PAT TOOLS FOR BIOLOGICS: CONSIDERATIONS AND CHALLENGES - Quality by Design for Biopharmaceuticals - page 238

12.8.2 Maximum Rate of Change
The maximum rate of change is the maximum value of (dC/dV) max . Decreasing (dC/
dV) max over multiple uses indicates transition spreading over a larger volume that is
indicative of integrity breaches.
V max is the corresponding volume to the (dC/dV) max . Any change in the location of
the maxima during transition is indicative of changes in bed integrity.

12.8.3 Breakthrough Volume
The breakthrough volume corresponds to the first value of V at which the normalized
solute signal is greater than 0.05 (for a step-up change). A decrease in the breakthrough
volume, that is, early breakthrough is a characteristic of integrity breaches.

12.8.4 Cumulative Error
The cumulative error is the difference between the actual solute signal and the signal
expected from a noninteracting plug flow. Factors that increase dispersion such as
column headspace or integrity breaches result in higher cumulative error.
I i ¼ R ABS(C-0)dV between V(0) and V(0.5)

I f ¼ R ABS(1-C)dV between V(0.5) and V(1.0)
Cumulative error E ¼ I i þ I f

12.8.5 Non-Gaussian HETP (HETP N )
The non-Gaussian HETP is calculated by the following formula:

2

2

HETP N ¼ Ls =ðM 1 =M 0 Þ

2 is the variance, and M 1 and M 0 are first and zeroth

where L is the length of the column, s

moments, respectively. Themoments equation for kth order is given byM k ¼ R V k (dC/dV)
dV. The variance for the distribution is calculated from the zeroth, first, and second
moments by using the following equation:

2

2

s ¼ðM 2 =M 0 Þ ðM 1 =M 0 Þ

This estimate of theoretical plates takes into account the entire transition without
assuming a Gaussian distribution. An increase in HETP N value correlates with decreased
efficiency of the column.

12.8.6 Gaussian HETP (HETP g )
HETP g expresses column efficiency as a ratio of the bed height to the theoretical number
of plates. The calculation estimates plate height with the assumption that the solute
response peak is a Gaussian distribution. The following formula is used to calculate