Variations of Hausdorff Dimension and Selected Textural Indices of Cheddar and Gouda Cheeses During Storage

Abstract

Two imported commercial cheeses (Cheddar and Gouda) were analyzed to characterize their textural changes during a storage period from 176 to 362 days at 4°C. Fractal dimension analysis was used to examine the structure of the cheeses, and a scaling model based on stress values was used to calculate fractal dimension (D_f) by the Hausdorff dimension count method. It was found that the variation of D_f of two cheeses showed a similar trend. When two cheeses were stored from 176 to 300 days, their D_f ranged from 2.37 to 2.43 for Cheddar or from 2.06 to 2.23 for Gouda cheeses. After that, the final values of D_f increased obviously to 2.87 (Cheddar cheese) or 2.63 (Gouda cheese) at the end of the storage period. The experiments carried out showed that the variation of D_f had a poor relationship with the changes of pH 4.6 soluble nitrogen and two instrumental textural indices (hardness and springiness) of the cheeses, especially in the late storage period, but the inhomogeneous cavities existing in the cheeses could be reflected by their D_f.

Keywords:

• Cheddar cheese
• Gouda cheese
• Fractal analysis
• Texture
• Hausdorff dimension

INTRODUCTION

During storage of cheeses, some biochemical changes occur and lead to the development of the typical textures for different cheese varieties,1 2 characterized by the facts, such as the body of the fresh cheese losing its firm, tough, and curd textures, and becoming softer.3–5 Instrumental texture profile analysis, uniaxial compression, and puncture test were widely used to assay cheese texture and provide information on both the deformation and fracture properties of the cheeses.6 7 However, these texture analysis methods are somewhat empirical,8 and the obtained assaying values were quite changeable with cheese varieties or batches, which made them difficult to become a universal or useful approach to monitor cheese quality.9–11

The microscopy methods are widely applied to analyze cheese microstructure. The development of complex instrumentation, such as transmission, scanning, cry-scanning and environmental electron microscopy, confocal laser scanning microscopy, dynamic light scattering, and nuclear magnetic resonance, have facilitated us to understand the structures and to predict functional properties of the cheeses during ripening.12–14 Unfortunately, most microscopic approaches are unable to qualitatively analyze the microstructure of the cheeses.15 Fractal analysis technique now is a powerful tool in protein science. The concept of fractal analysis was originally developed as a quantitative method to measure the complexity of structures that show a pattern of self-similarity.16 17 Protein aggregates can be characterized as a kind of fractal, thus it is feasible to determine its fractal dimension (D_f ) by using different methods, such as light-scattering, rheology, or microscopy.18–21 Shih et al. developed a scaling theory about the elastic properties of colloidal gels by considering the network structures of the gels as packed fractal flocks, and found that both the elasticity of the gels (E) and the limit of linearity (γ₀) exhibited a power-law behavior with respect to particle concentration (ψ), where the exponent was partly determined by D_f .22 Wu and Morbidelli23 proposed a revised version of the model of Shih et al.,22 through which the fractal dimension of colloidal gels could be estimated in any gelatin regime. Dávila and Parés24 evaluated the fractal structure of plasma protein gels with the rheological model and scanning electron microscopy. They determined D_f by the box count method, and found that the value of D_f ranged from 2.63 to 2.83. However, the current method of applying the box count method to scanning electron microscopy (SEM) binary images is not suitable to cheeses. The cheese gels were different from the colloidal gels (a homogeneous body).

In the present article, the textural changes of two cheeses, Cheddar and Gouda cheese, were determined during their storage period, and the fractal dimension of the cheeses' texture was evaluated. The stress values were used to calculate D_f by the Hausdorff dimension count method, which had been applied in other works.25–27 The main aim was to analyze the correlation between D_f and some macroscopic properties of the cheeses, and to show a new approach to predict some technological and textural properties of the cheeses with Hausdorff dimension.

MATERIALS AND METHODS

Materials

The Cheddar and Gouda cheese samples imported from Holland were obtained from a local supermarket in Harbin, and the pieces of Cheddar and Gouda cheeses with airtight packaging independently ranged from 176 to 362 days of maturation at 4°C. The chemical compositions of two cheeses were analyzed in triplicate with the methods below. All chemicals used were analytical reagents, and the water used was distilled water.

Analysis of Some Chemical Compositions

An appropriate sample size of the cheeses (about 50 g) was removed from various sections of the cheese blocks on the day following storage periods, and analyzed for fat, moisture, and protein content by the Babcock method 933.05, oven drying by method 926.08 and the Kjeldahl method 920.123,28 respectively. Meanwhile, pH 4.6 soluble nitrogen (pH 4.6 SN) was determined with the method of Kuchroo and Fox29 as indices of proteolysis. Twenty grams of cheese samples were mixed with 60 mL of distilled water and adjusted to pH 4.6 with 0.1 moL/L HCl, then homogenized for 10 min. The mixture was placed in a centrifuge tube, warmed to 40°C for 1 h, centrifuged at 3000× g for 30 min, and filtered through Whitman No. 4 filter paper. The residue was re-extracted with 60 g distilled water again. The extracts obtained were combined and subjected to nitrogen content analysis. All analyses were performed in triplicate.

Instrumental Texture Profile Analysis

Instrumental textural properties of the cheese samples were measured with an Instron Universal Testing Machine (QTS25 model, Instron Corporation, Stevens, UK). Cylinders of the cheese samples, 12.90 mm in diameter and 15.00 mm height, were prepared at 4°C from the inside of the cheese using a steel borer-type cutter, and left at room temperature (20 ± 1°C) for at least 2 h prior to testing. The test parameters applied were the same as those in the work of Pons and Fiszman.30 31 The samples were compressed to 66% of their original height at a crosshead speed of 1.0 mm/s; each test was repeated at least in triplicate.

Microstructure Observation

Cheese samples (in triplicate) were prepared and subjected for scanning electron microscopy analysis after different storage periods with the method of Madadlou et al.32 Cheese blocks were cut into approximately 5–6 mm³ cubes with a sharp razor and immersed in 2.5% gluteraldehyde for 3 h. The cubes were then washed with distilled water six times (1 min each), dehydrated in a graded (40, 55, 70, 85, 90, and 95%) series of ethanol for 30 min each, followed by defatting with chloroform for three times (10 min each). The samples were kept refrigerated and covered with ethanol until they were freeze fractured in liquid nitrogen to approximately 1-mm pieces. The pieces were mounted on aluminum stubs with silver paint, dried to critical-point and coated with gold for 6 min in a sputter-coater (type SCD 005, BalTec Inc., Balzers, Switzerland). The samples were viewed with a scanning electron microscope (model CM12, Philips, Eindhoven, the Netherlands) operated at 15.0 kV. Photomicrographs were recorded at 3000 and 5000 magnifications.

Calculation of Hausdorff Dimension from the Stress Values of the Cheese Samples

The cheeses stress (r) was determined by a uniaxial compression test with a TA-XT2 texturometer (Stable Micro Systems Ltd., Surrey, UK) using a cylindrical aluminium plunger of 50 mm diameter at 176, 205, 238, 268, 300, 330, and 362 days. The assaying was carried out at room temperature according to the following conditions: preload force of 10⁻³ N, force ramp rate of 5 mm/min. The samples evaluated were cut into cylindrical portions of 12.90 mm diameter and 15.00 mm height with a steel borer-type cutter. The measures were performed in triplicate. The stress values subsets were obtained as the plunger was compressing until samples were destroyed, and used to calculate D_f . The definition of the Hausdorff dimension was given in Eq. (1), where D_f is the Hausdorff dimension of r, N(r) denotes the numbers of D_f -dimensional Hausdorff object with r to measure, the parameter C is constant. Equation (1) can be expressed in logarithm form as Eq. (2):

(1)

(2)

(3)

The fractal dimension of the cheese is designed as D_f , which is the linear slope of Eq. (2). The parameter r represents the stress of the cheese samples, and its value typically ranges from 0.01 to 8.92 N for the cheese gels. N(r) represents the drop serial numbers of stress values.25–27 Since the determination of d_f by linear analysis is based on one dimension, it is necessary to add two extra dimensions to calculate value in the three-dimensional features of the cheese according to Eq. (3).24

Statistical Analysis

Analysis of variance (ANOVA) was conducted using SPSS version 12 (SPSS Inc., Chicago, IL, USA). Replicate means were considered significantly different at P < 0.05 unless stated differently. When significant differences were indicated by ANOVA, Tukey pair-wise comparisons were performed to indicate where the differences between properties existed.

RESULTS AND DISCUSSION

Chemical Composition of the Cheeses

Some chemical compositions of Cheddar and Gouda cheese samples during storage periods are shown in Table 1. No difference was found for moisture, proteins, or fat due to each sample being sealed airtight independently.

Table 1 Main chemical compositions of two cheeses during storage period studied

Cheese	Storage time (days)	Moisture content(g 100 g^−1)	Protein content(g 100 g^−1)	Fat content(g 100 g^−1)
Cheddar	176	44.10 ± 0.01	29.00 ± 0.03	24.20 ± 0.04
	205	44.09 ± 0.02	28.98 ± 0.04	24.19 ± 0.04
	238	44.09 ± 0.01	28.99 ± 0.03	24.19 ± 0.06
	268	44.08 ± 0.02	28.95 ± 0.05	24.17 ± 0.03
	300	44.05 ± 0.02	28.97 ± 0.02	24.16 ± 0.05
	330	44.06 ± 0.03	28.95 ± 0.03	24.15 ± 0.05
	362	44.05 ± 0.02	28.96 ± 0.03	24.15 ± 0.06
Gouda	176	39.60 ± 0.03	22.50 ± 0.03	30.40 ± 0.05
	205	39.56 ± 0.03	22.48 ± 0.01	30.36 ± 0.03
	238	39.55 ± 0.05	22.48 ± 0.03	30.35 ± 0.03
	268	39.55 ± 0.04	22.45 ± 0.06	30.32 ± 0.06
	300	39.52 ± 0.05	22.46 ± 0.04	30.34 ± 0.04
	330	39.50 ± 0.06	22.43 ± 0.04	30.35 ± 0.03
	362	39.51 ± 0.05	22.42 ± 0.06	30.34 ± 0.04

Hausdorff Dimensions of the Cheeses

Figure 1 gives the corresponding stress–strain curves for Cheddar and Gouda cheese samples stored at 176, 205, 238, 268, 300, 330, and 362 days. After the determination of the limit of linearity for each cheese sample, double-logarithmic plots of the serial numbers (N(r)) versus drop order stress values (r) were constructed, and the values were used to calculate D_f by the Hausdorff dimension count method using Eq. (2), which are given in Table 2. The values of D_f ranged from 2.37 to 2.43 for Cheddar or 2.06 to 2.23 for Gouda between 176 days and 300 days showing a slow variation, and then the values of D_f obviously increased to 2.87 (Cheddar) and 2.63 (Gouda) for the rest of the storage period.

The D_f , which reflected structure changes, occurred in the cheese samples and gave the information about cheese texture, depending on stress values. The stress–strain tests based on large destructive deformations was applied to study fracture properties of the cheese.33 Stress–strain and stress relaxation measurements had been successfully used to describe the textural changes of several cheeses during ripening.34–36 In the authors' work, all lines of stress–strain showed an upward trend, and the levels decreased progressively with an extension of the storage time, reaching the lowest values at 362 days (Fig. 1). However, the feature of D_f based on the stress values used to calculate fractal dimension by the Hausdorff dimension count method was quite different from stress–strain curves, it was distinctly increased at the end of the storage period. The results indicated that the Hausdorff dimension might be used to describe the structures of the protein aggregates and the cavities among them.

Table 2 pH 4.6 soluble nitrogen (SN) and two textural parameters of Cheddar and Gouda cheese samples during storage period related to corresponding Hausdorff dimension (D _f )

Cheese samples	Storage time (days)	pH 4.6 SN (%)	Hardness (N)	Springness	Df
Cheddar	176	26.90 ± 0.01^g	7.48 ± 0.22^a	0.77 ± 0.01^a	2.43 ± 0.08^b
	205	28.83 ± 0.01^f	6.67 ± 0.01^b	0.78 ± 0.01^a	2.41 ± 0.01^b
	238	31.14 ± 0.03^e	5.90 ± 0.12^c	0.77 ± 0.02^a	2.37 ± 0.07^b
	268	31.70 ± 0.01^d	5.12 ± 0.03^d	0.77 ± 0.01^a	2.39 ± 0.14^b
	300	32.40 ± 0.02^c	4.60 ± 0.02^e	0.75 ± 0.03^a	2.42 ± 0.14^b
	330	33.42 ± 0.01^b	3.55 ± 0.12^f	0.68 ± 0.01^b	2.73 ± 0.11^a
	362	33.59 ± 0.01^a	3.45 ± 0.02^f	0.66 ± 0.02^b	2.87 ± 0.05^a
Gouda	176	17.06 ± 0.02^g	12.24 ± 1.36^a	0.80 ± 0.01^a	2.06 ± 0.06^c
	205	18.76 ± 0.01^f	8.88 ± 0.51^b	0.74 ± 0.02^ab	2.06 ± 0.16^c
	238	24.25 ± 0.02^e	7.28 ± 0.37^c	0.69 ± 0.10^b	2.11 ± 0.09^bc
	268	25.75 ± 0.01^d	6.96 ± 0.41^c	0.78 ± 0.06^ab	2.13 ± 0.04^bc
	300	26.75 ± 0.01^c	4.60 ± 0.29^d	0.72 ± 0.03^ab	2.23 ± 0.05^b
	330	29.41 ± 0.02^b	4.27 ± 0.26^d	0.70 ± 0.02^b	2.58 ± 0.06^a
	362	31.63 ± 0.01^a	3.61 ± 0.30^d	0.67 ± 0.03^b	2.63 ± 0.03^a
One-way variance analysis by Tukey pair-wise comparisons was performed to evaluate the statistical difference between fitting parameters. Values in the same column of each cheese sample followed by different superscript letters differ significantly (p < 0.05).

Figure 1 Stress-strain curves for (a) Cheddar and (b) Gouda cheese samples stored at 176, 205, 238, 268, 300, 330, and 362 days. (Color figure available online.)

Correlations of D_f and pH 4.6 Soluble Nitrogen of the Cheeses During Storage

The pH 4.6-soluble nitrogen (SN) of the two cheeses during the storage period studied is shown in Table 2. Analysis indicated that pH 4.6-SN of both cheeses continued to increase during the whole storage period, the values of Gouda cheese increased from 17.6 to 31.63%, while that of Cheddar cheese increased from 26.90 to 33.59%. This result was due to continuous proteolysis in the cheeses, which was the most important reaction occurring during ripening of the cheeses.37–39 It is wildly believed that proteolysis is responsible for these textural changes in cheeses, and that their texture behavior could be influenced by changes in their degree of hydrate of proteins that interacted with the casein matrix in the curd.40 41 However, a poor relationship was found between the variations of the pH 4.6-SN during the storage period studied for the two cheeses and corresponding D_f , especially in the late storage period. When the cheese samples were stored in a time period of 300 to 362 days, their pH 4.6-SN increased at a slower rate (from 32.40 to 33.59% for Cheddar, or from 26.75 to 31.63% for Gouda), in contrast, the value of D_f increased clearly (from 2.42 to 2.87 for Cheddar, or from 2.23 to 2.63 for Gouda). This result meant that there was no clear relationship between proteolysis and D_f changes with storage time.

D_f and Instrumental Textural Characteristics of the Two Cheeses

Instrumental textural characteristics are widely applied indexes to quantify cheese texture, but there is often a poor relationship between these parameters and quality of cheeses.9 10 In this work, the variations of some textural characteristics of two cheeses during storage period are given in Table 2. An obvious and rapid decrease in hardness was found for two cheeses during a storage period of 176 to 300 days. Then, a slow decrease in hardness occurred during the storage period of 300 to 362 days. The variation of textural characteristics of two cheeses was different from their D_f , because D_f showed little or a slight increase during the first storage time from 176 to 300 days and then a clear increase for the rest of the storage period. There existed no relationship between the hardness (or springiness) and D_f of the cheeses. The results suggested that Hausdorff dimension probably was useful for indirectly probing the complex texture and quality of the cheese during storage, which made it impossible with any one of the two instrumental characteristics of the cheeses.

D_f and Microstructure of the Two Cheeses

Some SEM images of two cheeses at the different storage times are given in Fig. 2. The size of the aggregates and the cavities among clusters in two cheeses became larger and more inhomogeneous as the storage time increased. The correlation between microstructure and D_f of two cheeses was that the more inhomogeneous the cavities were, the higher values of Hausdorff dimension the cheese had. Compared to the Gouda cheese, the Cheddar cheese exhibited more inhomogeneous cavity, so its values of D_f were higher than that of Gouda cheese. As the storage periods increased, the two cheeses became more inhomogeneous, and their values of D_f exhibited an increasing trend. Based on this result, Hausdorff dimension might be a good alternative to reflect the microstructural changes of the cheeses during storage period.

Figure 2 The SEM photos for Cheddar cheese stored for (a) 240 and (b) 334 days, or Gouda cheese stored for (c) 241 and (d) 330 days.

CONCLUSIONS

The Hausdorff dimension count approach was used to qualitatively analyze microstructural characteristics of Cheddar and Gouda cheese during their storage period of 176 to 362 days, and a scaling model based on stress values was used to calculate fractal dimension (D_f ) by the Hausdorff dimension count method. The main compositions (moisture, proteins, and fat) of the two cheeses were kept stable, but the values of D_f of the two cheeses had a similar variation trend. When the storage period from 176 days increased to 300 days, the D_f of the cheeses changed slightly, after that their D_f showed a clear increase. The variations of the pH 4.6-SN, hardness and springiness of two cheeses showed no relation to the D_f , especially in the late storage period. The inhomogeneous cavities that existed in the two cheeses could also be reflected by Hausdorff dimension.

ACKNOWLEDGMENTS

This work was supported by the National High Technology Research and Development Program (“863” Program) of China (No. 2006AA10Z324) and Program for Innovative Research Team of Northeast Agricultural University (No. CXT007-1-3).