Why Use Level B or Level C Correlations?
A Level A in vitro–in vivo correlation (IVIVC) aims to map the in vitro dissolution profile to the in vivo absorption profile over time. In some development settings, the available formulations or data do not support that full mapping. Level B and Level C approaches summarize relationships at a coarser scale: they can help compare or rank formulations, but their summary measures do not generally recover the entire exposure time course.
The distinction matters when interpreting a high correlation. A strong relationship across formulations may indicate that a selected dissolution summary tracks a selected pharmacokinetic summary in the observed data. It does not, by itself, show that a dissolution profile can predict the plasma concentration–time curve, or that the relationship will hold for a new formulation. The introductory material on IVIVC levels establishes the broad distinction; here the focus is on how to calculate and critically interpret Level B and Level C summaries.
Level B: Comparing Mean Times
For a dissolution profile divided into intervals, let \(\Delta F_j\) be the fraction of the dose dissolved during interval \(j\), and let \(t_j^*\) be that interval’s midpoint. If the fractions represent the entire released dose, the discrete estimate of MDT is the fraction-weighted average of the interval midpoints. For data recorded as percentages, convert each increment to a fraction or divide the weighted sum by the total percentage released.
This calculation summarizes when release occurs on average; it discards the detailed shape of the profile. A common in vivo counterpart is MAT. Under linear, time-invariant pharmacokinetics with the same disposition behavior after oral and intravenous administration, MAT can be estimated as oral mean residence time minus intravenous mean residence time. When an appropriate intravenous reference is unavailable, an absorption-time summary may instead be obtained from a suitable absorption estimate. These conditions matter: the difference is not automatically interpretable as absorption time if disposition differs, pharmacokinetics are nonlinear, or the estimates are unreliable.
A Level B analysis pairs one MDT and one MAT for each formulation and examines how they vary together. The usual Pearson correlation describes the strength and direction of a linear association across those paired formulation summaries. It is not a pointwise comparison of dissolution and absorption at matching times. With few formulations, the correlation is particularly sensitive to one unusual formulation and should chiefly be treated as a descriptive result unless an appropriate inferential design supports more.
Worked Example: Calculating MDT and a Level B Correlation
Suppose four hypothetical formulations have cumulative dissolution increments in three intervals. The interval midpoints are 0.5, 1.5, and 3.0 hours. The increments below are fractions of the dose released in those intervals.
| Formulation | Increment 0–1 h | Increment 1–2 h | Increment 2–4 h | MAT (h) |
|---|---|---|---|---|
| A | 0.25 | 0.35 | 0.40 | 2.0 |
| B | 0.10 | 0.25 | 0.65 | 2.4 |
| C | 0.05 | 0.15 | 0.80 | 3.0 |
| D | 0.20 | 0.30 | 0.50 | 2.2 |
Each profile sums to 1.00, so the denominator in the MDT formula is 1. For formulation A, \( (0.5)(0.25)+(1.5)(0.35)+(3.0)(0.40)=1.850 \) h. For B, the calculation is \( (0.5)(0.10)+(1.5)(0.25)+(3.0)(0.65)=2.375 \) h. Similarly, for C it is \( (0.5)(0.05)+(1.5)(0.15)+(3.0)(0.80)=2.650 \) h, and for D it is \( (0.5)(0.20)+(1.5)(0.30)+(3.0)(0.50)=2.050 \) h.
Let \(x\) be MDT in hours and \(y\) be MAT in hours. The means are \(\bar{x}=2.23125\) h and \(\bar{y}=2.400\) h. The centered cross-product sum is \(\sum_i(x_i-\bar{x})(y_i-\bar{y})=0.4400\); the corresponding sums of squares are 0.37421875 h² and 0.5600 h². Therefore,
For these four hypothetical formulations, the two mean-time summaries have a strong positive linear association: formulations with larger MDT tend also to have larger MAT. This small set does not establish that the relationship is stable across formulations or predicts exposure in a new formulation. The calculation also assumes that the interval increments adequately represent release over the relevant profile; omitted late release or inconsistent normalization would change MDT.
Level C: One Dissolution Time or Several
A single-point Level C analysis selects a dissolution time in advance, such as the fraction dissolved at 2 hours, and relates that value across formulations to a pharmacokinetic parameter. A correlation coefficient can summarize linear association, while a regression equation expresses the fitted change in the chosen parameter per unit change in dissolution. The predictor’s scale must be stated: a fraction and a percentage differ by a factor of 100.
A multiple-point Level C analysis uses dissolution measurements at two or more selected times as predictors of a pharmacokinetic parameter. A typical model with two points is \(y_i=\beta_0+\beta_1x_{i1}+\beta_2x_{i2}+\epsilon_i\), where \(y_i\) is the parameter for formulation \(i\), and \(x_{i1}\) and \(x_{i2}\) are that formulation’s dissolution values at the chosen times. The fitted coefficients describe the model’s conditional linear relationship; they do not imply that each point independently causes a change in exposure.
For either regression form, the usual least-squares interpretation relies on a suitable linear mean relationship, independent formulation-level errors, and no severe influential observations. Conventional uncertainty calculations additionally require an appropriate error-variance model; exact small-sample tests often assume normally distributed errors. Dissolution values at different times may be strongly correlated, making multiple-point coefficients unstable even when fitted values look close. These assumptions concern the formulation-level model and are not made true merely by reporting a large correlation.
Worked Example: Single-Point Level C Correlation and Regression
Consider five hypothetical formulations. Let \(x\) be the fraction dissolved at 2 hours and \(y\) be \(C_{\max}\) in ng/mL.
| Formulation | \(x\): fraction dissolved at 2 h | \(y\): \(C_{\max}\) (ng/mL) |
|---|---|---|
| A | 0.20 | 8.1 |
| B | 0.35 | 9.4 |
| C | 0.50 | 11.0 |
| D | 0.65 | 12.1 |
| E | 0.80 | 14.4 |
The means are \(\bar{x}=0.500\) and \(\bar{y}=11.00\) ng/mL. From the five paired values, \(\sum_i(x_i-\bar{x})^2=0.2250\), \(\sum_i(y_i-\bar{y})^2=23.74\) (ng/mL)², and \(\sum_i(x_i-\bar{x})(y_i-\bar{y})=2.295\) ng/mL. Thus,
The least-squares slope is \(2.295/0.2250=10.20\) ng/mL per unit fraction dissolved, and the intercept is \(11.00-(10.20)(0.500)=5.900\) ng/mL. The fitted line is therefore \(\widehat{C}_{\max}=5.900+10.20x\). In this hypothetical dataset, the observed \(C_{\max}\) values increase closely with the 2-hour dissolution fraction. The slope is per full unit of fraction; a 0.10 increase in fraction corresponds to a fitted increase of \(10.20(0.10)=1.020\) ng/mL. This regression summarizes the selected parameter, not the concentration–time curve.
Worked Example: Multiple-Point Level C Regression
Suppose five hypothetical formulations have dissolution measured at 1 and 3 hours, in percentage points, and an \(AUC\) summary in hypothetical units of mg·h/L. Consider the model \(y=\beta_0+\beta_1x_1+\beta_2x_2+\epsilon\), where \(x_1\) and \(x_2\) are percent dissolved at 1 and 3 hours.
| Formulation | \(x_1\): % at 1 h | \(x_2\): % at 3 h | \(AUC\) (mg·h/L) |
|---|---|---|---|
| A | 20 | 70 | 42 |
| B | 30 | 60 | 44 |
| C | 40 | 55 | 47 |
| D | 50 | 40 | 48 |
| E | 60 | 30 | 50 |
The equation \(\widehat{y}=20.0+0.400x_1+0.200x_2\) reproduces every listed outcome: for A, \(20.0+(0.400)(20)+(0.200)(70)=42.0\); for C, \(20.0+(0.400)(40)+(0.200)(55)=47.0\); and for E, \(20.0+(0.400)(60)+(0.200)(30)=50.0\). The same substitution gives 44.0 for B and 48.0 for D. Because every residual is zero, the residual sum of squares is zero, its minimum possible value. The predictor columns are not exactly linearly dependent: for rows A, B, and C, the determinant of the matrix with rows \((1,x_1,x_2)\) is 50, not zero. Thus the coefficients are uniquely determined in this constructed example.
The coefficient 0.400 has units of mg·h/L per percentage point dissolved at 1 hour, conditional on \(x_2\); 0.200 has the corresponding units for 3 hours, conditional on \(x_1\). This deliberately exact example illustrates the calculation, not realistic precision or evidence of predictive performance. With only five formulations and three fitted coefficients, there are \(df=5-3=2\) residual degrees of freedom. Small changes in the data could substantially alter the fitted coefficients.
Justification, Diagnostics, and Limits
The Level B statistic follows directly from a weighted average. If each interval contributes \(\Delta F_j\) of the total released fraction at representative time \(t_j^*\), the average time associated with released material is total time-weighted released fraction divided by total released fraction: \(\sum_jt_j^*\Delta F_j/\sum_j\Delta F_j\). This is why MDT is a single summary rather than a reconstruction of \(F(t)\). Different release profiles can have the same weighted average while differing substantially in their early and late behavior.
Similarly, Pearson’s \(r\) is the centered cross-product sum divided by the product of the square roots of the two centered sums of squares. It ranges from \(-1\) to \(1\) and captures linear association, not every possible relationship. A curved relationship can have a modest \(r\), and a single influential formulation can make \(r\) appear very large. For a regression with an intercept and one predictor, \(R^2=r^2\); this identity does not turn a formulation-level fit into a mechanistic or time-course model.
Before interpreting either level, inspect the paired formulation values and a scatterplot. Check whether the association is approximately linear, whether the range of formulations is adequate, and whether one formulation dominates the result. For multiple-point Level C, also check whether dissolution predictors are redundant or highly correlated, whether the number of formulations is large enough relative to the number of coefficients, and whether residual spread and pattern are plausible. A high in-sample fit with many predictors and few formulations is especially weak evidence of transportable prediction.
The principal alternative is a Level A approach, which seeks a time-resolved relationship between dissolution and in vivo absorption and can support prediction of a concentration–time profile when appropriately developed and evaluated. Level B offers a mean-time comparison; Level C offers selected dissolution-to-parameter comparisons. These are simpler and may be useful for formulation ranking, but their simplicity is also their limitation. Ranking by \(C_{\max}\), \(AUC\), or a mean time does not establish accurate prediction of those measures outside the observed formulation set, nor does it ensure accurate prediction of \(T_{\max}\) or the full profile.
Choosing and Reporting a Summary
A defensible report states the formulations included, the dissolution metric and its time or interval definition, the in vivo parameter and units, and whether the result is a Level B or Level C analysis. For regressions, report the fitted equation with predictor scales, the number of formulations and coefficients, and relevant diagnostics. Avoid using a result outside the range of formulations and dissolution values from which it was estimated without additional justification.
The formulation-ranking use is most credible when the selected summaries reflect the development question, the observed data show no important departures from the model, and the ranking is not driven by one formulation. A ranking is still only a ranking: it should not be presented as a guarantee of bioequivalence, a quantitative exposure prediction, or proof that a change in dissolution caused a change in pharmacokinetics.
Check Your Understanding
Try each question, then compare your reasoning with the short solutions below.
- A profile releases 0.20, 0.30, and 0.50 of its dose in intervals with midpoints 0.5, 1.5, and 3.0 hours. Calculate MDT.
- What does a positive Level B correlation between MDT and MAT mean, and what does it not establish?
- A single-point Level C model uses fraction dissolved as \(x\), with fitted equation \(\hat{y}=4+8x\), where \(y\) is \(C_{\max}\) in ng/mL. What fitted change corresponds to a 0.05 increase in dissolved fraction?
- Why can a high multiple-point Level C \(R^2\) be misleading when only a few formulations are available?
- Which approach is intended to describe a time-resolved dissolution–absorption relationship: Level B, Level C, or Level A?
Solutions. (1) \(\mathrm{MDT}=(0.5)(0.20)+(1.5)(0.30)+(3.0)(0.50)=0.10+0.45+1.50=2.05\) h. (2) Across the observed formulations, larger MDT tends to accompany larger MAT in a linear pattern; it does not establish a full profile mapping or reliable prediction for new formulations. (3) The fitted change is \(8(0.05)=0.400\) ng/mL. (4) Many coefficients relative to the number of formulations can fit observed variation closely, while leaving few residual degrees of freedom and making coefficients unstable; an in-sample fit alone is not validation. (5) Level A seeks the time-resolved dissolution–absorption relationship.