Towards flexible management of postharvest variation in fruit firmness of three apple cultivars

Highlights

• Postharvest variability in flesh firmness of three apples cultivars was modeled.

• Fruit-to-fruit variability in flesh firmness increased during storage and shelf-life.

• Rapid softening apple cultivars showed very large postharvest variability in firmness.

• Parameters for softening depended on picking date.

Abstract

Stochastic modeling provides a useful tool in managing biological variation in the postharvest chain. In the current study, the fruit-to-fruit variability in the postharvest firmness of apples was modeled. Apples from three cultivars (‘Jonagold’, ‘Braeburn’, and ‘Kanzi’) were harvested at different levels of maturity, and stored at different temperatures and controlled atmosphere (CA) conditions. By using a kinetic model describing firmness breakdown as a function of time, temperature, controlled atmosphere conditions and endogenous ethylene concentration, the main stochastic variables were identified as the initial firmness and the rate constants for firmness breakdown and ethylene production. Treating these variables as random model parameters, the Monte Carlo method was used to simulate the propagation of the fruit-to-fruit variability in flesh firmness within a batch of apples during storage under different CA conditions and subsequent shelf-life exposure. The model was validated using independent data sets from apples picked in a different season. The model developed in this study can be used to predict the probability of having apples of certain firmness after long term storage for different scenarios of temperatures and CA conditions.

Introduction

Consumers prefer products with uniform quality, based on important quality indicators such as color, size, soluble solids content and firmness. Postharvest handlers are usually faced with the difficult task of dealing with large variations in product quality due to inherent biological variations. Biological variation is a consequence of several factors related to both pre- and postharvest treatments. For instance, apples picked on different dates will ripen differently during storage due to differences in maturity at harvest. Also, even for the same batch of produce, there is quite often fruit-to-fruit variation resulting from factors such as differences in shading, orientation of the fruit on the tree, and types and levels of fungicides (Kingston, 1991). As a result, there is always a large variation in product quality in a postharvest food chain. As illustrated by Carroll (2003) and Tijskens et al. (2003), biological variation cannot simply be treated as disturbances; rather it needs to be properly managed. It is, therefore, important to be able to predict how the initial variations in quality propagate along the whole chain.

Over the last decade, many authors have developed quality models incorporating biological variance to explain the propagation of biological variation in fruit (Tijskens et al., 2000, Hertog, 2002, Hertog et al., 2004, Hertog et al., 2007a, Hertog et al., 2007b, Scheerlinck et al., 2004, Nicolaï et al., 2006, Mziou et al., 2009). Several approaches have been used in developing these models, such as mixed effects models (Lammertyn et al., 2003a, De Ketelaere et al., 2004), stochastic kinetic models (Hertog, 2002, Schouten et al., 2004, Tijskens et al., 2005, Tijskens et al., 2008), Monte Carlo simulations (De Ketelaere et al., 2004, Hertog et al., 2009), numerically solving the Fokker–Planck equation (Scheerlinck et al., 2004) and the variance propagation algorithm (Nicolaï et al., 1998, Scheerlinck et al., 2004, Mziou et al., 2009). The use of mixed effects models and stochastic kinetic models depends on the availability of an analytical solution of the model equations describing the process. However, biological systems are usually so complex that even when the process is simplified, the resulting model equations usually do not have an analytical solution. Another prerequisite for obtaining an analytical solution of a set of differential equations is the assumption of constant boundary conditions, such as temperature and gas conditions, which is not realistic for a complete postharvest food chain. Numerically solving the Fokker–Planck equation allows calculation of the propagation of the probability density function of random variables of a stochastic system. However, for equations describing complex systems, such as most biological processes, it is nearly impossible to solve the Fokker–Planck equation, even using numerical methods. In the variance propagation algorithm, which is a first order approximation of the Fokker–Planck equation, only the propagation of the mean and variance is predicted. Further, a common practice is to assume that for a linear system, if the initial stochastic variables are normally distributed, then the distribution remains normal, such that a normal probability density function can be used to predict the propagation of the probability density function (Mziou et al., 2009, Gwanpua et al., 2012a). In most cases, neither the stochastic variables are initially normal, nor are the equations describing the system linear. The Monte Carlo method is a more robust method when it comes to modeling a stochastic system. It involves repeated simulation of a process characterized by a system with random model parameters, using new values for the random model parameter(s) for each run of the simulations. Although the Monte Carlo method demands a relatively high computation time, it has an advantage over the other methods in that it is neither limited by the number of stochastic variables involved, nor by the complexity of the model equations describing the process. This method has successfully been used extensively in food engineering (Nicolaï et al., 1999, Poschet et al., 2003, Pouillot and Delignette-Muller, 2010, Busschaert et al., 2011, Hoang et al., 2012). In postharvest science, Hertog et al. (2009) used the Monte Carlo method to model variability in the Hue color in tomatoes during different postharvest regimes, while De Ketelaere et al. (2004) used it to predict shelf-life of tomatoes. The Monte Carlo method was also used to predict the firmness of mangoes during storage by De Ketelaere et al. (2006) and to explain softening of individual avocado fruit by Ochoa-Ascencio et al. (2009).

Flesh firmness is one of the most important quality indicators used in apple grading. Like other quality aspects of biological products, there is also a large variation in flesh firmness of apples at harvest. It is very important to be able to understand and predict how this initial variation in the firmness propagates throughout the apple cold chain. Most of the current models for apple firmness are only able to explain changes in the mean firmness (Hertog et al., 2001, Johnston et al., 2001, Gwanpua et al., 2012b). The objective of this study is to use the Monte Carlo method to model and explain the fruit-to-fruit variability in flesh firmness within a batch of apples for three cultivars (‘Jonagold’, ‘Braeburn’, and ‘Kanzi’) during controlled atmosphere (CA) storage and subsequent exposure to shelf-life conditions. Furthermore, the model will be validated by independent data sets. Practical applications of the model will also be discussed.