SOP: Plotting of Thermal Death Time and Thermal Resistance Curves

GIFSQ — Global Institute for Food Safety & Quality

Principle

When the decimal reduction times (D values) determined at several temperatures are plotted on semi-log paper against temperature, a straight line is obtained — the thermal resistance curve (also called the phantom thermal death time curve). Its slope, measured as the number of °F required for the curve to pass over one log cycle, equals the z value of the TDT curve, and it characterizes the heat resistance of the organism by its D value at the reference temperature (usually 250 °F).

Materials required

• Semi-log paper, preferably having 3 cycles (logarithmic scale for D or TDT values, linear scale for temperature)
• D values determined at four or more temperatures (corrected for heating lag)

Procedure — Plotting the TDT (end-point) curve

Consider the end point of destruction at any given temperature as the shortest time in which no visible growth is observed in any of the replicate samples

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To plot the TDT curve on semi-log paper, note the time on the log scale and the temperature in °F on the linear scale; plot the corrected survival and destruction times

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As a rule, the TDT curve must lie above all survival points and below the largest possible number of destruction points; since even a single result indicating exceptional resistance may prevent adhering to this rule, some latitude is necessary

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The National Canners Association recommends drawing two straight lines that show the maximum divergence in slope, but are still above all survival points and below most of the destruction points; determine the slope of each by measuring the number of minutes required to traverse one logarithmic cycle, and average the z values obtained; average the F values for any temperature in the same way — the curve so constructed is called the “end-point” curve

Procedure — Thermal resistance (phantom TDT) curve

On the semi-log paper, plot the D values on the logarithmic scale and the temperature on the linear scale

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Draw the straight line of best fit through the points; this straight line gives the relationship between log D and temperature

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Characterize the thermal resistance by the D value at the reference temperature (usually 250 °F); measure the slope as the number of °F required for the curve to pass over one log cycle — this is the z value, the temperature necessary to bring about a ten-fold change in D value

Procedure — Fitting the line by least squares (when points do not fall on a straight line)

Calculate the slope b of the regression line, the intercept a on the y axis, and the coefficient of correlation r, using the following formulae:

      NΣ(xy) − (Σx)(Σy)
  b = ———————————————————————
      NΣ(x²) − (Σx)²

      Σy − bΣx
  a = —————————
          N

      NΣ(xy) − Σx·Σy
  r = ——————————————————————————————————————————
      √([NΣx² − (Σx)²][NΣy² − (Σy)²])

Where,

• N = number of observations of x and y
• x = temperature (°F)
• y = log of the D value (or TDT)

Note: the printed layout of these formulae is garbled in the available copy; the forms above are the standard least-squares expressions reconstructed from the book’s symbol definitions, and should be checked against the printed book.

The same least-squares procedure is used for plotting TDT curves and heat penetration data when the points do not fall on a straight line.

Calculation — TDT at any temperature from the curve

  250 − T

log(t/F) = ———————
z

or

            (250 − T)

t = F × antilog ———————————
z

Where,

• t = thermal death time at temperature T
• F = time in minutes required to destroy the organism at 250 °F
• z = slope of the TDT curve in °F
• T = temperature whose TDT is required

Note: the printed layout of this derivation is garbled in the available copy; the form above is reconstructed from the book’s definitions and verified against its worked example (TDT of Cl. botulinum at 250 °F = 2.78 min with z = 18 gives log(t/2.78) = 10/18, hence TDT at 240 °F = 9.99 ≈ 10 min, as printed; the exponential form below gives the same result), and should be checked against the printed book.

Precautions

• The z value of the thermal resistance curve has essentially the same value whether determined from a TDT curve or from a thermal resistance curve, under a constant set of conditions.

References

Ranganna, S., Handbook of Analysis and Quality Control for Fruit and Vegetable Products, 2nd edition.

Ball, C.O., Ind. Eng. Chem., 35, 71 (1943); Schmidt, C.F. & Bock, J.H., Food Res., 20, 606 (1955) — cited in the book.