail on the dipole-based calibration algorithm used to solve
Equation 8 and introduces a graphical tool for assessing the quality of a dipole
configuration.
2.3.1
Calibration Estimation Algorithm
This section describes how the calibration algorithm presented in this chapter works
and details two checks that were performed to ensure that the algorithm was correctly simulated. The estimation-based algorithm is summarized in the block diagram in Figure 6. It assumes a set of dimensions for the ellipsoid and determines
what the magnetometers would read with that set of dimensions when exposed to
a known set of magnetic dipoles. It then determines the error by comparing these
readings to the actual magnetometer measurements. The algorithm loops, each time
selecting a new set of ellipsoid dimensions in an attempt to minimize the square root
of the sum of the squares of the error (i.e., the least-squares error).
Figure 6: Block Diagram of Calibration Algorithm
As it was not possible to test this calibration algorithm with a real magnetometer
array attached to a conformal cap, it was necessary to simulate the system. MATLAB was used to construct the loop and to generate both the measurement model
and actual measurements. The optimization function was chosen to be MATLAB’s
builtin lsqnonlin because what was being optimized had the possibility of being
19
nonlinear, and lsqnonlin is optimized for solving least squares problems like the
one here. The estimation algorithm was constructed as a script in MATLAB. A
number of arrangements of magnetic dipoles (in the form of current-carrying coils)
were simulated in order to determine which would allow for the fastest and most
accurate calibration, the results of which will be described in Section 2.4.1.
A model seeks to approximate a system. No model can exactly replicate the
behavior of a system, as any real-life system is far too complex. Simplifying assumptions are made. It is important to perform some reality checks or to run some
test cases to verify that the model is a reasonable approximation of the system and
to ensure that no obvious mistakes have been incorporated into the model.
To ensure that various parts of the algorithm were constructed correctly, two
checks were performed. As the algorithm assumes that the dipoles are in the form
of current-carrying coils, the first check verifies that the equations used to model the
magnetic field strength at various axial and radial distances from a coil are correct.
The second check ensures that the loop depicted in Figure 6 produces zero error
when the correct head dimensions are input.
Magnetic Field Strength
To gain confidence that the MATLAB model used the
correct equations to calculate the magnetic field magnitude at each sensor position,
some plots were constructed. The magnetic dipoles used in calibration are modeled
by current-carrying coils, and the magnetic field produced by each coil varies as a
function of the radial and axial distances from the coil. The axial distance is the
distance along the axis about which each loop of the coil is centered, depicted by
the axis labeled “A” in Figure 7. The radial distance is the distance along a vector
that is normal to the coil’s axial vector. This radial axis is labeled as “R” in Figure
7.
20
Figure 7: Coil Axes
The magnitude of the magnetic field should fall off with the cube of the distance
between the dipole and the magnetometer [19]. If the calibration algorithm reflects
this, then the magnetic field magnitude multiplied by the cube of the distance at
which this magnetic field is measured will be equal to a constant. The equations
used to calculate the magnetic field as a function of axial and radial distance from
a current-carrying coil can be found in Appendix B. Figure 8 shows how the axial
and radial magnetic field magnitudes fall off with the distance between the source
and sensor. Figure 9 shows how the axial and radial magnetic field magnitudes fall
off with distance when multiplied by the cube of the distance. From this figure, it
is clear that the magnetic field magnitudes multiplied by the cube of the distance
are constant, showing that the axial and radial magnetic field magnitudes decrease
with the cube of the distance, and that this part of the algorithm is constructed
correctly.
21
Figure 8: Magnetic Field Magnitude vs. Displacement
Figure 9: Magnetic Field Magnitude Multiplied by the Cube of the Displacement
Self-Consistency of Estimator
An initial check to ensure that the estimator’s results
are sensible is to make sure that if the correct answer is input to the estimator,
the estimator returns the correct answer. If the estimator could not return the
correct answer when the correct answer is given as an initial guess, it would be clear
that some part of the estimator is not working correctly. There are infinitely many
22
possible sets of head dimensions, so it is impossible to show that the estimator is
self-consistent for all of these cases. Instead, demonstrating self-consistency for ten
random sets of head dimensions should be sufficient to show that the estimator is
self-consistent. In order to provide a more realistic check of the estimator, the head
dimensions were selected at random from between the 1st and 99th percentiles of
women’s head dimensions as listed in Appendix C. Table 1 shows the ten sets of
head dimensions that were tested. For all ten cases, the actual head dimensions
were input, and the output matched the input exactly.
Table 1: Data for Estimator Self-Consistency Check
ha (in)
2.8092
2.7259
2.7329
2.8027
2.7143
2.7220
2.8021
2.8640
3.0497
2.8374
2.3.2
hb (in)
3.8796
3.9600
3.9013
4.6192
4.2945
4.1115
3.8396
4.4404
3.9574
4.8333
hc (in)
3.5975
3.8393
3.4827
3.6379
3.4109
3.2822
3.8916
3.8333
3.7057
3.2108
GRaphical Estimation-quality Assessment Tool (GREAT)
The previous sections in this chapter have defined and described a calibration algorithm; this section introduces a graphical tool for assessing how well a dipole
configuration works with the calibration algorithm to come to a solution. This tool
not only requires less time than simply running the estimation algorithm repeatedly,
but in the event that a dipole configuration is low quality, this tool provides some insight into why that particular configuration is bad. This tool will be used in Section
2.4.1 to help determine the characteristics of an optimal dipole configuration.
In review, the calibration algorithm presented in Section 2.3.1 calculates a measurement model for a set of ellipsoid dimensions (ha,i , hb,i , hc,i ). This measurement
model consists of what each magnetometer would read if each magnetometer was sit-
23
uated about an ellipsoid with dimensions (ha,i , hb,i , hc,i ). As part of the calibration
algorithm, this measurement model is compared to the actual measurement. The
actual measurement is the readings from magnetometers situated about an ellipsoid
with dimensions (ha,0 , hb,0 , hc,0 ). The difference between the actual measurement
and the measurement model will be called the error, and the square root of the sum
of the squares of the error will be called the cost, or J.
The cost function can be calculated and compared for a number of different sets
of ellipsoid dimensions. For example, if (ha,0 , hb,0 , hc,0 ) represents the dimensions
of the actual ellipsoid, the cost function can be calculated for various combinations
of ha,0 + k1 α, hb,0 + k2 α, and hc,0 + k3 α, where k1 , k2 , and k3 are integers. Each
of these combinations will be called a location index. Figure 10 depicts a set of
location indices.
Figure 10: Location Indices
The tool introduced in this section calculates a discrete approximation of the
gradient of J. The cost function at each location index is subtracted from the cost
function of all adjacent location indices, producing ∆J1 , ∆J2 , ... ∆J6 , shown in
Figure 11 and defined in Equations 11 through 16. ∆J is not computed at the
boundaries of the set of location indices.
24
∆J1 (ha,0 , hb,0 , hc,0 ) = J(ha,0 + α, hb,0 , hc,0 ) − J(ha,0 , hb,0 , hc,0 )
(11)
∆J2 (ha,0 , hb,0 , hc,0 ) = J(ha,0 − α, hb,0 , hc,0 ) − J(ha,0 , hb,0 , hc,0 )
(12)
∆J3 (ha,0 , hb,0 , hc,0 ) = J(ha,0 , hb,0 + α, hc,0 ) − J(ha,0 , hb,0 , hc,0 )
(13)
∆J4 (ha,0 , hb,0 , hc,0 ) = J(ha,0 , hb,0 − α, hc,0 ) − J(ha,0 , hb,0 , hc,0 )
(14)
∆J5 (ha,0 , hb,0 , hc,0 ) = J(ha,0 , hb,0 , hc,0 + α) − J(ha,0 , hb,0 , hc,0 )
(15)
∆J6 (ha,0 , hb,0 , hc,0 ) = J(ha,0 , hb,0 , hc,0 − α) − J(ha,0 , hb,0 , hc,0 )
(16)
The smallest of ∆J1 , ∆J2 , ... ∆J6 will be called ∆Jmin , and each location index
will have a ∆Jmin value associated with it.
∆Jmin (ha,0 , hb,0 , hc,0 ) =
(17)
min(∆J1 (ha,0 , hb,0 , hc,0 ), ∆J2 (ha,0 , hb,0 , hc,0 ), . . . , ∆J6 (ha,0 , hb,0 , hc,0 ))
When considering the absolute value of the change in cost function for all sets of
adjacent location indices, the maximum of these will be called ∆Jmax . The scaled
∆Jmin value is defined as ∆Jmin,normalized .
∆Jmin,normalized = ∆Jmin /∆Jmax
25
(18)
Figure 11: Calculation of Cost Function Change
The ∆Jmin,normalized values for a set of location indices can be sorted in ascending
order and plotted. The negative values are the values of interest, as they describe
the rate of descent to any local minima of the cost function, and will be the only ones
plotted. The absolute value of these is taken to facilitate plotting on a log scale. To
make the plots easier to interpret, both the horizontal and vertical axes are on a log
scale. These plots will be called GREAT (GRaphical Estimation-quality Assessment
Tool) plots, and are used extensively in Section 2.4.1. An example GREAT plot is
shown in Figure 12. Because of the way they were constructed, the horizontal value
of the leftmost point on these plots equals the number of local minima in that set
of location indices, and the higher the vertical values, the faster the cost function
descends to local minima. An optimal magnetic dipole configuration will have few
local minima and a high rate of descent to local minima; in other words, an optimal
magnetic dipole configuration will on a GREAT plot have its far leftmost point have
a small horizontal axis value and all of its points have high vertical axis values. In
order for the least-squares estimation algorithm described in the previous section to
perform well, even in the absence of noise, the minimum values of ∆Jmin,normalized
should generally not be less than 10−10 . This value is based on empirical observation,
having run the algorithm on many cases.
26
Figure 12: Example GREAT Plot
2.4 Application of GREAT
This section gives numerous examples of GREAT. First, GREAT is used to assess
the quality of a few simple dipole configurations to provide insight into what may
constitute a good coil geometry. Second, a dipole configuration that is of increased
complexity and has been empirically shown to work well with the estimation algorithm is analyzed using GREAT to demonstrate that GREAT works as expected
and to enforce why this dipole configuration is a good one.
2.4.1
Magnetic Dipole Configuration Exploration
This section uses the graphical method described in Section 2.3.2 to compare various dipole configurations and understand what contributes to a superior dipole
configuration. The effects of the orientation, position, and quantity of dipoles are
investigated first, followed by an exploration of some more complicated dipole configurations. This section also includes information on how well these dipole arrangements allow the estimation algorithm to come to a correct solution.
The comparisons between the dipole configurations are made using the scaled
GREAT plots described in Section 2.3.2. As a reminder, these plots provide some
information about the cost function’s solution space. Optimization functions such as
27
MATLAB’s lsqnonlin typically search for a minimum in the vicinity of a starting
point. One thing that can cause an optimization function to think it has found this
minimum is if a change in the input variables in any direction causes too small of
a change in the output, thus it is desirable to have a large change in output, or a
steep rate of descent, to a minimum. On a GREAT plot, higher ∆Jmin,normalized
values correspond to a steep rate of descent. An optimization function may think it
has found the global minimum if it finds a local minimum. As such, the fewer local
minima in the cost function’s design space, the better. The number of local minima
that can be found in a set of location indices is equivalent to the horizontal value of
the leftmost point on a GREAT plot.
The exploration presented in this section fixes certain parameters and makes
some assumptions in order to provide a better comparison between the cases. All
magnetic dipoles used are in the form of current-carrying coils with three turns,
0.20 inch radius, and carrying 10−5 Amps. It is assumed that none of the magnetic
dipoles are of sufficient strength to “max out” any of the SERF magnetometers
attached to the conformal cap. It is assumed that the conformal cap is in the shape
of the top half of the head and is not rotated any in the head’s coordinate system.
The head size used is the 50th percentile for a woman: a = 2.85 inches, b = 3.55
inches, and c = 4.30 inches [32]. This design space exploration was performed using
only a limited number of simulated magnetometers, the positions of which are listed
in Appendix D. This is realistic in the sense that while all the magnetometers on
the conformal cap could be used for calibration, not all of them need to be used.
The dipole states used to generate the plots in this section are listed in Appendix
D.
Effect of Dipole Orientation
In understanding what dipole configuration will work
optimally with the calibration algorithm proposed in section 2.3.1, it is important
to understand what effect dipole orientation has. If the dipole, in the form of a
current-carrying coil, starts 2 inches in front of the head with its axis aligned with
the YH axis of the head (see Figure 13), it can rotate about two different axes: the
28
ZC axis, and the XC axis (the axis perpendicular to the YC and ZC axes). Figure
14 shows the GREAT plot for the coil rotating about the XC axis, and Figure 15
shows the GREAT plot for the coil rotating about the ZC axis.
Figure 13: Coil and Head Axes
Figure 14: Effect of Rotating Magnetic Dipole About X-Axis
Figure 14 shows the GREAT plot for the coil rotating about the XC axis in
15◦ increments. As the setup is symmetric, the coil needs only to be rotated up
to 45◦ . All four GREAT plots are very similar. The unrotated coil, shown on the
legend as “0 deg”, has the highest ∆Jmin,normalized values overall, and with 11 local
minima, has fewer local minima than the other three. Interestingly, none of these
coil configurations allow the calibration algorithm to come to reasonable solution.
29
Figure 15: Effect of Rotating Magnetic Dipole About Z-Axis
Figure 15 shows the GREAT plot for the coil rotating about the ZC axis in 15◦
increments. Once again, as the setup is symmetric, the coil needs only to be rotated
up to 45◦ . All four GREAT plots are very similar.
Effect of Dipole Location To find an optimal dipole configuration for calibration,
it will be beneficial to understand the effect of dipole position. The dipole starts
with its axis in line with the YH axis of the head (shown in Figure 16); this will be
called the base case. Following around the midline of the head (shown as the dotted
circle in Figure 16) in the counterclockwise direction, the GREAT plots are created
for the coil intersecting the YH axis, halfway between the XH and YH axes, and
intersecting the XH axis, labeled as “Front of Head,” “45 deg”, and “Side of Head”
on the legend in Figure 17. GREAT plots are also created for the case in which the
coil maintains the same orientation and distance from the center of the head, but
is placed directly above the head (XH = YH = 0), and at the very center of the
the ellipsoid, labeled “Top of Head” and “In Mouth” respectively in the legend in
Figure 17.
30
Figure 16: Coil Position with Respect to Head Axes
Figure 17: Effect of Magnetic Dipole Position
The “45 deg” case is clearly the worst, with more than half of the ∆Jmin,normalized
values being lower than for any of the other curves shown, and having approximately
80 local minima, the most local minima of any of the curves shown. The “In Mouth”
case, is the best of the curves shown, having the fewest local minima and generally
high ∆Jmin,normalized values. Unfortunately none of the coil configurations shown
allow the calibration algorithm to come to a correct solution.
Effect of Dipole Quantity
The quantity of dipoles used in calibration may have an
effect on how well the calibration algorithm can find a correct solution. GREAT
31
plots are created for different quantities of dipoles, equally spaced about the midline
of the head (shown as the dotted circle in Figure 18), with all coil axes aligned with
the YH axis of the head (shown in Figure 18). The GREAT plots are shown in
Figure 19.
Figure 18: Quantity of Coils Shown on Head Axes
Figure 19: Effect of Quantity of Magnetic Dipoles
The five and six coil cases appear to produce the best results, although only
slightly: they have the highest ∆Jmin,normalized values and the least local minima.
32
2.4.2
Recommended Geometry
This section gives detail on the best dipole configuration yet found. GREAT is used
to show why this dipole geometry works well.
Figure 20: Dipole Configuration
Figure 20 shows a promising coil arrangement from a few different angles. Its
GREAT plot is shown next to the GREAT plot of the base coil configuration case.
This promising coil configuration has reasonably high ∆Jmin,normalized values and
fewer than ten local minima. This coil configuration has been empirically shown to
work well with the estimation algorithm, finding an accurate solution fairly quickly.
The coil locations and orientations are listed in Appendix D.
Figure 21: Effect of Many Magnetic Dipoles in Varied Positions and Orientations
33
2.5 Discussion and Future Work
Future work can proceed by investigating the calibration algorithm’s susceptibility to
noise, by exploring variations on dipole arrangements, by expanding the capability of
the current calibration algorithm, and by investigating other calibration algorithms.
Susceptibility to Noise of Calibration Algorithm
This chapter simulates the MEG
calibration system in the absence of noise. To better understand the abilities and
limitations of the calibration algorithm and a particular dipole arrangement, noise
must be incorporated in the model. Different levels of noise should be simulated
to determine the noise level at which the calibration algorithm ceases to attain a
correct solution.
Other Magnetic Dipole Arrangements This chapter suggests one reasonably good
calibration dipole configuration, but an exhaustive exploration of dipole configurations has not been conducted. A natural step to take is to find a better magnetic
dipole arrangement, as determined by GREAT. Another possibility for future work
is to determine whether operating the magnetic dipoles at different frequencies could
strengthen the calibration algorithm presented in this thesis. A final idea for future work is to investigate a way to use the brain’s magnetic field, rather than the
external known magnetic dipoles, to calibrate.
Expand Capability of Calibration Algorithm
The calibration algorithm presented in
this chapter makes a number of simplifying assumptions, and it may be able to be
improved by altering some of these assumptions. It assumes that human heads are
ellipsoids; in reality, their shape is somewhat different, so to improve the calibration
algorithm, a more complex and realistic head model could be used. The calibration
algorithm assumes that the axes of the conformal cap are perfectly aligned with the
axes of the head. Typically, it will be impossible for the cap to be perfectly aligned
in MEG, so the calibration algorithm should be altered to allow for the cap to be
askew.
34
Other Calibration Algorithms Variations of the calibration algorithm presented in
this chapter may work well for calibration. One possibility is to leverage the considerable literature relating to the signal space separation method that is used in
traditional SQUID MEG setups, as it may have aspects that could be extremely
useful when transferred into the context of a conformal cap with an attached SERF
magnetometer array. An additional possibility is to develop a calibration algorithm
that does not require a head-shape model, that instead can solve for the position and
orientation of individual magnetometers without the measurements from other magnetometers. While this method seems more susceptible to noise than the method
presented in this thesis, averaging together several sequential measurements could
filter out the noise, and it has the advantage of not needing a large number of
iterations to arrive at a solution.
2.6 Summary
This chapter introduced a graphical tool (GREAT) for assessing the quality of magnetic dipole configurations for a particular calibration estimation algorithm. The
calibration estimation algorithm, a method for determining the positions and orientations of an array of SERF magnetometers situated on a conformal cap using a
configuration of magnetic dipoles with known parameters, was presented first. Next,
GREAT, a graphical tool for comparing magnetic dipole arrangments was detailed
and applied to explain the effect of position, orientation, and quantity of dipoles
on the ability of the calibration algorithm to find a correct solution. GREAT was
used to show why a particular magnetic dipole configuration of increased complexity allowed the calibration algorithm to come to a correct solution in the absence
of noise, thereby demonstrating that GREAT can help predict whether a magnetic
dipole configuration is likely to work well with the calibration algorithm.
35
3 Chapter 3: Thermal Design
This chapter investigates whether there exists a thermal insulation scheme for a
SERF magnetometer that allows the magnetometer to meet a set of temperature
and size requirements. A patient wearing a conformal cap containing an array of
these magnetometers must not experience thermal injury, and each magnetometer
on the cap must maintain the correct operating temperature and fit within a cube
that is 1cm on a side.
A multi-step analysis was conducted to answer this question. First, a quartermodel of a magnetometer from the array on the conformal cap (see Figure 22) is
simulated in the finite element analysis (FEA) package, ABAQUS, to obtain steadystate values for skin temperature and magnetometer operating temperature. Temperature values are found for a small quantity of thermal insulation arrangements.
Second, to relate the amount of thermal insulation to the temperature results, a
metamodel is devised. Finally, this metamodel is used to find a thermal insulation layout that satisfies all temperature requirements and allows for small overall
magnetometer dimensions.
Figure 22: Single Magnetometer from Conformal Cap, Quartered
36
3.1 Problem Statement
The overall goal is to keep the magnetometer package small while satisfying two
temperature requirements. One temperature requirement is to maintain the alkali
vapor in the glass cell at the right operating temperature. The other temperature
requirement is to keep the skin close to body temperature. These temperature
requirements can be met by making use of the thermal insulation properties of the
other device components (see Figure 23). This section first defines what is acceptable
for skin temperature and magnetometer operating temperature, then it defines the
partial differential equation (PDE) the model seeks to solve, followed by the PDE’s
loads and boundary conditions.
Figure 23: Alkali Vapor Chamber (White), Other Device Components (Light Gray),
and Skin (Dark Gray)
3.1.1
Requirements for Preventing Thermal Injury
This study will assume that the exterior skin temperture can never exceed 44◦ C.
An object in contact with skin, if warmer than the skin, causes a heat flux into the
skin, and depending on how quickly the body can carry heat away from that area,
there may also be a rise in skin temperature [33]. This heat flux will be different
for each person [34]. Thermal injury of the skin is complex, depending both on
the temperature the skin is exposed to and the time of exposure [35]. 44◦ C is a
pivotal temperature, being the temperature at which human skin begins to feel
pain [31]. Requiring about 6 hours to cause damage, it is the lowest temperature
that causes cutaneous burning [35], because at 44◦ C the rate of injury exceeds the
37
rate of recovery [35]. As such, it makes sense to layout the magnetometer insulation
such that the if it is attached to a latex cap, the inside surface of the latex cap never
reaches or exceeds 44◦ C.
3.1.2
Magnetometer Operating Temperature Requirement
This study will assume that the required operating temperature is 150◦ C. The rationale for this value is that the required magnetometer operating temperature depends
both on the alkali vapor being used (e.g. Cesium, Rubidium, or Potassium) and the
size of the cell. In this study, the cell will be a cube that is 5mm on a side. Assuming an optimized buffer gas of Neon is used, the required cell temperature would
need to be approximately 100◦ C for Cesium, 150◦ C for Rubidium, and 250◦ C for
Potassium [30]. The alkali vapor used is assumed to be Rubidium, so the required
operating temperature is 150◦ C.
3.1.3
Conduction PDE
This section defines the equation that must be solved for the magnetometer model.
As the temperature of the alkali vapor is relatively small, convection and radiation
are assumed to be negligible, so conduction is the only form of heat transfer that
is modeled. The following PDE relates temperature, T (K), time, t (s), thermal
conductivity, k (W/mK), density, ρ (kg/m3 ), specific heat capacity, cp (J/kgK),
and the heat generated by the heater, qheater (W/m3 ).
k∇2 T + qheater
∂T
=
∂t
ρcp
(19)
This PDE is the heat conduction equation with a heat source included. The model
seeks to solve this PDE subject to the following boundary conditions and load when
the system has reached steady-state ( ∂T
∂t = 0).
Loads There is one load, and it comes from the heater. It has a value of 300mW,
and it is applied to the top exterior surface of the glass cell, shown in Figure 24.
38
Figure 24: Surface to Which Load is Applied
Boundary Conditions The boundary conditions are as follows, with boundary conditions (a) through (e) corresponding to the darkened surfaces shown in Figures 25a
through 25e, respectively.
(a) The exterior faces of the magnetometer are maintained at room temperature
(20◦ C) (Figure 25a). It is assumed that convection in the ambient environment
will be sufficient for this to be the case.
(b) The bottom of the skin’s dermis is at body temperature (37◦ C) (Figure 25b).
(c) There is no heat flux through the interior sides of the glass cell (Figure 25c).
The justification for this is provided in Section 3.2.1.
(d) There is no heat flux through the sides of the conformal cap, epidermis, or
dermis (Figure 25d). The justification for this is provided in Section 3.2.1.
(e) There is no heat flux through the quarter model faces (Figure 25e). The justification for this is provided in Section 3.2.1.
(a)
(b)
(c)
(d)
Figure 25: Surfaces to Which Boundary Conditions Apply
39
(e)
3.2 Finite Element Analysis
Although the PDE in Equation 19 is simple in concept, it is very difficult to solve
for the SERF magnetometer. As such, it is solved using a finite element analysis
program. This section first describes the assumptions and materials used in the
finite element analysis. Second, the inputs and outputs of the analysis are defined.
Finally, this section gives more detail about the skin temperature and magnetometer operating temperature results obtained from the analysis. The FEA makes it
possible to determine if a specific magnetometer geometry meets temperature requirements, but it cannot directly solve for a magnetometer geometry given a set of
temperature requirements. Because of this, the FEA results will be used in Section
3.4 to construct a metamodel of the design space. Details of the finite element model
can be found in Appendix E.
3.2.1
Modeling Assumptions
In order to model the SERF magnetometer, several assumptions must first be made:
• Heater Power: The heater can be approximated as outputting its average
power. The heater that heats the magnetometer is set up to turn on and off to
control the temperature, outputting full power when it is on, and outputting
no power when it is off. For analysis purposes, it will be assumed that the
heater is on all the time at a power equivalent to the actual average power (for
example, if the heater outputs 100mW and is on 20% of the time, then it will
be assumed that the heater outputs 20mW and is on 100% of the time.)
• Alkali Vapor Temperature: The alkali vapor is assumed to have a uniform
temperature equal to the average temperature of the outer side walls of the
glass cell.
• Contact Resistance: It is assumed that there is no thermal contact resistance between any pair of contacting faces.
• Quarter Model: To speed up FEA processing time, it is assumed that the
40
model can be represented as one quarter of itself. This assumption is valid
because of symmetry: the sensor layout, boundary conditions, and loads are
all reflectively symmetric across the dotted lines shown in Figure 26, thus all
four quarters of the sensor will behave the same way. As little as an eighth of
the magnetometer could have been modeled. Because of symmetry, it is valid
to assume that there is no heat flux across the lines of symmetry (shown as
dotted lines in Figure 26).
Figure 26: Top View of Magnetometer, Shaded Portion Modeled in ABAQUS
3.2.2
Components, Materials, and Thermal Properties of Model
To account for the different thermal properties of the components, the thermal model
used to analyze the SERF magnetometer, conformal cap, and scalp will be divided
into seven materials. This model represents the dominant thermal pathways which
transport heat away from the magnetometer’s alkali-vapor containing cell. The alkali
R
vapor is enclosed in a glass cell, which is surrounded by Pyrogel
2250, a thermal
insulator. In the past, Corning 7740 (PyrexTM ) has been successfully used as the
material for the vapor-containing glass cell [29], so the models in this chapter will
assume the glass is PyrexTM . The alkali vapor, glass cell, and thermal insulation
are all enclosed in a case made from acetal, and these four components make up the
magnetometer. The magnetometer is attached to a conformal cap made of natural
rubber, which is in contact with the epidermis of the skin, and the skin’s dermis lies
beneath the epidermis. These components are shown in Figure 27. Although the
fiber optics and other device components mentioned in Chapter 1 are important to
magnetometer function, they comprise only a small fraction of the device volume,
so they will be neglected in the model.
41
Figure 27: SERF Magnetometer Components
To model the magnetometers as though they are attached to a conformal cap
positioned on the head, the conformal cap and layers of skin must also be included
in the models. Hair is not included because not all patients will have hair, and no
hair is a worse case of the model. If hair is more conductive than the conformal cap
or layers of skin, it will not make a difference, and if it is less conductive, it would
make for a better thermally insulated setup. Figure 27 shows a diagram of a SERF
magnetometer and the other components that are part of the model.
To further define the model for FEA, material thicknesses were assigned for all
R
components except for the Pyrogel
2250 insulation. The thicknesses of the glass
cell and acetal were chosen to be realistic, feasible numbers, and the natural rubber
was chosen to be the same thickness as a latex swim cap (a type of conformal cap).
R
The Pyrogel
2250 insulation thicknesses, labeled t1 , t2 , and t3 in Figure 28, will be
varied as part of the design space exploration. Table 2 shows the conductivities and
thicknesses of these engineering materials alongside those of the biological materials included in the model. All thermal conductivities are assumed to be spatially
uniform within each material.
42
Figure 28: Thermal Insulation Thicknesses
Human skin, a biological material, is also included in the model. Its susceptibility
to heat is complex and not yet fully understood; however, many of its thermal
properties are known, including the specific heat and thermal conductivities of the
epidermis, dermis, and subcutaneous layers [35, 36], so the skin can be broken up
into the epidermis and the dermis in the model. Skin ranges in thickness from 0.5
to 3.0mm, with the thickness of the epidermis ranging from 0.04 to 0.4mm, and
the thickness of the dermis ranging from 0.5 to 2.5mm [37]. The model seeks to
represent the thin skin of the scalp. The skin of the eyelids is the thinnest on the
body and can provide a lower bound and worst case for the thickness of the scalp.
As the eyelids are also in close proximity to the scalp, the scalp will be assumed to
have the same thickness as the eyelids.
Table 2: Thermal Conductivities and Component Thicknesses
Component
PyrexTM Glass
R
Pyrogel
2250
Acetal
Natural Rubber
Skin (Epidermis)
Skin (Dermis)
Thermal Conductivity (W/m-K)
1.4 [38]
0.0194 [39]
0.36 [40]
0.13 [41]
0.21 [36]
0.37 [36]
43
Thickness (mm)
0.5000
varies
0.2500
0.3175
0.0400
0.5000
3.2.3
Inputs and Outputs of the Finite Element Model
The finite element model has three input variables that affect two temperature
R
outputs. The three input variables are the top, side, and bottom Pyrogel
2250
insulation thicknesses: t1 , t2 , and t3 (see Figure 28). The first temperature output
is the average temperature of one of the exterior side faces of the glass cell. The
second temperature output is the average temperature of the side of the epidermis
that is contacting the conformal cap. It is important for these temperature outputs
to meet certain requirements, but it is also important for the overall magnetometer
dimensions to remain small, which is affected by the values for t1 , t2 , and t3 .
3.2.4
FEA Temperature Results
This section shows an example result of the FEA, explains which sets of insulation
thicknesses (t1 , t2 , t3 ) were input, and discusses how these results are used to find
a good set of insulation thicknesses.
Figure 29 shows an example result of the finite element analysis in ABAQUS.
This example case used 2mm as the top (t1 ), side (t2 ), and bottom (t3 ) thicknesses
(as shown in Figure 28) of insulation. The temperature bar shown in the figure
has units of Kelvin. For this case, the vapor temperature is 122◦ C, and the skin
temperature is 35.8◦ C. All sets of insulation thicknesses (t1 , t2 , t3 ) and the two
corresponding temperature results for each are listed in Appendix A.
44
Figure 29: FEA Temperature Results
Temperature results were found for a small number of combinations of insulation
thicknesses. In only two of the cases did the sensor operating temperature result
exceed 150◦ C (see Table 3); it was less than 150◦ C in all remaining cases. Additionally, in none of the cases did the value for the temperature of the top of the
skin exceed body temperature. The reason for this will be discussed in Section 3.5.
A change in t1 , t2 , or t3 produced a noticeable change in vapor temperature, but
an almost unnoticeable change in skin temperature. These results suggest that the
requirement for the skin to remain below 44◦ C will not be difficult to meet, so the
requirement for the magnetometer operating temperature is the active constraint.
Table 3: Cases for Which Magnetometer
Operating Temperature Exceeded 150◦ C
Heater
Power
(Watts)
0.3
0.3
Top
Thickness
(mm)
3.00
3.25
Side
Thickness
(mm)
3.00
3.25
Bottom
Thickness
(mm)
3.00
3.25
Average Cell Side
Temperature
(◦ C)
153.5346
159.9842
Top of Skin
Temperature
(◦ C)
35.9040
35.9196
3.3 Design Space Metamodel
As it is not possible for a FEA program to take in the temperature and size requirements and return the values of t1 , t2 , and t3 that best satisfy these requirements,
45
and trying a large number of combinations of t1 , t2 , and t3 to find a set that meets
all requirements can take a very long time, a different approach is taken. Temperature results are found for a number of sets of input values (t1 , t2 , t3 ) to probe the
design space. This section describes a metamodel that relates the three insulation
thicknesses t1 , t2 , and t3 to the vapor temperature values.
To probe the design space before creating the metamodel, ABAQUS was used to
calculate the vapor temperature for 67 different arrangements of thermal insulation,
the results of which can be found in Appendix A. To probe the design space evenly,
the first 64 of these cases were all possible combinations of t1 , t2 , and t3 , where
t1 , t2 , and t3 could take on values of 1.75mm, 2.00mm, 2.25mm, and 2.50mm. In
all cases, the heater power was 300mW. None of these 64 cases produced a vapor
temperature of at least 150◦ C, so in an effort to include some cases for which the
vapor temperature was at least 150◦ C, three additional cases were run: t1 = t2 =
t3 = 2.75mm, t1 = t2 = t3 = 3.00mm, and t1 = t2 = t3 = 3.25mm.
To create a metamodel of the design space, MATLAB’s function nlinfit was
used to fit the data to an equation. The equation to which the vapor temperature
data is fit has all possible first-, second-, and third-order combinations of t1 , t2 ,
and t3 in the numerator and the denominator (Equation 20). Tv is the average
temperature of an exterior side of the glass cell, t1 , t2 , and t3 are the thicknesses of
R
the Pyrogel
2250, and d1 through d16 are coefficients.
Tv =
d1 + d2 t1 + d3 t2 + d4 t3 + d5 t1 t2 + d6 t1 t3 + d7 t2 t3 + d8 t1 t2 t3
d9 + d10 t1 + d11 t2 + d12 t3 + d13 t1 t2 + d14 t1 t3 + d15 t2 t3 + d16 t1 t2 t3
(20)
MATLAB’s function, nlinfit, was used to find the coefficients d1 through d16
that best relate Equation 20 to the data. nlinfit is a function meant to find the
coefficients that best fit a user-specified function to a set of data. The coefficients
that allow Equation 20 to best fit to the data are those specified in Table 4.
46
Table 4: Magnetometer Operating Temperature Fit Equation Coefficients
Coefficient
d1
d2
d3
d4
d5
d6
d7
d8
d9
d10
d11
d12
d13
d14
d15
d16
Value
-140.4843
38.3474
99.0493
50.6945
-74.7016
-15.0364
-67.1685
110.5286
-0.8334
-0.2320
0.5332
-0.0600
-0.2296
0.7267
-0.1250
0.3617
3.4 Thermal Insulation Optimization
This section describes how the metamodel for magnetometer operating temperature was used to find a thermal insulation scheme that satisfies both temperature
requirements and allows for a small overall magnetometer package size. First the
metamodel for magnetometer temperature was rearranged to form an equation for
t1 as a function of Tv , t2 , and t3 (see Equation 31). Subsequently, an optimization
function was used to find a set of insulation thicknesses that satisfy the thermal
requirements and minimize the overall magnetometer dimensions. The optimized
thicknesses were then used in the ABAQUS model to verify that the temperature
requirements were met.
The goal was to solve the following optimization problem: Minimize J in Equation 21 subject to the constraints in Equations 22 through 26.
J = V olume(t1 , t2 , t3 )
47
(21)
Tvapor = 150◦ C
(22)
Tskin < 44◦ C
(23)
t1 < 0.5cm
(24)
t2 < 0.5cm
(25)
t3 < 0.5cm
(26)
A simplified version of that optimization problem was solved instead. This simplified
optimization problem is minimizing J as defined in Equation 27 subject to the
constraints shown in Equations 28 to 29.
J = t1 + 2t2 + t3
(27)
Tvapor = 150◦ C
(28)
Tskin < 44◦ C
(29)
This simplified optimization problem (Equation 27) is the equivalent of minimizing
the perimeter of the magnetometer, where the perimeter is defined as shown in
Equation 30, where P is the perimeter, and S1 and S2 are side lengths of the
magnetometer. S1 and S2 are defined in Equations 32 and 33 and depicted in
Figure 30.
P = 2(S1 + S2)
48
(30)
Figure 30: Overall Sensor Dimensions
The metamodel for magnetometer temperature was rearranged to form an equation for t1 as a function of Tv , t2 , and t3 (see Equation 31) in order to be able to apply
the constraint of Tvapor = 150◦ C and reduce the number of variables from three to
two. This equation allows one to solve for the required thickness of insulation on
the top of the magnetometer (t1 ) given a desired internal operating temperature
(Tv ) for the magnetometer and the thickness of the insulation on the sides (t2 ) and
bottom (t3 ) of the magnetometer.
t1 (Tv , t2 , t3 ) = −
(d1 − Tv d9 ) + t2 (d3 − Tv d11 ) + t3 (d4 − Tv d12 ) + t2 t3 (d7 − Tv d15 )
(d2 − Tv d10 ) + t2 (d5 − Tv d13 ) + t3 (d6 − Tv d14 ) + t2 t3 (d8 − Tv d16 )
(31)
The goal was for the magnetometer to fit in a cube that is 1cm on a side; whether
or not that is possible, the overall magnetometer dimensions must be minimized.
The glass cell and the acetal packaging for the magnetometer are equally thick on
all sides of the magnetometer, causing the thicknesses of thermal insulation, rather
than the other magnetometer components, to drive the overall dimensions of the
magnetometer. S1 is the height of the magnetometer and is defined in Equation 32.
S2 and S3 (labeled S2 and S3 in Figure 30) are the width of the magnetometer, and
they are defined in Equation 33. Dgc is the outer dimensions of the glass cell, and
tac is the thickness of the acetal case.
S1 = 2t1 + Dgc + 2tac
49
(32)
S2 = S3 = 2t2 + Dgc + 2tac
(33)
It is important to keep S2 and S3 small because if they get too large, too few
magnetometers will fit on the conformal cap. To fit enough magnetometers, the cap
size may be driven up, making the conformal cap solution infeasible for patients
with small heads, such as infants. If the S1 dimension is allowed to be too large, it
could cause one of two problems. If S1 is too large because t3 is too large, it may
cause the magnetometers to be positioned nonoptimally far from the head. If S1 is
too large, whether because t1 is too large or because t3 is too large, it may increase
the rotational inertia of the conformal cap with attached magnetometers; when the
patient’s head moves, there may be an increased lag in when the magnetometers
move, causing MEG measurement problems. As such, it is important for t1 , t2 , and
t3 to all remain small. Because of this, it was determined that t1 + 2t2 + t3 should
be minimized.
MATLAB’s optimization function fmincon was used to find a small value of
t1 + 2t2 + t3 . Using MATLAB’s fmincon, constraining t2 to be more than 2mm
and t3 to be more than 1mm, providing initial values of t2 =2.5mm and t3 =2.5mm
because the data suggest that a good answer may lie in this vicinity, an answer is
obtained. fmincon exited because the change in the function value was extremely
small. This optimization suggests that t1 =2.6mm, t2 =3.2mm, and t3 =1.9mm will
allow for the magnetometer to reach an internal 150◦ C operating temperature with
minimal insulation on the sides of the sensor.
3.4.1
Verification of Results
The FEA model was used to verify the optimal result obtained above. The ABAQUS
model was run with insulation thicknesses of t1 =2.6mm, t2 =3.2mm, and t3 =1.9mm,
the result of which is shown in Figure 31. From this model, the average temperature
of the exterior side face of the glass cell was found to be 150◦ C, and the average
temperature of the top of the epidermis was found to be 36◦ C, so the metamodel
50
produces the same answer as the ABAQUS model. These optimal thermal insulation thicknesses allow both temperature requirements to be met. Thus, the overall
magnetometer dimensions are 11.1mm tall and 12.9mm wide, for a total volume of
1.84cm3 per sensor.
Figure 31: FEA Temperature Results for Optimal Insulation Thicknesses
3.5 Discussion and Future Work
This chapter demonstrates that it is possible to meet both thermal requirements with
a SERF magnetometer that fits in a rectangular prism that is 13.0mm on two sides
and 11.1mm on the third side. While these dimensions are slightly larger than the
specified goal of no more than 1cm on any side, they are very close. Magnetometers
of this size may still be small enough to fit as part of the conformal cap setup and
are a thermally viable solution. While the thermal insulation scheme had a great
effect on the magnetometer operating temperature, it did not have a large effect
on the skin temperature. Furthermore, the temperature value of the exterior of
the skin was below body temperature for all cases tried in ABAQUS. This suggests
that the heat flux at the skin-device interface is away from the body rather than
toward it, which might mean that the body is losing heat to the air surrounding
the magnetometer. This seems unrealistic, and the model may be improved by
altering boundary conditions and assumptions. In particular, it may be wrong to
assume that the sides of the magnetometer remain at room temperature (20◦ C) at
51
all times. That assumption may only be correct when there is sufficient convection
outside of the magnetometer and when the magnetometer is not in close proximity
to any other magnetometers. Given the nature of the conformal cap, the spacing
between adjacent magnetometers is not consistent from one patient to the next, or
even between locations on the head. As such, the model presented in this chapter
may be a good representation of a case in which the magnetometers are spaced
farther apart. To improve the model, it may be necessary to expand the model
to include more magnetometers at varying distances from each other and the air
between the magnetometers. An array of magnetometers that is more dense may
make it possible to maintain the required operating temperature with less insulation.
An alternate way to improve the model may be to make some assumptions about
how fast heat can be removed from the surfaces of the magnetometer that are in
the ambient environment.
Future work also includes expanding the fit equations for the skin temperature
and magnetometer operating temperature to allow for different materials, thicknesses of materials, and for different heater configurations and powers. It may be
possible to use a higher heater power to reach the target operating temperature
without causing skin damage.
3.6 Summary
The goal of the chapter was to find a SERF magnetometer thermal insulation scheme
that would allow the magnetometer to fit in a 1cm cube while meeting a temperature
requirement for device operation and another temperature requirement for safety.
FEA, a metamodel, and optimization were used to find a thermal insulation layout
that met both temperature requirements, but fit in a 1.1cm × 1.3cm × 1.3cm rectangular prism. Although the package size does not meet the original specifications,
it is too large by only a small amount. This suggests that practical design of a SERF
magnetometer for MEG applications is thermally feasible.
52
4 Chapter 4: Conclusion
4.1 Thesis Contributions
This thesis addressed two challenges associated with applying an array of SERF
magnetometers arranged on a conformal cap to MEG. It addressed the need for a
tool to assess the quality of a configuration of magnetic dipoles used in the calibration of magnetometers situated on a conformal cap. It also addressed the thermal
requirements of a SERF magnetometer held in close proximity to human skin.
Development of a Tool to Assess the Quality of Dipole Geometry for Conformal Cap Calibration:
In calibrating an array of SERF magnetometers attached to a conformal
cap using their measurements of a known set of magnetic dipoles, some magnetic
dipole configurations will allow for quick, accurate calibration, and others will not.
This thesis introduced a graphical tool (GREAT) for determining whether an arrangement of magnetic dipoles would allow the calibration algorithm to quickly find
an accurate solution to the magnetometer layout. This tool used a discrete gradient
of the design space as part of a process to generate plots. These plots contained
two valuable pieces of information: the number of local minima in the design space
and the rate of descent to these local minima. These two characteristics were shown
to be decent predictors of how well a magnetic dipole configuration allowed the
calibration estimation algorithm to quickly attain an accurate solution. The best
magnetic dipole configuration yet found was also provided.
Design of Compact Thermal Packaging: SERF magnetometers, when held in close
proximity to human skin, present a risk of burning the skin. The magnetometers, without thermal insulation, may require significant power to maintain their
required operating temperature. This thesis presents an insulation scheme that
both prevents thermal injury to skin and allows the magnetometers to maintain
their required operating temperature with a reasonable amount of power, and the
suggested insulation layout allows for small overall dimensions of the SERF mag-
53
netometer. This demonstrates that it is possible to meet temperature requirements
in a package size small enough to be one of approximately 300 magnetometers on a
conformal cap. In addition to determining an acceptable thermal insulation layout,
this chapter presented a procedure that could be used in the future to find a good
thermal insulation layout should aspects of the magnetometer, like materials and
thicknesses of components, need to be altered.
4.2 Impact
The work in this thesis pertains directly to the medical industry, but may impact
other industries as well. The work in Chapter 2 was carried out with the thought that
it would apply to an array of SERF magnetometers attached to a conformal cap for
the purpose of MEG; however it was defined broadly enough that it could be useful
in other applications. As a new technology, SERF magnetometers have a number of
yet unexplored applications, and any that involve arrays of magnetometers arranged
about a full or partially-ellipsoidal surface may be able to make use of the calibration
algorithm proposed in this thesis. The thermal analysis work presented in this thesis
is also very important when it comes to applying SERF magnetometers to MEG,
especially when the magnetometers remain in close proximity to human subjects
during the MEG and when there is a limited amount of space available for thermal
insulation, as is the case with the conformal cap concept. Most importantly, the
thermal analysis work is relevant and useful for any SERF magnetometer application
for which living subjects may come close to the magnetometers and for applications
of the magnetometers that involve operation in room-temperature environments, to
prevent skin damage and to ensure an adequate operating temperature, respectively.
54
A Appendix A: FEA Results
Tables 5 and 6 present the inputs to and the results obtained from the ABAQUS
finite element model.
Table 5: FEA Data
Heater
Power
(Watts)
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
Top
Thickness
(mm)
1.75
1.75
1.75
1.75
1.75
1.75
1.75
1.75
1.75
1.75
1.75
1.75
1.75
1.75
1.75
1.75
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.25
2.25
Side
Thickness
(mm)
1.75
1.75
1.75
1.75
2.00
2.00
2.00
2.00
2.25
2.25
2.25
2.25
2.50
2.50
2.50
2.50
1.75
1.75
1.75
1.75
2.00
2.00
2.00
2.00
2.25
2.25
2.25
2.25
2.50
2.50
2.50
2.50
1.75
1.75
Bottom
Thickness
(mm)
1.75
2.00
2.25
2.50
1.75
2.00
2.25
2.50
1.75
2.00
2.25
2.50
1.75
2.00
2.25
2.50
1.75
2.00
2.25
2.50
1.75
2.00
2.25
2.50
1.75
2.00
2.25
2.50
1.75
2.00
2.25
2.50
1.75
2.00
55
Average Cell Side
Temperature
(◦ C)
112.6993
113.3581
113.8577
114.2070
118.9971
119.8651
120.4415
120.9000
124.4162
125.4279
126.2463
126.7890
128.9188
130.1386
131.0482
131.7798
114.5930
115.2787
115.8265
116.1305
121.1654
122.1316
122.7371
123.2316
127.0004
128.0754
128.8732
129.4897
131.8004
133.0621
134.0408
134.7386
115.9529
116.7107
Top of Skin
Temperature
(◦ C)
35.7210
35.6799
35.6491
35.6117
35.8205
35.7803
35.7419
35.7141
35.8412
35.7980
35.7572
35.7324
35.9083
35.8648
35.8275
35.7994
35.7299
35.6879
35.6551
35.6182
35.8325
35.7902
35.7521
35.7171
35.8504
35.8068
35.7684
35.7376
35.9164
35.8762
35.8371
35.8068
35.7355
35.6930
Table 6: FEA Data (Continued)
Heater
Power
(Watts)
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
Top
Thickness
(mm)
2.25
2.25
2.25
2.25
2.25
2.25
2.25
2.25
2.25
2.25
2.25
2.25
2.25
2.25
2.50
2.50
2.50
2.50
2.50
2.50
2.50
2.50
2.50
2.50
2.50
2.50
2.50
2.50
2.50
2.50
2.75
3.00
3.25
Side
Thickness
(mm)
1.75
1.75
2.00
2.00
2.00
2.00
2.25
2.25
2.25
2.25
2.50
2.50
2.50
2.50
1.75
1.75
1.75
1.75
2.00
2.00
2.00
2.00
2.25
2.25
2.25
2.25
2.50
2.50
2.50
2.50
2.75
3.00
3.25
Bottom
Thickness
(mm)
2.25
2.50
1.75
2.00
2.25
2.50
1.75
2.00
2.25
2.50
1.75
2.00
2.25
2.50
1.75
2.00
2.25
2.50
1.75
2.00
2.25
2.50
1.75
2.00
2.25
2.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
56
Average Cell Side
Temperature
(◦ C)
117.1625
117.5610
122.8085
123.7257
124.4474
124.9647
128.8857
130.0570
130.9000
131.4835
133.8993
135.2081
136.2599
137.0864
117.0250
117.8265
118.2926
118.7077
124.1680
125.1790
125.8202
126.2938
130.4169
131.6397
132.4393
133.1029
135.6860
137.0871
138.1059
138.9184
146.4154
153.5346
159.9842
Top of Skin
Temperature
(◦ C)
35.6575
35.6243
35.8363
35.7949
35.7568
35.7188
35.8556
35.8120
35.7728
35.7448
35.9234
35.8814
35.8452
35.8135
35.7407
35.6967
35.6593
35.6262
35.8440
35.7974
35.7607
35.7231
35.8604
35.8168
35.7784
35.7464
35.9297
35.8891
35.8504
35.8172
35.8671
35.9040
35.9196
B Appendix B: Magnetic Field Produced by Current-Carrying Coil
The following are the equations used to calculate the magnetic field as a function of
the axial and radial distance from the current-carrying coil [42]. CR is the radius of
the coil, CN is number of turns in the coil, and CC is the current through the coil.
u0 is the permeability of free space.
In the global coordinate system, along the axis that is in line with the coil’s axis,
CZ is the distance the coil is from the origin, and z is the axial distance from the
origin from which the magnetic field is measured. In the global coordinate system,
r is the radial distance from which the magnetic field is measured.
d1 = (CR + r)2 + (z − CZ )2
(34)
d2 = (CR − r)2 + (z − CZ )2
(35)
K is equal to the complete elliptic integral of the first kind of the expression in
Equation 36. E is equal to the complete elliptic integral of the second kind of the
expression in Equation 36.
4rCR /d1
(36)
Bax is the axial component of the magnetic field and Br is the radial component
of the magnetic field.
Bax =
Br =
2 − r 2 − (z − C )2 )
E(CR
CN CC u0
Z
√
(K +
)
d2
2π d1
2 + r 2 + (z − C )2 )
E(CR
CN CC (z − CZ )u0
Z
√
(−K +
)
d
2πr d1
2
57
(37)
(38)
C Appendix C: Head Dimensions
Head dimension data depicted in Figure 32 and Table 7 come from Reference [32].
Figure 32: Head Dimensions
Table 7: Head Size Data Corresponding to Figure 32
Percentile
1
50
Women
99
1
50
Men
99
A (in.)
5.2
5.7
6.3
5.6
6.1
6.7
58
B (in.)
7.6
8.6
9.7
8.0
8.7
9.9
C (in.)
6.4
7.1
7.8
7.1
7.7
8.4
D Appendix D: Magnetometer and Dipole States Used for Figures
Created Using GREAT
Table 8: Magnetometer Locations for All Plots in Section 2.4.1
X (in)
-2.0020
-2.7762
-2.5382
-1.1793
1.1462
2.5264
2.7802
2.0239
0.0197
-2.0459
-2.4281
-1.2820
1.2609
2.4198
2.0730
0.0094
-1.5096
-1.0672
1.0308
1.5147
0.0082
Y (in)
2.9754
0.6569
-1.8195
-3.8484
-3.8711
-1.8566
0.6170
2.9414
4.2398
2.0832
-0.6687
-3.1821
-3.2013
-0.7338
2.0212
3.7239
0.6800
-1.7485
-1.7975
0.6534
2.3769
Z (in)
0.5917
0.5917
0.5917
0.5917
0.5917
0.5917
0.5917
0.5917
0.5917
1.7750
1.7750
1.7750
1.7750
1.7750
1.7750
1.7750
2.9583
2.9583
2.9583
2.9583
2.9583
Table 9: Figure 14 Dipole States
Label in Figure
0 deg
15 deg
30 deg
45 deg
x (in)
0.0000
0.0000
0.0000
0.0000
y (in)
6.3000
6.3000
6.3000
6.3000
z (in)
0.0000
0.0000
0.0000
0.0000
59
θx (deg)
0.0
15
30
45
θy (deg)
0.0
0.0
0.0
0.0
θz (deg)
0.0
0.0
0.0
0.0
Table 10: Figure 15 Dipole States
Label in Figure
0 deg
15 deg
30 deg
45 deg
x (in)
0.0000
0.0000
0.0000
0.0000
y (in)
6.3000
6.3000
6.3000
6.3000
z (in)
0.0000
0.0000
0.0000
0.0000
θx (deg)
0.0
0.0
0.0
0.0
θy (deg)
0.0
0.0
0.0
0.0
θz (deg)
0.0
15
30
45
θy (deg)
0.0
0.0
0.0
0.0
0.0
θz (deg)
0.0
0.0
0.0
0.0
0.0
Table 11: Figure 17 Dipole States
Label in Figure
Front of Head
Top of Head
Side of Head
45 deg
In Mouth
x (in)
0.0000
0.0000
6.3000
3.1500
0.0000
y (in)
6.3000
0.0000
0.0000
3.1500
0.0000
z (in)
0.0000
6.3000
0.0000
4.4548
0.0000
60
θx (deg)
0.0
0.0
0.0
0.0
0.0
Table 12: Figure 19 Dipole States
Label in Figure
One Coil
Two Coils
Two Coils
Three Coils
Three Coils
Three Coils
Four Coils
Four Coils
Four Coils
Four Coils
Five Coils
Five Coils
Five Coils
Five Coils
Five Coils
Six Coils
Six Coils
Six Coils
Six Coils
Six Coils
Six Coils
x (in)
0.0000
0.0000
0.0000
0.0000
-5.4560
5.4560
0.0000
0.0000
6.3000
-6.3000
5.9917
0.0000
-5.9917
-3.7030
3.7030
5.4560
0.0000
-5.4560
-5.4560
0.0000
5.4560
y (in)
6.3000
6.3000
-6.3000
6.3000
-3.1500
-3.1500
6.3000
-6.3000
0.0000
0.0000
1.9468
6.3000
1.9468
-5.0968
-5.0968
3.1500
6.3000
3.1500
-3.1500
-6.3000
-3.1500
z (in)
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
θx (deg)
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
θy (deg)
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
Table 13: Figure 21 Dipole States
x (in)
-5.9917
-3.7030
3.7030
5.9917
0.0000
-3.7477
3.7477
3.7477
-3.7477
-3.7239
3.7239
0.0000
y (in)
1.9468
-5.0968
-5.0968
1.9468
6.3000
-3.7477
-3.7477
3.7477
3.7477
-2.1500
-2.1500
4.3000
z (in)
0.5000
0.5000
0.5000
0.5000
0.5000
2.0000
2.0000
2.0000
2.0000
3.5000
3.5000
3.5000
θx (deg)
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
61
θy (deg)
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
θz (deg)
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
θz (deg)
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
E Appendix E: Additional Finite Element Model Details
The finite element model was constructed in ABAQUS. It performed a heat transfer
steady-state analysis, considering only one mode of heat transfer: conduction. Mesh
size for each part was set to be approximately 1/15th of the part’s thinnest dimension. Using scripting, the following command was used to set the mesh controls:
setMeshControls( elemShape=TET,
regions= acetalCase.cells.getSequenceFromMask((’[#1 ]’,),),
technique=FREE)
Using scripting, the following command was used to set the mesh element type:
setElementType( elemTypes=( ElemType(elemCode=DC3D8,
elemLibrary=STANDARD),
ElemType(elemCode=DC3D6, elemLibrary=STANDARD),
ElemType(elemCode=DC3D4, elemLibrary=STANDARD)),
regions=(acetalCase.cells.getSequenceFromMask((’[#1 ]’,),),))
62
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