Which part of the curve carries which parameter
Five parts of the curve
![]()
Single dose with \(C_{\max} = 100 \times EC_{50}\). Dashed line: limit of quantification at 5% of baseline.
Two rates decide the shape
\[
\frac{d \ln B}{dt} = k_{\rm out} \left( \frac{B_0}{B} - 1 - E(C) \right)
\]
- \(B\) is pulled toward \(B_{\rm qss} = B_0 / (1 + E(C))\) at rate \(k_{\rm out}(1 + E(C))\).
- The drug falls at rate \(k_e\).
| Start, \(E = E_{\max}\) |
3.0 per day |
45 times faster |
follows the drug |
| End, \(E = 0\) |
0.02 per day |
0.3 times as fast |
lags, returns at its own pace |
The curve starts fast and ends slow, so its parts separate.
The fall carries \(k_{\rm out}(1 + E_{\max})\)
For \(t \ll 1/k_e\) the concentration is still near \(C_{\max}\), so the equation is linear with constant coefficient:
\[
B(t) = B_{\rm ss} + (B_0 - B_{\rm ss})\, e^{-k_f t},
\qquad k_f = k_{\rm out}\big(1 + E(C_{\max})\big)
\]
\[
t_{1/2,\rm fall} \approx \frac{\ln 2}{k_{\rm out}(1 + E_{\max})}
\quad \text{for } C_{\max} \gg EC_{50}
\]
- Here: 0.23 days, five and a half hours. Over by day 1.5.
- \(EC_{50}\) enters only through \(E(C_{\max}) = E_{\max}/(1 + EC_{50}/C_{\max})\), a 1% correction at \(C_{\max} = 100 \times EC_{50}\).
- This is the only place \(E_{\max}\) appears without \(EC_{50}\) beside it, and it is over before a weekly sample arrives.
- Published: the large-dose initial slope \(-k_{\rm out} B_0 E_{\max}\) (Sharma and Jusko 1996), and the fall curve as a lower bound at every dose that the response converges to (Peletier, Gabrielsson and Haag 2005).
The floor carries \(E_{\max}\), below the limit
After the fall, \(B\) tracks the quasi-steady state:
\[
B(t) \approx \frac{B_0}{1 + E\big(C(t)\big)} \approx \frac{B_0}{1 + E_{\max}}
\quad \text{while } C \gg EC_{50}
\]
- Here: 1.3 cells/µL. The large-dose nadir of Sharma and Jusko 1996.
- Above a limit at fraction \(f\) of baseline only if \(E_{\max} < 1/f - 1\): below 19 for a 5% limit, below 39 for 2.5%.
- The two published models that estimate \(E_{\max}\) put it near 155, so the floor is censored.
The climb back carries \(E_{\max}/EC_{50}\)
The count crosses \(f B_0\) when \(E(C) = (1 - f)/f\). With \(C = C_{\max} e^{-k_e t}\):
\[
t_f = \underbrace{\frac{1}{k_e} \ln \frac{f\, S_0\, C_{\max}}{1 - f}}_{S_0 \text{ only}}
+ \underbrace{\frac{1}{k_e} \ln\left(1 - \frac{1 - f}{f E_{\max}}\right)}_{E_{\max} \text{ alone}}
\]
- The second term is all the separate information on \(E_{\max}\) that the recovery holds.
- At \(f = 0.05\): \(-2.0\) days for \(E_{\max} = 150\), \(-7.2\) days for 50, \(-0.6\) days for 450.
- A few days on a crossing near day 100, read through 30% error from monthly samples.
Across doses: Levy’s dose-duration line
![]()
- Slope \(1/k_e\): each ten-fold dose adds \(\ln 10/k_e = 34.5\) days, 3.3 PK half-lives, whatever \(E_{\max}\) and \(EC_{50}\) are.
- Three sets with the same ratio fall on one line. Levy (1966, as cited in secondary sources) gave this line for direct effects; the quasi-steady state makes it hold here.
- Peletier et al. (2005) prove the general form: at large doses the recovery is a fixed shape travelling by \(\ln(C_{\max}/EC_{50})/k_e\).
The tail carries \(k_{\rm out}\); low doses carry \(S_0\)
Tail. Once \(S_0 C \ll 1\), the model is \(dB/dt = k_{\rm out}(B_0 - B)\):
\[
B_0 - B(t) \propto e^{-k_{\rm out} t} \qquad \text{(needs } k_{\rm out} < k_e\text{)}
\]
Doses that stay below \(EC_{50}\). \(E(C) \approx S_0 C\) for the whole curve, so the model holds \(E_{\max}\) and \(EC_{50}\) only as their ratio. For shallow depletion:
\[
\varepsilon_{\max} = 1 - \frac{B_{\min}}{B_0} = S_0 C_{\max} \left(\frac{k_{\rm out}}{k_e}\right)^{k_e/(k_e - k_{\rm out})}
\]
- A low-dose cohort informs the slope, not its two factors.
The area between curves follows \(S_0\) too
\[
\text{area} = \int_0^\infty (B_0 - B)\,dt \;\sim\; \frac{B_0}{k_e} \frac{E_{\max}}{1 + E_{\max}} \ln\left(1 + \frac{C_{\max}}{EC_{50}}\right)
\quad \text{(large dose)}
\]
| 1 |
50 |
0.33 |
21407 |
4077 |
| 1 |
150 |
1.00 |
21715 |
2066 |
| 1 |
450 |
3.00 |
21818 |
861 |
| 100 |
50 |
0.33 |
34999 |
16786 |
| 100 |
150 |
1.00 |
35481 |
13754 |
| 100 |
450 |
3.00 |
35631 |
10585 |
- Sharma and Jusko (1996) and Peletier et al. (2005): read \(E_{\max}\) from the slope of area against log dose. At \(E_{\max} = 150\) the slope factor is 0.993, so the slope carries almost nothing.
- At a given dose the area is the same for all three sets: it follows \(S_0\). The formula’s level is low because its omitted constant is large here.
What is identifiable and what is not
| Baseline |
\(B_0\) |
always |
✅ |
| Fall |
\(k_{\rm out}(1 + E_{\max})\) |
samples in the first one to two days |
❌ first sample at a week |
| Floor |
\(E_{\max}\) |
\(E_{\max} < 1/f - 1\) |
❌ below the limit |
| Climb back |
\(S_0\), plus a few-day \(E_{\max}\) term |
samples through recovery, PK known |
✅ |
| Tail |
\(k_{\rm out}\) |
samples until repopulation |
✅ |
| Low-dose cohort |
\(S_0\), \(k_{\rm out}\) |
\(C_{\max} \ll EC_{50}\) |
✅ but adds no \(E_{\max}\) |
- ✅ Always: \(B_0\), \(k_{\rm out}\), \(S_0 = E_{\max}/EC_{50}\)
- ⚠️ \(E_{\max}\): only from the fall, or a floor above the limit
- ❌ \(EC_{50}\) alone: from no part of the curve; identified exactly when \(E_{\max}\) is
The model is structurally identifiable. The loss is practical, caused by the quantification limit and the sampling schedule.
The textbook recipe needs the two readings a study loses
Sharma and Jusko (1998), Table 1, recover the parameters graphically:
- Baseline → \(B_0\)
- Large-dose nadir → \(E_{\max} = B_0/B_{\min} - 1\)
- Initial slope with \(E_{\max}\) → \(k_{\rm in}\), then \(k_{\rm out} = k_{\rm in}/B_0\)
- Concentration at the time of the nadir → \(EC_{50}\)
Design: two or more bolus doses, one high enough for full effect, and the baseline.
- ❌ Step 2: the nadir is 1.3 cells/µL, below the limit.
- ❌ Step 3: the initial slope is over in a day, before a weekly sample.
- What is left is the tail, for \(k_{\rm out}\), and the climb back, for \(E_{\max}/EC_{50}\).
Same ratio, same curve above the limit
![]()
Three-fold differences in \(E_{\max}\) and \(EC_{50}\) at fixed ratio. Above the limit the curves differ in the first two days and nowhere else.
