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QUALITY CONTROLCAPSULESwedish scientist Svante Arrhenius provided the first kinetic model to interpret the effect of temperature on reaction rate. Since then, pharmaceuticalscientists have often attempted to predict long-term chemical stability with insufficient understanding about the kinetic model, applyingthe practice in inappropriate situations. Understanding critical temperature relationships is the key to more accurately predicting long-term levelsfor a specific product.HIGH-TEMPERATURE STABILITYDo the Mathfor Shelf LifePredict stability usingdata collected at highertemperatures> By Michelle Duncan, PhD,and Irene Zaretsky, MSPharmaceutical scientists routinelypredict long-term chemicalstability at a lowertemperature using data generatedat a higher temperature over a shortertime period. The use of 40 degrees C chemicaldata (e.g., assay and related substances)to predict levels over long-term 25degrees C storage has become such commonpractice that the underlying theory isoverlooked or was never learned.There is disagreement about whetherthree months at 40 degrees C indicates expected25 degrees C levels for 12 monthsor for 24 months because of insufficientunderstanding or information about thekinetic model. More importantly, withoutsome theoretical understanding, thepractice is inappropriately applied to situationsthat do not fit the kinetic modelupon which it is based, resulting in erroneouspredictions.In addition to providing the backgroundof how these kinetic predictionswork, this paper will provide the key tounderstanding the temperature relationshipsand the ability to more accuratelypredict the expected long-term levels for aspecific product.Arrhenius Kinetics: WhereThis All BeganSwedish scientist Svante Arrhenius providedthe first kinetic model to interpret theeffect of temperature on reaction rate givenby Equation 1 (see info box). 1-3 The Arrheniusequation can be applied regardlessof the order (zero-order, first-order, etc.) ofthe reaction kinetics. 4 Equation 2 presentsthe linear form of the Arrhenius equationfor graphical presentation (y = mx + b).For many reactions, a linear relationshipcan be obtained between the inverse oftemperature (in degrees Kelvin) and thenatural log (Ln) of the measured rate constant(k), as shown in Figure 1.Figure 1.THE ARRHENIUS EQUATIONK = A exp (-Ea/RT) (Equation 1)k = reaction rate constantA = Arrhenius factor (y-intercept constant)Ea = the energy of activation for the reaction, cal/mole(1000 cal = 1 kcal)R = the ideal gas constant, 1.987 calories/deg moleT = the absolute temperature (degrees Kelvin), for 25° CT= 298° K and for 40°C T= 313° K)THIS EQUATION CAN BE WRITTEN IN SEVERALEQUIVALENT FORMS AS FOLLOWS:Ln k = -Ea/RT + Ln A (Equation 2, y = mx + b)Ln k2/k1 = -Ea/R*(1/T2-1/T1) (Equation 3)k2 = k1 * exp[-Ea/R*(1/T2-1/T1)] (Equation 4)The constants k1 and k2 are the rate constants at temperatureT1 and T2 (for example 25 degrees C and 40degrees C), respectively.Equation 3 presents a simplified formfor use with two temperatures. Equation 4expresses the relationship between thereaction rates and the corresponding temperatureswhen the activation energy (Ea)of the reaction is known. Equation 4 allowsfor the calculation of a reaction rate constantat a lower temperature when the activationenergy and the reaction rate at thehigher temperature are known.As shown by Figure 1, the Arrheniusmodel provides the ability to determine thereaction rate and, hence, predict stabilityat any temperature with knowledge of theactivation energy (Ea) and the reaction rateat another temperature.Limitations to Arrhenius’ ModelThe first requirement is a reaction (ongoing)where the reaction rate constant atArrhenius plot of Ln K against1/T. Slope = - Ea/R k = reactionrate constant, T = temperaturein degrees Kelvin Ea = activationenergy, R = ideal gas constant.© DREAMSTIME.COMPharmaceutical Formulation & Quality > April/May 2011
PHARMAQUALITY.COMa given temperature can be determined.When monitoring the generation of a primarydegradation product, the presenceof a secondary degradation reaction canintroduce error to the calculation of the primaryrate constant and any attempted Arrheniusmodel predicted rates. Likewise,if a component is consumed to the extentthat the reaction equilibrium changes, thereaction rate will not remain constant. TheArrhenius kinetic model requires a reactionrate constant (k).Figure 2.The zero-order kinetic model for 10% drug lossover 24 months at 25 degrees C shown as thepercent of drug remaining as a function of timeat 25 degrees C.The Arrhenius kinetic model can be utilizedacross the temperature range where aconstant relationship between the effect oftemperature on the reaction rate is maintained,where the graph is linear. The reactionmechanism should not change over thetemperature range studied. Conformanceto the model (linearity) is often lost whencrossing a phase change, such as freezingand the glass-phase transition of proteinsand peptides. Reactions where the rate isdependent upon oxygen, light (photochemical),diffusion, or microorganism-baseddecomposition may not demonstrate Arrheniuskinetics over any temperature range.Figure 3.Table 1.Months at 40degrees C to reach10% drug loss63Relationship of reaction rates at 40 degrees C and activationenergy (Ea) for drug lossk40 (drug %labelclaim per month)-1.6667-3.3333Finding Ea for the ReactionThe activation energy (Ea) for a reactioncan be determined by conducting stabilitystudies at several different temperaturesand applying the Arrhenius kinetic model.The slope of the line formed with Equation2 contains Ea, as demonstrated in Figure 1.Higher temperatures (e.g., 55 degrees C,70 degrees C) and corresponding shortertimes (e.g., weeks, days) can be employedfor this determination provided the Arrheniuskinetic model remains valid (see Limitationsto Using Arrhenius Kinetic Model).The Ea for drug decomposition will usuallyfall in the range of 12 to 24 kcal/mole, witha typical value of 19 to 20 kcal/mole. 5 Theactivation energy can be approximatedbased upon prior knowledge of the drugdecomposition kinetics. Once the Ea isknown, it usually remains valid for usethrough small concentration changes orslight formulation changes.Importance of Ea: TheoreticalModel for Drug LossThe following discussion demonstrates the relationshipof drug degradation kinetics at 40degrees C and 25 degrees C, where drug lossis the shelf life-determining parameter. Forthis illustration, acceptable product stabilityis based upon a lower limit of 90% label claimk25 (drug %labelclaim per month)-0.4167-0.4167Ea (kcal/mole)activation energy1726and the expiration date set at exactly 90% labelclaim. For this exercise, a zero-orderdegradation kinetics model (Δdrug/Δtime =—k) is applied to determine the rate for 10%of drug loss occurrence over 24 months at 25degrees C (Figure 2). The slope was calculatedand is equal to —0.4167 drug %labelclaim/ month, which is the reaction rate constantat 25 degrees C (k25). Hence, for the drugto remain within acceptance criteria for 24months at 25 degrees C, the rate of degradationat 25 degrees C must be less than 0.4167drug %label claim/month.In a similar manner, drug loss can bemodeled at 40 degrees C accelerated temperaturefor the two scenarios of 10% drug lossoccurring at three months and at six months(Figure 3). The slopes, which represent thereaction rate constant at 40 degrees C (k40),were calculated to be —1.6667 drug %labelclaim/month for the six months limit scenarioand —3.3333 drug %label claim/month for thethree months limit scenario.Using the Arrhenius equation (Equation3) and the reaction rates at 25 degrees C and40 degrees C, the Ea can be calculated asshown in Table 1. The Arrhenius equationcan be applied regardless of the order of thereaction kinetics. For the kinetic model thatassumes 10% drug loss at 24 months at 25degrees C (k25 = —0.4167 drug %labelTable 2.Relationship of reaction rates at 40 degrees C and activationenergy (Ea) for impurity generationMonths at 40degrees C for impurityto reach 1%k40 (%w/wimpurity per month)k25 (%w/wimpurity per month)Ea (kcal/mole)activation energy60.16670.041717Zero-order kinetic models for 10% drug lossover three months and over six months at 40degrees C.30.33330.041726April/May > Pharmaceutical Formulation & Quality
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