Why Use a Logarithmic Concentration Axis?
An individual concentration–time plot places one subject’s measured concentrations at their sampling times. On a linear concentration axis, equal vertical distances represent equal differences: the distance from 10 to 20 concentration units matches the distance from 90 to 100. This is useful for seeing absolute changes, but a few large concentrations can compress the low-concentration observations near the bottom of the graph.
A logarithmic concentration axis changes the visual comparison. Equal vertical distances represent equal ratios, so a doubling from 1 to 2 occupies the same distance as a doubling from 50 to 100. Low concentrations become easier to distinguish, and patterns of proportional change can be easier to inspect. The data remain concentrations in their original units; the axis changes how those values are spaced in the display.
A logarithmic axis is appropriate only for positive values. In addition, concentrations being compared should use a consistent unit, and the plot should make clear which observations are measured values and which are censored or otherwise not quantified. The choice of a log axis does not itself assert that the data follow a particular pharmacokinetic model.
What Changes Between Linear and Log Views?
A linear view emphasizes absolute concentration differences. It can make a high peak and its subsequent large absolute decline easy to see, while compressing small late concentrations. A logarithmic view emphasizes relative differences: a change from 2 to 1 is as large vertically as a change from 20 to 10. This can reveal whether low observations continue to change proportionally, whether the profile has a visible change in slope, and whether a small number of observations differ substantially in relative terms.
The display does not add information to the measurements. It reweights visual distances. A small vertical gap on a log axis can represent a large absolute difference at high concentrations, while the same gap at low concentrations represents a smaller absolute difference. Conversely, equal absolute changes can look very different at different concentration levels. Thus, the scale should be selected to support the question being examined, not because one view is universally superior.
Logarithms of concentrations are formally logarithms of dimensionless quantities. In practice, a plot’s position depends on the numeric concentration expressed in the chosen unit, or equivalently on the ratio of concentration to a reference unit. Changing from mg/L to µg/L multiplies every numeric concentration by the same factor. On a logarithmic axis, this shifts all observations vertically by the same amount; it does not change their relative spacing or the profile’s shape. For comparison across plots, however, use consistent units and state them explicitly.
Worked Example: Reading Multiplicative Spacing
Suppose a hypothetical subject has measured concentrations of 1, 2, 4, and 8 mg/L at 1, 2, 3, and 4 hours after a dose. Compare the vertical spacing of these points on a base-10 logarithmic axis with their spacing on a linear axis.
On a base-10 logarithmic axis, the vertical coordinate is proportional to \(\log_{10}(C)\). The values are:
The successive log differences are approximately 0.3010, 0.3011, and 0.3010. Each concentration doubles, so the points are equally spaced vertically, apart from rounding. On a linear axis, the successive differences are 1, 2, and 4 mg/L, so the vertical gaps grow. The logarithmic view makes the repeated doubling clear; the linear view shows that the absolute increases become larger.
This is a statement about the geometry of the plot, not a claim that the subject’s concentration rose according to a particular kinetic model. Four points with equal ratios are not, by themselves, sufficient to establish a mechanism or to justify extrapolation.
Log Axes and Exponential Decline
A common reason to inspect a log concentration axis is that an exponential concentration-time segment appears linear when concentration is plotted against time on a log scale. This is a geometric consequence of the exponential form, not proof that the observed profile follows that form.
The justification follows by taking the natural logarithm of both sides of the stated exponential expression:
This transformation explains why a straight-looking segment can be compatible with exponential decline. It does not establish that the assumptions behind the expression hold, nor does it establish that a straight segment identifies a unique process. Sparse sampling, measurement error, changing absorption or disposition phases, and the limited time range can all affect the visual pattern. Treat apparent changes in slope as features to investigate, not as definitive evidence for a specific model structure.
Worked Example: Checking a Straight-Line Pattern
Consider hypothetical concentrations from one subject during a declining portion of a profile: 80 mg/L at 2 hours, 40 mg/L at 4 hours, and 20 mg/L at 6 hours. Determine whether the log-scale spacing is consistent with a constant exponential rate over these intervals, and calculate the implied rate from the first interval.
The concentrations halve every 2 hours. For the first interval, if \(C(t)=C_0e^{-kt}\), the ratio is \(40/80=0.5\) over 2 hours. Thus:
The second interval has the same ratio, \(20/40=0.5\), and therefore gives the same rate, \(-\ln(0.5)/2\approx0.3466\ \text{h}^{-1}\). On a natural-log concentration axis, the vertical differences are \(\ln(40)-\ln(80)=\ln(0.5)\approx-0.6931\) and \(\ln(20)-\ln(40)\approx-0.6931\). Equal time intervals and equal log differences produce a straight line through these points.
The calculation describes the pattern in these hypothetical observations under the exponential assumption. Three points do not establish that the decline is truly exponential, and the rate should not be interpreted as a validated parameter without further modeling and assessment.
Constructing and Checking the Individual Plot
For an interpretable display, plot the subject’s observed concentration against the corresponding sampling time, use a logarithmic scale only for the concentration axis, and label the axis with the original concentration unit. Keep times in their recorded time unit and make the dose or relevant time origin clear. A log scale for time is a different display choice and is not required to make a concentration axis logarithmic.
If points are connected, the connecting segments are visual guides between sampled observations. They do not imply that concentrations were measured continuously or that a particular interpolation is valid. When sample times are irregular, retain their actual positions on the time axis; evenly spacing observations by their order would distort the apparent rate of change.
Before interpreting the plot, check the data and display:
- Confirm that positive concentrations are in a common unit and that the axis tick labels show that unit.
- Verify that plotted times and concentrations correspond to the same subject and the intended dose or study interval.
- Check for zero, negative, missing, or below-quantification records before applying a log scale.
- Inspect whether a few high values determine the visible range and whether the low values remain distinguishable.
- Compare with a linear view when absolute concentration differences or peak magnitude are important.
A log view can make relative outliers easier to notice, but it can also visually magnify small absolute differences among low values. It may hide the practical size of a concentration difference if the reader focuses only on relative spacing. A changing slope can be worth checking against sampling times, dose history, data handling, and the profile’s phase, but should not be assigned a mechanistic explanation from the graph alone.
Worked Example: Handling a Below-Quantification Observation
In a hypothetical profile, a subject has quantified concentrations of 12 mg/L at 2 hours and 3 mg/L at 6 hours. At 10 hours, the assay result is reported as below the lower limit of quantification (LLOQ), which is 1 mg/L. Explain what can and cannot be shown on a log concentration axis.
The two quantified observations are positive and can be plotted at 12 and 3 mg/L. Their ratio is \(3/12=0.25\), so on a base-10 log axis their vertical difference is:
The 10-hour result is not a measured concentration of 1 mg/L. It indicates that the concentration was below the stated quantification limit, not that it equaled zero or the LLOQ. It therefore cannot be placed at its exact measured concentration on the logarithmic axis. A display may mark the observation at the LLOQ boundary with a distinct symbol or downward arrow to indicate censoring, provided the convention is explained. That marker communicates the assay threshold; it must not be read as an exact value.
Replacing the record with zero is invalid for a logarithmic axis because \(\log(0)\) is undefined. Replacing it with an arbitrary positive value, such as LLOQ/2, invents a concentration and can alter the visual pattern. Preserve the below-quantification status and use a clearly identified censoring display rather than presenting a substitute as observed data.
Common Pitfalls and Alternatives
| Display or issue | What it highlights | What to check |
|---|---|---|
| Linear concentration axis | Absolute differences and peak magnitude | Low observations may be compressed near the axis |
| Logarithmic concentration axis | Ratios, multiplicative changes, and low positive concentrations | Zero is not displayable; absolute differences can be less intuitive |
| Log-transformed response values | Values expressed numerically as logarithms | Do not confuse transformed values with original-unit tick labels |
| Below-quantification records | Censoring relative to an assay threshold | Do not substitute zero or an invented measured value |
A common mistake is to say that a log axis “changes the concentrations” or that equal distances mean equal concentration differences. The correction is that the underlying observations remain the same, while the axis maps their positions according to logarithms; equal distances represent equal ratios. Another mistake is to interpret a straight decline on a log plot as proof of a particular elimination mechanism. Straightness is compatible with exponential change over the displayed range, but a visual pattern alone cannot establish the model or its cause.
The main alternative is to inspect the same individual profile on a linear concentration axis. Neither view replaces the other. Use the log view when low positive values and proportional changes matter; use the linear view when absolute changes and the scale of the peak matter. If the two views suggest different impressions, that is expected: they emphasize different aspects of the same measurements. A careful interpretation states which scale was used and what feature it makes easier to assess.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- On a base-10 log concentration axis, how do the vertical gaps for 5 to 10 mg/L and 20 to 40 mg/L compare?
- A profile falls from 64 to 32 mg/L over 3 hours. Under a constant exponential-decline assumption, calculate the implied rate \(k\).
- Why can an observation of zero concentration not be plotted on a logarithmic concentration axis?
- What does a straight segment on a log concentration-time plot support, and what does it not establish?
Solutions. (1) The gaps are equal because both intervals represent a doubling: \(\log_{10}(10/5)=\log_{10}(40/20)=\log_{10}(2)\approx0.3010\). (2) The ratio is \(32/64=0.5\), so \(k=-\ln(0.5)/3=\ln(2)/3\approx0.2310\ \text{h}^{-1}\). This is the rate implied by the stated assumption and these two values. (3) The logarithm of zero is undefined, so zero has no finite position on a log axis. (4) It supports that the displayed observations are compatible with an approximately exponential change over that range; it does not prove the model, uniquely identify a mechanism, or justify extrapolation.