Mapping the Meltdown Behavior of Frozen Dairy Desserts

The meltdown test is an efficient tool widely and commonly used to characterize structural changes in frozen desserts resulting from different ingredients and processing conditions. The meltdown is commonly determined by a gravimetric test, and it is used to obtain the onset ( M on ), rate ( M rate ), and maximum ( M Max ) meltdown. However, these parameters are calculated ambiguously due to the inconsistency in the methodology. This work aims at modeling the meltdown curves (weight vs time) of different commercial samples (36 commercial samples). Samples of commercial frozen desserts (40–60 g) was placed on a 304 stainless wire cloth (1.50 mm opening size and 52% open area) suspended about 15 cm above of an analytical balance, and the dripped portion of the melted ice cream was continuously recorded throughout the duration of the test. The meltdown test was conducted at room temperature. Each meltdown test generated more between 3000 to 4000 data points and was modeled using 4 equations: The logistic model, the Gompertz model, the Richard model, and the Hill model. All the melt-down curves were sigmoidal in shape, regardless of the type of frozen dessert. The experimental meltdown curves were adequately represented by the Logistic model, judging by several criteria (R 2 = 0.999, adjusted R Adj 2 = 0.999, Akaike probability = 6582, and F-value = 1.88 × 10 6 ). Thus, the Logistic model was shown to be an effective tool for predicting the meltdown curves of frozen desserts, and it can be used to define unambiguously the onset, rate, and maximum meltdown. More-over, a dimensionless response (meltdown behavior, M Be ) that combines M on , M rate , and M Max was developed and used for mapping the meltdown of different commercial frozen desserts.


INTRODUCTION
Hundreds of competing brands of frozen desserts are routinely displayed to consumers in supermarkets and hypermarkets, offering a variety of flavors, ingredients, and incursions in all kinds of sizes and presentations (Enteshari and Martínez-Monteagudo, 2020).The term frozen dessert is reserved for a group of desserts formulated with or without milk solids (e.g., milk fat, lactose, and whey proteins), frozen under shear, and consumed frozen (Goff and Griffiths, 2006).Examples of frozen desserts are ice creams (e.g., economy, regular, and premium), gelatos, frozen custards, nonfat, sugar-free, sherbets, soft-serve, and frozen yogurt.Legal definitions and compositional standards of frozen desserts can be found elsewhere (Goff, 2022).
Product hardness, shape retention, and resistance to melting are important characteristics of any frozen dairy desserts (Liu et al., 2022;Wildmoser et al., 2004).Ice cream hardness has been defined as the maximum force recorded during the first compression of a typical texture analysis profile (Soukoulis et al., 2010).An in-depth discussion on the interpretation of texture analysis for fresh and processed foods can be found elsewhere (Chen and Opara, 2013).Shape retention and resistance to melting are commonly evaluated by a gravimetrical method, known as the meltdown test, where a predetermined amount of ice cream is placed on a suspended wired mesh (Bolliger et al., 2000;Koxholt et al., 2001).The liquid leaving the solid matrix as the temperature of the sample increases is monitored for a given time.Overall, the meltdown of a given frozen dessert involves a lag phase or onset, a fast-melting phase or rate, and a plateau phase or maximum meltdown (Alvi and Martinez-Monteagudo, 2023;Liu et al., 2022).The meltdown of frozen dairy desserts is a complex process influenced by the fat content, type of hydrocolloids, homogenization, freezing, overrun, and number and size of ice crystals (Koxholt et al., 2001;Liu et al., 2022;Muse and Hartel, 2004).
The meltdown test was first introduced in 1999 by Baer et al. (1999), who studied the impact of gums on the structure and sensory properties of nonfat ice cream.Nearly 25 years have passed, and the method has been used for elucidating the role of fat destabilization (Bolliger et al., 2000), ice crystals (Muse and Hartel, 2004), overrun (Sofjan and Hartel, 2004), and mix viscosity (Freire et al., 2020) on melting characteristics and overall quality of ice cream.Moreover, several investigations have used the meltdown test for studying the feasibility of novel ingredients, including cellulose nanofibrils in low-fat ice cream (Velásquez-Cock et al., 2019), milk phospholipids in premium ice cream (Rathnakumar et al., 2023), and flexible and rigid polysaccharide in lowfat ice cream (Liu et al., 2023).The robustness of the meltdown test has facilitated the comparison between ice creams manufactured with different technologies, including hydrodynamic cavitation (Sim et al., 2021), power ultrasound (Alvi and Martinez-Monteagudo, 2023), and high-pressure jet (Voronin et al., 2021).
In summary, the meltdown test is an efficient tool widely and commonly used to characterize the meltdown behavior of frozen desserts including ice creams.However, the determination of key parameters (onset, rate, and maximum meltdown) depends on the test conditions (e.g., time intervals), and therefore, inherently subjective to possible errors of interpretation since different time intervals might provide different values and conclusions from the same sample.A more rigorous approach to accurately and unambiguously determine the key parameters derived from the meltdown test and aimed at providing an unbiased analysis is necessary.Meltdown curves reported in the literature (Velásquez-Cock et al., 2019;Sim et al., 2021;Liu et al., 2023;Alvi and Martinez-Monteagudo, 2023) resemble a s-curve, where a linear segment between 2 plateous are visaully indentified.S-curves also known as sigmoidal curves can be modeled b several equations, including logistic model (Equation (1)), the Gompertz model (Equation ( 2)), the Richard model (Equation (3)), and the Hill model (Equation ( 4)).Details of the different models are given in Table 1.Therefore, the objectives of this investigation are to determine the onset, rate, and maximum meltdown accurately and unambiguously through nonlinear regression, and to map the meltdown behavior of different frozen dairy and nondairy desserts.

Materials
36 commercial samples of ice cream were purchased from different local stores (Albertsons, Whole Foods Market, and Walmart, Las Cruces, NM, USA).Eight groups of frozen desserts were targeted, including low-fat, medium-fat, high-fat, dairy-free, high-protein, soft-serve, gelato, and sherbet.After purchase, samples were immediately transported to ACES Foods at New Mexico State University and stored in a chest freezer (FCM9SR, GE Appliance, Louisville, KY, USA).The temperature inside the freezer fluctuated between −18 to −20°C.Ice creams were kept in the freezer for at least 48 h before analysis.

Meltdown test
The meltdown of ice cream was measured using the gravimetric method, according to the methodology reported by Alvi and Martinez-Monteagudo (2023) with some modifications.Figure 1 illustrates the experimental rig used to measure the meltdown of ice cream.Overall, a scoop of ice cream (40-60 g and 3-5 cm height) was placed on a 304 stainless wire cloth (1.50 mm opening size and 52% open area, McMaster-Carr, Robbinsville, NJ, USA) suspended about 15 cm above of an analytical balance (FX-120i Precision Balance, A&D Weighing, Columbia, MD, USA).The test was conducted in ambient temperature and humidity (20-24°C and 45-55%, respectively), where external influences on the heat transfer were neglected.A 250 mL beaker located at the top of the balance was used to collect the dripped portion of the melted ice cream.The weight of the dripped portion was continuously re-

Hardness
The hardness of the commercial ice creams was determined with a TA.XT Plus Texture Analyzer (Texture Technologies Corp. and Stable Micro Systems Ltd., Hamilton, MA), according to the methodology reported elsewhere (Rathnakumar et al., 2023).Before the determination of hardness, a portion of the commercial ice creams (about 250 g) was transferred in a plastic cup (120 mm high and 70 mm diameter) and immediately stored in the chest freezer (−20°C).After 24 h, individual cups were placed on the platform of the texturometer, where the samples were vertically penetrated to a depth of 15 mm at a speed of 2 mm s −1 with a 3-mm diameter cylindrical probe.The retraction of the probe was conducted at 4 mm s −1 , and the test was conducted at 25°C.The maximum force (N) recorded during the compression was interpreted as hardness, and the entire test was conducted within 20 s.

Modeling meltdown
Each meltdown test generates between 3000 to 4000 data points and used to generate a meltdown curve (weight vs time).Then, the meltdown curves were modeled with 5 mathematical approaches, namely the logistic model (Equation ( 1)), the Gompertz model (Equation ( 2)), the Richard model (Equation ( 3)), and the Hill model (Equation ( 4)), Table 1.The adjustable parameters corresponding to the proposed models were calculated through nonlinear regression according to the determinant criterion (Knol et al., 2009) using a commercial software package (AthenaVisual Plus v21.1, AthenaVisual, Inc., Naperville, IL, USA).

Modeling performance and discrimination
The performance and discrimination among the models were carried out by a number of criteria, including the coefficient of determination (R 2 ), the adjusted coefficient of determination R Adj 2 ( ) , the mean absolute error (E%), the Akaike probability (AIC), residual sum of squares (RSS), and the F-value.The Akaike criterion accounts for the number of parameters by penalizing models with more parameters (Knol et al., 2009).The meltdown of ice cream was performed in triplicates, and all figures were made using Sigmaplot software V14.5 for Windows (SPSS Inc., Chicago, IL, USA).

Meltdown curves
Representative curves derived from the meltdown test are illustrated in Figure 2, where a sigmoidal behavior was observed for all tested samples.Overall, a sigmoidal behavior is characterized by a lag phase, a linear segment, and a plateau phase, from which, the onset of meltdown (M on ), the rate of meltdown (M rate ), and the maximum meltdown (M Max ) can be calculated, respectively.Sigmoidal meltdown curves have been reported in low-fat ice cream (Liu et al., 2023), ice cream enriched with phospholipids (Rathnakumar et al., 2023), and premium ice cream (Sim et al., 2021).
At the beginning of the meltdown curve, the amount of melted ice cream is negligible since the temperature of the ice cream has not yet reached the melting point of ice crystals.Eventually, the temperature of the ice cream reaches the melting point of ice crystals, and the melted crystals start to drift through the solid matrix (e.g., fat globules, proteins, and air pockets) by gravity.Subsequently, the liquid carrying soluble components (e.g., sugar and flavoring agents) as well as relatively small fat globules leave the solid matrix.The time re- quired for the first drop of liquid to leave the solid matrix is defined as the M on .As the test proceeds, the temperature of the ice cream passes the melting point of ice crystals, and more crystals melt and leave the solid matrix, which is represented by the linear segment within the sigmoidal curve.M rate (percentage of meltdown per s) is obtained from the slope of the linear segment.The transport mechanism during the fast-melting phase is likely Fickian diffusion, where the migration of the liquid phase is driven by the concentration gradient.The end of the linear segment physically represents when most of the ice crystals have melted, drifted, and left the solid matrix, and it also indicates the M Max of a given ice cream.The end of the meltdown is characterized by a plateau of constant weight, where some portion of ice cream dripped through the wire mesh (liquid), while another portion remains in the suspended wire mesh.It is worth mentioning that factors such sample mass and height as well as ambient temperature and relative humidity significant influence the overall heat and mass transfer during meltdown.

Modeling and model discrimination
The meltdown curves illustrated in Figure 2 exhibited a sigmoidal behavior, regardless of the brand and type of ice cream.Subsequent modeling was conducted in Brand B, where the two plateaus and linear segment were clearly visible.A number of models can be used to mathematically describe a sigmoidal behavior, such as the logistic model, the Gompertz model, the Richard model, and the Hill model (Table 1).The adjustable parameters and fitting performance of the tested models are shown in Table 2.All the tested models were seemingly adequate to describe the meltdown curves, judging by the R 2 and R Adj 2 values (≥0.998,Table 2).The R 2 measures the ability of the model to replicate the response, while R Adj 2 accounts for the influence of the number of parameters within the model.The values of M on , M rate , and M Max were about 634-636 s, 0.0031% s −1 , and 64-66%, respectively, regardless of the model.These candidate models were further discriminated by a wealth of statistical information that accounted for a number of parameters and fitting performances (Table 3).Details on the methodology for model discrimination can be found elsewhere (Stewart and Caracotsios, 2008).Overall, model discrimination is an optimization procedure where the candidate model that yields the lowest RSS and E(%), and the highest R adj 2 , AIC, and F-value is selected as the best model to describe the experimental data.The logistic model was selected as the best candidate model to represent the meltdown of ice cream.Model discrimination is a relative concept, not an absolute one since it only provides information about the statistically most plausible model, not necessarily the true one (Knol et al., 2009).
Figure 3 illustrates the location of the calculated parameters with the logistic model from a typical meltdown test.Inset zooms into the 0-650 s interval to highlight the onset of meltdown, where the first drop of the melted ice cream was recorded.Thus, the adjustable parameters of the logistic model precisely locate the onset, rate, and maximum meltdown.

Mapping the meltdown behavior
The M on , M rate , and M Max are relevant parameters that can be used for mapping the meltdown of different types of commercial frozen desserts.This idea was explored further by an additional set of experiments, where the hardness of a given frozen dessert was plotted against either the M on , M rate , or M Max (Figure 4a-c, respectively).Eight groups of commercial frozen desserts were tested, including (1) low-fat, (2) medium-fat, (3) high-fat, (4) dairy-free, (5) high-protein, (6) softserve, (7) gelato, and (8) sherbet.The error bars on the y-axis (hardness) correspond to the standard deviation of three replicates (n=3), while the 95% confidence interval was used as the error bars in the x-axis (onset, rate, or maximum).Figure 4a attempts to map frozen desserts based on their hardness and M on .Overall, the hardness-M on relation allowed for distinguishing six out of eight groups.The relationship hardness-M on failed to differentiate between ice creams marketed as low-, med-, and high-fat, where the different samples clearly overlapped ((1-3) in Figure 4a) in terms of hardness and M on .(8-19N and 2190-2730 s, respectively).Group 8 that includes sherbet was clearly mapped with relatively low values of hardness and M on (7.5-8.5 N and 195-284 s, respectively).Similarly, Group 7 that consists of samples marketed as gelato exhibited hardness values of 28 to 35 N and M on values of 550 to 710 s.Group 6 corresponds to soft-serve samples that exhibited hardness values of 0.03-0.05N and M on values of 1405 to 1625 s.The hardness-M on behavior of dairyfree samples, Group 4, was mapped within the medium range of hardness and M on (44-48 N and 2162-2694 s, respectively).Finally, Group 6 corresponds to highprotein ice creams that were mapped within the higher spectrum of hardness and M on (74-170 N and 3466-3855 s, respectively).
The values of M rate is often considered the most relevant parameter derived from the meltdown test (Liu et al., 2023;Muse and Hartel, 2004).Figure 4b maps the frozen desserts according to their hardness and M rate .By plotting the hardness vs M rate instead of the M on allowed to differentiate seven out of eight groups of frozen desserts.However, it failed to differentiate between frozen desserts marketed as med-fat and sherbet, where the values of M rate clearly overlapped ((2,8) in Figure 4b).The first group mapped at the lower spectrum of hardness and M rate corresponds to low-fat ice cream ((1) in Figure 4b) that exhibited hardness values of 10.5 to 18.6 N and M rate values 0.00061 to 0.00048% s −1 .Group 6 represents soft-serve samples with low values of hardness and M rate (0.52-0.89N and 0.00065 to 0.00081% s −1 , respectively).Ice creams marketed as high-fat ((3) in Figure 4b) were mapped within the low spectrum of hardness and medium M rate (12.16 to 15.88 N and 0.0015 to 0.0015% s −1 , respectively).Group 7 consists of gelatos having hardness values of 28.7 to 35.1 N and M rate values of 0.0013 to 0.0016% s −1 .Similar hardness-M rate was observed in dairy-free samples (Group 4), where the hardness values ranged from 44.6 to 47.89 N and M rate values ranged from 0.0013 to 0.0018% s −1 .Group 5 includes high-protein ice creams, where the samples exhibited relatively high values of hardness (80.36-159.64N) and fast meltdown (0.0012-0.0016% s −1 ).Additionally, the hardness was plotted against M Max , and this relationship allowed to differentiate seven out of eight groups of frozen des- serts (Figure 4b).The relation hardness-M Max failed to differentiate between frozen desserts marketed as med-fat and sherbet, where the values of M Max clearly overlapped ((2,8) in Figure 4c).Interestingly, the relation hardness-M rate provided the same output that the relation hardness-M Max .
As discussed earlier, the meltdown of frozen dairy desserts is a complex process, where a number of factors play a significant role, including the fat content and fat aggregates, type of hydrocolloids, homogenization, freezing, overrun, and number and size of ice crystals (Liu et al., 2022(Liu et al., , 2023)).In an attempt to map the commercial frozen desserts, a new metric coined as the meltdown behavior was used and it accounts for the M on , M rate , and M Max .M Be is a dimensionless response that describes the meltdown of a given frozen dessert by combining key parameters derived from the meltdown test.Moreover, the M Be expresses the meltdown as an effect and not as an explicit function that be correlated.Interestingly, the 8 groups of commercial samples were clearly mapped based on their meltdown behavior.Sherbet and soft-serve were mapped within the lower spectrum of hardness and meltdown behavior (( 8) and (6) in Figure 5).On the other hand, ice creams marketed as low-, med-, and high-fat were mapped within the low spectrum of hardness and medium spectrum of meltdown behavior ((1), (2), and (3) in Figure 5).Dairy-free and gelatos were mapped within the medium  spectrum of hardness and meltdown behavior (( 4) and (7) in Figure 5).Finally, high-protein ice creams were mapped within the high spectrum of hardness and meltdown behavior.

CONCLUSIONS
The meltdown curve (dripped weight vs time) obtained from several commercial frozen desserts exhibited a sigmoidal behavior that was modeled with four different equations (logistic, Gompertz, Richard, and Hill).Based on a wealth of statistical parameters (R 2 =0.999, adjusted R Adj 2 =0.999,Akaike probability = 6582, and F-value = 1.88 x 10 6 ), the logistic model was selected as the best candidate model for representing the meltdown curves and unambiguously calculating the onset, rate, and maximum meltdown.Moreover, a dimensionless response namely meltdown behavior (M Be ) that accounts for the M on , M rate , and M Max was develop and used for mapping the meltdown of different commercial frozen desserts.
corded by a data logger (AD-1688 Weighing Data Logger, A&D Weighing, Columbia, MD, USA) connected to the analytical balance.The meltdown temperature of the ice cream was recorded with a K-type thermocouple (Omega Engineering, Stam-ford, CT, USA) connected to a data logger (OM-CP-OCTPRO, Omega Engineering, Stam-ford, CT, USA).
Figure 2. Representative meltdown curves of commercial ice creams.The meltdown test was carried out at room temperature.
Figure 3. Illustration of the onset, rate, and maximum meltdown of a commercial ice cream.Inset represents a zoom into 0-650 s of the meltdown test.

Figure 4 .
Figure 4. Mapping the meltdown of commercial ice creams: (a) hardness vs onset of meltdown, (b) hardness vs rate of meltdown, and (c) hardness vs maximum of meltdown.

Figure 5 .
Figure 5. Mapping the meltdown behavior of different commercial ice creams.

Table 1 .
Alvi and Martinez-Monteagudo: MAPPING THE MELTDOWN OF FROZEN DESSERTS Equations used to model the meltdown of commercial ice creams 1 of meltdown at a given time; y L o -percentage of meltdown corresponding to the onset of meltdown for the logistic model; A L o -maximum meltdown for the logistic model; k L o ⋅ -rate of meltdown for the logistic model; t -time; t L o -time needed to achieve half the maximum meltdown for the logistic model; y G o -percentage of meltdown corresponding to the onset of meltdown for the Gompertz model; A G o -maximum meltdown for the Gompertz model; k G o ⋅ -rate of meltdown for the Gompertz model; t G o -time needed to achieve half the maximum meltdown for the Gompertz model; d R i -regression parameter for the Richard model; A R i -maximum meltdown for the Richard model; k R i ⋅ -rate of meltdown for the Richard model; t R i -time needed to achieve half the maximum meltdown for the Richard model; y H i -percentage of meltdown corresponding to the onset of meltdown for the Hill model; A H i -maximum meltdown for the Hill model; b H i ⋅regression parameter for the Hill model.

Table 2 .
Alvi and Martinez-Monteagudo: MAPPING THE MELTDOWN OF FROZEN DESSERTS Adjustable parameters of the different models used to describe the meltdown curves of commercial ice creams.Equations used to model the meltdown of commercial ice creams 1 -percentage of meltdown corresponding to the onset of meltdown for the logistic model; A L o -maximum meltdown for the logistic model; k L o ⋅ -rate of meltdown for the logistic model; t L o -time needed to achieve half the maximum meltdown for the logistic model; y G o -percentage of meltdown corresponding to the onset of meltdown for the Gompertz model; A G o -maximum meltdown for the Gompertz model; k G o ⋅ -rate of meltdown for the Gompertz model; t G o -time needed to achieve half the maximum meltdown for the Gompertz model; d R iregression parameter for the Richard model; A R i -maximum meltdown for the Richard model; k R i ⋅ -rate of meltdown for the Richard model; t R i -time needed to achieve half the maximum meltdown for the Richard model; y H i -percentage of meltdown corresponding to the onset of meltdown for the Hill model; A H i -maximum meltdown for the Hill model; b H i ⋅ -regression parameter for the Hill model; R 2 -coefficient of determination; R Adj

Table 3 .
Model comparison and model discrimination for the meltdown test 1