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Postharvest Biology and Technology 70 (2012) 42–50

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Postharvest Biology and Technology

journa l h o me pa g e: www.elsev ier .com/ locate /postharvbio

On the prediction of the remaining vase life of cut roses

Seth-Oscar Trompa,∗, Ruud G.M. van der Smana, Henderika M. Vollebregta, Ernst J. Wolteringa,b

a Food and Biobased Research, Wageningen University and Research Centre, P.O. Box 17, 6700 AA Wageningen, The Netherlandsb Horticultural Supply Chains group, Wageningen University and Research Centre, P.O. Box 630, 6700 AP Wageningen, The Netherlands

a r t i c l e i n f o

Article history:Received 18 January 2012Accepted 6 April 2012

Keywords:Vase lifeRoseTime-temperature sumCut flowersModeling

a b s t r a c t

The objective of the present paper was to examine the hypothesis that the time–temperature sum builtup during storage and transport at constant as well as stepwise changing temperatures is a good predictorof the remaining vase life of cut roses. Theoretical calculations and graphing of functions showed thatthe time–temperature sum closely approximated the more common approach to quality loss, involv-ing first order reaction kinetics with an Arrhenius temperature dependency. The time–temperaturesum approximation failed at temperatures below 2 ◦C, especially in the case of long storage times. Thetime–temperature sum approximation succeeded in the range 2–6 ◦C. For temperatures above 6 ◦C, thedegree-days model will underestimate the remaining vase life, depending on the storage time. The currentexperiment confirms these expectations from theory about the performance of the time–temperaturesum. In the experiment not only constant storage temperatures but also stepwise changing storage tem-peratures were applied. Because of its simple principle, the time–temperature sum has practical value,but we are now aware of its limitations.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Rapid technological developments in information and commu-nication technology such as RFID technology enable more elaboratemonitoring of distribution chain conditions. An example of such adevelopment is a wireless sensor platform to monitor the environ-mental conditions of perishable goods in the supply chain betweenproducer and consumer. The sensor platform is based on an intel-ligent RFID package in which multiple sensor technologies areincorporated. The idea is that by adding such devices to packagingsolutions (crates, containers, boxes, etc.) one can guarantee a prod-uct’s quality more effectively throughout the whole logistics chain.Moreover, by collecting and studying the data from these sensordevices, and by combining them with a product quality model,optimization of logistics can be achieved resulting in reduction ofwaste and preservation of the environment. Being able to predictthe remaining shelf life after distribution may assist in decisionmaking in the fresh produce business. This creates the demand formodels translating product history into reliable predictions of theremaining shelf life.

The (remaining) vase life of cut flowers is defined as the timethat flowers can be kept on the vase at room temperature, which isregularly assumed to be equal to 20 ◦C. Vase life ends according topredefined quality criteria. A regular quality criterion for cut roses

∗ Corresponding author. Tel.: +31 0 317 480204; fax: +31 0 317 483011.E-mail address: seth.tromp@wur.nl (S.-O. Tromp).

is the wilting of the flower due to natural senescence or due to othercauses such as vascular occlusion, which inhibits water supply tothe flowers (Mayak et al., 1974; Van Doorn, 1997). The developmentof occlusions is caused by various factors such as bacteria (Zagoryand Reid, 1986; Van Doorn et al., 1989), air emboli (Van Doorn,1990) and physiological responses of stems to cutting (Marousky,1969).

There is relatively little scientific literature on vase life mod-eling. The few papers on this topic have modeled vase life asa function of the time–temperature sum built up during stor-age and transport (Van Doorn and Tijskens, 1991; Van Meeteren,2007). In the cut flower trade, it is common practice to use thetime–temperature sum as a predictor for the vase life (Lill andDennis, 1986; Goedendorp and Barendse, 2010, 2011). For otherperishable products, it is more common to model quality changevia first order reaction kinetics, with temperature dependency fol-lowing Arrhenius (Tijskens and Polderdijk, 1996).

In this paper we show that for the vase life of cut roses, thetwo modeling approaches are not conflicting, albeit within a cer-tain temperature range. We develop a scientific basis for usingthe time–temperature sum as a predictor of the remaining vaselife after storage and transport in the distribution chain, both viatheoretical calculations and graphing of functions and via fittingexperimental vase life data, comprising both dry storage (in a box)and wet storage (in a bucket filled with water). The remaining vaselife after wet storage may be different from the remaining vase lifeafter dry storage (e.g. Rudnicki et al., 1986; Mor, 1989; Ichimuraand Shimizu-Yumoto, 2007).

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This paper is organized as follows. In Section 2, we first presentthe two modeling approaches, the time–temperature sum and first-order kinetics. In addition, we describe the set-up of the currentexperiment. In Section 3, we first show via theoretical calculationsand graphing of functions, that the time–temperature sum can beviewed as an approximation to first order kinetics. In addition, weanalyze the performance of both models through comparing modelpredictions with measurement data from the current rose storageexperiment and from experiments described in literature.

2. Material and methods

2.1. Quality change models

The time–temperature sum is a measure that can be calculatedfrom knowledge about historical storage time and temperature (e.g.after being recorded by a time–temperature logger). The experi-ence from practice seems to show that the time–temperature sumis indeed a rather good predictor of the remaining vase life (e.g.Goedendorp and Barendse, 2011). The so-called degree-days modelis a time–temperature sum model that assumes a cultivar depen-dent fixed vase life of non-stored flowers. According to this modelthe remaining vase life can be found by subtracting the recordedtime–temperature sum divided by the room temperature from thevase life of non-stored flowers. For example, if we assume a vaselife of non-stored flowers equal to 10 d at 20 ◦C (room temperature),while during the distribution chain the flowers are kept at 5 ◦C dur-ing 8 d, the recorded time–temperature sum equals 40 ◦C days, suchthat the model predicts s a remaining vase life of 8 d at 20 ◦C. Themodel implies that when storing the flowers during 12 d instead of8 d at 5 ◦C, the remaining vase life only decreases by another extraday. The assumption is made that wilting due to natural senescenceis the main cause for quality decay, and the model does not accountfor other quality decay causes such as infection with Botrytis cinerea,which is not a priori captured by a time–temperature sum model(Van der Sman et al., 1996).

The more common approach to quality change involves a modeltaking a first order reaction kinetics, with the temperature depen-dency following Arrhenius (Tijskens and Polderdijk, 1996). Thismodel assumes quality decay to be equal for flowers of the samecultivar (growers cut flowers at similar stage of development). Thehypothesis behind this first order kinetics model is that in generalthe quality decay is proportional to the cellular metabolism (respi-ration) of the produce, which has an Arrhenius type of temperaturedependency. The metabolic rate is directly related to ATP produc-tion, which drives biological processes (Adam et al., 1983; Jianget al., 2007).

2.2. Set-up of the current experiment

We generated vase life data using the cultivar ‘Red Naomi’, andfor different storage times and temperatures. Moreover, in the cur-rent experiment we applied stepwise temperature changes duringstorage. For stepwise changes of temperature, having the value ofTi (K) at time interval i of duration ti (d), the time–temperaturesum built up during storage is given by

∑i(Ti − 273.15)ti (◦C day).

The experiment is designed in such a way that several storagetreatments with either constant temperature or stepwise changingtemperature, have identical time–temperature sum. This providesa critical test for the degree-days model, which predicts that theremaining vase life of these treated flowers is equal.

Both dry and wet storage were applied. Dry storage was doneby packaging five bunches of 10 flowers into one box, while wetstorage was done by putting five bunches of 10 flowers into onebucket filled with tap water. The length of the cut flowers was

60 cm and the cut flower stems were re-cut after storage. Afterdry storage, the flowers were rehydrated by putting the flowersin tap water containing Chrysal Professional 2 T-bag and storingthem for 16 h at 5.5 ◦C. Vases were filled with tap water containingChrysal Professional 3. The storage conditions during vase life were20 ◦C and 60 ± 5% RH. Light intensity and day length during vase lifeas well as vase life assessment was 12/24 h light (12 �mol m−2 s−1

photosynthetically active radiation (PAR)).For each storage treatment condition, at maximum 50 flowers

(packed in 5 bunches of 10 roses each) were scored for their remain-ing vase life. Important incidents which may happen to roses arethe occurrence of petal browning due to severe B. cinerea infectionand bent neck due to vascular blockage (Van Doorn, 1997; Vrind,2005). Flowers which showed clear marks of such incidents werediscarded and omitted from the vase life analysis.

3. Results

3.1. Model development

We analyzed to what extent the degree-days model followedfrom a model taking first order reaction kinetics, with the tem-perature dependency following Arrhenius. According to the firstorder model, the vase life determining quality attribute q (which iswilting in case of cut roses) changes according to:

dq

dt= −kT q (1)

kT represents the temperature dependent quality decay rate(day−1). Assuming an Arrhenius temperature dependency of kT itfollows that

kT = kref eB((1/Tref )−(1/T)) (2)

B a product dependent parameter (K), T is the absolute temperaturein K, and kref is the quality decay rate at the reference temperatureTref which is 293.15 K (20 ◦C): the room temperature during the vaselife. In this paper the model according to Eqs. (1) and (2) is calledthe first order Arrhenius (FOA) model.

At constant temperature T the solution of the differential Eq. (1)is an exponential function:

q(t) = q0e−kT t (3)

with q0 the initial value of the determining quality attribute. It fol-lows that, when being stored at reference temperature Tref, the vaselife of non-stored flowers te (in days) equals the time it takes todecrease from the initial quality q0 to the minimum accepted valueof quality from consumers perspective qe:

te = ln(q0/qe)kref

(4)

When first having stored the flowers for the time t at a temper-ature T, the initial quality q0 has decreased to q0 · e−kTt, such thatthe remaining vase life VL (d) at temperature Tref will be equal to

VL = ln((q0 · e−kT t)/qe)kref

= ln(q0) − ln(qe) − kT t

kref

= ln(q0) − ln(qe)kref

− kT

kreft (5)

Hence, the dependency of the vase life on the storage time t islinear, by definition. We assume that (kT/kref) can be approximatedby the reduced temperature ((T − T0)/(Tref − T0)), with T0 equal to273.15 K (0 ◦C). This assumption will be made plausible in Section3.2. According to Eq. (5) the remaining vase life is by approximation

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44 S.-O. Tromp et al. / Postharvest Biology and Technology 70 (2012) 42–50

Fig. 1. The reaction rate normalized by the reaction rate at reference temperaturefor Q10 equals 1.5, 2 or 3, and the approximation of this normalized reaction rate by(T − 273)/20.

equal to:

VL = ln(q0) − ln(qe)kref

− kT

kreft ≈ ln(q0) − ln(qe)

kref− T − T0

Tref − T0t

= A − 120

(T − 273.15)t (6)

The model according to Eq. (6) represents exactly the degree-days model as introduced in Section 2.1, with A representing thevase life of non-stored flowers and (T − 273.15)t representing thetime–temperature sum in degree-days built up during storage.

3.2. Validity of the approximation

The Arrhenius temperature dependency according to Eq. (2) canbe characterized by the temperature dependent parameter Q10,which is defined as the increase factor of the reaction rate witha temperature increase of 10 ◦C. Q10 is mathematically defined by

Q10 = kT2

kT1

(7)

with T2 − T1 = 10 K and T1 and T2 in (K). It is known that for per-ishables Q10 as derived from measurements of the respiration rateat different temperatures has generally a value between 2 and 3(Platenius, 1942; Reid, 1991). For cut roses a value of Q10 around 3has been found (C elikel and Reid, 2005). Combining Eqs. (2) and (7)show that in the temperature range 273–283 K, Q10 = 2 correspondswith B = 5355 K. With this value of B, (kT/kref) is plotted in Fig. 1 asa function of the temperature T (K) according to Eq. (2).

For Q10 = 3 (at T1 = 273 K and T2 = 283 K corresponding toB = 8488 K), (kT/kref) is plotted in Fig. 1 too.

Fig. 1 shows that for cut roses, depending on the storage tem-perature T and depending on the product dependent parameterB, (kT/kref) can indeed be approximated by the reduced tempera-ture (T − T0)/(Tref − T0). Temperatures within cut roses distributionchains are preferably in the range of 2–15 ◦C. We conclude fromFig. 1 that in the case of Q10 = 2, the degree-days model appears tobe a good approximation of the FOA model in the range 8–15 ◦C.For temperatures below 8 ◦C, the remaining vase life is overesti-mated. In the case of Q10 = 3 the degree-days model appears tobe a good approximation of the FOA model in the range 2–6 ◦C.For temperatures in the range of 6–15 ◦C, the degree-days modelwill underestimate the remaining vase life. For temperatures below

2 ◦C, the remaining vase life is overestimated. It is noted that if tem-perature equals 0 ◦C, the degree-days model predicts an unlimitedremaining vase life, which is definitely not in agreement with thefinite vase life of roses.

3.3. Model fitting

The remaining vase life of each bunch of flowers was calculatedby taking the average remaining vase life of the flowers it is com-posed of and of which the vase life ends due to senescence. Modelfitting was undertaken on the basis of only those bunches of flow-ers which had five or more flowers of which vase life ends due tosenescence (so five or less flowers having been discarded due toincidents). The data set on which the models are fitted is given inTable 1.

Fitting was done using the function lsqcurvefit of the softwareMATLAB (R2009a) for solving nonlinear curve-fitting (data-fitting)problems in least-squares sense.

Fitting the vase life data according to the degree-days modelimplies estimating the parameter A according to Eq. (6). Step-wise changing temperatures data are included in model fitting. Inthe case of two subsequent stepwise changing temperatures theparameter A is fitted according to Eq. (8):

VL = ln((q0 · e−kT1t1 e−kT2

t2 )/qe)kref

= ln(q0) − ln(qe)kref

− kT1

kreft1 − kT2

kreft2 ≈ A − 1

20(T1 − 273.15)t1

− 120

(T2 − 273.15)t2 (8)

The vase life resulting from the FOA model is written as (basedon Eq. (5)):

VL = ln(q0) − ln(qe)kref

− kT

kreft = A − eB((1/Tref )−(1/T))t (9)

This shows that fitting the vase life data according to the FOAmodel implies estimating the two parameters A and B accordingto Eq. (9). Stepwise changing temperatures data are included inmodel fitting again. In the case of two subsequent stepwise chang-ing temperatures the parameters A and B are estimated accordingto Eq. (10):

VL = ln(q0) − ln(qe)kref

− kT1

kreft1 − kT2

kreft2

= A − eB((1/Tref )−(1/T1))t1 − eB((1/Tref )−(1/)T2)t2 (10)

For both models, a least squares fit is performed with the stor-age times and temperatures according to the experimental settings(the set point times and temperatures). Table 2 summarizes the fitresults of the current study.

3.4. Model validation

It is noticed from Table 2 that the estimated values of the param-eter B are very close to 8488 K, which is the value that correspondsto the value Q10 = 3 found for cut roses by C elikel and Reid (2005)albeit for a different rose cultivar (see Section 3.2).

3.4.1. Standard error and coefficient of determinationThe standard errors are small and within 2% (A) and 7% (B) of

the absolute values of the model parameters. Besides the standarderrors, the coefficient of determination (R2) is applied as a mea-sure for the explained variation. The FOA model is a two parameter

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Table 1Storage time (days), storage temperature (K) and vase life (days) for ‘Red Naomi’ for both wet and dry storage.

Bunch Storage time (d) Storage temperature (K) Wet storage Dry storage

t1 t2 T1 T2 Vase life (d)a SE n Vase life (d) SE n

1 3.5 3.5 278.7 288.7 6.1 0.1 10 6.5 0.2 102 3.5 3.5 278.7 288.7 6.7 0.1 10 6.5 0.2 103 3.5 3.5 278.7 288.7 6.4 0.2 9 6.9 0.1 104 3.5 3.5 278.7 288.7 6.5 0.2 10 7.3 0.6 105 3.5 3.5 278.7 288.7 6.4 0.2 9 6.3 0.1 10

6 3.5 3.5 288.7 278.7 6.5 0.2 8 6.5 0.2 107 3.5 3.5 288.7 278.7 6.1 0.1 9 6.1 0.1 108 3.5 3.5 288.7 278.7 6.2 0.1 9 6.2 0.1 109 3.5 3.5 288.7 278.7 6.2 0.2 6 6.0 0.0 10

10 3.5 3.5 288.7 278.7 7.0 0.0 6 6.1 0.1 10

11 7 7 273.7 273.7 11.4 1.0 8 7.6 0.5 512 7 7 273.7 273.7 8.4 1.3 5 7.6 0.4 813 7 7 273.7 273.7 8.6 0.5 8 9.0 0.7 1014 7 7 273.7 273.7 10.9 1.0 7 9.6 0.7 915 7 7 273.7 273.7 n.a.b n.a. n.a. 7.1 0.3 8

16 7 7 278.7 278.7 6.7 0.2 6 7.6 0.2 917 7 7 278.7 278.7 6.3 0.2 6 8.6 0.5 818 7 7 278.7 278.7 6.7 0.2 9 7.3 0.4 919 7 7 278.7 278.7 6.6 0.2 10 8.1 0.6 820 7 7 278.7 278.7 6.8 0.1 9 7.0 0.3 8

21 7 7 283.7 283.7 1.0 0.0 10 6.0 0.0 922 7 7 283.7 283.7 6.0 0.0 5 3.0 0.0 1023 7 7 283.7 283.7 6.0 0.0 6 3.0 0.0 1024 7 7 283.7 283.7 6.0 0.0 8 n.a. n.a. n.a.25 7 7 283.7 283.7 6.0 0.0 6 n.a. n.a. n.a.

26 7 7 273.7 283.7 6.6 0.4 5 n.a. n.a. n.a.27 7 7 273.7 283.7 6.8 0.4 6 n.a. n.a. n.a.28 7 7 273.7 283.7 5.5 0.5 8 n.a. n.a. n.a.29 7 7 273.7 283.7 5.1 0.6 8 n.a. n.a. n.a.30 7 7 273.7 283.7 7.0 0.2 6 n.a. n.a. n.a.

31 7 7 283.7 273.7 7.3 0.3 6 4.2 0.6 932 7 7 283.7 273.7 7.9 0.4 7 3.4 0.4 733 7 7 283.7 273.7 7.3 0.4 7 5.9 0.7 734 7 7 283.7 273.7 7.1 0.1 9 5.3 0.6 935 7 7 283.7 273.7 n.a. n.a. n.a. n.a. n.a. n.a.

36 7 7 288.7 278.7 3.0 0.0 6 6.0 0.0 537 7 7 288.7 278.7 n.a. 0.0 n.a. 6.0 0.0 1038 7 7 288.7 278.7 n.a. n.a. n.a. 6.0 0.0 839 7 7 288.7 278.7 n.a. n.a. n.a. 6.0 0.0 940 7 7 288.7 278.7 n.a. n.a. n.a. n.a. n.a. n.a.

41 10.5 10.5 273.7 273.7 7.4 0.4 8 7.0 0.0 542 10.5 10.5 273.7 273.7 8.5 0.6 6 8.1 0.5 943 10.5 10.5 273.7 273.7 8.2 0.7 6 7.0 0.0 544 10.5 10.5 273.7 273.7 8.3 0.6 7 7.0 0.0 645 10.5 10.5 273.7 273.7 7.5 0.5 6 n.a. n.a. n.a.

46 10.5 10.5 278.7 278.7 4.4 1.1 7 6.8 0.2 647 10.5 10.5 278.7 278.7 5.1 0.8 7 5.1 0.6 848 10.5 10.5 278.7 278.7 5.6 0.7 7 6.2 0.6 1049 10.5 10.5 278.7 278.7 6.7 0.2 6 4.8 0.7 950 10.5 10.5 278.7 278.7 6.5 0.2 6 5.4 0.9 5

51 10.5 10.5 283.7 283.7 1.0 0.0 6 n.a. n.a. n.a.52 10.5 10.5 283.7 283.7 1.0 0.0 7 n.a. n.a. n.a.53 10.5 10.5 283.7 283.7 n.a. n.a. n.a. n.a. n.a. n.a.54 10.5 10.5 283.7 283.7 n.a. n.a. n.a. n.a. n.a. n.a.55 10.5 10.5 283.7 283.7 n.a. n.a. n.a. n.a. n.a. n.a.

56 0 0 293.2 293.2 11.2 0.6 10 11.6 0.5 1057 0 0 293.2 293.2 11.3 0.7 9 11.0 0.6 958 0 0 293.2 293.2 11.4 0.7 9 12.1 0.5 1059 0 0 293.2 293.2 11.1 0.3 10 11.0 0.6 960 0 0 293.2 293.2 10.6 0.5 9 12.4 0.8 10

61 1 1 273.7 273.7 9.8 0.7 8 9.6 0.8 962 1 1 273.7 273.7 10.9 0.6 7 11.0 0.7 763 1 1 273.7 273.7 11.1 0.6 7 11.4 0.2 864 1 1 273.7 273.7 11.7 0.5 9 10.6 0.6 865 1 1 273.7 273.7 9.4 0.8 9 10.8 0.4 8

66 1 1 278.7 278.7 11.2 0.4 9 9.5 0.7 8

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Table 1 (Continued)

Bunch Storage time (d) Storage temperature (K) Wet storage Dry storage

t1 t2 T1 T2 Vase life (d)a SE n Vase life (d) SE n

67 1 1 278.7 278.7 10.9 0.6 9 9.3 0.7 768 1 1 278.7 278.7 11.6 0.2 9 9.6 0.7 869 1 1 278.7 278.7 10.7 0.5 9 9.6 0.7 970 1 1 278.7 278.7 10.8 0.5 10 9.7 0.7 9

71 1 1 283.7 283.7 10.3 0.6 10 10.2 0.6 972 1 1 283.7 283.7 9.4 0.6 9 8.7 0.6 1073 1 1 283.7 283.7 9.9 0.6 10 10.4 0.6 874 1 1 283.7 283.7 10.6 0.4 8 9.5 0.7 875 1 1 283.7 283.7 9.5 0.6 10 9.3 0.5 9

76 1 1 288.7 288.7 8.6 0.6 7 7.6 0.2 977 1 1 288.7 288.7 8.9 0.6 8 7.4 0.3 1078 1 1 288.7 288.7 8.9 0.6 9 7.0 0.3 979 1 1 288.7 288.7 7.6 0.2 9 7.0 0.3 880 1 1 288.7 288.7 7.8 0.2 8 7.5 0.2 10

81 2 2 273.7 273.7 9.6 0.2 10 12.9 0.9 882 2 2 273.7 273.7 11.9 1.0 9 11.1 0.7 983 2 2 273.7 273.7 10.9 0.5 10 12.6 0.9 984 2 2 273.7 273.7 12.2 0.9 10 10.3 0.4 985 2 2 273.7 273.7 11.9 0.9 8 10.1 0.4 986 2 2 278.7 278.7 8.9 0.6 8 8.3 0.7 787 2 2 278.7 278.7 8.7 0.4 7 8.5 0.5 688 2 2 278.7 278.7 8.8 0.5 6 9.0 0.3 1089 2 2 278.7 278.7 8.7 0.4 7 9.0 0.0 590 2 2 278.7 278.7 9.0 0.4 8 8.6 0.4 7

91 2 2 283.7 283.7 7.5 0.5 8 8.7 0.5 992 2 2 283.7 283.7 8.5 0.5 6 8.6 0.4 793 2 2 283.7 283.7 8.6 0.4 8 8.1 0.5 794 2 2 283.7 283.7 8.3 0.4 9 9.0 0.0 895 2 2 283.7 283.7 8.7 0.3 9 8.5 0.5 696 2 2 288.7 288.7 6.8 0.6 6 8.3 0.5 897 2 2 288.7 288.7 7.6 0.6 7 8.2 0.5 998 2 2 288.7 288.7 n.a. n.a. n.a. 7.0 0.7 599 2 2 288.7 288.7 n.a. n.a. n.a. 8.1 0.5 7

100 2 2 288.7 288.7 n.a. n.a. n.a. 7.9 0.5 8

101 3.5 3.5 273.7 273.7 8.6 0.4 10 8.1 0.4 10102 3.5 3.5 273.7 273.7 7.8 0.5 9 7.8 0.4 8103 3.5 3.5 273.7 273.7 8.3 0.6 8 10.1 0.8 9104 3.5 3.5 273.7 273.7 8.5 0.6 8 9.8 0.7 9105 3.5 3.5 273.7 273.7 8.0 0.6 9 8.0 0.5 7

106 3.5 3.5 278.7 278.7 8.8 0.7 9 7.9 0.5 9107 3.5 3.5 278.7 278.7 7.2 0.4 9 7.3 0.4 9108 3.5 3.5 278.7 278.7 8.7 0.8 9 9.7 0.7 10109 3.5 3.5 278.7 278.7 7.9 0.4 9 8.9 0.7 9110 3.5 3.5 278.7 278.7 8.3 0.4 7 8.2 0.7 9

111 3.5 3.5 283.7 283.7 7.0 0.0 8 6.5 0.2 10112 3.5 3.5 283.7 283.7 8.3 0.9 6 6.0 0.0 10113 3.5 3.5 283.7 283.7 6.4 0.6 8 6.0 0.0 10114 3.5 3.5 283.7 283.7 7.1 0.1 9 6.0 0.0 10115 3.5 3.5 283.7 283.7 6.4 0.5 9 5.6 0.4 9

116 3.5 3.5 288.7 288.7 5.2 0.0 5 3.0 0.0 10117 3.5 3.5 288.7 288.7 5.4 0.7 7 3.0 0.0 10118 3.5 3.5 288.7 288.7 n.a. n.a. n.a. 3.0 0.0 10119 3.5 3.5 288.7 288.7 n.a. n.a. n.a. 3.0 0.0 10120 3.5 3.5 288.7 288.7 n.a. n.a. n.a. 3.0 0.0 10

121 3.5 3.5 273.7 283.7 6.4 0.2 9 7.7 0.5 9122 3.5 3.5 273.7 283.7 6.8 0.1 10 9.2 0.5 10123 3.5 3.5 273.7 283.7 7.3 0.5 8 7.2 0.4 9124 3.5 3.5 273.7 283.7 7.1 0.2 10 7.1 0.2 9125 3.5 3.5 273.7 283.7 7.7 0.6 7 8.3 0.8 10

126 3.5 3.5 283.7 273.7 7.9 0.4 8 7.6 0.6 10127 3.5 3.5 283.7 273.7 8.2 0.4 10 8.0 0.6 10128 3.5 3.5 283.7 273.7 7.6 0.4 10 7.0 0.2 10129 3.5 3.5 283.7 273.7 7.7 0.7 9 7.1 0.2 10130 3.5 3.5 283.7 273.7 7.4 0.4 9 7.4 0.2 10

a Estimated mean of vase life of those flowers in the bunch of which the vase life ended due to senescence (and not due to Botrytis). Only flowers of which the vase lifeended due to senescence contributed to the average vase life of a bunch.

b n.a. = not available because for this bunch more than five flowers showed clear marks of Botrytis. Therefore the whole bunch was discarded from the data analysis.

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Table 2Estimated values and standard errors of parameters of the FOA model and the degree-days model for experiment with ‘Red Naomi’ (storage conditions given in Table 1). R2

adj

and R2 are calculated on the basis of the total unexplained variation.

Storage FOA model Degree-days model

A SEA B SEB R2adj

A SEA R2

Dry 10.00 0.20 8219 536 0.67 10.16 0.13 0.59Wet 10.30 0.17 8059 404 0.76 10.40 0.11 0.72

Table 3Pure error and lack of fit for both the FOA model and the degree-days model. R2

adjand R2 are calculated on the basis of the unexplained variation due to lack of fit only.

Storage FOA model Degree-days model

Pure error Lack of fit R2adj

(fit) Pure error Lack of fit R2 (fit)

Dry 46.4 130.3 0.76 46.4 174.3 0.68Wet 54.2 74.6 0.86 54.2 105.4 0.81

model whereas the degree-days model is a single parameter model.In order to correct for this difference, R2

adjinstead of R2 is calcu-

lated for the FOA model. The coefficients of determination are in therange of 0.59–0.76. The coefficient of determination of the degree-days model is slightly less than that of the FOA model. For bothmodels, the coefficient of determination is higher in the case ofwet storage.

3.4.2. Lack of fit versus pure errorThere were five bunches of 10 roses for each storage treatment

condition, which can be considered as replicates. Therefore, it ispossible for both models to partition the unexplained variationinto a ‘lack of fit’ and a ‘pure error’. The lack of fit represents thedifference between the local average of the duplicates and the fit-ted value. The pure error represents the difference between theobserved values of each duplicate and the local average, and canbe seen to represent the biological variance which exist betweendifference bunches of flowers. These values are given in Table 3.

It is noted that the unexplained variation due to pure error issmaller than the unexplained variation due to a lack of fit. MoreoverTable 3 shows recalculated coefficients of determination, now onlytaking into account the unexplained variance due to a lack of fit.This was done so in order to be able to compare the results of thecurrent experiment with results described in the literature (Section3.4.5).

3.4.3. Storage temperature dependent biasIn order to obtain insight into the model goodness of fit, we

did not restrict ourselves to R2 alone. As we have commented atSection 3.1, we expected the degree-days model to over or under-estimate the vase life depending on the storage temperature. Foreach model we calculate this bias as follows. For each storage treat-ment condition (a specific storage temperature T combined with aspecific storage time t), both a local average LAtT of at maximumfive observed vase live data and a model dependent fitted value FVtT

exist. The lack of fit LFtT for a specific storage treatment condition(t, T) is defined as

LFtT = FVtT − LAtT (11)

So this lack of fit is positive in the case of overestimation andnegative in the case of underestimation.

Let ST be the set of different storage times that are combinedwith the storage temperature T as a constant storage temperature.For example, regarding the current experiment, S273.7 = {2, 4, 7, 14,21}. Let NT be the number of elements (storage times) in this set ST.Then the temperature dependent bias BT (d) is defined as

BT = 1NT

∑t ∈ ST

LFtT (12)

For each model the bias BT is a measure for the over or underes-timation of the prediction generated by that model. BT representsthe average over or underestimation of all storage treatment con-ditions having a storage temperature T. This bias BT is computedand compared between models (Table 4).

Note that we did not consider the stepwise changing tem-peratures in this analysis of bias, because two different storagetemperature are involved, which it makes it difficult to comparebetween storage treatment conditions.

Table 4 shows that the relationship between the two models isas expected from our theoretical calculations and graphing of func-tions (Fig. 1). At 5.5 ◦C (278.7 K), the storage temperature which isthe closest to 4 ◦C, both models give more or less equal predictions.At the lower temperature of 0.5 ◦C (273.7 K) the degree-days modelpredicts longer vase lives than the FOA model and at higher tem-peratures the degree-days model predicts shorter vase lives thanthe FOA model. Moreover, it is noted that the FOA model shows atrend of underestimating the vase life at lower temperatures (0.5 ◦Cand 5.5 ◦C), whereas overestimating the vase life at higher temper-atures. The degree-days model overestimates vase life at 0.5 ◦C and15.5 ◦C, whereas it underestimates vase life at 5.5 ◦C and 10.5 ◦C.

3.4.4. Stepwise changing temperaturesTable 5 shows the results of several treatments with either

constant temperature or stepwise changing temperature, havingidentical temperature–time sums. The table shows that the FOAmodel explains some variation in the vase life data, which is notexplained by the degree-days model.

Table 4Storage temperature dependent bias BT (current study).

Temperature(K)

Dry storage Wet storage

FOA model Degree-days model FOA model Degree-days model

273.7 −0.6 0.7 −0.6 0.6278.7 −0.2 −0.5 −0.1 −0.3283.7 0.3 −0.4 0.2 −0.9288.7 1.1 0.7 0.5 0.1

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Table 5Four cases representing storage treatments with either constant temperature or stepwise changing temperature, having identical temperature–time sums.

Storage conditions Wet storage Dry storage

Time (d) Temperaturepart 1 (◦C)

Temperaturepart 2 (◦C)

Vase lifea (d) Predicted vase life(d) (FOA model)

Predicted vase life(d) (degree-daysmodel)

Vase life2 (d) Predicted vase life(d) (FOA model)

Predicted vase life(d) (degree-daysmodel)

7 5.5 5.5 8.2 8.5 8.5 8.4 8.4 8.37 0.5 10.5 7.1 8.3 8.5 7.9 8.1 8.37 10.5 0.5 7.8 8.3 8.5 7.4 8.1 8.3

7 10.5 10.5 7.0 7.4 6.8 6.0 7.2 6.57 5.5 15.5 6.4 7.1 6.8 6.7 6.9 6.57 15.5 5.5 6.4 7.1 6.8 6.2 6.9 6.5

14 5.5 5.5 6.6 6.7 6.6 7.7 6.7 6.314 0.5 10.5 6.2 6.3 6.6 n.a.b 6.3 6.314 10.5 0.5 7.4 6.3 6.6 4.7 6.3 6.3

14 10.5 10.5 5.0 4.5 3.1 4.0 4.5 2.814 5.5 15.5 n.a. 3.9 3.1 n.a. 3.8 2.814 15.5 5.5 3.0 3.9 3.1 6.0 3.8 2.8

a Local average vase life of bunches that have five or more flowers of which vase life ends due to senescence.b n.a. = not available because all flowers showed clear marks of Botrytis.

It can be calculated that regarding these storage treatments theaverage absolute bias of the FOA model is 0.59 (wet storage) and0.83 (dry storage), whereas the average absolute bias of the degree-days model is 0.60 (wet storage) and 0.98 (dry storage). Althoughthe degree-days model results into a higher bias on average, it isnot the case that the difference in vase life between a constantstorage temperature and a stepwise changing temperature is large,and that this difference is explained by the FOA model, whereas itis not explained by the degree-days model. The predictions of thedegree-days model are still rather close to the vase life data.

3.4.5. Vase life data from literatureWe compared the degree-days model with the FOA model

for three data sets from the literature (C elikel and Reid, 2005;Pompodakis et al., 2005; Goedendorp and Barendse, 2010). Nelland Leonard (2005) provided only three measurements per storagetreatment condition. For this reason that data set was not consid-ered in this comparison.

We fitted the data in the same way as we fitted the data from thecurrent experiment. Table 6 summarizes the results of the fittingprocess. Because only the average vase life per storage treatmentcondition could be read, only the lack of fit (and not the pure error)of both models could be calculated. Table 6 shows R2

adjcalculated

on the basis of only a lack of fit as error.Two out of three data sets taken from the literature showed that

the degree-days model shows even a higher coefficient of determi-nation than the FOA model. Only the data from Pompodakis et al.(2005) showed a higher coefficient of determination for the FOAmodel. Comparing Table 3 with Table 6 shows that the coefficients

of determination as estimated in the current study are comparableto values estimated based on vase life data found in the literature.

For each data set the temperature dependent bias BT was com-puted and compared between models (Tables 7–9).

These results are less consistent than those of the current study.Only the data from Pompodakis et al. (2005) confirmed our expec-tations that the degree-days model overestimates vase life at lowertemperatures and underestimates vase life at higher temperatures.

4. Discussion

Theoretical calculations and graphing of functions show thatthe degree-days model approximates the model taking a first orderreaction kinetic with an Arrhenius temperature dependency. Theapproximation is valid for Q10 = 3, a value which has been found forcut roses, and for storage temperatures that are applied in prac-tice. It is noted that especially in the case of the combination oflow storage temperatures (below 2 ◦C) and long storage times thedegree-days model fails: it is expected to overestimate the vase life.If the storage temperature is above 6 ◦C, the degree-days model isexpected to underestimate the vase life.

The current experiment shows that the coefficient of determina-tion of the degree-days model is only slightly less than the one of theFOA model, which confirms our theoretical calculations. Standarderrors of the parameters of both models are small (below 7%).

From theory it is expected that the degree-days model overes-timates the vase life below 2 ◦C and underestimates it above 6 ◦C.This expectation is also confirmed by the results from the currentexperiment. The degree-days model overestimates the vase life at

Table 6Fits of the FOA model and the degree-days model for four data sets from literature. R2

adjand R2 are calculated on the basis of the unexplained variation due to lack of fit only.

Source Storage times and temperatures Cultivar Storage FOA model Degree-daysmodel

A B R2adj

(fit) A R2 (fit)

C elikel and Reid(2005)

5 d0 ◦C, 2.5 ◦C, 5 ◦C, 7.5 ◦C, 10 ◦C, 12.5 ◦C and 15 ◦C

First Red Dry 9.1 5345 0.67 8.7 0.85Wet 9.5 5254 0.76 9.0 0.91

Pompodakis et al.(2005)

10 d1 ◦C, 5 ◦C and 10 ◦C

First Red Wet 13.0 2925 0.93 10.5 0.67Akito 14.1 2692 0.95 11.4 0.42

Goedendorp andBarendse (2010)

0, 3, 6, 10 and 20 d0.5 ◦C, 5 ◦C, 8 ◦C and15 ◦C

Catch Dry 12.8 8221 0.69 13.0 0.75Red Horizon 13.2 4965 0.77 11.9 0.74Upper Gold 15.9 4213 0.57 14.2 0.65Valentino 11.7 5660 0.52 10.8 0.69

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Table 7Storage temperature dependent bias BT (data set from C elikel and Reid (2005)).

Temperature(K)

Dry storage Wet storage

FOA model Degree-days model FOA model Degree-days model

273.2 −1.7 −0.8 −0.8 0.0275.7 −0.7 −0.1 −1.1 −0.6278.2 −0.4 −0.2 −0.4 −0.2280.7 0.3 0.2 0.6 0.6283.2 0.1 −0.2 0.8 0.5285.7 1.4 1.0 0.6 0.1288.2 1.9 1.4 1.0 0.5

Table 8Storage temperature dependent bias BT (data set from Pompodakis et al. (2005)).

Temperature(K)

First Red Akito

FOA model Degree-days model FOA model Degree-days model

274.2 0.0 1.8 0.4 2.6278.2 −0.8 −0.6 0.1 1.4283.2 0.9 −0.8 −0.9 −0.6

Table 9Storage temperature dependent bias BT (data set from Goedendorp and Barendse (2010)).

Temperature(K)

Catch Red Horizon Upper Gold Valentino

FOA model Degree-daysmodel

FOA model Degree-daysmodel

FOA model Degree-daysmodel

FOA model Degree-daysmodel

273.7 −0.9 0.3 −1.2 0.1 −3.6 −0.4 −4.8 −3.5278.2 −0.6 −0.7 −0.1 0.0 0.6 2.6 1.1 1.2281.2 −0.3 −1.1 −0.3 −0.8 1.1 2.5 2.4 1.9288.2 2.7 2 2.4 1.1 3.4 3.6 2.8 1.7

0.5 ◦C. At 5.5 and 10.5 ◦C, the degree-days model underestimatesthe vase life.

The fitted value of the reaction rate determining parameter B ofthe FOA model indicates that Q10 is about 3, a value which was alsofound by C elikel and Reid (2005) albeit for a different rose cultivar.This seems to confirm the validity of the FOA model. However, thecurrent experiment shows that the FOA model underestimates thevase life in the case of low temperatures (0.5 and 5.5 ◦C) and over-estimates the vase life in the case of higher temperatures (10.5 and15.5 ◦C). So, the current study shows that the FOA model under-estimates vase life in the case of low storage temperatures andoverestimates it in the case of high temperatures.

Comparing dry storage to wet storage, experimental data showa worse fit of both models in the case of dry storage comparedto wet storage. This could indicate that dry storage increases thevariability in vase life of individual roses within the same batch.

The degree-days model passes a critical test provided by severaltreatments with either constant temperature or stepwise chang-ing temperature, having identical time–temperature sums. Thedegree-days model predicts that the vase lives of these differentlytreated flowers will be equal. The FOA model predicts that stepwisechanging temperature compared to a constant temperature willinfluence the vase life negatively, regardless the sequence of thesubsequent storage temperatures. The current experiment showsthat the FOA model indeed explains some of the observed varia-tion in the vase life data, which is not explained by the degree-daysmodel. However, it is not the case that the differences in vase lifebetween a constant storage temperature and a stepwise changingtemperature are large, and that these differences are completelyexplained by the FOA model, whereas they are not explained by thedegree-days model. Therefore it is concluded that the degree-daysmodel passes this critical test.

The data sets from the literature show that regarding two outof three data sets from literature the degree-days model performs

even better (a higher coefficient of determination) than the FOAmodel. Only the data from Pompodakis et al. (2005) show a betterfit of the data by the FOA model. For these data, the value of theparameter B is low compared to the other datasets (Table 6), indi-cating that Q10 is smaller than 1.5. For this low value of Q10, theoryshows indeed that the FOA model cannot be approximated by thedegree-days model (Fig. 1).

4.1. Application and limitations of models in industry

The degree-days model provides in an easy way to predict theremaining vase life, which may be an advantage if calculationcapacity or time is limited. Moreover, the degree-days model needsonly one parameter (which is the vase life of non-stored flowers),whereas the FOA model needs an additional parameter which is noteasily measured. In this way the degree-days model may contributeto the development of monitoring systems based on technologysuch as RFID. However, there are some limitations.

In the current experiment an antimicrobial solution was used.Also the data sets from the literature come from experiments wherean antimicrobial solution was used. This means that our conclu-sions are only valid in the case of using such antimicrobial solutions,and we have no prediction of vase life available in cases with-out antimicrobial solutions. However, in practice, an antimicrobialsolution is regularly added, and in that, our research has practicalvalue.

The models developed assume the initial quality level (q0) isthe same for lines of flowers that go through different storagetreatments. Regarding the current experiment, this is a reasonableassumption because flowers origin from the same production (har-vesting) batch. However, in practice well defined protocols maylack for harvesting roses. Moreover, distribution batches may becomposed of different production batches. This will limit the appli-cability of the prediction models.

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The use of both prediction models may be limited if the bio-logical variance is high, because this will lead to a large predictionerror. It is noticed that the unexplained variation due to pure error(biological variance) is smaller than the unexplained variation dueto a lack of fit. A low biological variance will still give predictionerrors due to a lack of fit of the degree-days model in the case ofstorage temperatures below 2 ◦C or above 6 ◦C.

5. Conclusions

In the literature, the vase life of cut flowers is modeled as afunction of the time–temperature sum built up during storage andtransport. Moreover, in the cut flower trade it is common practiceto use the time–temperature sum (the degree-days model) as apredictor of the vase life. For other perishable products, it is morecommon to model quality change via a first order reaction kinetic,with temperature dependency following Arrhenius. We concludethat we have shown that for cut roses, the two modeling approachesare not conflicting, albeit within a certain temperature range. Weconclude that the time–temperature sum built up during storageand transport is a rather attractive predictor of the remaining vaselife of cut roses after distribution in case vase life ends by natu-ral senescence, and when storage temperatures are between 2 and6 ◦C. Because of its simple principle the time–temperature sum haspractical value, but we are now aware of its limitations. Based onthese positive results for cut roses, we recommend analyzing thepossibility of applying the degree-days model for other perishableproducts too.

Acknowledgements

The authors acknowledge the financial support from the projectPASTEUR under the CATRENE program (CT204-PASTEUR). Theauthors also acknowledge the reviewers for their insightful sug-gestions.

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