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Generalized n-dimensional biomechanical field analysis using statistical parametric mapping

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A variety of biomechanical data are sampled from smooth n-dimensional spatiotemporal fields. These data are usually analyzed discretely, by extracting summary metrics from particular points or regions in the continuum. It has been shown that, in certain situations, such schemes can compromise the spatiotemporal integrity of the original fields. An alternative methodology called statistical parametric mapping (SPM), designed specifically for continuous field analysis, constructs statistical images that lie in the original, biomechanically meaningful sampling space. The current paper demonstrates how SPM can be used to analyze both experimental and simulated biomechanical field data of arbitrary spatiotemporal dimensionality. Firstly, 0-, 1-, 2-, and 3-dimensional spatiotemporal datasets derived from a pedobarographic experiment were analyzed using a common linear model to emphasize that SPM procedures are (practically) identical irrespective of the data's physical dimensionality. Secondly two probabilistic finite element simulation studies were conducted, examining heel pad stress and femoral strain fields, respectively, to demonstrate how SPM can be used to probe the significance of field-wide simulation results in the presence of uncontrollable or induced modeling uncertainty. Results were biomechanically intuitive and suggest that SPM may be suitable for a wide variety of mechanical field applications. SPM's main theoretical advantage is that it avoids problems associated with a priori assumptions regarding the spatiotemporal foci of field signals. SPM's main practical advantage is that a unified framework, encapsulated by a single linear equation, affords comprehensive statistical analyses of smooth scalar fields in arbitrarily bounded n-dimensional spaces.
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Generalized n-dimensional biomechanical field analysis using
statistical parametric mapping
Todd C. Pata ky
Department of Bioengineering, Shinshu University, Japan
Abstract
A variety of biomechanical data are sampled from smooth n-dimensional spatiotemporal fields.
These data are usually analyzed discretely, by extracting summary metrics from particular
points or regions in the continuum. It has been shown that, in certain situations, such schemes
can compromise the spatiotemporal integrity of the original fields. An alternative methodology
called Statistical Parametric Mapping (SPM), designed specifically for continuous field analy-
sis, constructs statistical images that lie in the original, biomechanically meaningful sampling
space. The current paper demonstrates how SPM can be used to analyze both experimental
and simulated biomechanical field data of arbitrary spatiotemporal dimensionality. Firstly, 0-,
1-, 2-, and 3-dimensional spatiotemporal datasets derived from a pedobarographic experiment
were analyzed using a common linear model to emphasize that SPM procedures are (practically)
identical irrespective of the data’s physical dimensionality. Secondly two probabilistic finite el-
ement simulation studies were conducted, examining heel pad stress and femoral strain fields,
respectively, to demonstrate how SPM can be used to probe the significance of field-wide sim-
ulation results in the presence of uncontrollable or induced modeling uncertainty. Results were
biomechanically intuitive and suggest that SPM may be suitable for a wide variety of mechan-
ical field applications. SPM’s main theoretical advantage is that it avoids problems associated
with aprioriassumptions regarding the spatiotemporal foci of field signals. SPM’s main prac-
tical advantage is that a unified framework, encapsulated by a single linear equation, aords
comprehensive statistical analyses of smooth scalar fields in arbitrarily bounded n-dimensional
spaces.
1
1 Introduction
Many classes of biomechanical data share two mathematically non-trivial characteristics: (i)
spatiotemporal smoothness within (ii) regular discrete bounds. This is true of 1D trajectories like
vertical ground reaction forces (VGRF), and joint kinematics, 2D continua like contact pressure
and surface thermal distributions, and 3D continua like bone strain and cardiac flow velocity. It is
also true of simulated mechanical continua. All may be regarded as smooth scalar (or vector) fields
bounded by anatomy, temporal events, or both.
These data are smooth not only because they are sampled above the Nyquist frequency, but
ultimately because biological tissue viscoelasticity (Fung, 1981) causes biomechanical processes to
be spatiotemporally smooth by nature. Smoothness is statistically non-trivial because it implies
local data correlation and therefore that the number of independent processes is less, perhaps
far less than the number of sampled points. Regular spatiotemporal bounds are also non-trivial
because they imply registrability (Maintz and Viergever, 1998) and thus that a direct, continuous
comparison of multiple field observations may be possible.
Since experimental and probabilistically simulated nD fields can yield a large volume of data,
they are typically reduced through regionalization, by extracting multiple local VGRF optima
(e.g. Nilsson and Thorstensson, 1989) or by discretizing modeled anatomy (e.g. Radclie and Taylor,
2007), for example. Such procedures permit statistical testing but also unfortunately create an
abstraction: to understand tabular VGRF values, for example, one must mentally project these
data back to their reported regions in the original sampling space. Discretization can occasionally
also have statistical consequences, missing (Appendix D) or even reversing trends (Pataky et al.,
2008).
A methodology called Statistical Parametric Mapping (SPM) (Friston et al., 2007) can partially
oset these limitations by providing a framework for the continuous statistical analysis of smooth
bounded nD fields. It was originally developed for the analysis of cerebral blood flow in 3D PET
and fMRI images (Friston et al., 1991; Worsley et al., 1992) but it has since migrated to a variety of
diverse applications (Worsley, 1995; Chauvin et al., 2005) including a biomechanical one (2D pedo-
barography) (Pataky and Goulermas, 2008). SPM’s suitability for a wider range of biomechanical
2
applications has not yet been investigated.
The main goals of this study were to: (1) Review the mathematical foundations of nD SPM,
(2) Demonstrate how SPM can be utilized for the analysis of 0-, 1-, 2-, and 3D experimental data,
and (3) Demonstrate how SPM can be used in probabilistic simulations of biomechanical continua.
While each demonstration is, independently, narrowly focussed it is hoped that they collectively
reveal a broader utility.
2 Statistical Parametric Mapping (SPM)
2.1 General linear model (GLM)
The relation between experimental observations Yand an experimental design Xcan be sum-
marized using a mass-univariate GLM (Friston et al., 1995):
Y=X+"(1)
where is a set of regressors (to be computed), and "is a matrix of residuals. Yand "are (IK),
Xis appropriately scaled and (IJ), and is (JK), where I,J, and Kare the numbers of
observations, experimental factors, and nodes, respectively. The term ‘node’ is used here to refer to
the number of discrete measurement points, and an experimental observation is an n-dimensional
sampling of a scalar field that may be flattened into a K-vector. A full experiment yields Iflattened
K-vectors. Least-squares estimates of can be obtained by:
ˆ
=X+Y=(XTX)1XTY(2)
yielding errors:
"=YXˆ
(3)
where X+is the Moore-Penrose pseudo-inverse of X. Like the original dataset Y,themodelfits
Xˆ
are (IK). In general a large proportion of variability can be explained using this approach
3
(see Sect.4.1).
Having estimated parameters ˆ
and residuals ", the next task is to compute test statistic values.
The GLM (Eqn.1) aords arbitrary linear testing including: ANOVA, ANCOVA, etc., (Friston et
al., 1995), but for brevity only the generalized ttest will be considered presently. First nodal
variance 2
kis estimated as:
ˆ2
k=("T")kk
=("T")kk
Irank(X)(4)
where ("T")kk is the kth diagonal element of the (KK) error sum of squares matrix "T"and
where is the error degrees of freedom. The nodal tstatistic can then be computed as:
tk=cTˆ
k
ˆkpcT(XTX)1c(5)
where cis a (J1) contrast vector. The nodal values tkform a K-vector that can be reshaped
into the original nD sampling space and viewed in the context of the original data. For this reason
(5) is known as a statistical ‘map’ and is referred to in the literature as ‘SPM{t}’ (Friston et al.,
2007), a notation that shall be adopted henceforth. The contrast vector cassigns weights to the J
experimental factors and thus represents the experimental hypothesis (see Sect.3.1).
2.2 Statistical inference
SPM uses random field theory (RFT) (Adler, 1981) to assess the field-wide significance of an
SPM{t}. A technical summary of RFT procedures is provided in Appendix A. Briefly, for n>0,
RFT is charged with solving the problem of multiple comparisons. That is, one could expect to
observe higher tkvalues, simply by chance, when conducting multiple statistical tests. A Bonferroni
correction for Kmultiple comparisons is valid, but is overly-conservative (in general) because
spatiotemporal correlation (local smoothness) eectively ensures that fewer than Kindependent
processes exist. RFT takes advantage of this fact to conduct inference topologically, based on the
height and size of connected clusters that remain following suitably high SPM{t}thresholding (e.g.
t>3.0). Precise probability computations additionally depend on field smoothness (Appendix B)
4
and search space morphology (Appendix C). A key point is that a large suprathreshold cluster
is the topological equivalent of a large univariate tvalue. For clarity, a numerical 1D example is
provided in Appendix D.
3 Methods
3.1 Experimental dataset
A single subject (male, 30 years, 172 cm, 73 kg) from a previous study (Pataky et al., 2008), veri-
fied in post-hoc analysis to be representative of the mean subject’s statistical trends, was re-analyzed
here. Since population inference was not a goal a single subject was considered appropriate. The
subject performed twenty randomized repetitions of each of ‘slow’, ‘normal’, and ‘fast’ walking.
VGRF and pedobarographic data were sampled at 500 Hz (Kistler 8281B, Winterthur, Switzer-
land; RSscan Footscan 3D, Olen, Belgium). Walking speed was measured at 100 Hz (ProReflex,
Qualisys, Gothenburg, Sweden) and was treated as a continuous variable for statistical purposes.
Prior to participation the subject gave informed consent according to the policies of the Research
Ethics Committee of the University of Liverpool.
Maximal VGRF (0D), VGRF time series (1D), and peak (i.e. spatially maximal) pressure im-
ages (2D) were extracted from the original spatiotemporal (3D) pedobarographic dataset. Here
the 3D data are 2D time series (and the 1D data are 0D time series), but the data were treated as
mathematically 3D for volumetric smoothness, clustering, and topological probability computations
(Appendix A). The 1D data were registered via linear interpolation between heel-strike and toe-o.
The 2D data were registered to the chronologically first ‘normal’ walking image using mutual infor-
mation maximization through optimal planar rigid body transformation (Pataky and Goulermas,
2008). The 3D data were registered using the optimal 2D spatial transformation followed by linear
temporal interpolation, as above.
The data were modeled with four factors: main factor ‘speed’, an intercept, and linear and
sinusoidal time drift nuisance factors (see Appendix E); the nuisance factors were included both to
account for small baseline electronic drifts observed in pilot studies and to emphasize the flexibility
5
of the GLM for experimental modeling. The contrast vector: c=1000
T
(5) represents
the current hypothesis: that speed is positively correlated with the outcome measure (maximal
VGRF, etc.), and that the other factors are not of empirical interest. SPM analyses proceeded as
described in Sect.2. All analyses were implemented in Python 2.5 using Numpy 1.3 and Scipy 0.8
as packaged with the Enthought Python Distribution 5.0 (Enthought Inc., Austin, USA).
3.2 Simulation A
An axisymmetric finite element (FE) model of heel pad indentation was constructed following
(Erdemir et al., 2006) (Fig.1). The model was originally used to compute hyperelastic material
properties of 20 diabetic (D) and 20 non-diabetic (ND) subjects through inverse FE simulation.
Dierences in the material parameters (Fig.1 caption) between the two groups failed to reach
significance.
Here the reported variability is explored using Monte Carlo simulations to determine the relation
between univariate material parameter significance and field-wide stress significance as assessed
using SPM. Firstly, 1000 simulations were conducted for each group (D and ND) using the mean
material parameters and their reported variances. Indenter depth was 8.0 mm for all simulations.
Secondly, the mean Dparameter was varied between 7.0 and 7.7 in steps of 0.05 to span the
range of the reported value (D=7.02) and the univariate significance threshold (D=7.585).
For each D1000 simulations were repeated, holding variance constant.
Finally SPM was used to compare the probabilistic Von Mises stress fields resulting from each
(ND,D) combination using two-sample ttests (Appendix D). Only Dwas varied because: (a)
the implicit hypothesis of Erdemir et al. (2006) and related studies was that diabetic subjects
had stier heel pads, and (b) this parameter aects the material’s high-strain response, which is
potentially of greater clinical interest than low strain. All FE problems were solved using ABAQUS
6.7 (Simulia, Providence, USA).
6
3.3 Simulation B
A 3D human femur model (Fig.2) was used to explore strain field changes associated with hip
replacement pin placement (a simplification of Radclie and Taylor, 2007). Geometry was borrowed
from the third-generation standardized femur model (Cheung et al., 2004). Linear material param-
eters (Ramos and Sim˜oes, 2006), hip contact and abductor muscle forces (Radclie and Taylor,
2007), and two rigid circular pin postures were modeled (see Fig.2 caption). Rather than oer-
ing orthopedic realism, this pin scheme was meant to demonstrate that SPM follows mechanical
expectation.
Since hip forces are highly variable (Bergmann et al., 1993) Monte Carlo simulations were
conducted (1000 repetitions for each pin configuration) by varying all force components with a
standard deviation of 20% (R¨ohrle et al., 1984). Dierences in bone strain fields between the two
pin configurations were assessed using two-sample ttests (Appendix D) and irregular-lattice RFT
inference procedures (Worsley et al., 1999).
4 Results
4.1 Experimental data
Experimental data (Fig.3) exhibited systematic changes with walking speed. Statistical analy-
sis of the 0D dataset (Fig.3a) yielded t=25.1 (p=0.000), indicating significant positive correlation
between walking speed and maximal VGRF. The 1D VGRF data (Figs.3b,4) were positively corre-
lated with speed in early (0-30%) and late (75-100%) stance and were negatively correlated in mid
(35-70%) stance. 2D peak pressures (Figs.3c,5) were positively correlated with walking speed at
the heel and distal forefoot and negatively correlated over the distal midfoot and proximal forefoot.
All eects were significant (p=0.000).
Spatiotemporal 3D pressures (Figs.3d,6) were positively correlated with walking speed under
the heel, midfoot and proximal forefoot in early (0-25%) stance and under the medial forefoot and
phalanges in late (75-100%) stance. Negative correlation was found throughout mid- (35-85%)
stance for all areas excluding the phalanges. All three spatiotemporal clusters reached significant
7
(p=0.000). For reference, Table 1 lists the computed field smoothness and geometrical characteris-
tics of all experimental datasets.
4.2 Simulation A
While mean Non-Diabetic and Diabetic heel pad stress fields exhibited dierences (Figs.7a,b),
SPM supported the findings of Erdemir et al. (2006) by detecting no significant field-wide ef-
fects (Fig.7c); the maximum SPM{t}(t=0.8) was unsuitably low for thresholding. However, SPM
found significant broadly spanning stress responses (t>2.0, p=0.031; Fig.7d) considerably sooner
(D=7.300) than univariate parameter testing (D=7.585). This increased signal sensitivity (and
spatial detail evident in Fig.7d) resulted from SPM’s topological treatment of the field data (Friston
et al., 2007, Ch.19).
4.3 Simulation B
Mean bone strain fields were qualitatively dierent for the two pin configurations (Fig.8a), and
SPM found these dierences to be statistically significant (Fig.8b). The eects were limited to areas
surrounding the pins, were maximal under the pins (the direction of the largest force component),
and were biased away from the non-involved pin, as could be expected.
5 Discussion
This paper has demonstrated that SPM can be used to conduct statistical inference in a contin-
uous and field-wide manner on nD registered biomechanical datasets. While SPM has previously
been applied to 2D pedobarographic datasets (Pataky and Goulermas, 2008), the main new findings
were that: (i) SPM can also be applied to smooth biomechanical field data of any physical dimen-
sionality, and (ii) SPM can be used to probe probabilistic simulations of biomechanical continua.
5.1 Present results
The 0-2D experimental results (Figs.3-5) have been previously described elsewhere (Nilsson and
Thorstensson, 1989; Rosenbaum et al., 1994; Keller et al., 1996; Taylor et al., 2004; Pataky et al.,
8
2008), but the 3D results (Fig.6) have not. Specifically, pressures under the midfoot and proximal
forefoot appear to remain lower during slower walking throughout mid-to-late (35-85%) stance.
While these results are arguably new, and although post- hoc analyses revealed qualitative consis-
tency with the average subject, no population inferences are made because only a single-subject’s
data were presented (for clarity). The main points are that SPM appears to yield results that
are biomechanically consistent with other approaches and that SPM can handle data of arbitrary
dimensionality.
The present FE simulation results reveal, firstly, that SPM produces results that are consistent
with mechanical expectation (Fig.8b). Secondly, SPM revealed, non-trivially, that broad continuum
responses can reach statistical significance well prior to the point at which parameters governing
those responses reach significance. It may therefore be prudent for future investigations to consider
field-wide eects when interpreting mechanical parameter variance. These FE results collectively
imply that SPM provides a suitable framework for analyses of simulated continua, and thus that
it may be a useful compliment to existing probabilistic techniques (e.g. Dar et al., 2002; Laz et al.,
2007).
5.2 SPM
As a field analysis method SPM has a variety of scientific merits. It is highly generalized, and
thus highly flexible, and it aords both field-wide and spatiotemporally focussed hypothesis testing
(Friston et al., 2007). It is mathematically very well developed and has been validated extensively
(e.g. Worsley et al., 1992; Worsley, 1995). When compared with regionalization approaches, its
main advantage is that it avoids two potential sources of discretization-induced bias: (i) regional
conflation (Pataky et al., 2008), and (ii) aprioriassumptions regarding spatiotemporal foci (e.g.
Appendix D).
Another key advantage is that statistics are conducted on a field-wide basis, so investigators need
neither devise nor adapt regionalization schemes for or to particular problems, nor would particular
schemes require justification during scientific review. Along these lines, the existence of various
open-source SPM packages (e.g. SPM8, Wellcome Trust Centre for Neuroimaging, University
9
College London) promotes a greater degree of scientific transparency than is possible with ad hoc
or manual regionalization.
A final advantage is that statistical results lie directly in the original nD sampled continua.
For example, given the context of original VGRF time series (Fig.3b), it is straightforward to
interpret the corresponding SPM results (Fig.4), and one could argue that discrete extrema analysis
(e.g. Fig.3a) oers a somewhat incomplete impression of the field-wide changes associated with
experimental intervention.
As an aside, while SPM demands greater computational resources than regional techniques, it
should be noted that analyses can still be conducted with clinically feasible speed. The current
(non-compiled) Python implementation yielded statistical computation durations of: 1.48103,
2.95102, 3.31, and 14.3 s (0-3D experimental datasets, respectively); the 2- and 3D durations
would likely decrease substantially via optimized compiled implementation. These durations ex-
clude data organization coding, but a high-level automated interface could easily be constructed
(e.g. click on the five pre-surgery trials, click on the five post-surgery files, GO). Registration could
be performed automatically as data are collected; a recent 2D pedobarographic implementation
(Oliveira et al., in press) required only 50 ms per image pair. Therefore SPM’s computational
demand isn’t necessarily a practical limitation.
5.3 Limitations
SPM’s procedures, especially those of RFT-based inference (Appendix A), are mathematically
more complex than those of regional univariate (or discrete multivariate) approaches. These com-
plexities potentially pose barriers to general adoption in investigations of biomechanical continua.
However, both the availability of open-source SPM packages and the highly matured neuroimaging
lead (Friston et al., 2007) could help to lower these barriers if SPM is deemed to oer empirical
advantages.
From a biomechanical perspective SPM’s greatest limitation is potentially its requirement for
co-registration of 1D and higher dimensional datasets. One could argue, for example, that the
current VGRF registration scheme yielded a misregistration of early stance peaks (Fig.3b). In
10
this particular case the apparent misregistration had no biomechanical consequences because the
suprathreshold SPM{t}spanned broadly across early stance (Fig.4). There may be situations where
SPM{t}extent is not order-of-magnitude larger than registration inaccuracy.
In such cases nonlinear registration may help (Sadeghi et al., 2003; Goulermas et al., 2005),
but there are also undoubtedly situations where registration is not biomechanically feasible. Elec-
tromyographical signals with poorly defined temporal bounds or gross tissue deformity, for exam-
ple, may pose practical registration problems. Qualitative geometrical manipulations of FE models
(e.g. Lin et al., 2007) could also render simulation datasets unregistrable. Nevertheless, these are
limitations of registration and not of SPM per se. Continued biomechanical registration scrutiny
(Sadeghi et al., 2000; Sadeghi et al., 2003; Duhamel et al., 2004; Page and Epifanio, 2007) may
help to clarify SPM’s appropriateness for specific applications.
5.4 Summary
SPM aords topological statistical analysis of smooth, registrable n-dimensional scalar fields.
The present results suggest that SPM may be suitable for both laboratory and probabilistic simu-
lation studies involving a wide variety of biomechanical continua. SPM’s main advantages are that
statistical results lie directly in the original continuum and that potential problems associated with
ad hoc discretization are avoided.
Acknowledgments
Funding for this work was provided by Special Coordination Funds for Promoting Science and
Technology from the Japanese Ministry of Education, Culture, Sports, Science and Technology.
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Table 1: Smoothness (FWHM) and geometry (resel counts Rd) of the current experimental
datasets. The FWHM estimates assume isotropic and field-wide constant smoothness (see
Appendix A); in the 3D case the FWHM combines spatial (5 mm) and temporal (% stance)
dimensions from the current (57 23 100) (x, y, t) sampling lattice.
Dataset FWHM R0R1R2R3
0D - 1 - - -
1D 9.81 1 10.1 - -
2D 4.34 1 19.1 33.8 -
3D 7.59 2 24.9 108.5 67.7
14
Figure 1: Axisymmetric model of heel pad indentation (adapted from
Erdemir et al., 2006), 8mm indentation. The dashed rectangle depicts the
undeformed geometry. The authors reported Ogden hyperelastic material
parameters (means ±st.dev.) for non-diabetic (ND) and diabetic (D)
groups as: µND =16.45 (±8.27), µD=16.88 (±6.70) and ND=6.82 (±1.57),
D=7.02 (±1.43), respectively.
15
Figure 2: Three-dimensional femur model (adapted from Cheung et al.,
2004). The bone was modeled as linearly elastic (E=12.8 GPa, =0.4)
(Ramos et al., 2006). Two rigid pins were alternately placed in the
depicted positions. Modeled forces (averages) included hip contact acting
at the pin center (FH=[0.540,0.328,2.292] BW), and abductor force
FH(FA=[0.580,0.043,1.040] BW) for body weight of BW=800 N
(Radclie and Taylor, 2007). The femoral shaft was constrained from all
movement at its base.
16
Figure 3: Experimental data. A single subject performed 20 trials of each
of Slow, Normal, and Fast walking. (a) 0D raw dataset: maximal vertical
ground reaction force (VGRF), normalized by body weight (BW). (b) 1D
temporally registered dataset: VGRF time series. (c) 2D spatially
registered dataset (means): maximal (peak) pressure. (d) 3D
spatiotemporally registered dataset (means): pressure image time series.
17
Figure 4: SPM results, 1D experimental dataset, thresholded at t>3.5.
Probability (p) values indicate the likelihood that a suprathreshold cluster
of the same spatiotemporal extent could have resulted from a random field
process of the same smoothness as the observed residuals (Eqn.3).
18
Figure 5: SPM results, 2D experimental dataset, t>3.5.
19
Figure 6: SPM results, 3D experimental dataset, t>3.5.
20
Figure 7: Heel pad simulation results, undeformed geometry. (a) Mean
Non-Diabetic (ND) Von Mises ()field(ND =6.82). (b) Mean Diabetic
(D) field (D=7.02). (c) SPM{t}field for mean (ND =6.82, D=7.02);
SPM{t}max=0.8. (d) Inference image for D=7.300, t>2.0.
21
Figure 8: Femur simulation results. (a) Maximal principal strain fields
for the pin1 (left) and pin2 (right) configurations under mean force vector
loading. (b) SPM{t}field, (pin2-pin1), |t|>1.0. Inference results for
|t|>2.0 are noted.
22
Appendix A. Statistical inference
Random field theory (RFT) (Adler, 1981) provides the mathematical
foundation for conducting topological statistical inference on an SPM. Given
, the expected topological characteristics of an SPM depend on field smooth-
ness and search space geometry. Field smoothness can be estimated at each
node by first computing normalized residuals u(Kiebel et al., 1999):
ui="i
q"T
i"i
(A.1)
where iindexes the observations, then assembling an (In) gradient matrix
at each pixel (Worsley, 2007):
˙
uk
2
6
6
6
6
6
6
6
4
r(uk)1
r(uk)2
.
.
.
r(uk)I
3
7
7
7
7
7
7
7
5
=
2
6
6
6
6
6
6
6
4
@(uk)1
@1
@(uk)1
@2... @(uk)1
@n
@(uk)2
@1
@(uk)2
@2... @(uk)2
@n
.
.
..
.
..
.
.
@(uk)I
@1
@(uk)I
@2... @(uk)I
@n
3
7
7
7
7
7
7
7
5
(A.2)
where r(uk)iis the gradient of the ith residual’s kth node, and @(uk)i
@dis
the dth component of that gradient vector. Finally nodal smoothness can
be estimated as:
ˆ
Wk= (4log2)1
2|˙
uT
k˙
uk|1
2n(A.3)
Here ˆ
Wkestimates the full-width at half-maximum (FWHM) of a Gaus-
sian kernel that, when convolved with uncorrelated Gaussian random field
data, would produce the same smoothness as was observed in the normalized
residuals ui. As ˆ
Wkincreases the expected size of suprathreshold SPM{t}
clusters also increases, a fact that RFT exploits.
The expected topological characteristics of an SPM{t}also depend on
the geometry of the search space A,thenD space in which the data lie.
Assuming a 3D dataset, the first step is to count the number of nodes (0),
edges (d), faces (dd0), and cubes (123)inA. This task can be rapidly
implemented using morphological erosion (Nixon and Aguago, 2008):
0=|A B0|
d=|A Bd|
dd0=|A Bdd0|
123 =|A B123|(A.4)
where the Bmatrices are directional connectivity structuring elements (Ap-
pendix B). Having assembled these basic morphological characteristics of A,
its global geometry can now be summarized by ‘resel’ or ‘resolution element’
counts Rd(Worsley et al., 1996):
R0=0(1+2+3)+(12 +13 +23)123
R1=1
ˆ
Wh(1+2+3)2(12 +13 +23)+3123i
R2=1
ˆ
W2h(12 +13 +23)3123i
R3=1
ˆ
W3[123] (A.5)
For simplicity A.3 assumes isotropic smoothness and A.5 assumes position-
independent smoothness ˆ
W=ˆ
Wk/K, but these restrictions can easily be
lifted (Worsley et al., 1999). Each Rdis associated with an independent
probability density function (Worsley et al., 1996) (Appendix C) that di-
rectly depends only on the tthreshold.
These density functions can be used to compute a variety of topological
expectations, like the number of supra-threshold nodes and clusters, for
example (Friston et al., 1994). The final steps in RFT-based inference are
thus to threshold an observed SPM{t}at a suitably high value (e.g. t>3.0)
and then corroborate the observed topology with topological expectation,
computing pvalues for each cluster according to Friston et al. (1994), for
example. The logic of RFT is that smooth random fields are expected to
produce spatially broad suprathreshold clusters, but very broad and/or very
high clusters are expected to occur with low probability. The key message
is that a large suprathreshold cluster is the topological equivalent of a large
univariate tvalue.
Appendix B. Search space geometry
To rapidly compute the geometrical characteristic of an nD search
space defined by binary image A, one may use morphological erosion (Nixon
and Aguado, 2008):
=|A B|(B.1)
where the Bmatrices are structuring elements that describe the neighbor-
hood connectivity of interest. In three dimensions single nodes (B0), adja-
cent nodes (Bd), faces (Bdd0), and cubes (B123) are given by the sets:
B0={0,0,0}
B1={{0,0,0},{1,0,0}}
B2={{0,0,0},{0,1,0}}
B3={{0,0,0},{0,0,1}}
B12 ={{0,0,0},{1,0,0},{0,1,0},{1,1,0}}
B13 ={{0,0,0},{1,0,0},{0,0,1},{1,0,1}}
B23 ={{0,0,0},{0,1,0},{0,0,1},{0,1,1}}
B123 ={{0,0,0},{1,0,0},{0,1,0},{1,1,0},...
{0,0,1},{1,0,1},{0,1,1},{1,1,1}} (B.2)
Here A Bis an eroded binary image whose elements are ones if the B
pattern exists at a given node and zeros otherwise. |A B|is the set size
of A Bor, equivalently, the number of ones in the eroded image.
Appendix C. Euler characteristic densities
Each ‘resel count’ Rd(A.5) is associated with an independent proba-
bility distribution function, or Euler characteristic density, pd(t). To three
dimensions, directly from Worsley et al. (1996, Table 2), the densities are:
p0(t)=Z1
t
(+1
2)
⌫⇡1
2(
2)(1 + u2
)1
2(+1)du
p1(t)=(4log2)1
2
21+t2
1
2(1)
p2(t)=(4log2)
2
(+1
2)
(
2)1
2(
2)t1+t2
1
2(1)
p3(t)=(4log2)3
2
(2)21+t2
1
2(1) 1
t21
where is the degrees of freedom. Note that p0is the univariate Student’s t
distribution. The general nD form of these distributions is given in Worsley
(1994, Corollary 5.3).
Appendix D. Numerical example
Consider five fictional force trajectories from each of two experimen-
tal conditions ‘A’ and ‘B’. (Fig.D.a) on a normalized time interval 0-100%
(K=100). Condition A data were created by adding smooth Gaussian
noise (FWHM=10%) to yA(t) = 800 N. Condition B data were created
by first adding positive Gaussian signals to yB(t) = 800 N at t=75% and
15% (see Fig.D.b), and then subsequently adding filtered Gaussian noise
(also at FWHM=10%). These two simulated experimental conditions were
then compared using a two-sample ttest (Eqn.5, main manuscript) where
c=11
Tand:
X=2
6
4
1111100000
0000011111
3
7
5
T
(D.1)
After thresholding the resultant SPM{t}at t>3.0, the significance of
the suprathreshold clusters (Fig.D.c) was assessed using the RFT procedures
described above, where the average FWHM was estimated to be 10.4% using
Eqn.A.3. This simulation highlights three concepts: (i) SPM can be used to
analyze continuous field data in a topological manner. (ii) A Bonferroni ap-
proach (K=100, tcritical = 5.192) would fail to identify significance anywhere
in the temporal field. (iii) A discrete approach that focusses only on the re-
gion t=75% would fail to identify the other signal at t=15%. While this
example has been tailored to emphasize these concepts, the methodology (i)
and dangers (ii,iii) clearly also apply to real experimental data.
Figure D: Example 1D SPM analysis. (a) Simulated raw data for two experimental con-
ditions ‘A’ and ‘B’. (b) Mean curves with standard deviation clouds. (c) SPM{t}with
threshold t>3.0. The pvalues were computed according to Appendix A-C and indicate
the probability that the specific suprathreshold cluster could have occurred by chance.
Appendix E. Model visualization
The general linear model (GLM) (Eqn.1, main manuscript) consists of:
experimental observations (Y), experimental design (X), regression coe-
cients (), and model errors ("). Since these matrices can be quite large,
numerical probing of their elements is inconvenient. The elements may, how-
ever, be conveniently probed qualitatively using matrix visualization tech-
niques. The most important matrix to visualize is Xbecause it represents
the experimenter’s statistical modeling decisions: it describes all modeled
experimental factors, it reveals the randomness of the design, and together
with a contrast vector (Eqn.5, main manuscript) it explicitly describes the
experimental hypothesis.
Fig.E.1 depicts the design matrix that was used to analyze the current
pedobarographic data. Rows correspond to trials, and columns to modeled
experimental factors. The main factor of interest was ‘speed’, and the first
column of Fig.E.1 reveals that walking speeds were randomized. Three
factors of non-interest were also also modeled: an intercept ‘y0’ and two
low-frequency time-drift nuisance factors. Visualizing these factors as a
matrix image can be helpful to understand the experimental design, so X
renderings like Fig.E.1 are often presented in scientific papers.
It is also instructive to visualize the entire GLM (Fig.E.2). The obser-
vations Ymay be regarded as a bird’s eye view of Fig.3a. The model fits
(Xˆ
) closely resemble the experimental data (Y), with only relatively mi-
nor dierences ("). This indicates, anecdotally, that the GLM can explain
a large proportion of the experimental variability.
Figure E.1: Design matrix X(Eqn.1, main manuscript). The matrix is (604): 60 trials
and 4 modeled experimental factors. The color scale is normalized within columns.
Figure E.2: Grayscale renderings of the general linear model matrices (Eqn.1, main
manuscript) for the 1D VGRF dataset. Each matrix is (60100): 60 trials and 100
VGRF trajectory nodes per trial. Absolute errors |"|are presented.
References
Adler, R. J. 1981. The geometry of random fields, Wiley, Chichester.
Friston, K. J., Worsley, K. J., Frackowiak, R. S. J., Mazziotta, J. C., and Evans, A. C.
1994. Assessing the significance of focal activations using their spatial extent, Human
Brain Mapping 1, 210–220.
Kiebel, S. J., Poline, J. B., Friston, K. J., Holmes, A. P., and Worsley, K. J. 1999. Robust
smoothness estimation in statistical parametric maps using standardized residuals from
the general linear model, Neuroimage 10, 756–766.
Nixon, M. and Aguado, A. 2008. Feature extraction & image processing, second edition,
2nd ed., Academic Press.
Worsley, K. J. 1994. Local maxima and the expected Euler characteristic of excursion sets
of 2,Fand tfields, Advances in Applied Probability 26, 13–42.
Worsley, K. J. 2007. Random field theory. In: Statistical parametric mapping: the analysis
of functional brain images (K.J. et al. Friston, ed.), Elsevier/Academic Press, Amster-
dam.
Worsley, K. J., Andermann, M., Koulis, T., Macdonald, D., and Evans, A. C. 1999. De-
tecting changes in nonisotropic images, Human Brain Mapping 8, 98–101.
Worsley, K. J., Marrett, S., Neelin, P., Vandal, A. C., Friston, K. J., and Evans, A. C. 1996.
A unified statistical approach for determining significant signals in images of cerebral
activation, Human Brain Mapping 4, 58–73.
Appendix A. Statistical inference
Random field theory (RFT) (Adler, 1981) provides the mathematical
foundation for conducting topological statistical inference on an SPM. Given
, the expected topological characteristics of an SPM depend on field smooth-
ness and search space geometry. Field smoothness can be estimated at each
node by first computing normalized residuals u(Kiebel et al., 1999):
ui="i
q"T
i"i
(A.1)
where iindexes the observations, then assembling an (In) gradient matrix
at each pixel (Worsley, 2007):
˙
uk
2
6
6
6
6
6
6
6
4
r(uk)1
r(uk)2
.
.
.
r(uk)I
3
7
7
7
7
7
7
7
5
=
2
6
6
6
6
6
6
6
4
@(uk)1
@1
@(uk)1
@2... @(uk)1
@n
@(uk)2
@1
@(uk)2
@2... @(uk)2
@n
.
.
..
.
..
.
.
@(uk)I
@1
@(uk)I
@2... @(uk)I
@n
3
7
7
7
7
7
7
7
5
(A.2)
where r(uk)iis the gradient of the ith residual’s kth node, and @(uk)i
@dis
the dth component of that gradient vector. Finally nodal smoothness can
be estimated as:
ˆ
Wk= (4log2)1
2|˙
uT
k˙
uk|1
2n(A.3)
Here ˆ
Wkestimates the full-width at half-maximum (FWHM) of a Gaus-
sian kernel that, when convolved with uncorrelated Gaussian random field
data, would produce the same smoothness as was observed in the normalized
residuals ui. As ˆ
Wkincreases the expected size of suprathreshold SPM{t}
clusters also increases, a fact that RFT exploits.
The expected topological characteristics of an SPM{t}also depend on
the geometry of the search space A,thenD space in which the data lie.
Assuming a 3D dataset, the first step is to count the number of nodes (0),
edges (d), faces (dd0), and cubes (123)inA. This task can be rapidly
implemented using morphological erosion (Nixon and Aguago, 2008):
0=|A B0|
d=|A Bd|
dd0=|A Bdd0|
123 =|A B123|(A.4)
where the Bmatrices are directional connectivity structuring elements (Ap-
pendix B). Having assembled these basic morphological characteristics of A,
its global geometry can now be summarized by ‘resel’ or ‘resolution element’
counts Rd(Worsley et al., 1996):
R0=0(1+2+3)+(12 +13 +23)123
R1=1
ˆ
Wh(1+2+3)2(12 +13 +23)+3123i
R2=1
ˆ
W2h(12 +13 +23)3123i
R3=1
ˆ
W3[123] (A.5)
For simplicity A.3 assumes isotropic smoothness and A.5 assumes position-
independent smoothness ˆ
W=ˆ
Wk/K, but these restrictions can easily be
lifted (Worsley et al., 1999). Each Rdis associated with an independent
probability density function (Worsley et al., 1996) (Appendix C) that di-
rectly depends only on the tthreshold.
These density functions can be used to compute a variety of topological
expectations, like the number of supra-threshold nodes and clusters, for
example (Friston et al., 1994). The final steps in RFT-based inference are
thus to threshold an observed SPM{t}at a suitably high value (e.g. t>3.0)
and then corroborate the observed topology with topological expectation,
computing pvalues for each cluster according to Friston et al. (1994), for
example. The logic of RFT is that smooth random fields are expected to
produce spatially broad suprathreshold clusters, but very broad and/or very
high clusters are expected to occur with low probability. The key message
is that a large suprathreshold cluster is the topological equivalent of a large
univariate tvalue.
Appendix B. Search space geometry
To rapidly compute the geometrical characteristic of an nD search
space defined by binary image A, one may use morphological erosion (Nixon
and Aguado, 2008):
=|A B|(B.1)
where the Bmatrices are structuring elements that describe the neighbor-
hood connectivity of interest. In three dimensions single nodes (B0), adja-
cent nodes (Bd), faces (Bdd0), and cubes (B123) are given by the sets:
B0={0,0,0}
B1={{0,0,0},{1,0,0}}
B2={{0,0,0},{0,1,0}}
B3={{0,0,0},{0,0,1}}
B12 ={{0,0,0},{1,0,0},{0,1,0},{1,1,0}}
B13 ={{0,0,0},{1,0,0},{0,0,1},{1,0,1}}
B23 ={{0,0,0},{0,1,0},{0,0,1},{0,1,1}}
B123 ={{0,0,0},{1,0,0},{0,1,0},{1,1,0},...
{0,0,1},{1,0,1},{0,1,1},{1,1,1}} (B.2)
Here A Bis an eroded binary image whose elements are ones if the B
pattern exists at a given node and zeros otherwise. |A B|is the set size
of A Bor, equivalently, the number of ones in the eroded image.
Appendix C. Euler characteristic densities
Each ‘resel count’ Rd(A.5) is associated with an independent proba-
bility distribution function, or Euler characteristic density, pd(t). To three
dimensions, directly from Worsley et al. (1996, Table 2), the densities are:
p0(t)=Z1
t
(+1
2)
⌫⇡1
2(
2)(1 + u2
)1
2(+1)du
p1(t)=(4log2)1
2
21+t2
1
2(1)
p2(t)=(4log2)
2
(+1
2)
(
2)1
2(
2)t1+t2
1
2(1)
p3(t)=(4log2)3
2
(2)21+t2
1
2(1) 1
t21
where is the degrees of freedom. Note that p0is the univariate Student’s t
distribution. The general nD form of these distributions is given in Worsley
(1994, Corollary 5.3).
Appendix D. Numerical example
Consider five fictional force trajectories from each of two experimen-
tal conditions ‘A’ and ‘B’. (Fig.D.a) on a normalized time interval 0-100%
(K=100). Condition A data were created by adding smooth Gaussian
noise (FWHM=10%) to yA(t) = 800 N. Condition B data were created
by first adding positive Gaussian signals to yB(t) = 800 N at t=75% and
15% (see Fig.D.b), and then subsequently adding filtered Gaussian noise
(also at FWHM=10%). These two simulated experimental conditions were
then compared using a two-sample ttest (Eqn.5, main manuscript) where
c=11
Tand:
X=2
6
4
1111100000
0000011111
3
7
5
T
(D.1)
After thresholding the resultant SPM{t}at t>3.0, the significance of
the suprathreshold clusters (Fig.D.c) was assessed using the RFT procedures
described above, where the average FWHM was estimated to be 10.4% using
Eqn.A.3. This simulation highlights three concepts: (i) SPM can be used to
analyze continuous field data in a topological manner. (ii) A Bonferroni ap-
proach (K=100, tcritical = 5.192) would fail to identify significance anywhere
in the temporal field. (iii) A discrete approach that focusses only on the re-
gion t=75% would fail to identify the other signal at t=15%. While this
example has been tailored to emphasize these concepts, the methodology (i)
and dangers (ii,iii) clearly also apply to real experimental data.
Figure D: Example 1D SPM analysis. (a) Simulated raw data for two experimental con-
ditions ‘A’ and ‘B’. (b) Mean curves with standard deviation clouds. (c) SPM{t}with
threshold t>3.0. The pvalues were computed according to Appendix A-C and indicate
the probability that the specific suprathreshold cluster could have occurred by chance.
Appendix E. Model visualization
The general linear model (GLM) (Eqn.1, main manuscript) consists of:
experimental observations (Y), experimental design (X), regression coe-
cients (), and model errors ("). Since these matrices can be quite large,
numerical probing of their elements is inconvenient. The elements may, how-
ever, be conveniently probed qualitatively using matrix visualization tech-
niques. The most important matrix to visualize is Xbecause it represents
the experimenter’s statistical modeling decisions: it describes all modeled
experimental factors, it reveals the randomness of the design, and together
with a contrast vector (Eqn.5, main manuscript) it explicitly describes the
experimental hypothesis.
Fig.E.1 depicts the design matrix that was used to analyze the current
pedobarographic data. Rows correspond to trials, and columns to modeled
experimental factors. The main factor of interest was ‘speed’, and the first
column of Fig.E.1 reveals that walking speeds were randomized. Three
factors of non-interest were also also modeled: an intercept ‘y0’ and two
low-frequency time-drift nuisance factors. Visualizing these factors as a
matrix image can be helpful to understand the experimental design, so X
renderings like Fig.E.1 are often presented in scientific papers.
It is also instructive to visualize the entire GLM (Fig.E.2). The obser-
vations Ymay be regarded as a bird’s eye view of Fig.3a. The model fits
(Xˆ
) closely resemble the experimental data (Y), with only relatively mi-
nor dierences ("). This indicates, anecdotally, that the GLM can explain
a large proportion of the experimental variability.
Figure E.1: Design matrix X(Eqn.1, main manuscript). The matrix is (604): 60 trials
and 4 modeled experimental factors. The color scale is normalized within columns.
Figure E.2: Grayscale renderings of the general linear model matrices (Eqn.1, main
manuscript) for the 1D VGRF dataset. Each matrix is (60100): 60 trials and 100
VGRF trajectory nodes per trial. Absolute errors |"|are presented.
References
Adler, R. J. 1981. The geometry of random fields, Wiley, Chichester.
Friston, K. J., Worsley, K. J., Frackowiak, R. S. J., Mazziotta, J. C., and Evans, A. C.
1994. Assessing the significance of focal activations using their spatial extent, Human
Brain Mapping 1, 210–220.
Kiebel, S. J., Poline, J. B., Friston, K. J., Holmes, A. P., and Worsley, K. J. 1999. Robust
smoothness estimation in statistical parametric maps using standardized residuals from
the general linear model, Neuroimage 10, 756–766.
Nixon, M. and Aguado, A. 2008. Feature extraction & image processing, second edition,
2nd ed., Academic Press.
Worsley, K. J. 1994. Local maxima and the expected Euler characteristic of excursion sets
of 2,Fand tfields, Advances in Applied Probability 26, 13–42.
Worsley, K. J. 2007. Random field theory. In: Statistical parametric mapping: the analysis
of functional brain images (K.J. et al. Friston, ed.), Elsevier/Academic Press, Amster-
dam.
Worsley, K. J., Andermann, M., Koulis, T., Macdonald, D., and Evans, A. C. 1999. De-
tecting changes in nonisotropic images, Human Brain Mapping 8, 98–101.
Worsley, K. J., Marrett, S., Neelin, P., Vandal, A. C., Friston, K. J., and Evans, A. C. 1996.
A unified statistical approach for determining significant signals in images of cerebral
activation, Human Brain Mapping 4, 58–73.
... Meanwhile, Statistical Parameter Mapping (SPM) was used to analyze the kinematic and dynamics time-series curves, which has been widely used in biomechanical research [21][22][23]. SPM is a method for comparing the complete time series curves by considering the correlation of adjacent time points when calculating the appropriate significance threshold [24,25]. All kinematic kinetic variables in these two steps were extracted in the One-Dimensional Statistical Parametric Mapping (SPM1D) analysis. ...
... We used the open-source SPM1d script of the single-sample t-test for statistical analysis, 0.05 was the significance alpha level setting [24,25]. ...
Article
Research on dance lower extremity joint motion has been limited. Thus, the purpose of this study was to investigate the lower limb biomechanics differences between the side chasse step (SCS) and the bounce step (BS) of the second landing phase in Jive. Thirteen female recreational Latin dancers (Age: 22 2.5 years; Height: 1.65 0.05 m; Weight: 50 4.5 kg; Dance experience: 4 2 years) were involved in the experiment. The same music was used throughout the data collection period. We intended to determine whether these two steps generate different kinematic and kinetic data. The ankle, hip, and knee joint angle, moment, velocity, and ground reaction force were calculated for each step. Results demonstrated that the lower limb biomechanics of the two different steps showed significant differences. As a result, strengthening the lower limb muscles (gastrocnemius, Tibialis muscle, and quadriceps) is significantly important to balance the joint strength and prevent foot injury. According to the training time reasonably increasing the heel height should be recognized as important. The current study could provide new insights into reducing lower extremity injuries and improving dance performance.
... A statistical-processing technique that has allowed biomechanical variables to be examined in function of the time series of a task, is statistical parametric mapping (SPM) (Pataky, 2010;Pataky et al., 2016). This technique was first described by Friston et al. (1995), and then validated by Pataky based on kinematic and kinetic data (Pataky, 2010;Pataky et al., 2016). ...
... A statistical-processing technique that has allowed biomechanical variables to be examined in function of the time series of a task, is statistical parametric mapping (SPM) (Pataky, 2010;Pataky et al., 2016). This technique was first described by Friston et al. (1995), and then validated by Pataky based on kinematic and kinetic data (Pataky, 2010;Pataky et al., 2016). SPM is based on an inferential method, where hypothesis testing is directly applied to the signal (Pataky et al., 2016). ...
Article
The aim of this study was to apply statistical parametric mapping (SPM) to compare temporal changes in EMG amplitude between rearfoot (RF) and forefoot (FF) running techniques. Eleven recreational runners ran on a treadmill at a self-selected speed, once using a RF strike pattern and once using a FF strike pattern (randomized order). The myoelectric activity of five lower limb muscles [rectus femoris (RFe), biceps femoris (BF), tibialis anterior (TA), medial and lateral gastrocnemius (MG and LG)] was evaluated, using bipolar electromyography (EMG). EMG data from the RF and FF running techniques was then processed and posteriorly compared with SPM, dividing the analysis of the running cycle into stance and swing phases. The MG and LG muscles showed higher activation during FF running at the beginning of the stance phase and at the end of the swing phase. During the end of the swing phase the TA muscle's EMG amplitude was higher, when the RF running technique was used. A higher level of co-activation between the gastrocnemius and TA muscles was observed at the end of the swing phase during RF running. The myoelectric behaviour of the RFe and BF muscles was similar during both running techniques. These findings highlight the importance of SPM for the accurate assessment of differences in muscle activity during running and strongly suggest that these two running techniques predominately reflect adjustments of the leg and not the thigh muscles.
... This hip/knee strategy can be explained by the subject needing to rely more on the knees and hips to absorb forces and control the movement of the center of mass to safely and quickly complete the gait termination task due to the unexpected stimulus. Nevertheless, considering that discrete analysis might compromise the spatiotemporal integrity of the original field, the approach of checking the entire stance phase than discrete parameters proved to be more suitable for determining biomechanical differences [32]. Therefore, the present study introduced a variance equality test that compares the temporal kinematics variance along the entire waveform of lower extremity joints to detect significant differences in variance [13]. ...
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The efficacy of the variance equality test in steady-state gait analysis is well documented; however, temporal information on where differences in variability occur during gait subtasks, especially during gait termination caused by unexpected stimulation, is poorly understood. Therefore, the purpose of the current study was to further verify the efficacy of the waveform-level variance equality test in gait subtasks by comparing temporal kinematical variability between planned gait termination (PGT) and unplanned gait termination (UGT) caused by unexpected stimulation. Thirty-two asymptomatic male subjects were recruited to participate in the study. A Vicon motion capture system was utilized to measure lower extremity kinematics during gait termination tasks with and without unexpected stimulation conditions. The F-statistic for each interval of the temporal kinematic waveform was compared to the critical value using a variance equality test to identify significant differences in the waveform. Comparative tests between two types of gait terminations found that subjects may exhibit greater kinematics variance in most lower limb joints during UGT caused by unexpected stimulation (especially at stimulus delay and reaction phases). Significant greater variances during PGT were exhibited only in the MPJ sagittal and frontal planes at the early stimulus delay phase (4-15% and 1-15%). This recorded dataset of temporal kinematic changes caused by unexpected stimuli during gait termination is essential for interpreting lower limb biomechanical function and injury prediction in relation to UGT. Given the complexity of the gait termination task, which involves both internal and external variability, the variance equality test can be used as a valuable method to compare temporal differences in the variability of biomechanical variables.
... One-dimensional statistical parametric mapping (SPM1D) [5] is a Python/MATLAB library that recently introduced the statistical parametric mapping (SPM) statistical method to the biomechanics community. SPM1D has been employed in the analysis of time-varying human movement gait in both normal and pathological conditions (hemiplegic stroke gait, transtibial amputee gait, and knee OA) using biomechanical measurements (kinematics, moments, and EMG) [6][7][8]. ...
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SPM is a statistical method of analysis of time-varying human movement gait signal, depending on the random field theory (RFT). MovementRx is our inhouse-developed decision-support system that depends on SPM1D Python implementation of the SPM (spm1d.org). We present the potential application of MovementRx in the prediction of increased joint forces with the possibility to predispose to osteoarthritis in a sample of post-surgical Transtibial Amputation (TTA) patients who were ambulant in the community. We captured the three-dimensional movement profile of 12 males with TTA and studied them using MovementRx, employing the SPM1D Python library to quantify the deviation(s) they have from our corresponding reference data, using “Hotelling 2” and “T test 2” statistics for the 3D movement vectors of the 3 main lower limb joints (hip, knee, and ankle) and their nine respective components (3 joints × 3 dimensions), respectively. MovementRx results visually demonstrated a clear distinction in the biomechanical recordings between TTA patients and a reference set of normal people (ABILITY data project), and variability within the TTA patients’ group enabled identification of those with an increased risk of developing osteoarthritis in the future. We conclude that MovementRx is a potential tool to detect increased specific joint forces with the ability to identify TTA survivors who may be at risk for osteoarthritis.
... The sEMG time-series were significantly different if any values of statistical parametric mapping (SPM) over the entire gait cycle exceeded the critical threshold (alpha=0.05) 11) . ...
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... *Indicates significant differences between groups per an alpha value of p < 0.05. 50 . Two-way ANOVAs for NCs related to plantarflexion, dorsiflexion, and hip extension were performed. ...
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Stroke survivors often exhibit gait dysfunction which compromises self-efficacy and quality of life. Muscle Synergy Analysis (MSA), derived from electromyography (EMG), has been argued as a method to quantify the complexity of descending motor commands and serve as a direct correlate of neural function. However, controversy remains regarding this interpretation, specifically attribution of MSA as a neuromarker. Here we sought to determine the relationship between MSA and accepted neurophysiological parameters of motor efficacy in healthy controls, high (HFH), and low (LFH) functioning stroke survivors. Surface EMG was collected from twenty-four participants while walking at their self-selected speed. Concurrently, transcranial magnetic stimulation (TMS) was administered, during walking, to elicit motor evoked potentials (MEPs) in the plantarflexor muscles during the pre-swing phase of gait. MSA was able to differentiate control and LFH individuals. Conversely, motor neurophysiological parameters, including soleus MEP area, revealed that MEP latency differentiated control and HFH individuals. Significant correlations were revealed between MSA and motor neurophysiological parameters adding evidence to our understanding of MSA as a correlate of neural function and highlighting the utility of combining MSA with other relevant outcomes to aid interpretation of this analysis technique.
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The purpose of the study was to assess the influence of a preceding mountain ultramarathon on the impact between the foot and the ground and the resulting soft tissue vibrations (STV). Two sessions of measurements were performed on 52 trail runners, before and just after mountain trail running races of various distances (from 40 to 171 km). Triaxial accelerometers were used to quantify the foot--ground impact (FGI) and STV of both gastrocnemius medialis (GAS) and vastus lateralis (VL) muscles during level treadmill running at 10 km·h-1. A continuous wavelet transform was used to analyze the acceleration signals in the time-frequency domain, and the maps of coefficients as well as the frequency and damping properties of STV were computed. Fatigue was assessed from isometric maximal voluntary contraction force loss of knee extensors (KE) and plantar flexors (PF) after each race. Statistical nonParametric Mapping and linear mixed models were used to compare the means between the data obtained before and after the races. FGI amplitude and GAS STV were not modified after the race, while VL STV amplitude, frequency and damping significantly decreased whatever the running distance. A significant force loss was observed for the PF (26 ± 14%) and KE (27 ± 16%), but this was not correlated to the changes observed in STV. These results might reveal a protection mechanism of the muscles, indicating that biomechanical and/or physiological adaptations may occur in mountain ultramarathons to limit STV and muscle damage of knee extensors.Trial registration: ClinicalTrials.gov identifier: NCT04025138..
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Cyclists are exposed for a long period to continuous vibrations. When a muscle is exposed to vibration, its efficiency decreases, the onset of fatigue occurs sooner, and the comfort of the cyclist is reduced. This study characterised the vastus lateralis (VL) soft tissue vibrations for different input frequencies and different pedalling phases. Ten cyclists were recruited to pedal at 55, 70, 85, and 100 rpm on a vibrating cycle ergometer that induced vibrations at frequencies ranging from 14.4 Hz (55 rpm) to 26.3 Hz (100 rpm). The VL vibration amplitude was quantified with a continuous wavelet transform and expressed as a function of the crank angle. The pedalling cycle was split into four phases (downstroke, backstroke, upstroke, and overstroke) to express the mean vibration amplitude and frequency of each phase. Statistical analysis depicted that VL vibration frequency increased with the pedalling cadence and that the VL was exposed to up to 50% more vibration amplitudes during the downstroke phase at a slow cadence. The increase in the pedal vibration frequency, a higher vibration transmission due to greater normal force on the pedal, and strong activation of the VL during the downstroke phase were discussed to explain these results.
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Certain images arising in astrophysics and medicine are modelled as smooth random fields inside a fixed region, and experimenters are interested in the number of ‘peaks', or more generally, the topological structure of ‘hot-spots' present in such an image. This paper studies the Euler characteristic of the excursion set of a random field; the excursion set is the set of points where the image exceeds a fixed threshold, and the Euler characteristic counts the number of connected components in the excursion set minus the number of ‘holes'. For high thresholds the Euler characteristic is a measure of the number of peaks. The geometry of excursion sets has been studied by Adler (1981) who gives the expectation of two excursion set characteristics, called the DT (differential topology) and IG (integral geometry) characteristics, which equal the Euler characteristic provided the excursion set does not touch the boundary of the region. Worsley (1995) finds a boundary correction which gives the expectation of the Euler characteristic itself in two and three dimensions. The proof uses a representation of the Euler characteristic given by Hadwiger (1959). The purpose of this paper is to give a general result for any number of dimensions. The proof takes a different approach and uses a representation from Morse theory. Results are applied to the recently discovered anomalies in the cosmic microwave background radiation, thought to be the remnants of the creation of the universe.
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The maximum of a Gaussian random field was used by Worsley et al. (1992) to test for activation at an unknown point in positron emission tomography images of blood flow in the human brain. The Euler characteristic of excursion sets was used as an estimator of the number of regions of activation. The expected Euler characteristic of excursion sets of stationary Gaussian random fields has been derived by Adler and Hasofer (1976) and Adler (1981). In this paper we extend the results of Adler (1981) to χ 2 , F and t fields. The theory is applied to some three-dimensional images of cerebral blood flow from a study on pain perception.
Thesis
There are many interacting factors aecting the performance of a total hip replacement (THR), such as prosthesis design and material properties, applied loads, surgical approach, femur size and quality, interface conditions etc. All these factors are subject to variation and therefore uncertainties have to be taken into account when designing and analysing the performance of these systems. To address this problem, probabilistic design methods have been developed. A computational probabilistic tool to analyse the performance of an uncemented THR has been developed. Monte Carlo Simulation (MCS) was applied to various models with increasing complexity. In the pilot models, MCS was applied to a simplied nite element model (FE) of an uncemented total hip replacement (UTHR). The implant and bone stiness, load magnitude and geometry, and implant version angle were included as random variables and a reliable strain based performance indicator was adopted. The sensitivity results highlighted the bone stiness, implant version and load magnitude as the most sensitive parameters. The FE model was developed further to include the main muscle forces, and to consider fully bonded and frictional interface conditions. Three proximal femurs and two implants (one with a short and another with a long stem) were analysed. Dierent boundary conditions were compared, and convergence was improved when the distal portion of the implant was constrained and a frictional interface was employed. This was particularly true when looking at the maximum nodal micromotion. The micromotion results compared well with previous studies, conrming the reliability and accuracy of the probabilistic nite element model (PFEM). Results were often in uenced by the bone, suggesting that variability in bone features should be included in any probabilistic analysis of the implanted construct. This study achieved the aim of developing a probabilistic nite element tool for the analysis of nite element models of uncemented hip replacements and forms a good basis for probabilistic models of constructs subject to implant position related variability.
Book
In an age where the amount of data collected from brain imaging is increasing constantly, it is of critical importance to analyse those data within an accepted framework to ensure proper integration and comparison of the information collected. This book describes the ideas and procedures that underlie the analysis of signals produced by the brain. The aim is to understand how the brain works, in terms of its functional architecture and dynamics. This book provides the background and methodology for the analysis of all types of brain imaging data, from functional magnetic resonance imaging to magnetoencephalography. Critically,Statistical Parametric Mappingprovides a widely accepted conceptual framework which allows treatment of all these different modalities. This rests on an understanding of the brain's functional anatomy and the way that measured signals are caused experimentally. The book takes the reader from the basic concepts underlying the analysis of neuroimaging data to cutting edge approaches that would be difficult to find in any other source. Critically, the material is presented in an incremental way so that the reader can understand the precedents for each new development. This book will be particularly useful to neuroscientists engaged in any form of brain mapping; who have to contend with the real-world problems of data analysis and understanding the techniques they are using. It is primarily a scientific treatment and a didactic introduction to the analysis of brain imaging data. It can be used as both a textbook for students and scientists starting to use the techniques, as well as a reference for practicing neuroscientists. The book also serves as a companion to the software packages that have been developed for brain imaging data analysis. * An essential reference and companion for users of the SPM software * Provides a complete description of the concepts and procedures entailed by the analysis of brain images * Offers full didactic treatment of the basic mathematics behind the analysis of brain imaging data * Stands as a compendium of all the advances in neuroimaging data analysis over the past decade * Adopts an easy to understand and incremental approach that takes the reader from basic statistics to state of the art approaches such as Variational Bayes * Structured treatment of data analysis issues that links different modalities and models * Includes a series of appendices and tutorial-style chapters that makes even the most sophisticated approaches accessible.
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We present a unified statistical theory for assessing the significance of apparent signal observed in noisy difference images. The results are usable in a wide range of applications, including fMRI, but are discussed with particular reference to PET images which represent changes in cerebral blood flow elicited by a specific cognitive or sensorimotor task. Our main result is an estimate of the P-value for local maxima of Gaussian, t, χ2 and F fields over search regions of any shape or size in any number of dimensions. This unifies the P-values for large search areas in 2-D (Friston et al. [1991]: J Cereb Blood Flow Metab 11:690–699) large search regions in 3-D (Worsley et al. [1992]: J Cereb Blood Flow Metab 12:900–918) and the usual uncorrected P-value at a single pixel or voxel.
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Thirty healthy subjects were studied to determine whether and how changes in walking speed affect plantar pressure distribution and hindfoot angular motion. A capacitive pressure distribution platform and an electrogoniometer on the subjects' hind foot were used for data collection. There was a significant increase in peak pressure under the heel and the medial part of the forefoot and a significant decrease under the midfoot and lateral forefoot with increasing walking speed. This effect of a medialization of the loading pattern seems closely related to a more pronounced pronation motion as indicated by increased eversion of the hindfoot. The results emphasize the need for monitoring and controlling walking speed when comparing the foot loading characteristics of different groups of subjects.