Three-compartment hemodialysis modeling with logistic blood-tissue exchange and slow-pool dynamics

The classical two--compartment (2C) model is widely used to describe solute
kinetics during hemodialysis, yet it often misses two key features of real data: the
early concentration dip in the first dialysis hour and the long equilibration tail,
especially in patients with diabetic kidney disease. Building on our previous two--
compartment (2C) Monte Carlo work, we propose an extended three--compartment
model with logistic exchange (3C--Logistic) that adds a slow tissue pool representing
a slowly equilibrating tissue-associated kinetic compartment and allows the blood--tissue
exchange \(K_{cf}(t)\) to vary over time according to a logistic law.

The motivation is physiological but should be interpreted cautiously: the logistic
time dependence is used as an apparent kinetic representation of delayed tissue-
to-blood redistribution, for which progressive microvascular recruitment is one
plausible, but not directly measured, mechanism. Using Monte Carlo parameter screening followed by local derivative-free refinement,
we show that 3C--Logistic reduces systematic misfit in both the early and late parts
of the session and yields a nine-parameter model-derived kinetic description, from
which logistic-exchange and slow-pool descriptors
\((K_{c1}, K_{c2}, s, \tau, K_{fs}, V_s)\) may help characterize diabetic-like and
non-diabetic-like redistribution dynamics.
We applied the framework to intradialytic kinetics of both urea and phosphate.
For urea, the model reconciled the early trough and the long tail with
physiologically plausible parameter ranges. For phosphate, fits generally indicated a stronger slow-pool contribution,
consistent with slower equilibration from tissue stores. These parameters may serve as exploratory
kinetic fingerprints of redistribution capacity, providing a mechanistically
motivated, model-based rationale for future studies evaluating dialysis duration,
ultrafiltration strategies, and post-dialysis assessments. By capturing the
shared features of both solutes within one framework, 3C--Logistic may support
the future development of session-by-session personalization strategies after
external validation and could guide the design of non-invasive monitoring
protocols.