Enforcing Physiological Feasibility Constraints

Why Feasibility Must Be Built Into Patient Generation

A virtual patient is a vector of mechanistic parameters, not a collection of unrelated draws. A parameter set can satisfy every individual range and still describe an impossible patient: a free and bound concentration may not sum to the stated total, for example, or a clearance may exceed a physiological limit determined by another patient trait. Enforcing linked constraints means constructing parameter vectors whose components are jointly feasible.

The key decision is whether to generate values that satisfy the constraints by construction, rather than generate unrestricted values and repair them afterward. Direct construction avoids invalid vectors, but it can also change marginal distributions and dependence. The generated population must therefore be checked not only for constraint violations but also for unintended changes in the intended patient variability.

Constraints may be equalities, such as conservation of a total; inequalities, such as positivity; or linked bounds, where the allowable range for one parameter depends on another. The intended parameter ranges, dependence structure, and physiological meanings established in the earlier material on defining admissible parameter ranges and encoding correlated patient traits remain important. A feasibility construction does not by itself guarantee that the resulting population matches all those targets.

Constructing Feasible Vectors Directly

Suppose a patient has two nonnegative components \(a\) and \(b\) whose sum is a total \(T\). Drawing all three independently and then hoping that \(a+b=T\) is not a viable method: exact equality has probability zero under continuous independent draws. Instead, generate the total and a proportion, then derive the components:

The equality \(a+b=T\) now holds identically. If the components have additional bounds, the proportion must be restricted accordingly. For example, if \(a_{\min}\leq a\leq a_{\max}\), then, for \(T>0\),

The allowed interval for \(f\) can itself depend on \(T\). It must also intersect the physical range \([0,1]\); if it does not, that value of \(T\) cannot yield a feasible patient.

A ratio can be handled similarly. To impose \(r_{\min}\leq a/b\leq r_{\max}\) for positive \(b\), generate \(b\) and a ratio \(r\) in the specified interval, then set \(a=rb\). If \(a\) has its own bounds, the allowed range of \(r\) must be further restricted to values compatible with those bounds and the generated \(b\). The order of generation therefore matters: bounds on a derived quantity may restrict the possible values of the quantities generated first.

Linked Bounds and Conditional Construction

A parameter-dependent bound is often best represented as a conditional construction. Suppose a parameter \(k\) must satisfy \(0\leq k\leq b(z)\), where \(z\) is another patient trait and \(b(z)\) is a positive upper bound. Generate \(z\), then generate a unit-interval value \(u\), and set \(k=b(z)u\). For every \(u\in[0,1]\), this guarantees \(0\leq k\leq b(z)\).

This construction guarantees feasibility, but it also specifies a distributional choice. If \(u\) is uniform on \([0,1]\), then conditional on \(z\), \(k\) is uniform on \([0,b(z)]\). Its conditional mean is \(b(z)/2\), and its conditional variance is \(b(z)^2/12\). Therefore, even if \(u\) is generated independently of \(z\), \(k\) will generally depend on \(z\) through the bound. This is a mathematical consequence of the construction, not evidence that the model's physiological parameters have that particular conditional distribution.

More generally, if a conditional distribution for \(k\) is scientifically intended, use a transformation that has that conditional distribution while respecting the bound. The bound defines where \(k\) may occur; it does not determine how probability should be distributed within that range. The distinction is important when the intended patient population has specified marginal distributions or dependence, as discussed in the earlier material on copulas for non-Gaussian patient dependence.

Derivation: Why a Bounded Transformation Changes the Density

The effect of a linked bound can be made explicit. Fix \(z\) and suppose \(u\) is uniform on \([0,1]\), with \(k=b(z)u\) and \(b(z)>0\). For a value \(k\) between zero and \(b(z)\), the event \(K\leq k\) is equivalent to \(U\leq k/b(z)\). Since \(U\) is uniform,

Differentiating this conditional cumulative distribution with respect to \(k\) gives the conditional density:

Thus the transformation is not distribution-neutral: it induces a uniform conditional distribution. If that distribution is not appropriate, replace the uniform \(u\) with a variable whose transformation gives the intended conditional distribution. For example, a desired conditional cumulative distribution \(F_{K\mid Z}(k\mid z)\) can be represented through its inverse, \(k=F^{-1}_{K\mid Z}(u\mid z)\), provided the distribution is defined on the feasible interval.

Assumptions, Diagnostics, and Common Failure Modes

A valid construction depends first on the constraints being correctly specified. Units must be compatible in sums and ratios; a ratio of quantities with different units must be interpreted with its resulting units. Bounds must apply to the same parameter definitions used in the model. If a constraint is only an empirical tendency rather than a physiological impossibility, enforcing it as an absolute rule may improperly remove plausible patients.

After generating a population, assess both feasibility and population behavior. A useful audit includes:

• Evaluate each equality residual, such as \(a+b-T\), and confirm it is zero up to the expected numerical tolerance.
• Check every inequality, including parameter-dependent bounds, for every generated patient.
• Count empty feasible intervals encountered during generation; these indicate incompatible constraints or an incompatible upstream draw.
• Inspect marginal distributions and pairwise dependence after transformation, not only the distributions of the values sampled before transformation.
• Look for excess probability near boundaries. A construction can produce boundary pileups if its input distribution or transformation concentrates values there.
• Repeat checks using the same units, parameter definitions, and numerical precision used by the QSP model.

A common mistake is to sample all parameters independently, then clip any out-of-range value to the nearest allowed boundary. For example, replacing an invalid \(k\) with its maximum \(b(z)\) guarantees the inequality, but it creates an artificial concentration of patients exactly at the bound and can change the marginal distribution and dependence. The correction is to construct \(k\) from a distribution defined on its feasible range, and then diagnose the resulting population. Another common mistake is to enforce one constraint while overlooking another: deriving \(a\) to satisfy \(a+b=T\) does not ensure that \(a\) also lies within its own bounds.

Direct parameterization is not the only possible approach. Projection onto a feasible set can be useful when a valid vector is already available, but the result depends on the chosen distance and may alter distributions or place many patients on a boundary. Clipping is simpler but has the boundary problem described above. Generating unrestricted vectors and retaining only feasible ones can be inefficient when feasibility is rare and may change the represented distribution. These alternatives do not remove the need to state the intended distribution and verify the final population.