Why Sensitivity Matters
An optimal design is calculated using specified nominal model parameters. Those values may be uncertain, rounded, or revised after new information becomes available. Even when the change is small, it can move the sampling times that optimize a design criterion. The criterion value can also change, although it need not change at the same rate as the sampling times. Sensitivity analysis makes these two effects explicit.
The focus here is a local question: if a nominal parameter changes slightly, how does the optimum respond? This is different from averaging performance over a distribution of parameter values or selecting a design for a worst-case scenario. Those approaches answer different design questions. Local sensitivity is useful for identifying fragile schedules, checking numerical optimization, and deciding whether a small parameter revision merits a new optimization.
The utility may be an information-based criterion or another explicitly chosen design objective. For example, a design might maximize a function of a Fisher information matrix. The sensitivity formulas below apply to a smooth scalar utility; they do not require a particular information criterion. If the original goal is to minimize a smooth criterion, its negative can be treated as a utility to maximize.
Local Sensitivity of an Interior Optimum
Suppose \(d\) is an \(m\)-dimensional vector of design variables and \(\theta\) is a \(q\)-dimensional parameter vector. For the main result, assume \(U\) is twice continuously differentiable near the optimum; \(d^*(\theta)\) is an interior local maximum; the first-order conditions hold; and the Hessian with respect to the design variables is negative definite there. These conditions make the optimum locally strict and its Hessian invertible. Also assume the same local optimum branch remains relevant under the small perturbation being considered.
At an interior optimum, the gradient with respect to the design variables is zero. Write \(H\) for the Hessian of \(U\) with respect to \(d\), and \(G\) for the matrix of mixed derivatives, with entries \(G_{ij}=\partial^2 U/(\partial d_i\partial\theta_j)\), both evaluated at \((d^*,\theta)\). Differentiating the first-order conditions gives the local response of the design:
This is a local derivative, not a guarantee that the same formula describes a large parameter change. It gives a first-order approximation: for a small change \(\Delta\theta\), the design change is approximately \(-H^{-1}G\Delta\theta\). A parameter with a large mixed derivative can move the optimum, while a nearly flat criterion (a small-magnitude curvature in \(H\)) can make the optimum especially sensitive.
The optimized criterion value has a different first-order response. Define \(V(\theta)=U(d^*(\theta),\theta)\). Differentiating by the chain rule gives \(\nabla_\theta V=\nabla_\theta U+(\partial d^*/\partial\theta)^\mathsf{T}\nabla_d U\). At an interior optimum, \(\nabla_d U=0\), so the second term vanishes. Thus the first-order change in optimized utility is the direct parameter effect on the utility evaluated at the optimum. A design can therefore move appreciably even when its optimized value changes little.
Worked Example: Clearance and Volume in an Intravenous Design
Worked Example: Clearance and Volume in an Intravenous Design
Consider a hypothetical one-compartment intravenous bolus model with \(C(t)=(D/V)e^{-kt}\), where \(k=CL/V\). Suppose the design has one sampling time \(t>0\), the residual error is additive with known standard deviation \(\sigma\), and the target is log clearance while other parameters are held fixed. The single-observation information for log clearance is proportional to the squared concentration derivative divided by \(\sigma^2\). This specifies the criterion and its assumptions; it is not a general claim that this time optimizes every PK design.
Because \(\partial C/\partial\log CL=-Ckt\) when \(V\) is fixed, take the utility to be
For \(t>0\), maximizing \(U\) is equivalent to maximizing its logarithm. Differentiating that logarithm gives \(2/t-2k=0\), so the unique interior maximum occurs at \(t^*=1/k=V/CL\). The second derivative of the log utility is \(-2/t^2<0\), confirming a strict maximum.
Use hypothetical values \(D=100\) mg, \(V=20\) L, \(CL=4\) L/h, and \(\sigma=0.5\) mg/L. Then \(k=4/20=0.2\) h\(^{-1}\), and the optimal time is \(t^*=1/0.2=5.00\) h. At this optimum,
If clearance increases by 5% while volume is fixed, \(CL=4.2\) L/h and \(t^*=20/4.2=4.762\) h. This is a decrease of \(0.238\) h, or about 4.76%. The formula also shows why: the elasticity of optimal time to clearance is \(-1\). In this particular criterion, the optimized value does not depend on clearance, so its change is zero when only clearance changes.
If instead volume increases by 5% while clearance is fixed, \(V=21\) L and \(t^*=21/4=5.25\) h, a 5% increase. The optimized utility is proportional to \(V^{-2}\) with clearance fixed. Its new value is therefore \(13.53(20/21)^2\approx12.27\), about 9.30% lower. The sampling time moves later, while the optimized information decreases. Both conclusions follow from this stated model and criterion.
Justification and a Local Criterion-Loss Approximation
The sensitivity formula follows by differentiating the first-order condition. For a single parameter \(\theta_j\), the \(i\)th component of that condition is \(\partial U/\partial d_i=0\). Its derivative with respect to \(\theta_j\), along the optimum, is
Collecting these \(m\) equations gives \(H(\partial d^*/\partial\theta_j)+G_{\cdot j}=0\). Since \(H\) is invertible, multiplying by \(H^{-1}\) yields the stated formula. This argument also shows why the assumptions matter: if the Hessian is singular, this equation may not determine a unique derivative; if the optimum is on a boundary, the unconstrained first-order condition need not hold.
A second useful local result concerns using the old design after a parameter change. Let \(d_0=d^*(\theta_0)\), and suppose the new optimum is \(d_0+\Delta d\). A Taylor expansion in the design variables at the new parameter value gives, to second order, the utility lost by retaining \(d_0\) rather than moving to the new optimum:
Here \(H\) is evaluated at the original optimum, and the approximation is for small changes. Because \(H\) is negative definite, the right-hand side is nonnegative, as expected for a maximum. Substituting \(\Delta d\approx-H^{-1}G\Delta\theta\) estimates the performance cost of keeping the old schedule. This approximation concerns loss relative to the newly optimized design; it is not the total change in criterion value caused directly by changing parameters.
Worked Example: Absorption and Elimination Shift an Oral Sampling Target
Worked Example: Absorption and Elimination Shift an Oral Sampling Target
Consider the hypothetical one-compartment oral concentration model with first-order absorption and elimination:
For this example, define the sampling target as the time of maximum concentration, that is, maximize \(C(t)\) for \(t>0\). This is a response-peak objective, not a substitute for optimizing an information criterion when the scientific target is parameter precision. Assume \(k_a>k>0\). Setting the time derivative of concentration to zero gives \(k_a e^{-k_at}=k e^{-kt}\), and hence
With hypothetical values \(D=100\) mg, \(V=20\) L, \(CL=4\) L/h, and \(k_a=1.0\) h\(^{-1}\), \(k=0.2\) h\(^{-1}\). Thus \(t_{\max}=\log(5)/0.8=2.012\) h. Substitution into the concentration model gives
Now increase \(k_a\) by 10%, holding \(k\) fixed. The new peak time is \(\log(1.1/0.2)/(1.1-0.2)=1.894\) h. Substitution gives \(C(t_{\max})\approx(5)(1.1/0.9)(e^{-0.2(1.894)}-e^{-1.1(1.894)})=3.423\) mg/L. The peak shifts earlier by \(0.118\) h, and the peak concentration increases by about \(0.080\) mg/L.
Instead increase clearance by 10% with \(V\) and \(k_a\) fixed. Then \(k=4.4/20=0.22\) h\(^{-1}\), so \(t_{\max}=\log(1/0.22)/(1.0-0.22)=1.941\) h. The peak concentration is approximately \((5)(1/0.78)(e^{-0.22(1.941)}-e^{-1.0(1.941)})=3.263\) mg/L. Relative to baseline, the peak time is earlier by \(0.071\) h and the peak concentration is lower by about \(0.080\) mg/L. These finite perturbations show why the direction and size of a time shift must be calculated for the particular model and objective.
Worked Example: Coupled Sampling Times and Criterion Loss
Worked Example: Coupled Sampling Times and Criterion Loss
To illustrate coupled design variables, consider a hypothetical smooth utility measured in criterion points:
where \(d=(d_1,d_2)^\mathsf{T}\) contains sampling times in hours, \(m(\delta)=(2+0.4\delta,\ 6-0.2\delta)^\mathsf{T}\) hours, and \(\delta\) is a small dimensionless perturbation of a nominal parameter combination. The matrix \(Q\) is positive definite, so this utility has a unique maximum at \(d^*=m(\delta)\), with design Hessian \(H=-Q\). Therefore \(d^*(0)=(2,6)^\mathsf{T}\) h and \(\partial d^*/\partial\delta=(0.4,-0.2)^\mathsf{T}\) h per unit \(\delta\).
For a perturbation \(\delta=0.1\), the new optimum is \((2.04,5.98)^\mathsf{T}\) h. At the old design, the displacement from the new optimum is \(\Delta d=(0.04,-0.02)^\mathsf{T}\) h. Multiplying gives \(Q\Delta d=(0.06,0)^\mathsf{T}\) h, so the utility loss from retaining the old design is
The optimized utility remains 100 points in this constructed example, while the old schedule has utility 99.9988 points under the perturbed criterion. The example demonstrates that design movement and criterion loss are distinct quantities. Here the calculation is exact because the utility is quadratic; for a general criterion, the loss formula is a local approximation.
Diagnostics, Alternatives, and Limitations
A computed derivative should be checked rather than treated as a complete robustness assessment. A practical diagnostic is to perturb each nominal parameter in both directions, re-optimize, and compare the observed change with the derivative prediction. For a scalar parameter, use several decreasing step sizes \(h\) and compare the central finite difference \([d^*(\theta+h)-d^*(\theta-h)]/(2h)\) with the implicit derivative. Agreement as \(h\) shrinks supports the local calculation; disagreement can indicate numerical error, a step that is too large, or a change in the identity of the optimum.
Also inspect the curvature and the constraints. A nearly singular Hessian means a small change may produce a large time shift, and numerical inversion can be unstable. If a predicted time leaves its allowed range, the interior formula is no longer an adequate description of the constrained optimum. Multiple local optima can switch rank after a small perturbation, producing a discontinuous change in the selected design. In those cases, compare the feasible candidate optima directly and re-optimize under the perturbed parameters.
Local derivatives are efficient for ranking parameter effects and estimating responses to genuinely small changes. Finite-difference re-optimization is more direct and can reveal nonlinear behavior, but depends on optimizer reliability and the chosen perturbation size. Scenario-based or minimax approaches address performance across specified alternatives rather than the derivative at one nominal point. None removes the need to state the objective, feasible sampling times, and model assumptions. In particular, local sensitivity alone does not account for a probability distribution over parameters.
Check Your Understanding
Answer each question using the stated assumptions. Brief solutions follow the questions.
- An interior one-dimensional optimum satisfies \(U_d(d^*,\theta)=0\), with \(U_{dd}=-4\) and \(U_{d\theta}=2\). Find \(\partial d^*/\partial\theta\).
- A sampling time is \(t^*=V/CL\). If \(V\) is fixed and clearance increases by 2%, approximately what is the percentage change in \(t^*\)?
- For \(U(d,\theta)=100-(d-\theta)^2\), evaluate the optimized utility and the utility at the old design \(d=2\) after \(\theta\) changes from 2 to 2.1.
- Name two reasons why an implicit sensitivity calculation may fail to describe the selected design after a parameter perturbation.
Solutions. 1. The formula is \(\partial d^*/\partial\theta=-U_{dd}^{-1}U_{d\theta}\). Substituting gives \(-(-4)^{-1}(2)=0.5\) design units per parameter unit. 2. Since \(t^*=V/CL\), its elasticity to clearance is \(-1\); a 2% increase in clearance produces approximately a 2% decrease in time. 3. The new optimum is \(d^*=2.1\), with utility \(100\). At the old design, \(U(2,2.1)=100-(2-2.1)^2=99.99\), so retaining the old design loses 0.01 utility units. 4. Examples include a boundary becoming active, a nearly singular Hessian, multiple optima switching rank, or a perturbation large enough that the local linear approximation is inaccurate.