Assessment by MRI of local porosity in dough during proving. theoretical considerations and experimental validation using a spin-echo sequence

Magnetic Resonance Imaging 21 (2003) 1071–1086

Assessment by MRI of local porosity in dough during proving. Theoretical considerations and experimental validation using a spinecho sequence A. Greniera,b,*, T. Lucasa, G. Colleweta, A. Le Bailb a

Cemagref Food Process Engineering Research Unit, Rennes Cedex 35044, France b ENITIAA-GEPEA (UMR CNRS 6144), Rennes Cedex 35044, France

Received 15 November 2002; received in revised form 22 April 2003; accepted 23 April 2003

Abstract Proving is a key stage in the development of the final structure of bread, as invasive measurements may provoke dough collapse. Therefore, better understanding and better control of the nucleation and the growth of bubbles require the development of non-invasive methods of measurement. In the present work, a non-invasive method is presented for the measurement of local dough porosity from MR image analysis. For this, a direct relation between the gray level of a voxel and its gas fraction was established in the absence of heat and mass transfer. At whole dough scale for a one-dimensional expansion, the porosity estimated from the gray level was compared with the porosity estimated from total dough volume measurements in a range of [0.10, 0.74 m3 of gas/m3 of dough]. For short proving times (⬍30 min), MR image analysis underestimated porosity by a maximum of 0.03 m3 of gas/m3 of dough, but otherwise the difference between the two means of measurement was within the standard error of total dough measurements (⫾0.01 m3 of gas/m3 of dough). Maps of local porosity in dough during proving are also presented and discussed. © 2003 Elsevier Inc. All rights reserved. Keywords: MRI; Bread dough; Bread proving; Dough volume; NMR relaxation

1. Introduction The final character of most bakery products depends to a significant extent on the creation and control of gas bubble structures in the unbaked matrix and the retention of these gas bubbles in a suitable form until the matrix becomes set or baked. In baked products, the small holes in the crumb, commonly referred as the “crumb cell structure”, contribute to texture, eating quality, mechanical strength and perceived product freshness as well as to visual appearance. To obtain this foam structure, it is necessary to process the ingredients in an appropriate way starting with mixing and continuing through to leavening (proving), cooking and cooling. Simply speaking, during the mixing process, the main ingredients, i.e., flour and water, are stirred together to form the dough with strong mechanical properties arising from the formation of a three-dimensional network due essentially to water adsorption. Air may also be incorpo* Corresponding author. Tel.: ⫹33-2-23-48-21-21; fax: ⫹33-2-23-4821-15. E-mail address: [email protected] (A. Grenier). 0730-725X/03/$ – see front matter © 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0730-725X(03)00194-2

rated at this stage; its volume fraction commonly ranges from 4 to 15% depending on the mixing process and the dough formulation, and may reach up to 20% in very specific cases [1–3]. Such air bubbles are assumed to be the initial nuclei for the development of gas bubbles [4]. The second stage of the process is proving, when rising agents react. Roughly speaking, the metabolism of the micro-organisms chemically transforms the polysaccharides into carbon dioxide and ethyl alcohol. Carbon dioxide then diffuses through the paste, reaching the initial air bubbles, which then grow. The final result is an increase of the specific volume of the dough as the elastic properties of the dough network help in retaining the gas bubbles. The general trend could be described as follows. During short proving times (t ⬍ 20 min), volume increases no more than 10%. This corresponds to a period of latency in gas production and/or diffusion through the liquid phase of dough to air nuclei [5]. Afterwards, the volume increases linearly. For long proving times the expansion rate decreases progressively, depending on flour quality and the process parameters. In fact, the dough can no longer retain the gas additionally produced

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during proving and the dough volume converges to its maximum value for prolonged proving [6– 8]. The end of proving is marked by an expansion up to three or four times its initial volume [9], which is equivalent to porosities of 0.70 and 0.78 respectively (in m3 of gas per m3 of dough, calculated with an initial porosity of 0.1). Once this volume is reached, the dough is introduced into an oven and the product is baked. A dramatic volume increase is observed due to the thermal increase of the water pressure and to an increase in the rate of carbon dioxide production. After a certain temperature, the latter contribution disappears as a consequence of the thermal inactivation of the metabolic reactions. When the baking is concluded the bread is cooled and then packed. In the following, only the proving step will be discussed. As developed above, proving is a key stage in the development of the final structure of bread. As invasive measurements may provoke dough collapse, the characterization of the expansion process has been reduced to a number of global volumetric parameters (dough volume, CO2 volume, etc.) [10]. But better understanding and better control of the nucleation and growth of bubbles require the development of non-invasive methods of measurement, with the characterization of bubble size and distribution [11]. Its feasibility at different scales of observation has been recently investigated with: X-ray tomography [12], confocal microscopy [8] and Magnetic Resonance Imaging (MRI) [13]. Many of these previous works give only qualitative information. Whitworth et al. [12] focus on the internal deformations of the dough on expansion by analyzing the movement of zones of increased density and the changes in shape of large bubbles. Takano et al. [13] relate the intensity of the MRI signal to the proportion of CO2 without any form of demonstration and discuss the structure of bubbles (size, connections) as discerned from visual analysis. No quantitative data has been presented from the latter work. A quantitative approach has been developed using confocal microscopy images (640 ⫻ 640 ␮m), yielding the number of bubbles and the total area occupied by them [8]. Only gas cells close to the dough surface can be observed with such a technique, which is a first limitation as the behavior of bubbles at the dough surface may not be typical of the bubbles in the bulk of the sample. Another limitation is encountered when attempting to extend the conclusions drawn at a microscopic scale to the dough scale; especially with the omission of bubbles of large size. The present work aims at evaluating the performance of MRI and is focused on its possibilities for quantifying local porosity in dough.

2. Determining dough porosity with MRI intensity: theoretical considerations and procedure for experimental validation The three-dimensional referential of the MRI equipment is used. The x- and y- axes refer to respectively the hori-

Fig. 1. Location of the MRI slice and view from above.

zontal and vertical positions in the 2D image; the z- axis refers to the image depth (see Fig. 1). The dough system is considered as gas bubbles contained in a paste. The dough volume measured by MRI, noted V(t), is a slice characterized by its thickness ⌬z and its area Axy(t). The latter changes with the proving time noted t, with t ⫽ 0 corresponding to the end of mixing. The volume V(t) is divided into n(t) voxels, defined by a square section (⌬x ⫽ ⌬y) and a thickness ⌬z. Each voxel is represented by a number i, i ⑀ IN. The MRI slice volume V(t) is defined by: V(t) ⫽ n(t) Vi ⫽ n(t) ⌬x2 ⌬z

(1)

where Vi is the volume of the voxel i. During proving, the voxel volume is composed of a gas fraction (mainly CO2) and a paste fraction: Vi(t) ⫽ Vi,gas(t) ⫹ Vi,pa(t)

(2)

where Vi,gas(t) and Vi,pa(t) define the volume occupied by gas and paste respectively in the voxel i. The gas volume fraction in the voxel i is called porosity and defined in m3 of gas at t per m3 of dough at t: ␧i(t) ⫽

Vi,gas(t) Vi(t)

(3)

By deduction, the paste volume fraction is 1 ⫺ ␧i(t). A gray level value noted GLi(t) is attributed to each voxel i. In the case of a spin-echo sequence and considering that the voxel i is full of paste, the gray level, noted GLi,pa(t), is then proportional to: GLi,pa(t) ⬵ K ␳(t)Vi C⬜,i TE

冉

␥2ប 2B0,iI(I ⫹ 1) 4kB␪ TR

冊

䡠 e⫺T2(t) 1 ⫺ e ⫺T1(t) ⫹ B(t)

(4)

where K is the gain, C⬜,i the antenna sensitivity at position i, ␳(t) the proton density, ␥ the gyromagnetic coefficient, ប the Plank constant (6.6256 10⫺34 J.s), B0,i the permanent field, I the quantum spin number, kB the Boltzmann constant (1.3806 10⫺23 J.K⫺1), T2(t) the transversal relaxation time (ms), T1(t) the longitudinal relaxation time (ms) and ␪ the temperature (K) and B(t) the noise inside the MR image. ␳(t) Vi represents the quantity of protons contained in the voxel i. The NMR properties of the dough, ␳(t), T2(t) and

A. Grenier et al. / Magnetic Resonance Imaging 21 (2003) 1071–1086

T1(t), may vary during proving as its temperature, its component concentrations and even its composition may vary under enzymatic and/or metabolic reactions and heat or mass transfer with its environment. At last, Eq. (4) takes into account that the same paste (same NMR properties) in two voxels with different spatial co-ordinates may not give the same gray level as B0,i and C⬜,i may vary in the threedimensional referential. These could be generated by different sources of inhomogeneity including B0 inhomogeneity, bandwidth filtering of data, RF transmission and reception, RF standing waves and RF penetration effects [14]. In the following of this section, we attempt to establish a direct relation between the gray level (GLi) of a voxel i (composed of both paste and gas) and its gas fraction (␧i). This is equivalent to trying to establish a relation between the gray level and the paste volume fraction, as the latter is deduced from the gas volume fraction. Likewise, the gray level of the paste is known to be proportional to the paste volume fraction contained in the voxel in question (see Eq. (4)) provided that all other contributions are known or kept constant during proving. It is focused on this very last point in the next paragraphs. The absence of interaction between the gas and the MRI signal of paste is assumed (H1). A centered Gaussian distribution of the electronic noise of ␴Noise dispersion is assumed (H2). This is verified if [15]: GLBackground(t) GLi(t) ⱖ 5 with ␴Noise(t) ⫽ ␴Noise(t) 冑␲/2

(5)

where the electronic noise mean, GLBackground(t), is evaluated from the background of the MR image. In the following, the contribution of electronic noise to the MRI signal will thus be neglected (see Eq. (4)). Assuming hypotheses H1 and H2, the gray level of a voxel i composed of both paste and gas, whatever the proportion of gas, is proportional only to the proton material contained in the voxel. During proving, the upwards expansion of the dough due to the development of bubbles implies that paste will move from one voxel to another (see Fig. 2). Such movement cannot be predicted. The paste is then considered to be homogeneous through the whole dough (H3) but can still vary uniformly according to time t. It is additionally assumed that the MRI signal does not vary spatially due to magnetic field inhomogeneities, non-linearity of the gradients, the sensitivity of the antenna, etc. (H4). Subject to the validity of the hypotheses mentioned above (H3, H4), the MRI signal given by a voxel full of paste is uniform through the whole dough at a given time t: @i, @t ⱖ 0, GLi,pa(t) ⫽ GLpa(t)

(6)

The gray level of the voxel i composed of both gas and paste can now be defined as: GLi(t) ⫽ [1 ⫺ ␧i(t)]GLpa(t)

(7)

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Fig. 2. Schematic development of bubbles observed through a voxelshaped window of which spatial co-ordinates are fixed whereas the dough is allowed to expand in the upward direction. Dashed lines symbolise the interface between gas and paste which cannot be observed through the defined window at time t. Arrows symbolise the hypothetical motion of paste due to the volume expansion of dough located below this window of observation and to the compression forces exerted by the growing bubbles. This means that the paste observed at an early time in the window will have partially moved at upwards locations as proving proceeds (and in much less proportion to left-handed or right-handed locations). For prolonged proving times, the paste observed in the window is then a mix of the paste observed at early times and of new paste previously located below the window.

In the right-hand term of Eq. (7), [1 ⫺ ␧i(t)] decreases with the proving time, which induces a decrease in the gray level of the voxel i. GLpa(t) takes into account the variation in the NMR properties of paste (T2(t), T1(t) and ␧(t) in Eq. (4)) resulting from changes in temperature, (water) concentration or composition (enzymatic reaction) which may possibly occur during the proving process. It is worth noting that it is virtually impossible to assess the gray level of paste (GLpa) in the complete absence of bubbles as the mixing step always incorporates air bubbles. If it is now assumed to remain constant with time i.e., temperature or concentration variations or enzymatic reactions do not modify the MRI signal of paste (H5): @t ⱖ 0, GLpa(t) ⫽ GLpa

(8)

then the determination of local porosity from the gray level GLi no longer depends on the gray level of paste, but on the knowledge of GLi and ␧i at a reference time: 1 ⫺ ␧i(t) GLi(t) ⫽ GLi(tref) 1 ⫺ ␧i(tref)

(9)

The porosity of each voxel i during proving can then be estimated as follows: ␧i(t) ⫽ 1 ⫺ [1 ⫺ ␧i(tref)]

GLi(t) GLi(tref)

(10)

In practice, the end of mixing (t ⫽ 0) is used as the reference time. The issue is the estimation of the distribution of reference porosity over the whole dough. The voxel volume (⬎ 1 mm3) is assumed to be sufficiently great with respect to bubble size, which is between 10 and 100 ␮m in diameter at the end of mixing, assuming bubbles of spherical shape [16,17]. This results in assuming that at the MRI scale of observation the gas fraction is uniformly distributed at the reference time (H6):

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A. Grenier et al. / Magnetic Resonance Imaging 21 (2003) 1071–1086

@i, ␧i(tref) ⫽ ␧(tref)

(11)

This new assumption also yields to: @i, GLi(tref) ⫽ GLpa[1 ⫺ ␧(tref)]

(12)

@i, GLi(tref) ⫽ GL(tref)

(13)

Finally, Eq. (9) becomes: 1 ⫺ ␧i(t) GLi(t) ⫽ GLi(tref) 1 ⫺ ␧(tref)

(14)

So provided that hypotheses H1-6 are satisfied and that the initial porosity ␧(tref) is known, the porosity of a voxel i can be estimated from its gray level during proving: ␧i(t) ⫽ 1 ⫺ [1 ⫺ ␧(tref)]

GLi(t) GLi(tref)

(15)

Eq. (15) cannot be validated easily, if ever. A volume balance can be applied only at dough scale i.e., if the whole dough system is considered. At this scale, the mean porosity ␧៮ (t) can easily be estimated by means of volumetric measurements: ␧៮ i(t) ⫽

V(t) ⫺ V(tref)[1 ⫺ ␧(tref)] V(t)

(17)

Replacing Eq. (16) into Eq. (17) yields to V(tref) GL(t) ⫽ GL(tref) V(t)

MR image. Indeed an error on the number of voxels may in turn induce an error on the calculated mean gray level:

冘 GL (t)

i⫽n(t)

i

(16)

where V(t) is the dough volume and V(tref)[1 ⫺ ␧(tref)] is the volume occupied by the dough in the complete absence of gas bubbles. V(t) ⫺ V(tref)[1 ⫺ ␧(tref)] then represents the volume occupied by gas at time t in agreement with Eq. (3). Here the volumes V(t) and V(tref) are estimated from MRI measurements (see Eq. (1), but volume can be assessed by any other measurement technique. Then, only the generalisation of Eq. (15) to the total dough volume gives the opportunity to validate experimentally the pertinence of the gray level parameter for determining dough porosity. Still provided that initial porosity is assessed, the mean gray level at time t noted GL(t) is directly related to the mean porosity at time t: 1 ⫺ ␧(t) GL(t) ⫽ GL(tref) 1 ⫺ ␧(tref)

Fig. 3. Experimental device.

(18)

This new relation is of major importance as it will allow the validation of the present method without having to estimate the reference porosity. Validation of the present method will be processed step by step. The first step (see Estimation of the number of voxels in the dough MRI slice by thresholding the MR images in the Results and Discussion Section) will be to verify the accuracy of the number of voxels assigned to the dough once the segmentation process has been performed on the original

@t ⱖ 0, GL(t) ⫽

i⫽1

n(t)

(19)

If the gray level value is uniform in all voxels in the dough, the assignment of excess voxels from the background will tend to decrease the mean gray level value artificially whereas omitting voxels really belonging to the dough will have no effect. In the case of proving, the gray levels are distributed around the mean value. The effect on the gray level mean value of omitting voxels from the dough or of adding them to it becomes more complex and almost impossible to predict as it influences both the n(t) and

冘

i⫽n(t)

GLi(t) terms.

i⫽1

The pertinence of the segmentation method can be achieved by comparing the number of voxels assigned to the dough to other volumetric measurements. As the slice thickness, FOV and matrix parameters remain constant, the volume of the dough slice measured by MRI is dependent only on the number of voxels (see Eq. (1)). Assuming that the matrix grid perfectly superimposes on the dough outlines (no partial volume effect in the voxels of the outlines) and assuming a perfect one-dimensional expansion of dough in a cylindrical device (H7), as illustrated in Fig. 3: @x, @z, @t, H(x,z,t) ⫽ H(t)

(20)

then the selected volume V(t) can be also expressed: V(t) ⫽ ⌬z 2RH(t)

(21)

Dividing the MRI slice volume at t by the MRI slice volume at tref yields to:

A. Grenier et al. / Magnetic Resonance Imaging 21 (2003) 1071–1086

H(t) V(t) ⫽ V(tref) H(tref)

(22)

Considering now the total volume of the dough of cylindrical shape noted VT(t): VT(t) ⫽ ␲R 2H(t)

(23)

and still assuming one-dimensional expansion (H7), dividing the total dough volume at t by the total dough volume at tref gives: H(t) VT(t) ⫽ VT(tref) H(tref)

(24)

Assuming there is one-dimensional expansion of the dough, the volume ratio calculated between times t and tref from MR images (see Eq. (22)) is thus representative of the volume ratio from total dough volume measurements taken at the same times (see Eq. (24)). This will be a key point in the validation of the segmentation method. In the second step (see Constancy of the gray level of paste (validation of hypotheses H1 to H5) in the Results and Discussion Section), the validity of the assumptions made when developing the method (hypotheses 1 to 5) will be verified. Eq. (18) is further developed below to highlight one of its properties, which will make possible the achievement of this second objective in a quantitative way. Replacing into Eq. (18) the expressions of V(t) and of V(tref) given by Eq. (1) yields to: n(tref) GL(t) ⫽ GL(tref) n(t)

(25)

non-conservation of the sum of gray levels and will be demonstrated in this way. In the last step (see Determination of the gas content from the MRI signal during proving in the Results and Discussion Section), the mean porosity calculated from the mean MRI gray level (deduced from Eq. (17)): ␧៮ (t) ⫽ 1 ⫺ (1 ⫺ ␧(tref))

(28)

冑

冘冋冉⭸X⭸␩ 冊 u 册 2

i⫽k

uY⫽

2 Xi

i

i⫽1

(29)

Applied to Eq. (29) and assuming that all parameters are independent, it yields to:

冑

冉 冋

u␧៮ (t) ⫽

GL(t) GL(tref)

冊

2

u␧2៮ (tref) ⫹

冋 册

1⫺␧៮ (tref) GL(tref)

[1⫺␧៮ (tref)]

GL(t)

GL2(tref)

2

册

2

2

uGL(t)

(30)

2

uGL(tref)

The mean gray level uncertainty uGL(t) depends on GLi(t) and n(t) uncertainties (see Eq. (19)): @t ⱖ 0, uGL(t)

GL(t) n(t) ⫽ GL(tref)n(tref)

(26)

⫽

where GL(t) n(t) is the sum of gray levels for all voxels

冘 GL (t) and the gray level in dough

冉冤 冉 冘 冊冊 ⭸ ⭸n

i

i⫽1

at the reference time is defined in Eqs. (12) and (13). Eq. (26) becomes:

冘 GL (t) ⫽ (1 ⫺ ␧(t

ref))GLpa

GLi(t)

i⫽1

@t ⱖ 0,

n(tref) (27)

Thus, provided that the number of voxels is accurate, the sum of gray levels is expected to remain constant throughout the proving process. As n(tref) and ␧(tref) are undoubtedly fixed, such constancy relies only on hypotheses 1 to 5, which assume that the gray level of paste is constant in space and time. In other words, any changes in the relaxation behavior of the paste or any inhomogeneity of the magnetic fields occurring during proving should result in

i

⫺

i⫽1

n2(t)

1 2 u n(t) GLi(t)

冥

2

u2n(t) (31)

冉 冉 冘 冊冊 ⭸ ⭸n

i⫽n(t)

GLi(t)

i⫽1

冘 GL (t)⫺ 冘

i⫽n(t)

i⫽1

冘 GL (t)

i⫽n(t)

The following approximation is made:

i⫽n(t)

i

i⫽n(t)

n(t)

⫹

i⫽n(t)

GL(t) n(t) ⫽

GL(t) GL(tref)

will be compared to the mean porosity calculated from total dough volume (see Eq. (16)), and the different contributions to its uncertainty will be evaluated from the following equations. Given Y ⫽ ␩(X1, X2,. . ., Xk) with X1, X2, . . . , Xk independent, the uncertainty of the parameter Y noted uY is given by:

Isolating the time-dependent parameters in the left-hand term of Eq. (25) gives:

attributed to dough:

1075

⬇

i⫽n(t)⫺1

i

i⫽1

@t ⱖ 0,

冉 冉 冘 冊冊 ⭸ ⭸n

GLi(t)

(32)

i⫽1

i⫽n(t)

GLi(t) ⬇GLn(t)

(33)

i⫽1

The nth voxel would undoubtedly belong to the outline voxels. It assumed that its gray level value is in average half of the value of the mean gray level (due to partial volume effect) (H8):

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A. Grenier et al. / Magnetic Resonance Imaging 21 (2003) 1071–1086

冉

冉 冘 GL (t)冊冊 ⫽ 0.5 GL(t)

⭸ ⭸n

i⫽n(t)

i

(34)

i⫽1

This yielded to: @t ⱖ 0, uGL(t)

冤

⫽ ⫺0.5

冘 GL (t)

i⫽n(t)

i

i⫽1

n2(t)

flask. At the same time, the last flask was placed in the measurement area for MRI, the temperature of which was controlled at 26.0 ⫾ 0.3°C. At the end of proving, the dough temperature was checked and the pots were weighed (m(tf)). The experiment was repeated twice (3 runs).

冥

3.3. Data measurements and analysis 2

1 2 u2n(t) ⫹ u n(t) GLi(t)

(35)

3. Materials and methods 3.1. Dough preparation and description of the experimental device All experiments were carried out with the same flour (⬇11.3% of protein, 15% of water and 0.66% of ash). The dough samples were obtained by mixing 500 g of flour, 290 g of water, 12.5 g of salt and 8 g of dry baking yeast in a Kenwood mixer for 4 min at 40 rpm followed by 11 min at 80 rpm. Before incorporation, the yeast was hydrated for 15 min at 35°C in 290 g of water previously mentioned. Just after mixing, dough temperature and water content were checked. They were 24.0 ⫾ 0.4°C and 45.3 ⫾ 0.2% wet basis (wb) respectively. After mixing, four cylindrical flasks were filled with dough (40.1 ⫾ 0.2 g), closed, weighed (m (0)) and placed in a temperature-controlled environment for proving. All these operations were carried out at ambient temperature (21°C) within 9 min. The experimental device is described in Fig. 3. When each flask was closed, a rigid cover was gently placed on top of the dough sample, so that the dough surface stuck to it and was flattened to form as horizontal a plane as possible. The mass of the cover was the same for all samples (11.4 ⫾ 0.1 g). The cover had a twofold purpose: a) to limit dehydration by isolating the dough from contact with the air and b) to ensure onedimensional expansion of the dough, as a rod was connected to it and passed through the lid of the flask, forcing the cover to remain in a horizontal position. For NMR measurements, a tube (Ø ⫽ 8 mm, 15 cm height) was filled with dough (3 mm high) and weighed. The initial height of the sample was adjusted to ensure that at the end of proving it would not exceed the threshold height (10 mm) above which the magnet homogeneity could no longer be guaranteed. 3.2. Experimental procedures At t ⫽ 9 min after the end of mixing, four flasks were placed in their respective controlled environments for proving. Three of them were placed in a temperature controlled laboratory room. Temperature was regulated at 25.6 ⫾ 0.1°C. At different times of proving, the total dough volume noted VT(t) was measured by reading graduations on the

3.3.1. Water content The dry matter was determined on 5 g of dough in a drying oven at 104°C for 24 h. The measurement was replicated twice (3 replicates). 3.3.2. Water loss Water loss during proving was expressed in g of water per 100 g of initial dough: WL ⫽

m(tf) ⫺ m(0) m(0)

(36)

3.3.3. Temperature The temperature of the proving rooms at different times during proving and the dough temperature at the end of proving were collected using a HP Data Acquisition Unit 34970A connected to a PC. Temperatures were recorded every 30 s, with calibrated T-type thermocouples (0.1 mm in diameter). 3.3.4. NMR A Bruker PC 120 Minispec (Brucker SA F-67166, Wissembourg), with a 0.47T magnetic field operating at a resonance frequency of 20 MHz was used for NMR measurements. The probe was temperature-controlled (25.6°C) by circulating fluorocarbon (FC Fluorinert from 3M). At different times of proving, the relaxation signal was acquired with a Carr-Purcell-Meiboom-Gill (CPMG) sequence for the T2 and a saturation recovery (SR) for the T1. At the end of proving the sample was weighed (m(tf)). Sequence parameters were: relaxation delay 3 s, respectively 845 and 100 points for the CPMG and the SR sequences. Phase sensitive detection (PSD) was used. 3.3.5. MRI The MR images were acquired on a SIEMENS OPEN 0.2T imager and with a spin-echo sequence with the following parameters: TE ⫽ 11 ms, TR ⫽ 300 ms, number of averages ⫽ 2, FOV ⫽ 175 ⫻ 175 mm2, matrix size ⫽ 96 ⫻ 128 (rectangular FOV: 6/8), slice thickness ⫽ 12 mm. This implied a pixel resolution of 1.37 mm (FOV/matrix size). The FOV was selected to cover the dough height until the end of proving. The acquisition time was 1 min 01 s. The proving time attributed to the MR images was centred with respect to the sequence duration and calculated as: tref ⫽

␦t ⫹ ␦t1 2

@tk ⬎ tref, tk ⫽ tk⫺1 ⫹

␦t ⫹ ␦t2 2 (37)

A. Grenier et al. / Magnetic Resonance Imaging 21 (2003) 1071–1086

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where ␦t, ␦t1, ␦t2 and tk⫺1 were respectively the sequence duration, the delay between the end of mixing and the first load of the sequence, the delay between the end of the k⫺1th acquisition and the very beginning of the kth acquisition, and the time of the previous MR image acquisition.

porosity uncertainty can be estimated according to the set of equations below:

3.3.6. Calculation of the number of voxels in the dough slice by thresholding the MR images To select the area corresponding to the object under study (the dough), the Otsu method was applied [18]. This method is a nonparametric and unsupervised method of automatic threshold selection for picture segmentation. According to the gray level histograms, the Otsu method makes it possible to dichotomise the voxels into two classes: the background and the object, with a threshold separating these two classes. The choice of threshold value relies on minimization of within-class variance and maximization of between-class variance. If the value of a voxel under consideration was superior or equal to the threshold value, the voxel was taken into account, otherwise it was masked. The selected area was then submitted to a dilation-erosion, making it possible to close the dough outlines.

with uGLn(t) given by:

3.3.7. Normalization of the dough MR images using a phantom Spin echo MR images can be corrected for nonuniformity by dividing them by a phantom MR image of oil acquired with a TR greater than or equal to 1000 ms and approximately the same TEs [14,19]. For the phantom MR images, a cylinder (Ø ⫽ 100 mm, 12 cm height) was filled with vegetable oil stabilized at 16°C for one day. The phantom MR images were acquired with a spin echo sequence with the following parameters: TE ⫽ 11 ms, TR ⫽ 1000 ms, number of averages ⫽ 10. The slice position and the FOV were identical to those used for the dough MR images. The so-called normalized gray level was calculated from the original gray level value as follows: @t ⱖ 0, GLi,n(t) ⫽ GLc,p

GLi(t) GLi,p

(38)

where GLi,n(t) is the normalized gray level of the voxel i, GLi,p is the gray level of the phantom voxel at the same location as the voxel i in the dough MR image and GLc,p is the gray level of the central voxel in relation to the antenna, which is chosen as a reference (i.e., minimal heterogeneities) and GLi(t) the gray level of the voxel i in the original dough MR image.

u␧៮ (t) ⫽

冑冋 冉

GLn(t)

冊

2

u␧2៮ (0)⫹

冋

1⫺␧៮ (tref)

册

2

2

uGL (t) n GLn(tref) 2 GL (t) n 2 ⫹ [1⫺␧៮ (t )] uGL (t ) ref n ref GLn2(tref) GLn(tref)

册

(39)

2

2

@t ⱖ 0, uGLn(t)

冉 冊 冉 冉 冘 冊 冉 冘 冊

冊

2

2

GLn(t) 2 GLn(t) 2 ⫽ uGLc,p ⫹ ⫺0.5 un(t) GLc,p n(t) 2

GLc,p i⫽n(t) 1 ⫹ u2 n(t) i⫽1 GLi,p GLi(t) 2

1 i⫽n(t) GLi,n(t) 2 ⫹ uGLi,p n(t) i⫽1 GLi,p

(40)

The uncertainty of the gray level in a voxel i is calculated as follows, whatever the image: @t ⱖ 0, @i, uGLi(t) ⫽

GLBackground(t)

冑␲/2

(41)

where GLBackground(t) is the mean gray level of the MR image background. The uncertainty of the initial porosity u␧៮ (tref) is arbitrary (no data reported in literature) and was set at 20%. The uncertainty of the number of voxels is estimated as follows: @t ⱖ 0, un(t) ⫽

冉

冊

Max兩nˆ (t) ⫺ n៮ (t)兩

冑3

(42)

where nˆ (t) is the estimated number of voxels assigned to the dough once the thresholding method has been performed on the original MR image of dough and n៮ (t) is the number of voxels expected on the basis of the known dimensions of the dough object: @t ⱖ 0, n៮ (t) ⫽

2VT(t) ␲R ⌬x2

(43)

where VT(t) is the total dough volume from volumetric measurements. Eq. (42) assumes that each value comprised within the interval Max兩nˆ (t) ⫺ n៮ (t)兩 has the same probability. 4. Results and discussion

3.3.8. Estimation of porosity uncertainty The calculation of porosity uncertainty has been developed in the Theoretical Considerations Section in the absence of inhomogeneities. When division by a phantom is used to correct any inhomogeneities (see Eq. (38)) and still assuming that the different parameters are independent,

Fig. 4 presents the MR images in gray level during proving. The experimental validation of the method will exploit the mean gray level from these images. The validation process previously presented in the Theoretical Considerations Section will be followed step by step.

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Fig. 4. Cartography of normalized gray levels in dough during proving. Gray levels values are represented in false colors for an illustration purpose and their range of variation is given by the scale located on the right of the images.

4.1. Estimation of the number of voxels in the dough MRI slice by thresholding the MR images The aim was to evaluate the accuracy of the number of voxels assigned to the dough once the thresholding method had been performed on the MR images. This was done by comparing the total dough volume and the MRI slice volume; it was shown in the Theoretical Considerations Section that assuming a one-dimensional expansion and a perfect adjustment between the matrix grid and the dough outlines, the total dough and MRI slice volume ratios were comparable (see Eq. (22) and Eq. (24)). The time-course changes in the volume ratio V(t)/V(tref) by MRI and volumetric methods are compared in Fig. 5. Reference time was the first MRI acquisition which occurred at t ⫽ 9 min after the end of mixing. The general trend in volume evolution was similar for both methods of measurement (Fig. 5) and agreed with the description given in the Introduction section. During the first twenty minutes of proving, volume increased no more than 10%. This corresponds to a period of latency in gas production and/or diffusion from yeast to the bubble nuclei. After this latency time and up to 100 min, volume increased linearly with a mean slope of 1.4% per minute. Beyond 100 min, the expansion rate decreased progressively. With the exception of some short proving times (t ⬍ 40 min), the relative error between the two volume ratios obtained from the two methods was included in the standard error calculated for the total dough volume ratio from nine

repetitions (⬍ 3.4%) (Fig. 6) and was thus judged nonsignificant in the present case. So by comparing these two volume ratios and their errors, the segmentation method could be validated and the estimation of the numbers of voxels at t was judged satisfactory. To assist further discussion, Fig. 7 presents the evolution of the gray level distribution in the original MR image (dough and background) during proving, on which the threshold value has been superimposed. Fig. 8 presents the evolution of the relative error between the number of voxels

Fig. 5. Comparison between MRI and volumetric measurements: evolution of the mean dough volume normalized to its reference volume during proving (respectively mean of 3 runs for MRI and mean of 9 runs for volumetric measurements). Their corresponded standard deviations are also represented. Reference time was the first MRI acquisition which occurred 9 min after the end of mixing.

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Fig. 6. Evolutions of the standard error of the total dough volume (9 runs) and the relative error between MRI and volumetric measurements during proving. The relative error was calculated from the mean MRI volume evolution (mean of 3 runs) and also from each run (runs 1, 2 and 3).

assigned to the dough after thresholding and the number of voxels expected on the basis of the known dimensions of the dough object (see Eq. (43)). For short proving times (t ⬍ 40 min), the MRI volume ratio was inferior to the total dough volume ratio (up to 3.9% difference, Fig. 6). For such short times, the mean gray level was 15 to 28 times higher than the background mean gray level (Fig. 7). But due to the partial volume

effect, the voxels of the dough outlines exhibited a gray level value intermediate between the gray level of the background and the mean gray level of dough, but they had little weight in the statistical analysis. As a consequence the threshold value was high compared to the gray level value in the voxels of the dough outlines and the latter voxels were not all taken into account for the dough. This is illustrated in Fig. 7 for t ⫽ 35 min 54s of proving. The mean

Fig. 7. Evolution of the gray level distribution in the MR image (dough ⫹ background) during proving (run 2). Reference time was the first MRI acquisition which occurred 9 min after the end of mixing.

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Fig. 8. Evolution of the difference between the number of voxels assigned to the dough after thresholding and the number of voxels determined from the device dimensions. The difference is normalized by the number of voxels determined from the device dimensions and expressed in percentage.

gray levels in the dough and in the background were 759.1 a.u. and 33 a.u. respectively and the threshold value was automatically selected at 390 a.u. This led to the omission of 26 voxels with gray levels of between 209 and 374. This underestimation resulted in an underestimation of the volume at time t and thus of the MRI volume ratio. For longer proving times (t ⬎ 40 min), as the mean porosity kept on increasing, both the mean gray level in the dough and the threshold value decreased (Fig. 7) whereas the gray level in the background remained quite constant. In this new stage, the estimation of the number of voxels in the dough depended on the rate at which the porosity developed in the voxels of the dough outlines. At the end of mixing, when the flask was filled with dough, air bubbles could be incorporated in an uncontrolled number between the flask walls and the dough and act as nuclei for preferential bubble development along the walls. If the porosity developed in the voxels of the dough outlines more slowly than in the bulk voxels, the gray level in the voxels of the dough outlines was then incorporated in the gray level distribution of bulk voxels in the dough and the relative number of voxels omitted from the dough outlines was reduced (run 2 in Fig. 8). As a consequence, the MRI volume ratio was better estimated than at short proving times and was within the standard error calculated for the total dough volume ratio from nine repetitions (run 2 in Fig. 6). An extreme consequence of this configuration was the overestimation of the MRI volume ratio compared to the total dough volume ratio (run 2 in Fig. 6). Ideally assuming that the MRI volume at t was now accurate, the error on the MRI volume at tref still remained and its underestimation,

discussed above, now yields to an overestimation of the MRI volume ratio. If the porosity developed in the voxels of the dough outlines at the same rate as in the bulk voxels or faster, their gray level decreased in the same proportion and the threshold value was still too high to permit their assignment to the dough. In such a case, the gray level in the voxels of the dough outlines was rapidly incorporated in the gray level distribution of the background. The number of omitted voxels from the dough outlines did not then decrease to a large extent (run 3 in Fig. 8) and the MRI volume ratio remained underestimated as compared to the total dough volume ratio (run 3 in Fig. 6). 4.2. Constancy of the gray level of paste (validation of hypotheses H1 to H5) At the end of mixing, the dough temperature was 24.0 ⫾ 0.4°C. The temperatures of the proving environments were respectively 25.6 ⫾ 0.1°C and 26.0 ⫾ 0.3°C in the MRI area. At the end of proving, the core temperature of samples was in equilibrium with the ambient temperature. MRI measurements performed with the same sequence on dough formulated without yeast showed that the gray level of paste Table 1 Weight loss during 130 min proving (expressed in g/100g of dough)

run 1 run 2 run 3

MRI

Outside of MRI

⫺0.28 ⫺0.30 ⫺0.30

⫺0.13 ⫾ 0.02 ⫺0.08 ⫾ 0.03 ⫺0.15 ⫾ 0.01

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Table 2 Time-course changes in the NMR intensity of a transversal relaxation signal (CPMG) at 11 ms during proving (mean of 3 runs) Proving time (min)

4

16

20

30

37

41

50

60

70

75

U (V/g of dough) ␴ (V/g of dough)

5.99 0.05

6.11 0.01

6.15 0.21

6.17 0.18

6.16 0.11

6.23 0.20

6.19 0.17

6.31 0.17

6.34 0.12

6.36 0.11

decreased by 0.5% per degree Celsius in the temperature range [10; 30°C] [20]. As this rate of variation is low and the temperature gradient did not exceed a few degrees in the present study, the contribution of temperature variations to the time-course changes in the MRI signal was neglected. The total weight loss during proving is presented in Table 1. The weight loss was inferior or equal to 0.3% of the initial weight so it could be neglected. NMR measurements carried out on dough samples at different times during the proving process showed that the signal intensity at 11 ms on the CPMG relaxation curve (RD ⬎ 5 T1) (Table 2) did not present significant variations (variance analysis p ⬍ 0.05). The T1 values did not vary either. So the hypothesis that the enzymatic or metabolic reactions did not interfere with the spin-echo signal (TE ⫽ 11 ms) was also validated. As the relaxation signal was not affected by either heat and mass transfer or biochemical reaction (H5) and provided that the gray level was not affected by inhomogeneities in the magnetic field (H4), the sum of gray levels assigned to the dough was now expected to be constant in time (see Eq. (27)) in the Theoretical Considerations Section). Its evolution with the proving time is reported in Fig. 9. Without any image correction from inhomogeneities, the sum was constant and close to its reference value during the first 60 min (V(t)/V(tref) ⱕ 1.90). But afterwards, the sum progressively decreased to reach 93%, 91% and 91% of its initial value for runs 1, 2 and 3 respectively at the end of proving (illustrated in Fig. 9 for run 2). These last timecourse changes in the sum of gray levels were greatly reduced when correcting MR images of the dough with MR images of a phantom (see Materials and Methods Section). This result tends to partially attribute the previous deviation to the spatial inhomogeneities. Fig. 10 presents the spatial distribution of gray levels in an oil phantom on which the outlines of the dough at the reference and final times of proving have been superimposed. The gray level of oil was quite homogeneous within the dough outlines taken at the reference time. This was still the case as long as the dough volume only doubled from its initial value. As the dough continued to expand, its upper parts crossed over areas of less magnetic homogeneity. The mean gray level decreased from 2324.8 ⫾ 31.6 a.u. to 2239.3 ⫾ 67.2 a.u. between the reference time and the final proving time. However, the sum of normalized gray levels still deviated from unity for very long proving times (t ⬎ 100 min or V(t)/V(tref) ⱖ 2.8); this was attributed to an under-estimation of n(t) (Fig. 8), as the histogram of the dough was in places superimposed on that

of the background at its left extremity and the voxels from the dough outlines were attributed to the background (Fig. 7). Imperfections in correcting the magnetic inhomogeneities could also be under concern. As image analysis and the hypotheses had been validated (previous sections), the mean gray level ratio was now expected to equal the reciprocal of the total volume ratio (see Eq. (18) in the Theoretical Considerations Section). Before beginning the third validation step, deviation in mean gray level during proving is discussed. This step is transitory as both the number of voxels and the gray level sum contribute to the calculus of the mean gray level (see Eq. (19)) and the latter contribute to the calculus of porosity from MRI data (see Eq. (18)). The gray level ratio is represented as a function of the volume ratio in Fig. 11. The reference time was the first MRI acquisition, which occurred 9 min after the end of mixing. The volume considered in Fig. 11 is the one calculated from the number of voxels assigned to the dough (the so-called MRI slice volume). Linear regression applied to these mean values (three repetitions) showed a slope equal to unity and an offset equal to zero with a correlation coefficient of 0.9995. For V(tref)/V(t) ⱕ 0.35, the mean gray level ratio may deviate slightly from this regression (from 3 to 6% whatever the run). This corresponded to a proving time of longer than 100 min or a mean porosity value of over 68% (m3 of gas per 100 m3 of dough). Here again the limitations of the present method were encountered. It has previously been shown that beyond this point the sum of the normalized gray levels was no longer conservative (Fig. 9) and that the number of voxels assigned to the dough after thresholding might be underestimated (Fig. 8). Although both of these parameters contribute to the calculation of the mean gray level (see Eq. (19)), manifestly the underestimation of the sum of the normalized gray levels prevailed and resulted in an underestimation of the mean gray level. Here too, the deviation is slight (⬍ 6%) and confined to very long proving times (t ⬎ 100 min). Eq. (18) (equality of the mean gray level ratio to the reciprocal of the volume ratio) was then validated by the results presented in Fig. 9, as was the present method at whole dough scale. 4.3. Determination of the gas content from the MRI signal during proving Once Eq. (18) had been validated, dough porosity could be deduced from the mean gray level from Eq. (28) (see

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Fig. 9. Evolution of the gray level sum and the normalized gray level sum during proving (run 2). Each sum was normalized respectively to its reference value. Reference time was the first MRI acquisition which occurred 9 min after the end of mixing.

Theoretical Considerations Section). The reference was still the first MRI acquisition (9 min after the end of mixing). As gas production and/or diffusion is commonly retarded up to 20 min [5] after the end of mixing, porosity at tref, ␧៮ (tref), was assumed not to have changed from its initial value (at the end of mixing, t ⫽ 0) and was set at 10% (m3 of gas per 100 m3 of dough) [2]. 4.3.1. Determination of the mean porosity from the MRI signal during proving Fig. 12 presents the evolution during proving of the mean porosity calculated either from the mean gray level or from the total dough volume measurements. The general trend could be deduced from the trend of the total dough volume, which has already been discussed (Fig. 5). For short proving times, the relative error between the two methods of calculation was over 5% (Fig. 13); the maximum difference was 11.4, 14.2 and 23.2% for runs 1, 2 and 3 respectively, higher than the relative error of porosity calculated from the total dough volume measurements (⬍ 8.5% for short proving times). For proving times of over 40 min, the relative error between the two methods of calculation was under 5% and was not significant compared to the relative error on porosity calculated from the total dough volume measurements (nine repetitions) (Fig. 13). 4.3.2. Uncertainty on the mean porosity Let us now examine this question in greater depth. Fig. 14 presents, according to Eqs. (39)–(43), the evolution of the different contributions: GL uncertainties, ␧(tref) uncertainty, n(t) uncertainties, to the mean porosity uncertainty. The contribution of the uncertainty on a given parameter

to the uncertainty on the mean porosity consists of the uncertainty of the parameter in question multiplied by a factor which is in certain cases time-dependent. The GLi(t) contribution was constant with time as its uncertainty as well as the related factors were constant with time (Fig. 14). As far as the ␧(tref), n(tref), GLi(tref) and GLp contributions are concerned, their respective uncertainties were constant with time (U␧(tref) ⫽ 0.02, Un(tref) ⫽ ⫺15, UGLi(tref) ⫽ 25.85, UGLp ⫽ 36.07) but the related factors decreased with time as a function of 1/n(t) (Fig. 14). As a consequence they

Fig. 10. Spatial distribution of gray level in an oil phantom. Gray levels values are represented in false colors and their range of variation is given by the scale located on the right of the images. The outlines of the dough at the reference and final times of proving have been superimposed for comparison purposes.

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Fig. 11. Evolution of the mean gray level normalized to its reference value as a function of the reciprocal of the dough volume normalized to its reference value (three runs).

all decreased as a function of 1/n; their values for t ⬎ 44 min (or ␧៮ (t) ⬎ 0.38), for instance, were inferior to half of their initial values. The time-course changes in the n(t) contribution were twofold (Fig. 14): as with the previous parameter, the factor decreased as a function of 1/n(t), but its uncertainty also varied with time (see Fig. 8). Over approximately the first 44 min, the uncertainty on n(t) increased from 3.12% to 8.16% and then decreased to approximately 4.7% (Fig. 8). For longer proving times, the uncertainty on n(t) could be considered constant (Fig. 8) and was set to 4.7% (the cited values refer to mean values). For porosities close to the initial porosity, the main uncertainty contributions are GLi(t), GLi(tref) and GLp (⬎ 20%). Uncertainty contribution of ␧(tref), n(tref), n(t) were not as great (7 to 15%) but can hardly be neglected (Fig. 15). Thus at this stage, the uncertainty on the mean porosity can be improved by reducing each contribution. The uncertainty on n(t) could be reduced by improving the thresholding method. It would be difficult to reduce the uncertainty on GLi(t), as increasing the number of averages would lead to a poorer time resolution and increasing the voxel size would make it impossible to distinguish local heterogeneities (the ultimate purpose of such a method!). The uncertainty on GLp could be reduced by increasing the number of averages (already high in the present work). An increase in the number of averages would also reduce the uncertainty on GLi(tref), as a period of latency allows a longer acquisition time. The uncertainty on ␧(tref) was arbitrarily fixed and was undoubtedly overestimated, while the uncertainty on ␧(tref) could be reduced by more accurate measurements. For porosities of between 0.15 and 0.40, the uncertainty on n(t) increased and it was the main contribution at ␧(t) ⫽

0.30. This undoubtedly explains the excessively high uncertainty observed on the mean porosity in Fig. 13, which could be reduced by improving the thresholding method. At ␧(t) ⬎ 0.40, all the time-dependent contributions had decreased further to half their initial values. In addition, n(t) was better estimated and its uncertainty was between 3.5 and 5.8%. As a result, the uncertainty on the mean porosity reached “reasonable“ levels (⬍10%). In this case, GLi(t) was the most important contribution to the porosity uncertainty. At ␧(t) ⫽ 0.65, for instance, it contributed 61% of the porosity uncertainty whereas all other contributions were under 15%. So a further reduction in the porosity uncertainty at the end of proving could if necessary be achieved only by reducing the uncertainty on the gray level in dough (by improving the signal to noise ratio in the original dough image).

5. Conclusion A new non-invasive method for measuring local porosity by means of the MRI gray level value has been presented and validated at whole dough scale. For a one-dimensional expansion, porosity estimated from the gray level was compared with porosity estimated from total dough volume measurements in a range of [0.10, 0.74 m3 of gas/m3 of dough]. For porosities of below 0.40 m3 of gas/m3 of dough, MR image analysis underestimated the porosity by a maximum of 0.03 m3 of gas/m3 of dough, but otherwise the difference between the two means of measurement was within the standard error of total dough volume

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Fig. 12. Evolution of the mean porosity estimated from volumetric measurements and mean gray level during proving. The mean porosity was calculated from 9 runs for volumetric measurements and from 3 runs for MRI.

measurements (⫾ 0.01 m3 of gas/m3 of dough). For porosities of over 0.70 m3 of gas/m3 of dough, the porosity estimated from the gray level deviated from total dough volume measurements (⫹ 0.01 m3 of gas/m3 of dough). In practice, during proving, the total dough volume increases by between 3 and 4 times the initial volume. This corresponds to a mean porosity of between 0.66 and 0.75 m3 of gas/m3 of dough. So the actual method allowed to observe the proving process with little

bias except for some very specific industrial conditions. Optimizing the thresholding method could help in validating the relation between the gray level of a voxel and its gas fraction for porosities of over 0.70 m3 of gas/m3 of dough. A key point in the validation process was the absence of T2 or T1 changes with the enzymatic and metabolic reactions taking place during proving. This result also contributed to the originality of this work. Likewise, a limitation of the method was the omission of

Fig. 13. Evolution of the relative error between the porosity calculated from MRI gray level (mean of 3 runs) and total dough volume measurements (mean of 9 runs). Comparison with the standard error on porosity calculated from the total dough volume measurements.

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Fig. 14. Estimation of uncertainty on the mean porosity estimated from the gray level in function of the uncertainties on each parameter used for its calculation. The uncertainty on the initial porosity is assumed to be equal to 20%.

heat and mass transfer which often takes place at the industrial scale. As well, coupling heat and mass transfer during proving with a study of local porosity in the whole dough will be a new challenge for future investigations. Moreover, the results of the present work also showed the necessity of image normalization from magnetic deformation. For a spin-echo sequence, it has been shown that up to 9% of the total signal could be lost. It is worth

noting that such a method could be systematically applied when testing other sequences for the study of the proving process. Although it was limited to observation of the mean porosity, it made it possible to observe the mapping of local porosity during proving (as illustrated on gray levels in Fig. 4). Future research will also focus on the exploitation of the local porosities in MRI maps to describe the underlying mechanisms of proving.

Fig. 15. Estimation of uncertainty, expressed in percent, on the mean porosity estimated from the gray level in function of the uncertainties on each parameter used for its calculation. The uncertainty on the initial porosity is assumed to be respectively equal to 20%.

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