Live Rohu samples were collected (irrespective of the sex ratio) from a local fish farm pond (near IIT Kharagpur, West Bengal, India) having same cultural practices and rearing environment and then collected sample were transported to the lab in polybag containing farm pond water. Fish were kept free from stress before stunning death. The fish were killed by percussive stunning (involves hitting the fish's head with a wooden) the head for sudden death on 0-day means starting day of storage (Abbas et al., 2005; Jain et al., 2007). Random sampling was done such that three fishes were taken from all the lots, and their average length and weights were further measured and found to be 29.53 ± 2.32 cm and 620 ± 30 g, respectively. Rohu samples were covered by a layer of aluminum foil and kept at different storage temperatures (5 ± 1, 0 ± 1°C) for the study. The fish sample was stored at the required temperature for the study in the refrigerator, which was domestic (Whirlpool, FF2D258C/2016, Gurugram, India), A deep freezer (Voltas, 205 CF Metal Top, Delhi, India), and a quick microprocessor freezer (REMI, RQFVM-205D, Vasai, India) for 24 days. A multi-probe thermocouple (digital thermometer) was used to ensure the consistency of the required set temperature in every case.

The volatile compound, TVB-N were estimated by the method executed by Prabhakar et al. (2019) i.e., the Conway micro-diffusion method referred by FSSAI (2015) manual (Chouhan et al., 2015) with partial modification. However, some other routine methods may be used to check the TVB-N limit, such as the direct distillation method described by Conway, Antonacopoulos (1968) and the distillation of an extract deproteinized trichloroacetic acid reported by FAO/WHO through FAO/WHO (1968). As per FSSAI (2015) manual, 5 grams of fish muscle taken from stored fish was homogenized with 45 mL of distilled water. 5 ml of this homogenate were added with an equal volume of 10% trichloroacetic acid (TCA) (w/v) and then filtered through Whatman's filter paper no. 1. 1 ml of this extract and saturated potassium carbonate each was pipetted into the outer ring of the micro-diffusion unit containing 1 ml inner ring solution. Inner ring solution was prepared by mixing 2% boric acid (92 ml), 0.1% alcoholic bromocresol (4 ml), and 0.1% alcoholic methyl red (4 ml). The covered and sealed unit was rotated gently to mix the solutions of the outer ring. The unit was incubated for 180 mins at 37°C and cooled at room temperature. 0.02 N sulfuric acid was used for titration of the inner ring solution until it turned pink. Then, 1 ml of 10% trichloroacetic acid was taken for a blank test instead of sample extract. The same procedure was adopted for TMA-N determination as TVB-N, but 1 ml of formalin was added before adding potassium carbonate to the extract to react with ammonia, allowing only the TMA-N to diffuse over the unit. Each experiment was replicated three times for every storage condition, and the obtained TVB-N and TMA-N results were expressed in mg/100 g of sample. These TVB-N and TMA-N obtained data were used for its kinetic study.

In general, the number of spoilage bacteria (total bacterial count or specific bacterial) and level of some biochemical indexes (TVB-N) are used to evaluate the freshness of fish. However, as per Chouhan et al. (2015), Fan et al. (2008), and Prabhakar et al. (2020), the total viable count (TVC) was considered to be calculated to explore the microbiological quality of fish and fish products. Therefore, a microbiological quality analysis was done as per Downes and Ito (2001) with minor modification. 10 g of Rohu muscle was weighed aseptically into a sterile sample dish and homogenized with 90 ml of sterile physiological saline (0.854% NaCl) for 3 mins. using a sterile pipette, 1 ml of the supernatant was aseptically transferred into a 9 ml saline tube and mixed well-using vortex mixer. Similarly, further dilutions were prepared for the inoculation. 1 ml each of the appropriate dilutions was made and poured onto agar by spread plate technique. The plates were incubated for 48 h at 37°C in an inverted position. Total viable count (TVC) was enumerated and expressed as log10 cfu/g of the sample. The experiment was conducted at an interval of 4 days.

Many biological reactions in fish, such as volatile formations (TVB-N, TMA-N, and other biogenic amines) and vitamins loss during isothermal storage and processing operation, probably follow fixed order of kinetics (Peleg, 2016). The formation of the TVB-N and microbial population in stored Rohu was modeled using different models (Table 1) that all set of algebraic equations (empirical and phenomenological models). In the present study, all analyses (TVB-N and microbial biomass) were done in triplicates, and readings were expresses as average values ± SD. The biomass mass data (TVB-N and microbial population) were fitted to different models (Table 1) using non-linear regression analysis through the ORIGIN 2021 software (Origin Lab Corporation, Northampton, USA). The suitability and fitness of models were examined based on the coefficient of determination (R²), chi-square (χ²), and root mean square error (RMSE). However, secondary modeling, i.e., the Arrhenius model was not for the temperature sensitivity analysis of microbiological and chemical changes in the stored fish because the study has been made at only two temperatures.

Where, n is the number of experiments; Y_i,e is the experimental value of the ith experiment; Y_i,p is the predicted value of the nth experiment by model; and Y_e is the average value of experimentally determined values.

Along with the above statistical analysis, models were also discriminated using statistical analysis of the residuals and Akaike information criterion (AIC) and corrected Akaike information criterion (AIC_c), which were calculated from Equations 5, 6 (Van Boekel, 2009; Heising et al., 2014a). AICs is AIC with a correction for small sample sizes specially when the sample size is more than the number of model parameters. The lowest values of AIC and AICs are preferred for selecting the best fit model among all models.

Where, SS_r is the residual sum of squares, n is the number of observations, and p is the number of parameters.

Bayesian information criterion (BIC) is also a criterion for model selection among a finite set of models. It penalizes the complexity of the model, which refers to the number of parameters in the model. The lowest value is preferred for the selection of the model. BIC can be calculated by using Equation 7.

The results obtained in terms of TVB-N and microbial population from the storage study were used to consider existing empirical models (growth/kinetic) from literature (Table 1) to explore its suitability for TVB-N and Microbial biomass development. Thereafter, parameters were estimated and compared with other validated models from the literature, and the temperature effect was also explored. TVB-N formed in fish during storage is due to some microbial metabolites responsible for off-flavors and off-odors (Cheng et al., 2015; Prabhakar et al., 2019). These metabolites may be TMA-N, ammonia, and DMA. The total volatile base-nitrogen (TVB-N) index combines these amines into a unique indicator used to assess the spoilage of fresh fish. In fact, the content of TVB-N is the only biochemical index presently included in EU regulations to discriminate fit from unfit fish in commerce or an official inspection. Neither TVB-N nor trimethylamine (TMA) content can be used as indexes of quality in the early stages of storage (Dalgaard et al., 1993; Howgate, 2010a). The Codex Alimentarius Committee (CAC) and FAO/WHO (1968) have recommended TVB-N as a regulatory compound for deciding the freshness of different fish species. TVB-N is also associated with the activity of endogenous enzymes (Kyrana et al., 1997; Vareltzis et al., 1997). According to Connell (1990), spoiled fish exhibit a TVB-N level above 35 mg N/100 g. The formation of TVB-N is also the function of fish species. The European Commission (European Union Law 95/149/EC, 1995) has specified TVB-N limits for different species, such as 25 mg/100 g for Sebastes spp. (Helicolenus dactylopterus, Sebastichthys capensis), 30 mg/100 g for species belonging to the Pleuronectidae family (except halibut: Hippoglossus spp.), and 35 mg/100 g for Salmo salar. On the other hand, storage conditions (time and temperature) affect the shelf life and quality of fish (Fan et al., 2008) and can increase the formation of TVB-N, which is a significant cause of spoilage of Rohu fish (Mehta et al., 2014). According to Connell (1990), the permissible limit of the TMA-N level should be lower than 35 mg N/100 g, although this limit varies from species to species. Malle and Poumeyrol (1989) reported TMA-N values of 10–15 mg/100 g in aerobically stored fish and at a level of 30 mg TMA-N/100 g in packed cod. O'zogul et al. (2004) reported that the concentrations of TMA-N in numerous fatty fish never reached the limit of 5 mg TMA-N/100 g. However, the rejection limit in fish flesh is usually higher. Therefore, TMA-N formation in stored fish depends upon species, fish composition, storage temperature, duration, and local environmental composition. However, some studies have found significant correlations between TVB-N, TMA, and storage time (Baixas-Nogueras et al., 2001; Ruiz-Capillas and Moral, 2001). Storage conditions like temperature and duration influence the volatiles formation and other biochemical changes in fish caused by either microbial or enzymatic activities. Spoilage bacteria are selected primarily due to the physical and chemical conditions in the products; however, seafood spoilage involves the growth of the microorganism to a high number (>10⁶-10⁷cfu/g) of antagonisms or symbiosis between the different groups of microorganisms may influence their growth and metabolism (Gram and Dalgaard, 2002).

The majority of quality and shelf-life models in the literature focus on the most probable prediction. There is a huge literature on fish spoilage, technological, economic, and safety implications, and a continuous attempt has been made to monitor the spoilage in quantitative terms. Microbial counts are probably the most direct method, which is generally exercised using a traditional technique that involves incubation. It is very time-consuming and logistically inconvenient (Peleg, 2016). A mathematical tool can be applied to replace this direct method for rapid determination of microbial population, which predicts the shelf life (Heising et al., 2014a; Prabhakar et al., 2019). The various models (for TVB-N and microbial biomass) selected based on literature and the data pattern were analyzed for experimental data obtained in the present study. A variety of mathematical models has described microbial growth. Their properties and how well they fit and predict and predict experimental growth data are discussed by Peleg and Corradini (2011). Most of the empirical models are based on physical and biochemical principles of mass transport in stored fish. Adapted Howgate model uses a mathematical model framework for sensor response of intelligent packaging. Therefore, indicating that the modeling of TVB-N dynamics could be used to predict shelf life. For the volatile formation, a wide range of models starting from complete exponential to asymptotic behavior has been used to explain the dynamics (Van Boekel, 2008; Heising et al., 2014b; Garcia et al., 2017). Furthermore, models having asymptotic nature such as logistic, modified logistic, Gompertz, and modified Gompertz were found to be extensively used in the literature (Baranyi and Roberts, 1994; Giannuzzi et al., 1998). Based on this, these models were also considered for the present study with a hypothesis that asymptote is expected.

The experimental TVB-N values during storage of Rohu fish at an interval of 4 days were modeled to find the best empirical models. The most common TMA-N or TVB-N formation model is usually an exponential model, i.e., Equation 8 (Tsironi et al., 2009; Tsironi and Petros, 2010; Yao et al., 2011; Garcia et al., 2017).

Where, C stands for TVB-N content, k is the velocity of the reaction, and t describes the time elapsed since the fish arrived for the market (days). C₀ is the TVB-N content at t = 0, i.e., when the fish arrives to the storage. The experimental data of TVB-N were fitted to an exponential model (Equation 8). Model and statistical parameters were represented in Table 2. As per the fitted exponential model, the predicted initial concentration (C₀) of TVB-N was found to be closer to the sample stored at 5°C (5.69 ± 0.3102 mg/100 g) than 0°C (5.29 ± 0.2825 mg/100 g), which showed that it was independent of temperature. Figure 1 showed the exponential model curve having coefficient of determination (R² = 0.9887 and 0.9796), Chi-square (χ² = 0.3976 and 0.2782). The values of R² are quite high, and χ² are low, indicating its goodness of fit for the dynamical understanding of TVB-N under experimental conditions of the present study. The reaction rate, i.e., k-values, were 0.053 ± 0.003 and 0.042 ± 0.003 day⁻¹ for the TVB-N formation of the Rohu sample stored at 5°C and 0°C, respectively. k values indicated that lower storage temperature would have arrested the reaction, slowing the reaction velocity. Figure 1 showed that there was a continuous increasing pattern of TVB-N formation throughout the storage period, irrespective of the demarcation of slow and advanced spoilage, without reaching and explaining a maximum asymptote. Howgate (2010a); Heising et al. (2014b) critically pointed out that the critical drawback of the conventional model of TVB-N content is their inability to produce the initial dwell and the exponential behavior.

Heising et al. (2014b) postulated that the volatile formation in fish and fish products stored in ice or refrigerated conditions from the harvesting/catch often starts with a period in which the volatiles like TMA-N was not increased. It was followed by another phase with a rapid increase in the TMA-N concentration, followed by a third phase in which the volatile formation rate diminishes. The TVB-N is comprised of TMA-N, DMA-N, and NH₃ gas, and its formation is a function of enzymatic and bacterial actions influenced by storage conditions (Huss, 1995; Prabhakar et al., 2020). Nevertheless, the exponential model is sufficient enough to simulate the TVB-N formation in all phases and stages. Though the fitting of exponential mode is good, as attributed by R² and χ² values. Keeping these facts, another model, such as the modified exponential model (Equation 9), was considered for modeling the dynamics of TVB-N formation.

Moreover, this model was suggested by Howgate (2010b) for the TMA-N formation in fish stored on ice. The parameter “d” represents the dwell (days), and C₀ describes the TVB-N concentration at time t = 0. A logistic model, which is an advanced form of the exponential model, was found to have a good fit with the present TVB-N experiment values as (R² = 0.9246 and 0.9255), and chi-square (χ² = 3.3328 and 1.2697) these statistical parameters showed less fit in comparison to exponential model (Equation 8).

Modified exponential model (Equation 9) attributed a dwell period of 4.94 × 10⁻¹⁵ days at 5°C, and a 6.22 × 10⁻¹⁴ day at 0°C, which was negligible, which can be understood through the logistic model curve (Figure 2), the corresponding k values were found to be 0.11283 ± 0.006 and 0.09483 ± 0.005 day⁻¹. The present findings are in line with the trend of fishy odor formation in packed codfish filets at varied storage temperatures (0, 5, 10, 15°C) which were reported by Heising et al. (2014a).

It was inferred that this modified exponential model (Equation 9) doesn't include and describe a parameter for maximum concentration of TVB-N. Thus, the model increases exponentially even under the advanced spoilage phase (Howgate, 2010b; Heising et al., 2014b). This equation only uses for the initial phase of volatile formation. For minimum and maximum asymptotic concepts, other models such as the Howgate (Equation 10) and adapted Howgate model (Equation 11) were considered for fitting analysis under present experimental conditions. The parameters C_max represents upper asymptote concentration (mg TVB-N per 100 g fish muscle), C_min lower asymptote concentration (mg TVB-N per 100 g fish). This equation gives the asymptote value of C_max-C_min for a higher value of t. It will give a lower value of C_TVB−N at the same time t and C_max since the value of the numerator decreases with increasing C_min. The data were modeled using Equations 10, 11 considering C_min > 0, k > 0, t_d > 0, and t > 0; otherwise, it will make an unrealistic situation. The data of TVB-N under both storage conditions and the C_max were parameterized for Howgate and adapted Howgate model as 20 ± 3.6452 and 20 ± 3.4707 mg/100 at 5 °C which are very close to each other and, for 0°C, C_max values were estimated to be 19.99 ± 5.8026 and 18.94 ± 4.8631 mg/100 g for the respective model.

The points of inflection (t_d) ranged between 9.2161 ± 3.313 to 14.0912 ± 7.6868 days in the Howgate model (Equation 10), whereas it was ranged from 9.2079 ± 3.1947 to 12.6494 ± 6.5675 days for the adapted Howgate model (Equation 11). The estimated statistical parameters like R² ranged from 0.96338 to 0.98518 for the Howgate model. For the adapted Howgate model, R² values were found to be 0.9640 and 0.9840. The k-values of the Howgate model were 0.1394 ± 0.0486 and 0.0823 ± 0.0241 day⁻¹ whereas, for the adapted Howgate model, it was found 0.14245 ± 0.0450 and 0.08677 ± 0.0252 day⁻¹. The graphical pattern of TVB-N formation in the model curve of Howgate and adapted Howgate model was present in Figures 3, 4. As a result, the exponential model was found to be the best fit model for predicting TVB-N biomass in rohu fish stored at 0°C and 5°C. The exponential model fitting showed R² (0.9796, 0.98887) Adj. R² (0.9755; 0.9865), the lowest χ² value (0.3976, 0.2782), RMSE (0.52741, 0.6306), AIC (−7.3122, −4.8106), AIC_c (−0.5122, 1.9894) and BIC (−7.4204, −4.9188) and thus this can be selected as the best fit model for TVB-N prediction in rohu fish stored at 0 and 5°C, respectively. After a complete interpretation, it was found that the TVB-N curve was not captured fully the whole storage phenomena, where it was hypothesized that an asymptote is expected. The asymptote model could be a better fit for this study, although no asymptote is observed.

Predictive microbiology is now a possible alternative to traditional microbiological assessment of food safety and quality. Various environmental factors on the growth of microorganisms have since been quantified in laboratory media, and kinetic models have been developed (Dalgaard and Huss, 2020). However, for most fish products, the microorganisms responsible for spoilage are not known. Furthermore, the spoilage microflora is also likely to change when storage conditions (e.g., temperature and atmosphere) are changed over wide ranges. This dynamic nature of the microbial spoilage process indicates that specific spoilage organisms' spoilage domains must be determined before kinetic models can be developed. Sometimes, interpreting growth curves and growth parameters of non-specific microbial groups is always a complex undertaking since they represent the sum of growth curves of a mix of different microorganisms. Nevertheless, in the present study, specific spoilage bacteria were not determined; instead, total viable count (TVC) was calculated for dynamical analysis. At the first stage, three-parameter models (logistic, modified logistic, and Gompertz) and after that, four-parameter models such as modified Gompertz were fitted to microbial experimental data of stored Rohu. Figures 5–8 represented the logistic, modified logistic, Gompertz, and modified Gompertz model curve. All the model equations are appended below:

The logistic model (Equation 12), initially proposed by Verhulst in the 18th centuries, is been a widely used model of organisms' growth dynamics in a finite resource habitat. It has opened the framework of more elaborative models like Baranyi and Roberts (1994) and others (Peleg et al., 2007). It represents the momentary growth rate of a microbial population in a virgin environment. There is a lot to explore about the habitat and resources. Thus, this study considered this model as a comparative mode and further exploitation of multi-parametric models as deemed fit to explore the microbial growth dynamics in stored rohu fish.

Parameter “a” represents the difference of microbial biomass at lower and higher asymptote of Equation 12. The k is for growth rate, and c describes the carrying capacity of the biomass dynamics, and M is the microbial population after time t. Another modification like the modified logistic model is conceptualized by introducing another parameter like “b” (b > 0) depends upon temperature and other environmental factors; c without carrying capacity.

A general Gompertz model expression (Equation 14) of three parameters was used to understand the microbial biomass during rohu fish storage as the Gompertz model and Baranyi and Roberts (1994) have been extensively used in the literature. The further extension of the Gompertz model in the form of modified Gompertz equation, as reported previously (Zwietering et al., 1991). Equation 15 represents the modified Gompertz model having some response factors such as lag phase duration (t_l) and maximal growth rate (μ_max). This model has four parameters that can explain the storage conditions having an influence on the rohu fish.

The data presented in Table 3 imply the model and statistical parameters estimated after data fitting to different models (Equations 12, 13, 14, and 15). In light of this literature, discussion, and criticisms, all these models were considered curve fitting analysis to understand the dynamics of microbiological assisted volatile formation to find the best fit model for predicting the freshness of stored Rohu fish under limited controlled refrigerated storage. The values range varied from 9.5031 ± 0.3533 to 5.8469 ± 0.4564, 9.5031 ± 0.3534 to 8.5469 ± 0.4564, 10.0799 ± 0.4297 to 9.4356 ± 0.9688 and 5 ± 3.190 to 5 ± 2.455 for logistic and Gompertz and modified Gompertz model. The Gompertz model's carrying capacity (c) was found to be ranged from 3.3349 ± 0.4799 and 4.4769 ± 1.2541. The maximal growth rate (μ_max) values were 0.4602 ± 0.41115 and 0.3631 ± 0.1772 day⁻¹ for the rohu samples stored at 5 and 0°C. The statistical parameters of model fitting ranged from 0.0823 to 0.0769, 0.0823–0.0769, 0.0537–0.1071 and 2.0169–0.6329 (χ²) and 0.9919–0.9899, 0.2291–0.9899, 0.9947–0.9859 and 0.8518–0.9379 (R²). From the analysis of model parameters, the Gompertz model was found to be the best fit model for microbial biomass prediction in rohu fish stored at 5°C (R² = 0.9947, χ² = 0.0537), having lowest AIC (−18.379), AIC_c (−6.3792), and BIC (−18.542). In contrast, both of logistic and modified logistic models were best suited at 0°C storage condition (R² = 0.9919, χ² = 0.0823, RMSE = 0.28683) as it showed the lowest AIC (−16.342), AIC_c (−3.4013) and BIC (−14.123) values. It has been evident from the literature that most researchers have modeled the microbial growth of specific species and strains. Antunes-Rohling et al. (2019) studied modeling of microbial growth in terms of different microbial groups such as Aerobic Mesophiles, Anaerobic Mesophiles, Aerobic Psychrotrophes, Anaerobic Psychrotrophes, Pseudomonas, Shewanella, Lactic Acid Bacteria, Enterobacteriaceae, Molds, and Yeasts. He examined these analyses in modified-atmosphere-packed hake (Merluccius merluccius) filets stored at different temperatures ranging from 1–10°C. The present study's model curve and statistical parameters were found in line with Antunes-Rohling et al. (2019). Moreover, it is recommended to explore further growth modeling of a specific group of microorganisms under more range of storage temperatures. Overall, as a future study, it is recommended to go for higher no of data points (days of storage) to make the modeling process more robust and highly significant.