## Mathematica* 9 files for Predicting Nutrient Degradation in Storage from Two Successive Concentration Measurements

This webpage offers five programs developed with the support of NASA (Under Project SA-14-042) to extract the kinetic degradation parameters of nutrients in stored foods (e.g., vitamins in space-foods) and use them to predict future degradation levels.

All five programs are based on the following assumptions:

1. The nutrient's degradation follows fixed order kinetics, n, which is known a-priori or can be assumed.
2. The food's past temperature history is non-isothermal, i.e., T(t) ≠ constant.
3. The process's rate constant temperature dependence follows the exponential model k(T(t)) = kTref exp(c (T(t))-Tref)) where k(T(t)) is the momentary degradation rate, kTref is the rate constant at a reference temperature Tref and c a constant having temperature reciprocal units. (It can and has been shown that this simple model can replace the more complicated Arrhenius equation without sacrificing the fit.)

Where these conditions are satisfied, one can show that, in principle at least, two successive concentration or concentration ratio measurements are sufficient to estimate the magnitudes of kTref and c by numerically solving two simultaneous rate equations. Once kTref and c have been estimated in this way, they can be used to reconstruct the entire degradation curve for the particular temperature history and predict future degradation levels.

The five programs, labeled A to E, are all written in Mathematica* 9. They are presented below as Mathematica* notebook (.nb) files, which a user having the installed Mathematica* software can open, view, modify, print and interact with. They are also presented as Computable Document Format (.cdf) files which can be opened, viewed, printed and interacted with (but not modified) using the free Wolfram CDF Player* application. They are also presented as Portable Document Format (.pdf) files which can be opened, viewed and printed (but not modified or interacted with) using the free Adobe Acrobat Reader DC+ application.

Program A: The user starts by generating two concentration or concentration ratios for a temperature history (profile), T(t), entered as an algebraic expression, which can contain "If" statements. The user then calculates two concentrations or concentration ratios to which "experimental errors" can be added. The second step is an attempt to retrieve the generation parameters, reconstruct the entire degradation curve, and predict and test the prediction of a future concentration or concentration ratio.

Program B: The same as Program A except that the user enters two concentration ratios, presumably or actually experimentally determined.

Program C: The same as Program B except that the temperature profile is not entered as an algebraic expression but is imported as a set of numerical time-temperature data values. The program converts these data into a continuous interpolated function which is used in all subsequent calculations.

Program D: Successful calculation of kTref and c with programs A and B requires close initial guesses of these parameters, which might not always be easy to obtain. Program D facilitates the process by allowing the user to manually match the two entered points on the screen with a degradation curve whose position can be varied using separate slider controls for kTref and c. The kTref and c estimates so obtained can be used as initial guesses in program A or B, or serve as the parameter values themselves.

Program E: Successful calculation of kTref and c with program C requires close initial guesses of these parameters, which might not always be easy to obtain. Program E facilitates the process by allowing the user to manually match the two entered points on the screen with a degradation curve whose position can be varied using separate slider controls for kTref and c. The kTref and c estimates so obtained can be used as initial guesses in program C, or serve as the parameter values themselves. Note that the difference between program E and program D is that in the latter, the temperature profile is imported as a numerical data set and not entered as an algebraic expression.

Micha Peleg and Mark D. Normand
Department of Food Science
University of Massachusetts
Amherst, MA 01003

Use this page to download five Mathematica* 9 files (programs A, B, C, D & E) either as Mathematica* notebook (.nb), Computable Document Format (.cdf) and Portable Document Format (.pdf) files. If you have Mathematica* version 9 or newer and the Wolfram CDF Player* already installed then you can view the files immediately in a window of your web browser. (If you clicked on the .cdf file of either the D or E program then you will be able to operate the controls of the included Manipulate panel in the window's display. If you clicked on the .nb file of either the D or E program then the Manipulate panel will not be visible or useable until the file is saved and opened in Mathematica*.) In either case you should save the file by clicking on the window of downloaded text and choosing "Save Page As..." from your browser's File menu. If you have installed Mathematica* you will be able to open, view, modify, print and interact with the saved .nb or .cdf files in their proper format. If you have only installed the Wolfram CDF Player* then you should download only the .cdf files which may be opened, viewed and printed but not modified. However, in the case of programs D and E you will be able to interacte with and operate the controls of the Manipulate panels in either a browser window or in the Wolfram CDF Player. If you only want to open, view and print the files and you have the Adobe Acrobat Reader program installed you should download the .pdf files.

### Download five Mathematica* 9 notebook (.nb) files (programs A, B, C, D & E).

The notebook (.nb) files may be opened, viewed, printed, modified and interacted with if you have Mathematica* 9 or newer.

### Download five Mathematica* 9 Computable Document Format (.cdf) files (programs A, B, C, D & E).

The Computable Document Format (.cdf) files may be opened, viewed, interacted with and printed (but not modified) using the free Wolfram CDF Player* application.

### Download five Adobe Acrobat Portable Document Format (.pdf) files (programs A, B, C, D & E).

The Portable Document Format (.pdf) files may be opened, viewed and printed (but not modified or interacted with) using the free Adobe Acrobat Reader DC+ application.

* Mathematica® and Wolfram CDF Player® are registered trademarks of Wolfram Research, Inc.
You can get more information about Mathematica by visiting the Web site of Wolfram Research at: http://www.wolfram.com/