Three-dimensional Quantitative Analysis of Bread Crumb by X

JFS
E: Food Engineering and Physical Properties
Three-dimensional Quantitative Analysis of
Bread Crumb by X-ray Microtomography
PASQ
UALE M. FAL
CONE, ANT
ONIET
TA BAIANO, FRANC
O ZANINI, LUCIA MANCINI, GIULIANA T ROMBA, D
O
ASQU
ALC
NTONIET
ONIETT
RANCO
RANCESCO
D.. DREOSSI, FRANCESC
M ONT
ANARI, NIC
OL
A SCUOR, AND MAT TEO A. DEL NOBILE
ONTANARI
ICOL
OLA
Introduction
I
t is well known that the consumer perception of bread quality
strongly depends on the appearance (Scanlon and Zghal 2001)
and crumb mechanical properties (Aguilera and Stanley 1999).
These parameters are highly correlated with the crumb cellular
structure, also on a microscopic scale (Scanlon and Zghal 2001). The
relationship between crumb structure and appearance is self-evident due to the interactions between structure and light (Kamman 1970; Pyler 1988). On the other hand, crumb mechanical properties can be evaluated in a more complex way by means of both
sensory and instrumental analysis. As for the other porous solids
(Roberts and Garboczi 2002), the bread crumb mechanical properties depend on the mechanical properties of the solid matrix, the
void volume fraction and the morphology of the cell structure.
Bread crumb is usually defined as a spongy material characterized
by the presence of both closed and open cells. In the former the
solid phase is located both in the faces and edge, whereas in the
open cells the solid phase is located only in the edge (Scanlon and
Zghal 2001). In each bread sample, the cell walls can have different
length, thickness, and orientation. These structural parameters
have to be measured for a quantitative description of the crumb
microstructure. Anisotropy (often called “fabric”) is a typical structural characteristic of foamed materials (Marie and others 2003). In
foamed materials, anisotropy depends on both the microstructure
of the solid phase and cell spatial arrangement. Anisotropy is a
measure of the three-dimensional asymmetry, which is a preferen-
MS 20040518 Submitted 8/3/04, Revised 9/29/04, Accepted 12/2/04. Authors
Falcone, Baiano, and Del Nobile are with Dept. of Food Science, Univ. of
Foggia, Via Napoli, 25, 71100 Foggia, Italy and Istituto per la Ricerca e le
Applicazioni Biotecnologiche per la Sicurezza e la Valorizzazione dei Prodotti
Tipici e di Qualità, Univ. degli Studi di Foggia, Via Napoli, 25, 71100 Foggia,
Italy. Authors Zanini, Mancini, Tromba, Dreossi, and Montanari are with
Sincrotrone Trieste S.C.p.A., Basovizza (TS), Italy. Author Scuor is with Dept.
of Materials Engineering and Applied Chemistry, Univ. of Trieste, Trieste,
Italy. Direct inquiries to author Del Nobile (E-mail: ma.delnobile@unifg.it).
© 2005 Institute of Food Technologists
Further reproduction without permission is prohibited
tial alignment of the cells along a certain axis or the degree of the
solid phase dispersion around the main direction (Odgaard and
others 1997). Porosity (void volume fraction) is the 1st measure of
the material distribution in foamed foods; nevertheless, this parameter does not include information about the cell favorite direction (Cowin 2004). As a consequence, porosity is not able to describe
the microstructure of anisotropic foamed materials, and there is a
great interest in quantifying the anisotropy index.
The bread crumb elastic modulus strictly depends on the direction of the compression test (Hibberd and Parker 1985) as well as
on the cell shape, the degree of anisotropy (Gibbson and Ashby
1997; Whitworth and Alava 1999), the cell size distribution (Gibson
and Ashby 1997; Zghal and others 2001), and the cell wall thickness
distribution (Gibson and Ashby 1997).
If compared with the sensory analysis, the image analysis could
represent a more convenient approach to the assessment of bread
quality, because with it, one can obtain objective information about
food structure (Chan and Batchelor 1993). Recently, a suitable and
reliable experimental methodology for a nondestructive investigation of the bread cellular structure has been presented (Falcone
and others 2004): the phase-sensitive X-ray computerized microtomography (PS-XRM). This technique allows to obtain a three-dimensional (3D) representation of the inside structure of a sample
from a set of projection measurements recorded from a certain
number of points of view. Their ability for the contrast-enhanced
imaging without any sample preparation allows overcoming typical artifacts in the visualization of food structure.
The objective of this work was to characterize quantitatively the
inner structure of the bread crumb by analyzing numerically their
3D tomographic images. To this purpose, an algorithm able to perform 3D stereological calculations was used to process digital data.
To evaluate the applicability of the proposed image analysis technique in bread structure evaluation, the same crumb samples used
to acquire the digital images were submitted to compression tests.
Vol. 70, Nr. 3, 2005—JOURNAL OF FOOD SCIENCE E265
Published on Web 4/28/2005
E: Food Engineering & Physical Properties
ABSTRA
CT
hase-sensitiv
e X-r
ay micr
otomogr
aphy was used as a nondestr
uctiv
e imaging technique for the comABSTRACT
CT:: P
Phase-sensitiv
hase-sensitive
X-ray
microtomogr
otomography
nondestructiv
uctive
puterized reconstruction of three-dimensional (3D) images of bread crumb microstructure. After image acquisition,
numer
ical algor
ithms w
er
e used for the slice rreconstr
econstr
uction. S
uccessiv
ely
er
e rrender
ender
ed and quantitanumerical
algorithms
wer
ere
econstruction.
Successiv
uccessively
ely,, 3D images w
wer
ere
endered
tively analyzed. Stereological analyses were performed to determine the void volume fraction and some directionally
dependent par
ameters
face
or
eparameters
ameters,, such as cell wall number
number,, cell wall thickness
thickness,, cell wall spacing, and specific sur
surface
face.. M
Mor
oreov er
or
ite or
ientations of the cell str
uctur
e and the degr
ee of anisotr
op
y rrelated
elated to the arr
angement of the
er,, the fav
favor
orite
orientations
structur
ucture
degree
anisotrop
opy
arrangement
solid phase around the main directions were also determined. Results from statistical analysis suggest that in
addition to the void volume fraction, the parameters describing anisotropy or the favorite orientation of cell structure are crucial in understanding the differences observed in crumb architecture. Indeed, the latter parameters can
explain more than 50% of whole structural variance existing between bread samples. The ability to reflect the bread
crumb compression behavior proved the effectiveness of the microstructural descriptors, as obtained from image
analysis, to give representative information on the inner architecture of bread crumb.
Keywor
ds: anisotr
op
y, br
ead, micr
ostr
uctur
e, ster
eological analysis
ay computer
iz
ed micr
otomogr
aphy
eywords:
anisotrop
opy
bread,
microstr
ostructur
ucture
stereological
analysis,, X-r
X-ray
computeriz
ized
microtomogr
otomography
Analysis of bread crumb microstructure . . .
Results from mechanical tests were compared with those from image analysis.
rithm (k = 2) ( JMP v. 4.0, SAS Inst. Inc., Cary, N.C., USA). The segmentation process converted the images into a binary format that
allowed quantification of the structural indices.
Material and Methods
Image analysis of the crumb anisotropy
Samples choice
Two crumb samples, 25 mm height, 25 mm width, and 12 mm
thickness (they will be henceforth called “sample 1” and “sample
2”), were cut from the slice of a commercial loaf of white pan bread.
The advantage of this choice is that bread samples have an initial
moisture content that is practically the same. Indeed, mechanical
behavior of bread crumb strictly depends on both the moisture
content in solid phase and cell architecture.
Data collection
E: Food Engineering & Physical Properties
The sample image acquisition was carried out by means of PSXRM technique at the SYRMEP line (SYnchrotron Radiation for
MEdical Physics) of the Elettra Laboratory (Trieste, Italy).
Each sample was hermetically closed in a Plexiglas tube to avoid
dehydration, mounted on the rotation stage, and analyzed by
means of synchrotron radiation for the image acquisition. The experimental conditions were optimized to enhance both the contrast and resolution according to Falcone and others (2004). In particular, the distance sample-detector was 200 mm, the radiation
intensity 12 keV, and the exposure time 1 s. For each sample, 1440
radiographs were acquired for equally spaced rotation angles over
a total rotation of 180°.
Once the scan data were collected, the slices (cross-sections) of
each sample were mathematically reconstructed and saved in a 32bit TIFF format by means of a set of routines written with the interactive data language (IDL) and based on the back-projection procedure (Montanari 2003). The reconstructed field of view for the
microtomographic images was 28 × 28 × 28 mm, and the nominal
resolution was 14 ␮m along each edge of the voxels in the 3D arrays. The reconstructed images were then processed using the
ImageJ software (ver.1.29, NIH 2003) to render the bread microstructure as two-dimensional (2D) stacks and 3D images.
Image processing
All the images were reduced from a 32-bit to an 8-bit format: this
step was accomplished by adjusting the brightness level so that the
histograms were uniformly distributed within 256 gray scale. In this
way, the final density of the digital information was equal to 714.48
pixels per cm.
Volumes of inter
est ((V
VOI
s)
interest
OIs)
A systematic volume of interest (VOI) selection method was
used to extract cubic volumes from the reconstructed microtomographic images. With this aim the ImageJ software (version 1.29,
NIH [2003]) was used to select the square cross-sections and, later
on, to extract the related cubic volumes within the original 3D arrays. A macro command created by prerecording a set of processing
operations allowed random extraction of sections from the slices.
Crumb cell segmentation
Before calculating the structural indices, a segmentation process
based on the threshold-value method was carried out (Falcone and
others 2004). The segregation of the gas and solid phases was simply obtained by using a single threshold value because the gray
level distribution is represented, for each microtomographic images, by a bimodal-histogram. The optimum threshold values were
determined on the entire set of the digital data in 2D stacks by
applying an iterative cluster analysis, based on the k means algoE266
JOURNAL OF FOOD SCIENCE—Vol. 70, Nr. 3, 2005
Any current method suitable for performing the anisotropy analysis from images consists in a numerical approach based on the
stereological calculus. The 1st step consisted in obtaining some
direction-dependent measurements (basic quantities) by probing
the segmented images by means of appropriate volumetric array
(3D testing sphere) of parallel test lines. According to Inglis and
Pietruszckak (2003), the directional data were used to derive an
anisotropy descriptor, that is, the mean intercept length vectors
(MILv). The 2nd step consisted in developing a 3D distribution
function useful for the visualization and interpretation of the above
fabric descriptor. With this aim, a multivariable linear least square
fitting technique was used to fit an ellipsoid to the MILv data. Finally, a MIL tensor based on the ellipsoid coefficients was defined according to Harringan and Mann (1984), and their eigenvalues and
eigenvectors were determined to derive some derivative quantities
related to the anisotropy and favorite orientations of cell structure.
Stereological analysis of bread images determined the following
structural descriptors: standard morphological indices of the
crumb microstructure such as the porosity (expressed as void volume fraction), cell wall thickness, number of cell walls and specific
cell surface; anisotropy descriptors of the cellular structure such as
the isotropic index, the elongation index, relative degrees of anisotropy, and the fractional anisotropy. The isotropic index represents
the degree of randomization of cell wall orientation and describes
the degree of deviation of 3D distribution function of MILv from a
spherical shape. The elongation index and the relative degrees of
anisotropy represent the degree of dispersion of the solid phase
around the preferential orientation of the crumb structure. The
fractional anisotropy is a more complex index that takes into account both the isotropic index and elongation index.
The above-mentioned indices were obtained from 3D images.
The porosity was also calculated on the bi-dimensional images by
using the ImageJ software.
The structural descriptors were determined by applying a suitable numerical procedure. It consisted in the adaptation of the “fabric tensor model” proposed by Harringan and Mann (1984). By
modifying an algorithm provided by Hipp and others (1996), the
anisotropy indices were automatically computed for each selected
VOI. The original algorithm has already been applied to other
spongy materials (Hipp and others 1996; Simmons and Hipp 1997),
and it is available from the site: http://www.kin.ucalgary.ca/isb.
Morphological parameters were calculated on the basis of a parallel
plate model, according to which the solid phase is made of cell walls
interconnecting pores; the cell walls have parallel sides (some of
them are open whereas other ones are closed), and the solid phase
is unevenly distributed both in the faces and the edges. As reported
in literature, this is the most suitable physical model for the representation of the bread crumb structure (Gibson and Ashby 1982).
All structure descriptors were calculated by using the equations
listed in Table 1.
Statistical analysis
The t test was performed to assess the significance of the differences detected between the considered samples. The STATISTICA
6.0 software (Stat Soft Inc. 1984-2001, Tulsa, Okla., U.S.A.) was used.
Furthermore, principal component analysis (PCA) was performed by using the above-mentioned software to cluster the
structure descriptors in a reduced number of new variables as a
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Analysis of bread crumb microstructure . . .
Symbol
Unita
Equation
Morphological descriptors
P
(*)
W_N
mm–1
W_Th
mm
W_Sp
mm
SS
b
mm2/mm3
Fabric descriptors
PrinMILi
(*)
stress-strain curves. Because both used samples have practically
the same moisture content just as the mechanical tests were run,
their different mechanical behaviors have to be attributed only to
their different cell architecture.
Compression tests were carried out by using a cylindrical probe
at a crosshead speed of 10 mm/min. The elastic modulus could be
evaluated from the initial slope of the stress-strain curve. However,
because the crumb surfaces are not perfectly parallel, the estimation of the storage modulus is generally affected by an error that
might be quite significant. For this reason, a mathematical model
capable of describing the entire stress-strain curve was used (Del
Nobile and others 2003):
(1)
where ⑀ is the engineering strain, that is, the absolute deformation divided by the initial height of the sample, ␴ is the engineering stress, EC is the elastic modulus, and Ki are some constants and
have to be regarded as fitting parameters. The goodness of fit was
evaluated by means of the relative percent difference, or mean
relative deviation modulus (Boquet and others 1978),
to the following expression:
according
Is_Ix
(*)
El_Ix
(*)
(2)
DA
(*)
where N is the number of observations, Mp the predicted values,
and
,the experimental values.
DA1
(*)
DA2
(*)
DA3
(*)
FA
(*)
a (*) indicates they are nondimensional parameters .
b S is the number of voxels corresponding to the solid phase within 3D
v
image, and T v is the total number of voxels within the same volume.
Results and Discussion
F
igure 1 is a picture of one of the slices of the white pan bread
used in this investigation. To evaluate the internal structure of
bread crumb, 3D images, and stress-strain data were obtained and
analyzed. The adopted strategy includes (1) nondestructive imaging of crumb samples by means of the PS-XRM technique; (2) evaluation of the effectiveness in data processing of a suitable numerical procedure able to perform stereological calculations; (3) image
analysis of the acquired 3D images with the aim of determining the
void volume fraction, cell wall number, cell wall thickness, cell wall
spacing, specific surface, favorite orientations of the cell structure,
and the degree of anisotropy of the solid phase around the main directions; (4) statistical analysis of the results from image analysis;
(5) compression tests of the same bread samples used for image ac-
function of their ability to describe the variance in the crumb structure. In this way, it was possible to characterize the structure descriptors that better discriminated the examined bread samples. To
this aim, the raw data were transformed in a correlations matrix.
The use of the correlation matrix allowed standardization of the
variance of raw data that were expressed on different scales. A 3factor rotation based strategy was performed to optimize PCA.
Finally, the classification analysis was used as a classification
technique, so that the relations among the original variables and
the VOIs were highlighted by plotting them in a 2D factor plane
formed by pairs of principal components chosen from the 3 principal components.
Determination of the elastic modulus
After the image acquisition, the crumb samples were submitted
to destructive compression tests by using a Universal Testing Machine (Instron mod. 4301, maximum load 100 N) to obtain the
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Figure 1—Slice of white pan bread used in the experiment
Vol. 70, Nr. 3, 2005—JOURNAL OF FOOD SCIENCE
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E: Food Engineering & Physical Properties
Table 1—Mathematical definition of both the crumb morphology and anisotropy descriptors as obtained by stereological calculations
Analysis of bread crumb microstructure . . .
quisition; and (6) analysis of the mechanical properties at continuum level as a function of their morphological characteristics.
In the following, each step in bread evaluation after image acquisition is presented.
Table 2—Results of the anisotropy analysis applied to the
VOI400 (a cubic volume 400 pixels for side, 1 pixel = 14 ␮m)
and to the VOI100 (a cubic volume 100 pixels for side, 1 pixel
= 14 ␮m) both extracted from sample 1
Evaluation of computer code
ability to calculate the porosity
P
W_N
W_Th
W_Sp
SS
␶1
␶2
␶3
To assess the effectiveness of the numerical procedure that was
used to correctly quantify the structural descriptors from images, a
validation test was carried out. Porosity was calculated on each
cross-section of a 400 × 400–pixel 2D stack related to sample 1 by
using ImageJ software and also on the corresponding 400 × 400 ×
400–pixel VOI by using our computer code. The 2 porosity levels
were then statistically compared by performing a t test. As can be
inferred from results (data not shown), the mean value of the porosity calculated by using ImageJ software on the 2D stacks did not
yield statistically different (P < 0.05) results from those calculated
on the corresponding 3D array by using our algorithm, suggesting
that the proposed numerical algorithm can be used for further applications.
Data processing
E: Food Engineering & Physical Properties
Due to the inability for the currently available software to process
digital arrays larger than 1 Gb, structural descriptors and their variation in bread images were evaluated by processing small VOIs
extracted from the reconstructed 3D arrays.
To evaluate the accuracy of the used methodology to derive representative information of a given volume from different sub-volumes extracted from it, a validation test was carried out based on a
statistical approach. A 400 × 400 × 400–pixel VOI ( VOI 400) was
cropped from the original reconstructed volumes for each crumb
sample. Then sixteen 100 × 100 × 100–pixel sub-volumes (VOI100)
were extracted from each VOI400. All VOIs were processed to calculate
the structural indices and their variations within each VOI400. Figure
2a and 3a show the reconstructed 2D images (XRM) of a cross-sections of VOIs400 corresponding to sample 1 and sample 2, respectively. These cross sections had a 400 pixel-side. In Figure 2a and 3d, the
subdivision in smaller cross-sections having a 100 pixel-side are
reported. Figure 2b and 3b show the 3D views of VOIs400 rendered
with 481 cross-sections of the investigated bread images. As one
would expect, the 2 bread samples show an apparently similar porous structure. Nevertheless, the effective void volume fraction
result is equal to 0.884 ± 0.058 and 0.743 ± 0.012 for sample 1 and
sample 2, respectively. A summary of the descriptive statistic of the
morphological descriptors for VOI400 and VOI100 related to sample 1
is reported in Table 2. Results demonstrated the effectiveness of
the proposed approach to quantify structural indices. In other
words, the evaluation of the structural descriptors made on a certain volume gives the same results obtained by processing different sub-volumes extracted from it. Similar results were obtained on
sample 2 (data not shown). As a consequence, there are possibilities to derive representative information on internal 3D structure
of a whole crumb slice or bread loaf by processing a set of sub-volumes extracted from it.
Relationship between the 2 bread
samples in terms of structure
With the purpose to obtain representative information of the
whole reconstructed arrays, 68 and 71 VOIs, having the same pixel
dimension (14 mm) and the same number of pixels for each side
(100), were randomly selected and analyzed to determine the structural indices for sample 1 and sample 2, respectively. Figure 4 shows
the map of the VOIs randomly selected within sample 1. Note that
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JOURNAL OF FOOD SCIENCE—Vol. 70, Nr. 3, 2005
Mean_VOI100
0.866
3.180
0.328
0.043
7.540
29.859
19.772
8.283
SDa
VOI400
0.068
1.207
0.193
0.014
3.427
22.318
17.712
8.144
0.860
3.016
0.285
0.046
7.008
23.406
19.771
6.636
a SD is standard deviation.
the back-projection algorithm was effectively applied to mathematically reconstruct the map of the crumb bread X-ray absorption
profile. The microtomographic images by themselves constitute
quantitative data on microstructure because the intensity of each
voxel is directly related to the nature and quantity of the material
contained in that volume.
Table 3 reports the results obtained by applying a descriptive
statistical analysis on the calculated structural indices for both sample 1 and sample 2. As can be inferred from data, some morphological descriptors such as porosity, cell wall number, cell wall thickness, specific surface of solid phase as well as some anisotropy
descriptors such as isotropy index, elongation index were statistically different between 2 investigated bread images (P < 0.05).
Relationships among the structure indices
By comparing the data from image analysis, it was possible both
to characterize the crumb microstructure and to evaluate the contribution of each structural index to the whole variability in the
bread crumb images. In particular, structural descriptors were conveniently grouped in 3 new artificial independent variables (or factors) by means of PCA of the original indices.
Table 4 reports the eigenvalues resulting from PCA and the percentage of variance that they explained. As can be inferred from
data, more than 70% of the whole variance in crumb microstructure
was explained by the 1st 3 principal components. In particular, the
1st principal component (PC1) explained the largest percentage of
the data variability (39.09%). The 2nd principal component (PC2)
explained a variance of 16.83%, and it is not correlated with the PC1.
Finally, the 3rd principal component (PC3), explained a residual
14.03% of the whole variance. By considering the weight that the
original variables have in each principal component, PC1 and PC3
have been considered as the Global Fabric and Orientation Descriptors (GFOD) and PC2 as the Global Morphological Descriptors
(GMD) of the crumb microstructure.
Figure 6 shows the projection of the original variables on the
GMD-GFOD factor plane. Based on the distance of each small circle from the specific factor axes, it was possible to evaluate the
weight of each original variable on the GMD and GFOD factors. The
sign of the factor coordinates highlights the way in which they influence the new variables. Concerning the GMD factor axis, porosity, cell wall number, and specific surfaces of solid phase showed
the highest weights. In particular, porosity and cell wall thickness
influence the amount of variability explained by the GMD factor in
an opposite way with respect to cell wall number and specific surfaces. Concerning the GFOD factor, isotropy index, elongation index, fractional anisotropy, and relative degree of anisotropy
showed the highest weights.
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Analysis of bread crumb microstructure . . .
It was evident that the examined VOIs100 were better discriminated by the GFOD factor and, as a consequence, the anisotropy
and the orientation original descriptors must be evaluated to carefully characterize the crumb structure.
Figure 6 shows the stress-strain data recorded during the compression tests performed on the 2 crumb samples. Equation 1 was
fitted to the stress-strain data to evaluate the values of model’s
parameters; the results are listed in Table 5. The calculated values
are 5.40 and 9.26 for sample 1 and sample 2, respectively, sugof
gesting that the proposed model satisfactorily fits the data. As can
be inferred from the data listed in Table 5, there is a statistically
significant difference (P < 0.01) between the 2 sets of fitting parameters including the elastic modulus.
Because the moisture content was practically the same for the 2
crumb samples, the different mechanical behavior can be ex-
Figure 2—(a) Reconstructed XRM image of a cross-section
of the VOI400 (a cubic volume 400 pixels for side, 1 pixel =
14 ␮m) extracted from sample 1. In evidence, there is the
grid used to extract subvolumes of interest (100 pixels for
side). (b) XRM 3D image of the VOI400 extracted from sample
1, rendered with 481 bi-dimensional cross-sections having a 14-␮m thickness. The threshold value used for the
cell segmentation was equal to 196.
Figure 3—(a) Reconstructed XRM image of a cross section of the VOI400 (a cubic volume 400 pixels for side, 1 pixel
= 14 ␮m) extracted from sample 2. In evidence, there is
the grid used to extract subvolumes of interest (100 pixels
for side). (b) XRM 3D image of the VOI400 extracted from
sample 2, rendered with 481 bi-dimensional cross-sections
having a 14-␮m thickness. The threshold value used for
the cell segmentation was equal to 196.
E: Food Engineering & Physical Properties
Relationships between structure indices
and mechanical properties
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Vol. 70, Nr. 3, 2005—JOURNAL OF FOOD SCIENCE
E269
Analysis of bread crumb microstructure . . .
Table 3—Results obtained by applying the t test to evaluate the statistical difference existing among the structure
indices of the 2 white pan bread samples
Mean
(sample 1)
Morphology descriptors
P
0.884
W_N
2.677
W_Th
0.658
W_Sp
0.044
SS
6.255
Fabric descriptors
IS_Ix
0.479
EL_Ix
0.427
DA
6.808
FA
0.540
DA1
0.171
DA2
–0.286
DA3
–0.335
Orientation descriptors
OR1_␪
59.100
OR1_j
128.321
OR2_␪
73.741
OR2_j
91.900
OR3_␪
138.049
OR3_j
68.796
SDb
(sample 1)
SDb
(sample 2)
Mean
(sample 2)
t value
0.798
4.606
0.248
0.044
11.835
8.265
–8.426
2.467
0.151
–8.791
136
136
136
136
136
0.000
0.000
0.014
0.879
0.000
0.058
1.256
1.338
0.014
3.224
0.062
1.425
0.376
0.007
4.158
0.524
0.382
4.238
0.511
0.148
–0.246
–0.287
–2.266
2.317
2.031
1.493
2.563
–2.148
–2.165
136
136
136
136
136
136
136
0.025
0.021
0.043
0.137
0.011
0.033
0.032
0.135
0.135
10.158
0.1200
0.063
0.121
0.157
0.094
0.090
2.901
0.101
0.039
0.091
0.099
63.914
116.929
69.356
92.612
144.508
59.344
–0.712
1.570
0.528
–0.153
–0.791
1.113
136
136
136
136
136
136
0.477
0.118
0.597
0.877
0.428
0.267
38.130
42.213
49.603
28.429
50.234
50.072
41.080
42.995
47.830
25.957
45.550
49.688
df
Pa
a If P < 0.05, mean values are significantly different with a confidence level equal to 95%
b SD is standard deviation.
E: Food Engineering & Physical Properties
plained with the differences in the arrangement of crumb. By comparing results from mechanical tests and those from image analysis, it was possible observe that crumb elastic properties could be
closely related to the structural characteristics. In particular, sample
2, which had the highest elastic modulus, was characterized by the
lowest porosity (or the highest density), the highest values of cell
wall number, and specific surface of solid phase. Moreover, both
samples showed an orthotropic anisotropy that is a preferential
distribution of the solid phase in the space. In particular, all VOIs
extracted from sample 2 showed the highest anisotropy degree as
expressed by the elongation and the isotropic indices. Figure 7 reports a bivariate plot of the primary versus tertiary eigenvalues
relative to the MIL tensor. It is evident a significant and different
trend in anisotropy of crumb samples corroborating a preferential
and different orientation of the solid phase between 2 bread images. For sample 2, the preferential orientation that resulted aligned
to that of the compression test.
Conclusions
T
he 2D imaging techniques allow determination of simple geometric parameters only, such as the void volume fraction, the
porosity, and the specific surface area, but are not able to evaluate
Figure 4—Scheme of the sampling made to extract VOI100
(a cubic volume 100 pixels for side, 1 pixel = 14 ␮m) from
sample 1 images.
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JOURNAL OF FOOD SCIENCE—Vol. 70, Nr. 3, 2005
Figure 5—Bivariate fit of the original variable coordinates.
GMD is the global morphological descriptor whereas GFOD
is the global anisotropy descriptor of the crumb structure.
URLs and E-mail addresses are active links at www.ift.org
Analysis of bread crumb microstructure . . .
Table 4—Eigenvalues of the correlation matrix and results
of the PCA analysis applied to the microstructure descriptors
Eigenvalue
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
6.6460
2.8612
2.3857
1.8110
1.1595
0.6644
0.4535
0.4251
0.2305
0.1295
0.0955
0.0538
0.0381
0.0295
0.0123
0.0027
0.0009
% Total
variance
39.0946
16.8309
14.0338
10.6530
6.8209
3.9084
2.6679
2.5008
1.3562
0.7617
0.5620
0.3167
0.2245
0.1737
0.0724
0.0161
0.0056
Cumulative
eigenvalue
6.6460
9.5073
11.8931
13.7041
14.8636
15.5281
15.9816
16.4068
16.6373
16.7668
16.8624
16.9162
16.9544
16.9839
16.9963
16.9990
17.0000
% Cumulative
explained
variance
39.0943
55.9253
70.0593
80.6122
87.4334
91.3418
94.0097
96.5106
97.8668
98.6286
99.1906
99.5074
99.7320
99.9058
99.9782
99.9943
100.0000
Table 5—Results of fit performed on stress-strain data a
Parameter
EC [Pa]
K1
K2
K3
K4
Sample 1
Pb
Sample 2
3.23 × 105
6.43 × 105
[2.96 × 105, 3.50 × 105] [6.06 × 105, 6.81 × 105]
2.60
6.46
[2.05, 3.11]
[6.12, 6.76]
–1.35
–2.71
[–1.65, –0.978]
[–2.79, –2.61]
7.76 × 10–3
1.35 × 10–2
[6.64 × 10–3, 9.71 × 10–3] [1.27 × 10–2, 1.4 × 10–2]
1.23 × 107
1.68 × 107
[1.09 × 107, 1.40 × 107] [1.62 × 107, 1.75 × 107]
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
a The values in brackets represent the confidence interval, which was
calculated as mean ± 1.96 × SE (where SE is the standard error).
b If P < 0.05, mean values are significantly different with a confidence level
equal to 95%.
structure trough image analysis. As for the other porous materials,
an interesting long-term perspective of the 3D imaging of foamed
foods is the possibility of predicting some functional characteristics
(elastic properties, for example) directly from their structure, by
means of the finite element analysis of the digital arrays for example.
P
W_N
W_Th
W_Sp
SS
␻I
Figure 6—Stress-strain curves showing the observed and
calculated mechanical behavior of the sample 1 and
sample 2 under the compression tests. 䊊, sample 1; 䉱,
sample 2; ——, best fit of Eq. 1 to the sample 1 data; – – –
–, best fit of Eq. 1 to the sample 2 data.
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Number of solid voxels. It corresponds to the solid phase
(crumb) within the 3D testing sphere used in the
stereological calculus.
Number of void voxels. It corresponds to the void phase
(air) within the 3D testing sphere.
Total number of voxels. It was calculated by summing
and .
Crumb density (solid volume fraction). It was calculated
as ratio between and .
Crumb porosity (void volume fraction)
Cell wall number
Cell wall thickness
Cell wall spacing
Specific surface. It is the solid phase surface per unit volume of crumb.
Sampling directions of 3D testing sphere used in stereological calculus.
Figure 7—Trend in the anisotropy degree of the VOIs within
sample 1 (red line) and within sample 2 (blue line).
Vol. 70, Nr. 3, 2005—JOURNAL OF FOOD SCIENCE
E271
E: Food Engineering & Physical Properties
List of symbols
3D information, such as the anisotropy indices of inner structure.
The 3D analysis represents a new approach to the study of the
foamed food structure. In addition to the crumb porosity, some directionally dependent parameters strictly influenced the mechanical behavior of the bread crumb. The proposed numerical procedure effectively allows determination of the void volume fraction,
some anisotropy descriptors, and the preferential orientation of the
cellular structure. A quantification of these additional 3 dimensional indices is necessary to characterize the crumb microstructure. The
anisotropy and favorite orientation descriptors explain at least the
53.12% of whole structural variance. Moreover, as can be inferred
from the stress-strain data, all structural parameters change consistently with the mechanical behavior of samples.
These findings indicate that there are possibilities to find application for the proposed methodology to evaluate the bread micro-
Analysis of bread crumb microstructure . . .
N(␻i)
E: Food Engineering & Physical Properties
Number of interfaces between solid and void phase along
the ␻i within the 3D testing sphere. This parameter is a
directionally dependent basic quantity obtained from
the stereological analysis, and it was calculated as the
number of intersections between parallel test lines and
solid phase as represented in digital arrays.
L
Total length of test lines within the 3D testing sphere.
This parameter is not a directionally dependent basic
quantity.
Primary eigenvalue of the MIL tensor. This parameter is
␶1
related to the primary axis magnitude of the ellipsoid
used to describe the 3 dimensional distribution of the
solid phase in the bread images.
Secondary eigenvalue of the MIL tensor. This parameter
␶2
is related to the secondary axis magnitude of the ellipsoid.
Tertiary eigenvalue of the MIL tensor. This parameter is
␶3
related to the tertiary axis magnitude of the ellipsoid.
PrinMIL1 Magnitude of primary MILv vector individuated by the
ellipsoid in volumetric arrays. This parameter is related
to the amount of solid phase along the primary favorite
orientation of cell structure.
PrinMIL2 Magnitude of secondary MILv vector individuated by
the ellipsoid in a volumetric arrays. This parameter is related to the amount of solid phase along the secondary
favorite orientation of cell structure.
PrinMIL3 Magnitude of tertiary MILv vector individuated by the
ellipsoid in a volumetric arrays. This parameter is related
to the amount of solid phase along the tertiary favorite
orientation of cell structure.
PrinMILm Mean value among 3 PrinMIL.
Is_Ix
Isotropic index. This parameter indicates the deviation
from the spherical case of the distribution function (ellipsoid).
El_Ix
Elongation index. This parameter indicates the flatting
degree of the distribution function (ellipsoid) along the
tertiary favorite orientation of cell structure.
DA
Relative degrees of anisotropy. This parameter indicates
the flatting degree of the distribution function (ellipsoid) along the secondary favorite orientation of cell
structure.
DA1
Relative degrees of anisotropy. This parameter indicates
the mean dispersion degree of the solid phase along the
primary favorite orientation of cell structure.
DA2
Relative degrees of anisotropy. This parameter indicates
the mean dispersion degree of the solid phase along the
secondary favorite orientation of cell structure.
DA3
Relative degrees of anisotropy. This parameter indicates
the mean dispersion degree of the solid phase along the
tertiary favorite orientation of cell structure.
FA
Fractional anisotropy. It is a complex index of anisotropy
and was defined according to Matusani and others
(2003). This parameter takes into account both the isotropic index and elongation index.
OR1_␪ Colatitude [␪] of the primary favorite orientations of cell
structure with respect to a polar coordinate system.
E272
JOURNAL OF FOOD SCIENCE—Vol. 70, Nr. 3, 2005
OR1_j
OR2_␪
OR2_j
OR3_␪
OR3_j
Ki
Ec
N
Longitude [j] of the primary favorite orientations of cell
structure with respect to a polar coordinate system.
Colatitude [␪] of the secondary favorite orientations of cell
structure with respect to a polar coordinate system.
Longitude [j] of the secondary favorite orientations of cell
structure with respect to a polar coordinate system.
Colatitude [␪] of the tertiary favorite orientations of cell
structure with respect to a polar coordinate system.
Longitude [j] of the tertiary favorite orientations of cell
structure with respect to a polar coordinate system.
Fitting parameters
Elastic modulus
Mean relative deviation modulus
Number of observations
Experimental values
Predicted values
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