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Chapter 37: Partial Foot Amputations and Disarticulations: Surgical Management
in diabetic patients. Foot Ankle Clin
2010;15(3):501-507. Medline DOI
29. Bowker J: Partial foot amputations
and disarticulations: Surgical aspects.
J Prosthet Orthot 2007;19(3s):62-76.
DOI
30. Krause FG, Pfander G, Henning
J, Shaghi M, Weber M: Ankle
dorsiexion arthrodesis to salvage Chopart’s amputation with
anterior skin insuciency. Foot
Ankle Int 2013;34(11):1560-1568.
Medline DOI
31. Yonclas P, O’Donnell CJ: Prosthetic
management of the partial foot
amputee. Clin Podiatr Med Surg
2005;22(3):485-502. Medline DOI
32. Tosun B, Buluc L, Gok U, Unal C:
Boyd amputation in adults. Foot
Ankle Int 2011;32(11):1063-1068.
Medline DOI
33. Ng V, Berlet G: Amputations of the
foot and ankle, in Parekh S, ed: Foot
and Ankle Surgery. New Delhi, India,
JP Medical, 2012, pp 377-388. DOI
34. Scaduto A, Bernstein R: Syme
and Boyd amputations for bular
deciency, in Weisel S, ed: Operative
Techniques in Orthopaedic Surgery. Philadelphia, PA, Lippincott
Williams & Wilkins, 2011, pp
1295-1303.
35. Pinzur MS, Gold J, Schwartz D, Gross
N: Energy demands for walking in
dysvascular amputees as related to
the level of amputation. Orthopedics
1992;15(9):1033-1036, discussion
1036 -1037. Medline
36. Roukis TS, Singh N, Andersen CA:
Preserving functional capacity as
opposed to tissue preservation in the
diabetic patient: A single institution experience. Foot Ankle Spec
2010;3(4):177-183. Medline DOI
© 2016 American Academy of Orthopaedic Surgeons Atlas of Amputations and Limb Deciencies, Fourth Edition
461
Chapter 38
Prosthetic Management Aer Partial Foot Amputation
Michael P. Dillon, PhD Stefania Fatone, BPO(Hons), PhD
Abstract
Partial foot amputation is becoming more common despite high rates of skin breakdown
and the need for subsequent amputation surgery. A wide variety of prosthetic, orthotic,
and footwear interventions are available to minimize complications and restore premorbid mobility and quality of life. As more knowledge is gained on the eectiveness of these
interventions, many long-held beliefs are becoming less certain. Further research is needed
to determine which interventions are the most eective for a particular patient population.
Keywords: balance; energy expenditure; gait; mobility; partial foot
amputation; plantar pressure; prosthesis; quality of life
Introduction
The understanding of partial foot amputation (PFA) and the effects of prosthetic
and orthotic intervention have dramatically changed during the past decade.
Recent research suggests that the incidence of PFA increased linearly from
2000 to 2010 and that, if current trends
continue, the incidence of PFA will triple
during the first half of the 21st century.1
Unfortunately, only a few studies have
reported detailed incidence data on PFA,
and further work is necessary to build
confidence regarding current trends
and their generalizability to a global
population.
that the incidence of transfemoral and
transtibial amputation declined during
the same time period.
incidence of PFA appears to be proportional to the decline in transtibial and
transfemoral amputations. Although the
types of amputation procedures have
Dr. Dillon has received research or institutional support from Össur and serves as a board member,
owner, ocer, or committee member of the Australian National Member Society of the International
Society for Prosthetics and Orthotics, the Australian Orthotics and Prosthetics Association, and the
American Academy of Orthoti sts and Prosthetists. Dr. Fatone ha s received research or institutional
support from Myomo and Ultraex Systems and serves as a board member, owner, ocer, or committee member of the American Academy of Orthotists and Prosthetists, the Australian National
Member Society of the International Society for Prosthetics and Orthotics, the Orthotics Prosthetics
Research Foundation, and the Archives of Physical Medicine and Rehabilitation.
2-5
There is good agreement
1,2,5 -9
The increased
changed, the incidence of lower limb
amputation has remained steady.1 Public
health initiatives do not appear to have
led to a decreased incidence of lower
limb amputation, but improvements
in revascularization surgery, earlier assessment at specialized high-risk foot
clinics, and better management of diabetes at the community level may have
decreased the severity of vascular disease, making PFA a feasible alternative
to more proximal amputation.
1,10 -14
Approximately 75% of all PFAs af-
fect the toes.
1,2, 9,15
In comparison, there
are few ray resections, transmetatarsal,
tarsometatarsal (Lisfranc), or transtarsal
(Chopart) amputations (Figure 1). The
proportion of PFAs affecting only the
toes may be surprising to prosthetists
and orthotists, who usually treat individuals with a more proximal amputation.
Most individuals with an amputation
affecting only the toes probably receive
follow-up care through a high-risk foot
clinic rather than a prosthetic-orthotic
center.
Approximately 30% to 50% of patients with a PFA have a complication
such as dehiscence, ulceration, or failure
of the wound to heal.
16-20
It is difficult
to determine the amputation level that
is most likely to allow optimal healing,
particularly in patients with serious
vascular compromise or a complex comorbidity such as end-stage renal disease, hypertension, or diabetes.
21,22
The
longer-term challenges are to manage
progressive equinovarus contractures
and moderate forefoot plantar pressures
that are high compared with the contralateral limb or appropriately matched
control subjects.
23-29
As a result of such
complications, approximately one-third
of patients with a PFA require a secondary amputation on the same limb,
regardless of the level of the initial
17,22 ,30 -32
PFA.
Rates of complications are
not markedly different in those with diabetes, which suggests that the presence
of advanced systemic disease is not the
only influential factor.
17,21,22
A wide variety of prostheses and orthoses is available to mitigate the risk
of complications and restore premorbid
function.
33-35
The types of devices have
changed little over time and include
toe fillers, insoles, silicone cosmetic
prostheses, ankle-foot orthoses, and
above-ankle prostheses.
Recent research has challenged longheld views about the effectiveness of
current prostheses and orthoses.36 For
example, a device can be used to restore
the effective foot length, but it is unclear
whether normalizing gait is important
to reestablish the premorbid level of
© 2016 American Academy of Orthopaedic Surgeons Atlas of Amputations and Limb Deciencies, Fourth Edition
463
Section 3: Lower Limb
Figure 1
tion of the third, fourth, and f th metatarsals and toes (the rays). C, Disarticulation of all ve toes at the metatarsophalangeal joint. D, Transmetatarsal
amputation. E, Tarsometatarsal (Lisfranc) amputation. F, Midtarsal (Chopart) amputation.
mobility.
able to determine the interventions most
likely to reduce rates of complications
and reamputation or to improve quality
of life for individuals living with a PFA.
Emerging research is guiding efforts to
understand how prosthetic and orth otic
interventions can benefit individuals
with a PFA.
Schematic draw ings showing partial foot a mputations at the most common l evels. A, Amputation of the fourth and fth toes . B, Resec-
37, 38
Little information is avail-
example, an anterior-shell carbon-fiber
ankle-foot orthosis can be used with an
insole and toe filler to restore effective
foot length, normalize gait, and distribute pressure away from the distal end of
the residuum to provide protection and
improve comfort during standing and
walking (Figure 2).
In general, an individual with a rel-
atively proximal amputation requires a
Prosthetic, Orthotic, and
Footwear Interventions
A wide variety of custom prosthetic, orthotic, and footwear interventions can
be used to treat individuals with PFA.
35,39
These can be categorized as belowankle or above-ankle interventions.
Common below-ankle interventions
include toe fillers, insoles, and silicone
cosmetic prostheses; above-ankle interventions include ankle-foot orthoses
and above-ankle prostheses. Their use
depends on the amputation level and the
treatment objectives. Interventions are
chosen to fulfill specific treatment goals,
including making standing and walking
more comfortable, restoring premorbid
mobility, cosmetic restoration, minimizing interface pressures on the distal end
of the residuum, preventing equinovarus contracture, or reducing the risk of
ulceration and skin breakdown.
Devices can be used in combination to
achieve multiple treatment goals. For
33,34,39
correspondingly substantial device. An
individual with a metatarsophalangeal
disarticulation might be provided with
an insole and/or toe filler (Figure 3), but
33-
someone with a Chopart amputation
might be provided with an above-ankle
24,34
prosthesis that encloses the residuum
and leg in a solid laminated shell (Fig-
ure 4). An individual with a midfoot
amputation (a transmetatarsal amputation or ray resection) may receive a
variety of interventions, ranging from
an insole to a silicone cosmetic prosthesis or ankle-foot orthosis. The types
of interventions provided to individuals
with a PFA vary by country and health
jurisdiction depending on the stipulations of funding bodies, the professional
disciplines involved in treatment, and
the expertise and experience of the
treating clinicians. In Australia, the use
of pedorthotics (custom or customized
footwear) is less common than in parts
of the United States, Japan, and many
Figure 2
BlueRocker (Allard) ankle-foot orthosis combined with an ethylene vinyl acetate insole.
Photograph of a ToeOFF
European countries, where the formal
training and expertise of pedorthotists
are comparatively well recognized in
healthcare practice.
Toe Fillers and Insoles
Toe fillers and insoles are commonly
used for those with a distal forefoot
amputation such as a toe amputation,
metatarsophalangeal disarticulation, or
ray resection (Figure 3). An individual
with a transmetatarsal amputation also
may be provided an insole. Toe fillers
are used alone or attached to an insole to
Atlas of Amputations and Limb Deciencies, Fourth Edition © 2016 American Academy of Orthopaedic Surgeons
464
Chapter 38: Prosthetic Management After Partial Foot Amputation
Figure 3
ller. An ethylene vinyl acetate insole supports
the hindfoot and arch. An open-cell polyurethane foam protects the sensitive distal inferior
end of the residuum. The toe ller material is
an expanded low-density closed-cell polyethylene foam.
Photograph of an insole and toe
fill the cavity in normal-length footwear.
Insoles serve as a bed for the residuum
and can be designed to maintain alignment of the residuum and redistribute
pressure away from the sensitive distal end to minimize the likelihood of
skin breakdown. Toe fillers and insoles
are designed to be worn with footwear.
Because toe fillers and insoles do not
encompass the residuum, they rely on
the shoe to maintain their position with
respect to the residuum. Extra-depth
shoes or low-top boots often are used to
accommodate the orthosis and provide
adequate suspension.
Toe fillers and insoles often are made
from a closed-cell foam, which resists
compression and thinning. Materials
with different mechanical properties
can be incorporated, depending on the
defined treatment goals. For example,
closed-cell polyethylene foam might be
used under the heel and arch to correct foot alignment, with a low-density
material used to protect the distal end
of the residuum against shear forces.
Some toe fillers incorporate vertical
cuts through the dorsum (as used in the
Shape & Roll prosthetic foot developed
by Northwestern University)
37,4 0
for the
purpose of reducing the stiffness of the
filler and facilitating forefoot bending
Figure 4
above-ankle prosthesis (a clamshell prosthesis)
for a Chopar t amputation residuum. Th e socket
material is a laminated glass-ber composite. A
prosthetic foot has been bonded to the socket
to replace the lost forefoot. The bivalve socket
allows donning , given the bulbous distal end o f
the residual limb.
Photograph of a bivalved,
during walking. Other devices take the
opposite approach and incorporate a
carbon-fiber footplate to help prevent
buckling of the orthosis across the distal
end of the residuum when loaded. Research has not established the efficacy
of either type of orthosis.
Silicone Cosmetic Prostheses
A silicone cosmetic prosthesis can
provide a cosmetically acceptable restoration of a partially amputated foot,
usually when the amputation is at or
distal to the midfoot. It is possible to
match the skin color, shape, and alignment of the toes and nails to those of
the contralateral limb (Figure 5). The
design of the prosthesis depends on
characteristics of the residuum. For
an individual with a transmetatarsal
amputation, the prosthesis might take
the form of a slipper socket that encompasses the entire residuum. A foot
with resection of the medial two rays
requires a distal opening to allow the
lateral toes to protrude through the end
of the prosthesis.
Figure 5
cosmetic prostheses. A, A p rosthesis for a transmetatarsal residuum, with matching of skin
color, hair, and nails. B, A prosthesis for a rst
and second medial ray resection in which the
third, fourth, and fth toes protrude through
the opening at the distal end of the prosthesis.
Photographs of two silicone
Silicone prostheses are made to intimately fit the residuum; they require
the use of a water-based lubricant and
a shoehorn for donning. The intimacy
of fit provides suction suspension that
allows the prosthesis to be worn with
open sandals. Although a silicone prosthesis can be worn without shoes, the
usual recommendation is to wear shoes,
especially when outdoors, to prolong the
life of the prosthesis.
Ankle-Foot Orthoses
An ankle-foot orthosis encompasses all
or a part of the foot and extends proximal to the ankle. An ankle-foot orthosis
is often provided after a transmetatarsal
or tarsometatarsal (Lisfranc) amputation (Figures 2 and 6). Variations in
materials, footplate lengths, trim lines,
and articulation at the ankle mean that
ankle-foot orthoses with a similar appearance can serve quite different mechanical functions and treatment goals.
An ankle-foot orthosis is often used in
conjunction with other interventions to
© 2016 American Academy of Orthopaedic Surgeons Atlas of Amputations and Limb Deciencies, Fourth Edition
465
Section 3: Lower Limb
Figure 6
carbon-ber ankle-foot orthosis with an anterior leg shell, reinforced bilateral uprights, and
a full-length footplate.
Photograph of a custom-made
achieve complex treatment goals. For
example, an insole that distributes pressure away from the sensitive distal end
of the residuum can be used in combination with an anterior-shell ankle-foot
orthosis to control the progression of
the leg over the stance foot (Figure 2).
Above-Ankle Prostheses
An above-ankle prosthesis is usually
provided after a proximal forefoot amputation such as a Chopart amputation.
The device typically encloses the residuum and leg segments in a rigid socket
to eliminate ankle motion (Figure 4).
The narrow dimension of the leg just
proximal to the ankle and the relatively
bulbous distal residuum often necessitate a bivalved socket to facilitate donning. Other design variations include a
medial opening window or a built-up
liner. The shape of the residuum and
leg make it relatively simple to achieve
self-suspension of the prosthesis.
To replace the missing forefoot, a
conventional prosthetic forefoot or
carbon-fiber footplate usually is bonded
to the socket. These carbon-fiber footplates are specifically designed for this
purpose; that is, they are stiff enough to
support body weight and can be bonded
to the socket without failing under the
large external moments applied during
the late stance phase of walking. The
socket can be designed to distribute a
portion of body weight away from the
residuum and onto the leg, as is often
necessary because the residuum has an
inadequate surface area for comfortable
distribution of interface pressures.
Individuals with a PFA often do not
like wearing an above-ankle prosthesis.
It is difficult to create a slim, cosmetically
acceptable prosthesis that can easily fit
into conventional footwear.41 A socket
design that eliminates ankle motion may
seem counterproductive to retention of
the anatomic ankle joint. Prosthetic design variations that allow ankle motion,
such as separate foot and leg shells with
external ankle joints, allow the range
of motion of the anatomic ankle to be
preserved.
37
Effectiveness Research
Increasing interest in the effectiveness of
interventions for individuals with a PFA
has led to research into the benefits and
limitations of the available devices. The
research is best characterized as emerging. Most of the studies are small in scale
and simply describe what is observed
when patients use their own prosthesis
or orthosis. The observational studies
are an important step toward experimental studies designed to compare the
effectiveness of different interventions.42
The current evidence on interventions
for individuals with a PFA is related to
gait, balance, energy expenditure, plantar pressure, community mobility, and
quality of life.
Gait
Most research into function after PFA
has focused on laboratory-based measures of gait. Regardless of the cause of
amputation, individuals with a PFA have
several well-characterized gait anomalies, including a reduced center of pressure excursion beneath the residuum,
reduced ankle power on the amputated
side, and increased power generation
at either or both hips.
37, 38,4 1-5 3
Individuals with a PFA secondary to diabetes
or peripheral vascular disease walk at
approximately two-thirds the speed of
their healthy counterparts.
42,48-52
This
reduction in walking speed is not observed in individuals who had a traumatic PFA.
38,4 4,53
No marked differences
in walking speed have been observed
among individuals with a PFA at different levels or with different prosthetic
and orthotic interventions, but these
studies were not specifically designed
to answer such questions, and further
research is needed.
42,44,5 4,55
Retention of the metatarsal heads
appears to be essential to the use of
the ankle joint and calf musculature
to generate ankle power during late
44-47
stance.
If the metatarsal heads are
compromised, power generation across
the ankle is negligible during gait, regardless of the length of the residuum
or the type of prosthetic or orthotic intervention.
44 ,47,55- 57
Lack of power generation may serve as a useful adaptation
to avoid pressure on the distal end of
the residuum or to reduce shear forces caused by ankle plantar flexor contraction.44 Increased power generation
at either or both hips compensates for
the lack of ankle power generation and
contributes to a gait pattern similar to
that of individuals with a transtibial
amputation.
36,42 ,58
Observational studies suggest that
below-ankle devices, such as a toe filler,
an insole, or a slipper socket, do not nor
malize the center of pressure excursion;
instead, the center of pressure remains
proximal to the end of the residuum
until weight is shifted to the unaffected
limb at contralateral heel contact.
38,4 4
Devices that extend above the ankle,
such as an above-ankle prosthesis or an
ankle-foot orthosis designed to restrict
dorsiflexion, can normalize the center of
pressure excursion.
37, 38,4 4
The ability of
a device to restore effective foot length
-
Atlas of Amputations and Limb Deciencies, Fourth Edition © 2016 American Academy of Orthopaedic Surgeons
466
Chapter 38: Prosthetic Management After Partial Foot Amputation
is believed to require three design features: a suitably stiff forefoot capable of
supporting body mass, a socket or an
anterior leg shell capable of comfortably
distributing to the leg and remaining
foot the interface pressures caused by
loading the toe lever, and a relatively
stiff connection between the foot and
leg segment to moderate the moments
caused by loading the toe lever.43 A rigid
ankle, a free joint with a dorsiflexion
stop, or the type of stiffness inherent in a
ToeOFF BlueRocker (Allard) ankle-foot
orthosis can be used to normalize the
center of pressure excursion.
38,41,42
Balance
Few studies have analyzed balance in
individuals with a PFA.
41,5 9
Standing
balance was reported to be more compromised in individuals with a PFA
than in those with diabetic neuropathy
alone. Balance was compromised to the
same extent in individuals with a PFA,
transtibial amputation, and diabetic foot
ulceration.59 This finding was based
on an assessment of anterior-posterior
center of pressure excursion and was
attributed to loss of ankle control. No
changes were observed in mediolateral
center of pressure excursion, which is
primarily controlled by the hip. Because
the risk of falling increases with diabetic neuropathy, and more so with PFA,
balance training was recommended for
individuals with a PFA.59 Only one study
has reported on the effect of prosthetic
and orthotic interventions on balance;
no difference was found in dynamic balance during walking with a below-ankle
or an above-ankle device.
41
Energy Expenditure
Classic research found a reduction in
oxygen cost with progressively distal levels of amputation.
60,61
It often is
assumed that walking with a PFA re
quires less energy expenditure than
walking with a lower limb amputation
performed at a higher level.
17,32,62- 64
The
available evidence does not support this
assumption and instead suggests that
energy expenditure is similar in individ
uals with a transtibial amputation or a
48,5 4,65
PFA.
This assertion is supported by
the observation that, unlike those with
a hip disarticulation or a transfemoral
amputation, individuals with a transtibial amputation or a PFA have a similar
gait pattern after the metatarsal heads
are compromised.36 Net oxygen cost is
likely to be similar in individuals with a
transtibial amputation or a PFA if, as is
the case in others with a lower limb amputation, individuals with a PFA modify
their walking speed to keep the rate of
oxygen uptake within normal limits.
Plantar Pressure Distribution
There is some indirect evidence that
devices can redistribute pressure away
from the distal end of the residuum to
other parts of the foot or leg.
38,42
work suggested that total-contact insoles
can reduce plantar pressure in individuals with diabetes who have a first ray
amputation.66 However, no evidence exists to recommend any particular pressure reduction technique for individuals
with a PFA.
41,42
This limitation is important because of the high rate of complications in individuals with a PFA.36
No studies have determined whether
prosthetic and orthotic interventions are
effective in minimizing rates of complications, surgical revision, or more
proximal amputation. Factors such as
systemic health, vascular supply, and
diabetes control may be more important
for minimizing the risk of complications
than the choice of de vices. In the future,
it may be possible to design devices that
can mitigate the risk of complications.
Community Mobility
The extent to which normalizing gait,
balance, energy expenditure, and plan-
-
tar pressure are important to individuals
with a PFA remains unclear. Achieving
independent community mobility may
be more important. Walking speed often is used as a marker of functional
36
Pilot
mobility, although it appears to be com-
-
parable in individuals with different levels of PFA or transtibial amputation.36 A
study of patients undergoing lower limb
amputation secondary to peripheral artery disease or diabetes found that while
ambulation improved after surgery, it
did not return to premorbid levels after
12 months.
67, 68
This finding was similar
in individuals with a PFA or transtibial
amputation.
Quality of Life
Because few studies have formally
evaluated quality of life, insights into
the experience of living with a PFA are
limited.
36,69
Many individuals believe
that quality of life is improved by PFA
compared with transtibial amputation
because it is easier and safer to ambulate
short distances without using a prosthesis (as when going to the toilet at
18,20,65
night).
It is not clear whether living
without pain or being able to participate in recreational activities has a more
profound influence on quality of life
than being able to walk short distances
without a prosthesis.69 The available
descriptive data suggest that quality of
life is comparable in individuals with a
PFA or a transtibial amputation, and this
finding has been corroborated by a comparison of quality of life in those with
vascular disease and a PFA or transtibial
amputation.
69-7 2
This research suggested
that quality of life is substantially influenced by age, the number of years
living with diabetes, and the presence
of complications such as retinopathy,
but that amputation level does not affect
quality of life.
69
Clinical Implications
Many long-held beliefs are being questioned with increasing knowledge of
PFA and the influence of prosthetic
and orthotic interventions. For example,
PFA has long been preferred to a more
proximal amputation because it was believed that outcomes were better.
The emerging evidence suggests that
20,22,73
© 2016 American Academy of Orthopaedic Surgeons Atlas of Amputations and Limb Deciencies, Fourth Edition
467
Section 3: Lower Limb
individuals with a PFA or transtibial
amputation have a similar gait pattern,
energy expenditure, mobility, and quality of life, but that PFA leads to much
higher rates of serious complications,
surgical revision, and more proximal
secondary amputation.
36,58
Innovative prosthetic and orthotic
interventions are needed to improve
device effectiveness and address issues
important to individuals with a PFA.
This is exemplified in a case study of
PFA after traumatic injury that suggests that the use of a nonarticulated
above-ankle prosthesis with vacuum-assisted suspension is a viable means of
improving comfort, function, and re
sidual limb health.
74
Despite increasing understanding of
how aspects of gait can be affected by
interventions, it is not known whether normalizing gait is important for
community mobility, participation in
recreational or vocational pursuits, or
restoration of premorbid quality of life.
Similarly, the ability of prosthetic and
orthotic interventions to reduce the
high rates of complications and reamputation is not known. Future research
efforts should focus on defining the
most important outcomes and the extent to which prosthetic and orthotic
interventions are effective in achieving
those outcomes. For example, if minimizing the risk of complications and
secondary amputation is found to be
of primary importance to individuals
living with a PFA, it will be important
to determine which devices are effective
at accomplishing these treatment goals.
Minimizing the high rates of complications and secondary amputation
can be considered particularly important from personal health and economic
perspectives. Individuals with a PFA describe a more persistent, pervasive fear
of further amputation than those living
with more proximal levels of limb loss.75
This experience is believed to contribute
to the depression and anxiety reported
by individuals with a PFA.75 From an
economic perspective, the burden of
PFA is staggering. Approximately 50%
of all PFAs do not heal, and efforts to
achieve wound healing after a PFA occur
over many months, with costs in the
United States from $27,000 to $36,000
per person.
17,76,77
In the United States,
the total annual cost of wound care after
PFA is estimated to exceed $600 million,
based on an incidence of 20 per 100,000
individuals and a modest complication
rate consistent with the proportion of
individuals requiring secondary amputation.
31,78
If the cost of secondary
amputation surgery is included in the
estimate, the total annual cost exceeds
-
$1 billion.
79
Summary
As more is learned about PFA and the
effects of current prosthetic and or thotic
interventions, gaps in knowledge are being identified and used to guide additional research. A better understanding
of the needs of individuals living with
a PFA will lead to innovations and research to ensure that interventions effectively meet those needs.
Acknowledgments
The authors thank APC Prosthetics,
Sydney, Australia, for providing the silicone cosmetic prostheses and the Prosthetics and Orthotics Department, Royal
Melbourne Hospital, Melbourne, Australia, for providing the other devices
appearing in this chapter. We also thank
Matthew Quigley and Christopher Robinson for their thoughtful reviews and
comments on the draft manuscript.
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