Choosing a Scale for the Calculation
A pharmacokinetic state can be represented as an amount, a concentration, or a transformed version of either. The useful representation depends on the calculation. Concentration is often the reported quantity; amount may be the state naturally associated with compartment transfer; and cumulative exposure is an integral over time, not an instantaneous concentration. A transformation can simplify one calculation while making another less transparent.
Throughout, interpret the stochastic differential equations below in the Itô sense. For a scalar state \(X_t\) satisfying \(dX_t=b(X_t,t)\,dt+s(X_t,t)\,dW_t\), a twice continuously differentiable transformation \(Z_t=g(X_t)\) satisfies Itô's formula, up to the time the state leaves the domain of \(g\):
The term involving \(g''\) is essential: choosing a new scale changes the drift as well as the diffusion. For a vector of compartment states, derivatives and cross-diffusion terms must also be accounted for; the same choice criteria apply, but a scalar calculation cannot replace the full multivariate formula.
Start With the Target Quantity and State Domain
Before transforming, identify the quantity the calculation needs. If concentration is merely amount divided by a fixed volume, a linear rescaling is usually the clearest choice. If the calculation requires a second moment, a power may expose a useful equation. If the target is cumulative exposure, adding an integral state preserves its definition directly. A logarithm is useful for positive states when relative changes or multiplicative variability are central, but the logarithm itself is not concentration or AUC.
The mathematical domain matters as much as the target. The logarithm requires a strictly positive state. A square root requires a nonnegative state and has an unbounded derivative at zero. Even when a formula is valid before a boundary is reached, it may not describe what happens at that boundary. For a general transformed SDE, assume the coefficients give a solution up to exit from the relevant domain and apply Itô's formula on that interval. Stronger claims, such as that the transformation is valid for all times, require that the process remain in the domain.
Worked Example: Converting Amount to Concentration
Suppose a one-compartment amount \(A_t\), in mg, follows \(dA_t=(R-kA_t)\,dt+\sigma A_t\,dW_t\), with constant infusion \(R=12\) mg/h, elimination rate \(k=0.25\) h\(^{-1}\), and \(\sigma=0.4\) h\(^{-1/2}\). Let the distribution volume be the fixed value \(V=20\) L. Find the concentration SDE.
Set \(C_t=A_t/V\), so \(g(A)=A/V\), \(g'(A)=1/V\), and \(g''(A)=0\). Itô's formula therefore gives \(dC_t=dA_t/V\). Substituting \(A_t=VC_t\) yields \(dC_t=(R/V-kC_t)\,dt+\sigma C_t\,dW_t\).
Here \(R/V=12/20=0.6\) mg L\(^{-1}\)h\(^{-1}\). At a current concentration of \(5\) mg/L, the drift is \(0.6-(0.25)(5)=-0.65\) mg L\(^{-1}\)h\(^{-1}\), and the diffusion coefficient is \((0.4)(5)=2\) mg L\(^{-1}\)h\(^{-1/2}\). The linear scaling introduces no Itô correction. This conclusion depends on \(V\) being fixed; if volume varies stochastically, dividing by it is a nonlinear transformation of a multivariate state and requires the corresponding additional terms.
When a Logarithmic State Helps
For \(C_t>0\), the transformation \(Z_t=\log C_t\) places multiplicative fluctuations on an additive scale. This can make relative changes easier to interpret and can simplify simulation or analytic work. It does not remove the need to check positivity, and the curvature of the logarithm contributes a negative drift correction when the diffusion is nonzero.
Worked Example: Logging a Positive Concentration Process
Suppose \(dC_t=-kC_t\,dt+\sigma C_t\,dW_t\), with \(C_0=10\) mg/L, \(k=0.2\) h\(^{-1}\), and \(\sigma=0.3\) h\(^{-1/2}\). For \(C_t>0\), let \(Z_t=\log C_t\). Since \(g'(C)=1/C\) and \(g''(C)=-1/C^2\), Itô's formula gives
The correction is \(-\sigma^2/2=-0.045\) h\(^{-1}\), so the log drift is not simply the original elimination drift \(-0.2\). Integrating gives \(Z_t=\log(10)-0.245t+0.3W_t\), and exponentiating gives \(C_t=10\exp(-0.245t+0.3W_t)\), which is positive at every finite time. This explicit solution verifies that the logarithm remains defined for this model.
For comparison, the mean concentration is \(E[C_t]=10e^{-0.2t}\), not \(10e^{-0.245t}\). The latter is the deterministic part of the log state. The Brownian exponential contributes the factor \(e^{\sigma^2t/2}\) to the mean of \(C_t\), offsetting the Itô correction in the exponent.
Power Scales and Cumulative Exposure
A power transformation can be chosen when the quantity of interest involves a power of concentration. For \(g(C)=C^p\), where \(p\) is such that the function is twice differentiable on the state domain, the transformed drift includes both \(pC^{p-1}b(C,t)\) and \(\tfrac12p(p-1)C^{p-2}s^2(C,t)\). This may give a direct equation for a moment, but it need not simplify the entire model.
Worked Example: Squaring Concentration to Track a Second Moment
Use the geometric concentration model \(dC_t=-0.2C_t\,dt+0.3C_t\,dW_t\), with \(C_0=5\) mg/L, and set \(Q_t=C_t^2\). Here \(g'(C)=2C\) and \(g''(C)=2\). Thus
The initial value is \(Q_0=25\) (mg/L)\(^2\). Taking expectations in the explicit geometric solution, or solving this linear moment equation, gives \(E[Q_t]=25e^{-0.31t}\). At \(t=2\) h, this is \(25e^{-0.62}\approx13.45\) (mg/L)\(^2\). This is the second raw moment \(E[C_t^2]\), not the variance: the variance is \(E[C_t^2]-(E[C_t])^2\). Squaring is appropriate for this target, but it does not make \(Q_t\) an exposure measure.
Cumulative exposure has a different mathematical form. If the target is \(Y_T=\int_0^T C_s\,ds\), retain concentration and add the state equation \(dY_t=C_t\,dt\), \(Y_0=0\). This is an exact representation of the AUC over \([0,T]\), and it does not require taking a logarithm or power of concentration.
Worked Example: Expected AUC Under Multiplicative Noise
For the same positive geometric model with \(C_0=10\), \(k=0.2\) h\(^{-1}\), \(\sigma=0.3\) h\(^{-1/2}\), define \(Y_T=\int_0^T C_s\,ds\). The explicit solution gives \(C_s=10\exp[(-0.2-0.045)s+0.3W_s]\). Since \(W_s\sim N(0,s)\), the normal moment-generating formula gives
Because \(C_s\geq0\), Tonelli's theorem permits exchanging expectation and time integration, including before finiteness is established. Hence \(E[Y_T]=\int_0^T E[C_s]\,ds=10(1-e^{-0.2T})/0.2\). For \(T=5\) h, \(E[Y_5]=50(1-e^{-1})\approx31.61\) mg h/L. The exposure state is still random; this calculation gives its mean, not its whole distribution. In particular, \(\log C_t\) is not \(\log Y_t\), and neither is AUC.
Justifying and Checking a Transformation
The general scalar rule gives a practical justification: substitute the chosen function and its first two derivatives, check every coefficient, and state the interval on which the calculation is valid. For fixed-volume conversion, the rule also establishes exactly why no additional drift appears.
Decide whether the calculation needs amount, concentration, a power or moment, a relative-change scale, or an accumulated exposure.
Confirm that the state lies where the transformation is defined and smooth. Specify what happens at a boundary if the process can reach one.
Compute both derivatives and include the second-derivative drift term. For multiple states, include cross-diffusion terms.
Verify the transformed coefficients, units, initial condition, and whether the output is actually the target quantity.
A reliable check is to compare the transformed equation with a direct calculation in a case where an explicit solution is available. Other warning signs include a log applied to zero or negative simulated concentrations, an undefined square-root derivative at zero, and an AUC calculation that reports an instantaneous state instead of a time integral. At dosing times with jumps, apply the transformation to the state values on either side of each jump; a continuous-path Itô calculation alone does not specify the jump update.
Choosing Among Alternatives
| Choice | Useful when | Main check or limitation |
|---|---|---|
| Keep amount | Compartment amounts and transfers are the direct target | Convert to concentration only with an appropriate volume model |
| Scale to concentration | Concentration is the reporting or calculation scale and volume is fixed | A varying volume requires additional stochastic terms |
| Log concentration | Positive multiplicative variation or relative changes are central | Requires positivity and includes an Itô drift correction |
| Use a power | A power or moment is the target | May complicate coefficients and can have boundary issues |
| Augment with AUC | Cumulative exposure over a specified interval is the target | Produces an additional state; its distribution may still require work |
No choice is universally best. A log can yield a simple additive-noise equation but hide the arithmetic concentration mean. A power can directly describe a moment but may not simplify the state dynamics. Keeping the original state avoids unnecessary transformations, while augmenting for exposure preserves the definition at the cost of tracking one more variable. The selection is a modeling and calculation choice, not a claim that one state scale is inherently correct.
Common Mistakes and Limitations
- Dropping the Itô correction. Replacing \(\log C_t\)'s drift by \(b(C_t,t)/C_t\) omits \(\tfrac12g''s^2\). For geometric noise this misses the term \(-\sigma^2/2\).
- Logging a state that can be zero or negative. The formula for \(\log C_t\) is not valid at those values. A numerical simulation that crosses zero is a warning about the model, discretization, or domain, not permission to continue taking logarithms.
- Confusing a transformed state with the target. \(C_t^2\) is not AUC, and \(\log C_t\) is not cumulative exposure. Define exposure through its time integral.
- Assuming fixed-volume scaling still applies to a random volume. If both amount and volume vary, \(A_t/V_t\) requires a transformation of both states and their joint diffusion structure.
These transformations do not repair a misspecified PK model or guarantee analytic tractability. Boundary behavior, dose jumps, random volume, and parameter uncertainty can require additional modeling decisions. A useful transformation is one whose domain and interpretation fit the calculation and whose resulting equation can be checked.
Check Your Understanding
Try each question, then compare with the brief solutions below.
- If \(C_t=A_t/V\) for constant \(V>0\), what is the Itô correction for this transformation, and why?
- For \(dC_t=-0.1C_t\,dt+0.2C_t\,dW_t\), what is the drift of \(Z_t=\log C_t\)?
- For \(dC_t=-kC_t\,dt+\sigma C_t\,dW_t\), how should a cumulative AUC state be defined?
- Why is \(\log C_t\) not the same quantity as \(\log Y_t\), where \(Y_t=\int_0^t C_s\,ds\)?
- For a square-root transformation, what domain issue should be checked before applying Itô's formula at zero?
Solutions. (1) There is no correction because \(g''(A)=0\) for \(g(A)=A/V\). (2) The drift is \(-0.1-\tfrac12(0.2)^2=-0.12\). (3) Set \(Y_0=0\) and \(dY_t=C_t\,dt\), so \(Y_T=\int_0^T C_s\,ds\). (4) They are functions of different quantities: the first is the logarithm of instantaneous concentration, while the second is the logarithm of accumulated concentration over time, when \(Y_t>0\). (5) The derivative of \(g(C)=\sqrt C\) is unbounded at zero, so smoothness there fails; boundary behavior must be handled rather than applying the interior formula at zero without justification.