Empirical modelling of postharvest changes in the firmness
of kiwifruit
Jason R. Benge
a,*, H. Nihal De Silva
b, Nigel H. Banks
a, Peter B. Jeffery
aaInstitute of Food,Nutrition and Human Health,College of Sciences,Massey Uni
6ersity,Pri6ate Bag 11222, Palmerston North, New Zealand
bThe Horticulture and Food Research Institute of New Zealand,Pri
6ate Bag 11030,Palmerston North, New Zealand Received 29 November 1999; accepted 21 February 2000
Abstract
Several different types of empirical mathematical models were used to characterise the softening behaviour of ‘Hayward’ kiwifruit (Actinidia deliciosa(A. Chev) C.F. Liang et A.R. Ferguson) during storage at 0°C. Our purpose was to determine whether or not the softening behaviour of fruit conformed to a limited number of patterns. If this were so, such models, coupled with measurements made in a short period after harvest, might form the basis of a quantitative tool that would allow the industry to segregate batches of fruit with differing storage potentials. Initially, three simple models were used to characterise firmness data: a Complementary Michaelis – Menten type (CMM), Exponential (EXP), and Complementary Gompertz (CG). However, these were unable to characterise firmness changes with sufficient accuracy, either in the early, middle or latter stages of storage. Instead, the firmness data were better characterised by two more complex models that were identified during the course of the study: a segmented Jointed Michaelis – Menten type (JMM) and Inverse Exponential Polynomial (IEP). With the JMM model, different functional relationships were assumed for different regions of the time domain although its parameters were difficult to estimate accurately when the number of data points for a region of the time domain was limited. The IEP model best characterised the firmness data although its ability to predict the softening behaviour of fruit was poor given a limited window of data. None of the equations that were studied provided a standard curve that could be useful as a predictive model for firmness in storage. Nevertheless, the more complex equations did accurately characterise our firmness data and have potential value for comparing treatment effects in experimental programmes. © 2000 Elsevier Science B.V. All rights reserved.
Keywords:Actinidia deliciosa; Flesh firmness; Prediction; Segmented models; Storage potential
www.elsevier.com/locate/postharvbio
1. Introduction
Firmness is a key criterion by which the re-maining storage potential of kiwifruit is assessed. At harvest, the flesh firmness of kiwifruit is gener-ally in the range of 60 – 110 N (Newtons) but they
* Corresponding author. Present address: 19 Hassard St., Manaia, Taranaki, New Zealand. Tel.: +64-6-3569099.
E-mail address:[email protected] (J.R. Benge).
are not eating-ripe until the firmness is in the range of about 4 – 8 N (MacRae et al., 1990). Therefore, a large decrease in firmness occurs after harvest before the fruit is ready to eat. Generally, fruit softening accelerates over the first period of stor-age. Subsequently, the rate of softening slows, with the inflection point between the two phases occur-ring at a time after storage when firmness drops to about 40 N. In some instances, the rate of softening may increase again towards the end of storage while in others, there may be a lag phase after harvest where there is very little fruit softening (MacRae et al., 1989, 1990).
Although the general nature of softening in kiwifruit is well established, very little has been published concerning the modelling of kiwifruit softening. This is surprising given the potential of modelling to provide objective comparisons of treatment effects on fruit softening in experimental studies. Furthermore, if fruit softening could be characterised by a limited number of simple rela-tionships with time in storage, there could be potential to predict subsequent storage behaviour from measurements made soon after harvest. This would mean that fruit that are likely to soften excessively might be separated from other fruit early in storage and sent to market earlier, thereby facilitating inventory management and reducing fruit losses. In this paper, a number of models were fitted to firmness data for batches of kiwifruit from different growers and seasons. Our purpose was to determine whether or not a model with robust parameter estimates could be identified that would characterise the softening behaviour of fruit from different sources.
2. Material and methods
2.1. Data
The source data used for modelling changes in the firmness of ‘Hayward’ kiwifruit came from studies in 1991 and 1994 which assessed the storage behaviour of fruit from the same four randomly selected orchards in the Bay of Plenty region, New Zealand. In the 1991 season, fruit were collected in May on two occasions from each grower, i.e. May 1 (early harvest period) and May 10 (main harvest
period). This provided two sets of firmness data for each grower in the 1991 season. In 1994, fruit were collected only once from each grower on May 11 (main harvest period).
Fruit from all three harvests were of export grade (count size 36) and commercially packed into single-layered trays with polyliners. The fruit were stored at 0°C for 6 – 8 months with firmness as-sessed at approximately fortnightly intervals using a drill-mounted Effegi penetrometer fitted with a 7.9 mm plunger. Two measurements were made per fruit, at right angles to each other on areas pared free of skin. On each sampling occasion, 10 to 50 fruit per grower batch were assessed. Across all growers and seasons, measurements of firmness, using the penetrometer, had associated with them standard deviations between fruit that ranged from 10 (measurements made just after harvest) to 40% (measurements made towards the end of storage) of the means. Hence the contribution of this to the error of batch means based on 10 to 50 fruit sample sizes ranged from (as CV) 1.5 – 3% (measurements made just after harvest) and 6 – 13% (measurements made towards the end of storage).
2.2. The models
The functions selected for modelling the firmness data above have the following features. Firstly, they are monotonic, i.e. generate curves that are always decreasing which is consistent with our subject matter knowledge of fruit getting softer and never firmer during storage. They are also consistent with the data under analysis on the entire time domain. The models also contain as few parameters as possible needed to characterise the given sets of data.
Initially, the following three empirical models were identified as being likely to characterise our firmness data:
Complementary Michaelis–Menten type(CMM)
FF=A
1− tv+t
(1)Exponential decay (EXP)
and
Complementary Gompertz(CG)
FF=A0+B
1−1
exp(bexp(−kt))
(3)where FF is the mean flesh firmness att number of days in storage and the Roman and Greek symbols represent parameters in the model. In the CMM model,Ais a scale parameter equivalent to initial firmness andva half-time parameter repre-senting the time taken for firmness to drop to half the initial value. In the EXP model,A0represents
the lower asymptotic value andA1 represents the
drop in firmness from the initial value to the lower asymptote; lrepresents the relative rate of decline with time.
Functions (1) and (2) have only one shape parameter, which implies that they can describe only one curvature in a firmness versus time plot. In comparison, the CG model can describe two curvatures. The A0 in this model represents the
lower asymptotic value of firmness, andBa scale parameter, which together, with b, a horizontal shift parameter, determines the initial firmness value. The parameter k determines the rate at which firmness declines with time.
One approach to modelling data such as firm-ness data with changing curvature is to use a segmented model where two or more functions are used in different regions of the time domain. Continuity and smoothing constraints are im-posed on the equations to ensure they meet in the appropriate way at ‘joint’ points. The advantage of this approach is that individual functions can be kept simple to ensure subject matter relevance of model parameters. The following segmented model based on additions to the CMM, was therefore formulated to characterise the firmness data presented here:
Jointed Michaelis–Menten type (JMM)
FF=
With this model, the relationships between firm-ness and time in both the first and second phases are described by Michaelis – Menten type func-tions. The point of inflection, at which the change from one function to the other occurs, is defined by the parameter, u, on the time axis. The model contains five parameters: a, a1, u, v1 and v2.
Estimation of all five parameters puts no con-straints on how the two functions meet at the ‘joint’ point. The two equations must meet at
t=u (continuity constraint) and have the same slope at this point (smoothing constraint). Impos-ing these two constraints gives the followImpos-ing four-parameter model:
The functions discussed so far (Eqs. (1) – (3), (4a) and (4b)) might be expected to describe the soft-ening behaviour of kiwifruit where only two phases of softening are evident. However, where a third phase of softening (i.e. an increase in the rate of decline towards the end of storage) is evident, other models that accommodate a third curvature, such as a general cubic polynomial would be more appropriate. However, this model will not ensure monotonicity of the fitted curve. Hence, the following exponential polynomial model, derived from a general cubic polynomial and whose rate is always of the same sign, was considered:
In6erse exponential polynomial(IEP)
FF= d
condi-tion to ensure non-zero rate of decline on the entire time domain.
2.3. Model fitting
All of the above models were fitted to each of our data sets by the method of non-linear least squares using the NLIN procedure of the SAS statistical software (SAS, 1996), with the estima-tion of parameter statistics (i.e. parameter esti-mates, standard errors and correlations between estimates). The curves presented here for each grower are fits to the overall mean firmness values of all fruit on each sampling occasion.
Initially, the five models were compared by fitting them to the three sets of firmness data for one grower (Grower 1). Then the two best models were compared in more detail by fitting them to the three sets of data for all four growers.
There-after, the best model was identified and then some useful properties and deductions of that model were explored in more detail. These included the estimated initial firmness of fruit and the time taken for the firmness of that fruit to reach a range of threshold levels including the export threshold of approximately 10 N.
The estimators of non-linear models, such as those above, only achieve the properties of linear models asymptotically, that is, as the sample size approaches infinity, and it is therefore important to have some measure of the non-linear behaviour of such models i.e., how close their properties are to the properties of linear estimators (Ratkowsky, 1990). For the purposes of this paper, correlations between parameter estimates were used to com-pare the non-linear behaviour of models with high correlations interpreted as indicating strong non-linearity.
Fig. 2. Fits of the jointed Michaelis – Menten type (JMM) and inverse exponential polynomial (IEP) models (columns) to data for fruit harvested from growers 2 – 4 (rows) during the main harvest in 1991.
The JMM and IEP models both characterised the softening behaviour of fruit well (Figs. 1 – 3). Of the two, the IEP was the superior as it de-scribed the entire softening behaviour well whereas the JMM tended to underestimate firm-ness during the latter stages of softening. The relative standard errors of the parameter estimates of the JMM model were consistently greater than those of the IEP model (Tables 1 and 2). In particular, estimates of the first rate parameter in the JMM model (v1) had very large errors.
Across growers, the estimates and errors of the parameters in the IEP model were reasonably consistent, as were the correlations between them. In contrast, the estimates and errors of the parameters in the JMM model varied more across growers, especially the v1 parameter (indicating
that this parameter is related to ‘at random’ dif-ferences between batches caused by different growing conditions and harvest maturity).
Fig. 3. Fits of the jointed Michaelis – Menten type (JMM) and inverse exponential polynomial (IEP) models (columns) to data for fruit harvested from growers 2 – 4 (rows) during the main harvest period in 1994.
3. Results
3.1. Goodness of fit of models
Table 1
Parameter estimates and their relative standard errors (Rel. S.E.=standard error/parameter estimate) obtained from fitting the jointed Michaelis–Menten type (JMM) and inverse exponential polynomial (IEP) models to the data for fruit harvested from each of the four growers during the main harvest period in 1991
Parameter Grower 1 Grower 2 Grower 3 Grower 4
Rel. S.E. Estimate Rel. S.E. Estimate Rel. S.E. Estimate Rel. S.E. Estimate
JMM
0.093 53.1
a 44.21 0.185 47.46 0.143 51.6 0.071
0.496 281.24*
v1 45.98* 3.983 92.62* 1.536 29.69* 0.511
0.072 35.05 0.154 34.21
19.71 0.121
v2 25.56 0.067
0.097 27.64
u 30.64 0.295 25.97 0.237 25.32 0.093
IEP
0.019 102.35
d 90.21 0.055 84.41 0.050 88.08 0.031
0.067 −2.025
b0 −2.880 0.162 −2.140 0.096 −2.320 0.089
0.060 0.0862 0.109 0.0886
0.1210 0.065
b1 0.0950 0.066
b2 −0.00095 0.095 −0.00065 0.161 −0.00069 0.095 −0.00070 0.101
* Estimates have large 95% confidence intervals that include 0.
In nearly all cases, all the parameters within each of the JMM and IEP models were highly correlated with each other (Table 3). Correlations between parameters in the JMM model were slightly more variable than those in the IEP model, especially the correlations of the parame-ter v2 with the parameters a and u.
3.2. Estimation and prediction using the IEP model
Since the IEP model provided the best descrip-tion of trends in firmness in storage, reladescrip-tionships between the estimates of various properties and deductions of the curves (e.g. initial firmness and
Table 2
Parameter estimates and their relative standard errors (RSE; standard errors/parameter estimates) obtained from fitting the jointed Michaelis–Menten type (JMM) and inverse exponential polynomial (IEP) models to the data for fruit harvested from each of the four growers during the main harvest period in 1994
Grower 2 Grower 3 Grower 4
Parameter Grower 1
Rel. S.E. Estimate Rel. S.E.
Estimate Rel. S.E.
Rel. S.E. Estimate
Estimate JMM
0.148 54.02 0.117 50.485 0.132 48.956
a 44.02 0.166
34.399
72.89 1.303* 66.64 1.141* 1.126* 38.17 1.348*
v1
0.118
v2 30.78 0.126 35.51 0.106 38.26 34.43 0.146
28.28
0.194 38.47 1.904
30.15 0.183
u 0.179 33.638
IEP
0.053
d 85.69 0.073 93.57 0.061 79.19 81.646 0.089
−1.979 0.208 −2.226 0.171
b0 −2.801 0.143 −3.108 0.210
0.134
0.0796 0.0839 0.113 0.0860 0.109 0.0927 0.169
b1
−0.00059 0.155
−0.00063 0.202
−0.00060
b2 0.148 −0.00062 0.237
Table 3
Correlations between parameter estimates obtained from fitting the jointed Michaelis–Menten type (JMM) and inverse exponential polynomial (IEP) models to the data for fruit harvested from each of the four growers during the main harvest period in 1991
IEP JMM
v1 v2 u Parameter d b0 b1
Parameter a b2
Grower 1
time to reach a given level of firmness) generated by that model were examined with the purpose of identifying any which might reasonably predict the storage life of fruit.
Firstly, we estimated the time taken for the flesh firmness of fruit to reach the industry export threshold level of c. 10 N (t10
est) using parameter
estimates obtained from fitting the model to each full data set. In order to calculate the time taken for firmness to drop to a given value ofx N (txest)
such as 10 N, Eq. (5) needs to be first written as a cubic function of time. For given values of parameters,b0,b1, b2 andd, and a given level of
firmness, FFL, the roots of the function provide a
solution to tx est
The time to reach 10 N ranged from 123 to 203 days. Thet10estvalues were then plotted against the
corresponding estimates of initial firmness (d/
(1+exp(b0)); Fig. 4). The two parameters were
poorly related, with a non-significant (P=0.625) linear regression, which accounted for only 3% of the total sum of squares in t10
est. We also plotted
Fig. 4. Relationship between the estimated initial flesh firmness of fruit and the estimated time taken for the flesh firmness of that fruit to reach the export threshold level for firmness of 10 N (t10est), as determined using the IEP model. Each point
Fig. 5. Relationship between the estimated time taken for the flesh firmness of fruit to reach a given level (tx
est) and the
estimated time taken to reach 10 N (t10est), as determined using
the IEP model. Each point represents the average estimate for a grower batch. Two outliers have been omitted.
data set while rearrangement of Eq. (5) provided an estimate of b0 as:
b.0=ln5 n
i=1
FFi−FF0
FF0−FFiexp
b1ti+b2ti2+b2 2 ti
3
3b1
1/n(7)
Subsequently, the correlations betweent10 pred
andt10 est
values (Figs. 5 and 6) were found to be non-signifi-cant though they did increase in strength the less data sets were curtailed (i.e. from 30 – 90 days).
4. Discussion
The softening of kiwifruit occurs in a character-istic manner as demonstrated by the data presented
Fig. 6. Relationships between the predicted (t10
pred) and
esti-mated (t10
est) times for the flesh firmness of fruit to reach 10 N,
as determined using the IEP model. Estimated times were obtained using parameters obtained from fitting the model to full data sets. Predicted times were estimated using parameters obtained from fitting the model to data sets that were curtailed to only contain firmness values measured up to 30 (A); 60 (B); or 90 N (C) days of storage. Each point represents the average estimate for a grower batch. Outliers have been omitted. t10est against the estimates of d, b0, b1, and b2
obtained from fitting the IEP model to each data set (data not shown) but onlyb2was significantly
correlated (R2
=0.47;P=0.029); as it increased,
t10
estincreased linearly.t 10
estwas also plotted against
the estimated times to a range of different threshold firmness levels (i.e. 20, 40 and 70 N; Fig. 5). The only significant correlation occurred with time to 20 N (R2
=0.81,P=0.0004).
For real time predictions, models need to be robust with as few unknown parameters as possi-ble. Generally, these unknown parameters have to be estimated using data from a limited time window set by the prediction origin (i.e. the point up to which data is available and from which future behaviour is predicted). To test the real time predictive capacity of the IEP model, for each data set, prediction origins were set to 30, 60 and 90 days and then the predicted times for the firmness of fruit to reach 10 N (t10
pred
) were calculated using each of the curtailed data sets (Note, subsetting of data has to be done with time rather than firmness, which is not fixed and subject to error; although firmness is the classifying variable and not time, for practical purposes, time is used as the selection criterion). To ensure robust predictions, b1 and b2 were set to
here. A mechanistic approach to modelling this behaviour would probably provide a better under-standing of the softening process and subsequently form the basis of a quantitative tool with consider-able predictive power. However, the objective of the current work was to identify empirical models that would characterise the softening behaviour of fruit and describe the variation between batches from different sources, i.e. growers and seasons. The objective was not to understand the softening process through mathematical modelling.
At the onset of this study, a number of models that were expected to reasonably characterise the softening pattern of kiwifruit were identified. The first of these models, CMM, has already been used successfully to characterise the softening behaviour of some batches of fruit (Benge et al., unpublished data; Davie et al., unpublished data). However, of all the models explored, the CMM was the least accurate at characterising the firmness data pre-sented here and in particular lacked the capacity to reflect curvature during the early stages of soften-ing. Previous success with this function may be attributed to the study of fruit already well into the initial acceleration phase of softening. When the data sets in the current study were curtailed (e.g. starting at a fixed firmness of about 50 N), the CMM did reasonably characterise the softening curves, although still not as well as the other models (data not shown). The CMM could be successfully used to characterise the softening behaviour of fruit that are more advanced in softening.
The EXP and CG models examined in this paper both characterised the firmness data considerably better than the CMM, although neither character-ised firmness across the entire time domain with sufficient accuracy. Both models contain more parameters than the CMM model, which appears necessary to accurately characterise more complex curves; those parameters have no obvious biologi-cal meaning thereby limiting the predictive capacity of the models. Nevertheless, they still characterise the softening behaviour of kiwifruit well, although it is important to note that both contain a parame-ter value for a horizontal asymptote; according to these models, fruit firmness would never reach 0 which is probably unrealistic. Hence, any predic-tions must be limited to the storage times shown in the data.
Both the JMM and IEP models constructed in this study accurately characterised the firmness data that they were applied to. However, the estimates of the parameters obtained for both of these models were highly correlated indicating strong non-linear behaviour, although there are examples of badly behaved non-linear models in which correlations are not particularly high and vice versa (Ratkowsky, 1990). Therefore, the high correla-tions obtained here may not necessarily imply that the estimators have poor distributional prop-erties.
Generally, the IEP model fitted the data best as indicated by the smaller relative errors associated with its parameters. Noticeably, the v1parameter
in the JMM model had very large standard errors, indicating that a large range of combinations of parameter values in that model would be capable of describing the softening behaviour of fruit from each grower. This is because that parameter is associated with the early part of each curve and was therefore estimated from just a few data points. The accuracy of this model in characterising the soften-ing behaviour of kiwifruit might be enhanced by making a greater number of initial firmness mea-surements, providing the measurement error is not too large, although it may be more sensitive to small changes in initial values because of interactions between parameters, i.e. more data might not increase the robustness of the parameter-isation.
While the JMM and IEP models may character-ise firmness data well, like the EXP and CG models, they contain a number of parameters with no obvious biological meaning, Because they are em-pirical and not functions of underlying factors affecting firmness (e.g. temperature), they have limited use for making predictions under varying conditions. The greater uncertainties associated with the estimates of the parameters in the JMM model, especially the v1 parameter, further limits
the predictive capacity of this model.
Batches of fruit are rejected for export if the average firmness for those batches is c. 10 N or less. While the most accurate models for charac-terising the softening behaviour of kiwifruit may be empirical, they may still have useful properties or deductions that may provide some indication of how long fruit can be stored before being unsuitable for export. However, further explo-ration of the curves generated by the IEP model in this study failed to reveal any useful relation-ships between properties or deductions that could be used to predict fruit storage life reiterating the limited predictive capacity of this model.
5. Conclusion
In conclusion, this paper has demonstrated the major advantages and disadvantages of empiri-cally modelling biological systems such as fruit softening. While empirical models can accurately characterise a system, they largely contain parameters that have no obvious biological mean-ing and therefore have limited predictive capacity. Nevertheless, empirical models can be used to objectively compare biological systems when full
sets of data are available and would be particu-larly useful when comparing the effects of treat-ments in experimental programmes. However, for accurately predicting biological phenomena, more mechanistic approaches are necessary, although this requires a good understanding of the pro-cesses underlying those phenomena, something that is currently lacking in the fruit softening area.
Acknowledgements
We would like to acknowledge Dr Alistair Hall for his mathematical input during the develop-ment of the models presented in this paper.
References
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