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 − t
v + t
1 Exponential decay
EXP FF = A
+ A
1
exp − lt 2
and Complementary Gompertz
CG FF = A
+ B
1 − 1
expbexp − kt 3
where FF is the mean flesh firmness at t number of days in storage and the Roman and Greek
symbols represent parameters in the model. In the CMM model, A is a scale parameter equivalent to
initial firmness and v a half-time parameter repre- senting the time taken for firmness to drop to half
the initial value. In the EXP model, A
represents the lower asymptotic value and A
1
represents the drop in firmness from the initial value to the
lower asymptote; l represents 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 A in this model represents the
lower asymptotic value of firmness, and B a 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 = Á
à Í
à Ä
a + a
1
u − t
v
1
+ u − t
t 5 u a
v
2
v
2
+ t − u
t \ u 4a
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, a
1
, u, v
1
and v
2
. 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 following four- parameter model:
FF = Á
à Í
à Ä
a + av
1
v
2
u − t
v
1
+ u − t
t 5 u a
v
2
v
2
+ t − u
t \ u 4b
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
1 + exp b
+ b
1
t + b
2
t
2
+ b
2 2
t
3
3b
1
5 Again, d is a scale parameter and the other
parameters determine the shape of the curve. Indi- vidual parameters of this model have no direct
biological interpretations. The function is con- strained to be monotonically decreasing, but it
flattens out to zero rate at the beginning of the third phase. In situations where this cannot be
assumed, this function can be slightly modified with an extra parameter and a boundary 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. 1. Fits of five different models columns to firmness data for fruit harvested from grower 1 during the early harvest period in 1991 row A, the main harvest period in 1991 row B and the main harvest period in 1994 row C. CMM, Complementary
Michaelis – Menten type; EXP, exponential decay; CG, complementary Gompertz; JMM, jointed Michaelis – Menten type; IEP, inverse exponential polynomial.
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 v
1
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 v
1
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
