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Transit compartment models, standard for pharmacokinetic/pharmacodynamic delays, impose artificial constraints on delay distribution mean and variance due to integer compartment requirements, necessitating complex mixed-integer optimization. This work reformulates these models as hypoexponentially generalized distributed delay differential equations, enabling direct estimation of real-valued mean and variance parameters without increasing model complexity. The hypoexponential approach yields comparable or superior fits to Erlang transit models across biological examples while significantly reducing computational burden by eliminating compartment number grid searches.
Biologically meaningful delay distributions can now be estimated directly from data, bypassing the artificial mean-variance coupling and computational complexity of traditional transit compartment models.
Transit compartment models are widely used in pharmacokinetics/pharmacodynamics to represent delayed biological processes such as absorption, maturation, cellular proliferation, and treatment-induced toxicity. This transit compartment formulation imposes an artificial relationship between the mean and variance of the delay distribution, as the number of transit compartment must be integer-valued number of transit compartments. Consequently, estimating the number of compartments from experimental data leads to mixed-integer optimization problems that artificially constrain model dynamics and bias parameter inference. However, these transit compartment models are mathematically equivalent to distributed delay differential equations with Erlang delay kernels. We leverage a hypoexponential generalization of these distributed delay differential equations that preserves the familiar ordinary differential equation structure while allowing the mean and variance of the delay distribution to be estimated directly as real-valued parameters. This approach does not increase the number of inferred parameters and can be immediately implemented as a straightforward modification of existing transit compartment models. We demonstrate the approach using models of viral eclipse phases, lymphocyte proliferation, and chemotherapy-induced neutropenia. Across these examples, the hypoexponential formulation provides comparable or improved fits relative to Erlang transit models and substantially reduces computational cost by avoiding grid search over compartment numbers. This framework provides a practical approach to estimate biologically meaningful delay distributions in quantitative systems pharmacology (QSP) models.