Why Phase Diagnosis Is a Sampling Problem
After an infusion, a measured cell count may rise, reach a maximum, decline, and eventually remain detectable at a low level. Those words describe features of a time course; they do not by themselves identify the biological mechanism producing it. With sparse measurements, the practical question is narrower: which phases are supported by the observed counts, and which remain unresolved between sampling times?
The earlier material on cell persistence through survival and homeostatic support provides mechanisms that can affect late counts. Here the focus is on diagnosing patterns in the measurements, not selecting among those mechanisms. In particular, a blood count is a measurement in a specified sample and compartment; its trajectory alone does not establish the total number of cells in the body or prove that a change was caused by cell proliferation or loss.
These are operational descriptions. Two measurements can establish the direction of net change between their sampling times, but they cannot show whether the count changed monotonically between them. A sampled maximum establishes a maximum among the observed values, not the exact time or height of a continuous-time peak.
Interval Log-Slopes
Raw differences show absolute change, but the same difference can have very different significance at different count levels. A useful diagnostic summary is the interval log-slope, also called the average exponential rate over that interval. It describes fractional rather than absolute change per unit time.
This formula follows from a local exponential representation, \(N(t_{i+1})=N(t_i)\exp(r_i\Delta t)\), where \(\Delta t=t_{i+1}-t_i\). Dividing by \(N(t_i)\), taking natural logarithms, and dividing by \(\Delta t\) gives the stated expression. More generally, even when the trajectory is not exponential, the expression remains a concise description of the net fractional change between the two observations.
Under a constant-rate exponential model, the sign of \(r\) gives the direction of change throughout the interval. Without that assumption, the sign only describes the endpoints. This distinction matters most when a peak or trough may have occurred between samples.
Worked Example: Expansion Followed by Contraction
Suppose a hypothetical assay reports engineered-cell counts of 120 cells/µL on day 0, 480 cells/µL on day 7, and 360 cells/µL on day 14. Calculate the interval log-slopes and describe what the observations support.
From day 0 to day 7, $$ r_1=\frac{\ln(480/120)}{7} =\frac{\ln(4)}{7} =\frac{1.38629}{7} =0.1980\ {\rm day}^{-1}. $$
The positive rate indicates net expansion between those measurements. From day 7 to day 14, $$ r_2=\frac{\ln(360/480)}{7} =\frac{\ln(0.75)}{7} =\frac{-0.287682}{7} =-0.04110\ {\rm day}^{-1}. $$
The negative rate indicates net contraction over the second interval. Day 7 is a sampled maximum because 480 cells/µL exceeds both neighboring observations. The data therefore support expansion before the day-7 sample and contraction afterward at the level of interval endpoint changes. They do not determine the exact continuous-time peak, nor prove that the count rose or fell monotonically within either interval.
What Sparse Sampling Can and Cannot Establish
A peak is more convincingly localized when measurements show an increase leading up to a high value and a decrease afterward. Even then, the observation times only bracket the peak: if the highest sample is on day 7, the continuous-time maximum could have occurred before, on, or after that day, depending on the unobserved trajectory. When the samples are widely spaced, the bracket can be broad.
If all observed counts decline, the sampled values support contraction across the observed period, but an unobserved temporary increase cannot be ruled out purely from those endpoints. If the first and last counts are similar, the net interval log-slope is near zero; that does not establish a flat trajectory. The count could have risen and then fallen, or fallen and then recovered.
Worked Example: Equal Endpoints Do Not Rule Out a Peak
In a hypothetical sparse series, the measured count is 100 cells/µL on day 0 and 100 cells/µL on day 28, with no measurements between. The endpoint calculation is $$ r=\frac{\ln(100/100)}{28}=0\ {\rm day}^{-1}. $$
This supports zero net endpoint change over 28 days. It does not demonstrate persistence at a constant level. For example, a possible unobserved trajectory could rise to 400 cells/µL on day 14 and return to 100 cells/µL by day 28. A piecewise exponential trajectory with those values would have rates $$ r_{0,14}=\frac{\ln(400/100)}{14}=0.09902\ {\rm day}^{-1}, \qquad r_{14,28}=\frac{\ln(100/400)}{14}=-0.09902\ {\rm day}^{-1}. $$
That trajectory and a trajectory remaining at 100 cells/µL throughout have identical observed endpoints. Since no intermediate values distinguish them, the phase pattern is not identifiable from these two samples. More frequent measurements around the plausible expansion-to-contraction transition would be needed to detect such a peak.
A late phase of persistence is often described when cells remain detectable and the observed changes are small relative to the time scale and measurement precision. There is no universal numerical slope threshold that separates persistence from slow contraction: the relevant tolerance depends on assay performance, sampling schedule, and the scientific decision. Report the observed rate and uncertainty where available rather than presenting a qualitative label as a demonstrated mechanism.
Worked Example: A Slowly Declining Late Count
Consider hypothetical late counts of 1,000 cells/µL on day 0, 920 cells/µL on day 60, and 850 cells/µL on day 120. The first interval rate is $$ r_1=\frac{\ln(920/1000)}{60} =\frac{\ln(0.92)}{60} =\frac{-0.0833816}{60} =-0.001390\ {\rm day}^{-1}. $$
For the second interval, $$ r_2=\frac{\ln(850/920)}{60} =\frac{\ln(0.923913)}{60} \approx\frac{-0.07913}{60} =-0.001319\ {\rm day}^{-1}. $$
Across the full 120 days, the average log-slope is $$ r_{0,120}=\frac{\ln(850/1000)}{120} =\frac{\ln(0.85)}{120} =\frac{-0.162519}{120} =-0.001354\ {\rm day}^{-1}. $$
The counts remain detectable and decline slowly in both sampled intervals. This is consistent with persistence accompanied by slow contraction, not with a strictly constant count. If, for illustration, each count has about 10% relative measurement variability, changes of roughly 8% between adjacent observations may be difficult to distinguish from that variability without an explicit error model or replicate information. The three counts alone do not establish whether the true late slope differs meaningfully from zero.
If the full-period rate were assumed to continue as a constant exponential rate, the corresponding half-life would be $$ t_{1/2}=\frac{\ln(2)}{|r_{0,120}|} =\frac{0.693147}{0.001354} \approx512\ {\rm days}. $$ This is a conditional calculation, not evidence that the same rate will persist for 512 days. A later change in the rate would invalidate that extrapolation.
A Practical Diagnostic Sequence
Phase labels are most useful when they follow a transparent sequence of checks. The following workflow separates what the measurements show from what a model or biological explanation would add.
Confirm sampling times, count units, assay consistency, and whether values are above the quantification limit. A zero or below-quantification result cannot be inserted into a logarithm as if it were an exact positive count.
For each pair of positive adjacent counts, calculate the raw difference and the interval log-slope. Record the duration of each interval so that rates are compared on a common time basis.
Use increasing and decreasing intervals to identify sampled evidence for expansion and contraction. Call an observation a sampled maximum only when it exceeds its neighboring sampled values.
Ask whether sampling could have missed a turning point and whether the observed changes exceed expected assay and biological variability. If they do not, describe the phase as unresolved rather than forcing a label.
State what the measured compartment supports, then consider whether additional measurements or a mechanistic model are needed to explain the pattern. A count trajectory by itself does not identify why the count changed.
Diagnostics, Alternatives, and Limitations
The log-slope is not a substitute for inspecting the counts against time. Plot the observations with their units, show the actual sampling times, and retain the observed values rather than displaying only a fitted curve. A smooth curve can visually imply a well-localized peak even when only two widely separated observations constrain it. If replicate measurements or assay precision estimates are available, use them to judge whether small changes are distinguishable from measurement variation.
Raw differences and log-slopes answer different questions. A raw difference reports the absolute change, such as an increase of 360 cells/µL; the log-slope reports average fractional change per day and is more comparable across different count levels. Neither alone establishes the within-interval trajectory. A mechanistic or flexible time-course model can interpolate between observations and quantify uncertainty, but with sparse data its conclusions depend strongly on model structure and assumptions. A fitted peak is not automatically identified merely because software returns a peak time.
Important failure modes include comparing slopes computed over unequal intervals without accounting for duration; treating a single noisy high count as definitive evidence of a peak; interpreting a near-zero endpoint rate as proof of no dynamics; and extrapolating a late rate far beyond the observation period. Do not take logarithms of zero or replace a below-quantification count with an arbitrary small positive number without a justified measurement model. Such substitutions can create large, misleading rates.
Finally, a measured blood count is not necessarily a direct measure of the whole-body cell population. Movement into or out of the sampled compartment, sample handling, and assay characteristics can affect the observed series. The present diagnostic classification describes the data; resolving those explanations requires evidence beyond the sparse count pattern.
Check Your Understanding
Answer each question from the definitions and examples above. Solutions follow the questions.
- A hypothetical count rises from 200 cells/µL on day 0 to 600 cells/µL on day 10. Calculate the interval log-slope and state what it supports.
- Counts are measured only on day 0 and day 30, and are identical. What can be concluded, and what cannot?
- Three measurements are 80, 150, and 110 cells/µL on days 0, 5, and 10. Is day 5 a sampled maximum? Does that establish the exact continuous-time peak?
- Why should a below-quantification result not be replaced by an arbitrary small number before calculating a log-slope?
- Two late counts show a small decline. Name one reason this may not establish slow biological contraction.
Solutions. (1) \(r=\ln(600/200)/10=\ln(3)/10=0.1099\ {\rm day}^{-1}\). The positive rate supports net expansion between the measurements, not necessarily monotonic increase at every intervening time. (2) The endpoint log-slope is zero, so there is no net endpoint change. A peak, trough, or other intervening dynamics remain possible. (3) Yes: 150 exceeds both neighboring observations, so it is a sampled maximum. The continuous-time peak is not located exactly by these three measurements. (4) An arbitrary substitution changes the log ratio and can produce an artificial rate; a censored measurement requires an appropriate treatment rather than being treated as an exact count. (5) The decline may be comparable to assay or biological variability, so it may not be distinguishable from measurement variation without precision information or replication.