Combined PK/PD and TK/TD Modeling
While the level of sophistication of PK/PD and TK/TD models has continued to improve as research characterizing translational biological cascades of beneficial and adverse/toxic effects continues, PK/PD models have tended to be “agency”-driven, and TK/TD relationships have been much broader in scope. Yet, PK/PD and TK/TD models are simultaneously modeling input and outcomes (Figure 5.8).38 PK/PD modeling strategies have been the focus up to this point because of regulatory requirements for new drug entities and the wealth of literature coverage. On the other hand, while TK/ TD literature is more difficult to find due to its breadth of systems, many of the PK/PD modeling strategies can be easily applied to address TK/TD challenges.
The broader scope of TK/TD literature, which includes phytochemical and TK/TD models in animals39-41 and general modeling strategies,42-44 demonstrate that defining of a toxicological endpoint is difficult; creating models to describe those difficult endpoints is even more difficult; and determining
Figure 5.8 Strategy and scope of toxicokinetic/toxicodynamic modeling. Information from ref. 38.
model parameter estimates is even more difficult. Two models of TK/TD are highlighted in this section to illustrate an essential element for any well-designed TK/TD model and/or experiment; both link the dose or environmental disturbance (usually concentration of pollutant) time-course to adverse or toxic effects to the entity exposed to the drug or disturbance. The TK/ TD model’s ability to predict outcomes outside the experimental design are highly dependent on the TK/TD model’s linking of the time-course input of dose or disturbance to the resulting outcome.
Lobo and Balthasar39 created three TK/TD time-dependent models from time-static models, for mice exposed to the cancer chemotherapy agent methotrexate. To preserve the dose relationship of the static models, the time derivative was obtained for each of the static models to create the time-dependent models. The adverse/toxicological event of concern was the nadir of the time-, dose-, and exposure-dependent nadir of percentage body weight. Their study is an excellent example of PK/PD and simultaneous modeling of TK/TD for a drug with complex time-independent and time-dependent measurable outcomes.
As described earlier, TK/TD tends to be broader in scope than PK/PD with regards to the system under examination. Ashauer and co-workers41 modeled the influence of environmental chemical on reduced fish growth. This work is analogous to in vitro-in vivo correlation (IVIVC) in PK/PD modeling. The IVIVC aspect of this model creates a more time-static dosing regimen as input, but the linking of in vitro to in vivo is reasonably done and points out the dose-dependency incorporated into the TK/TD model. Eqn (5.25) gives the relationship between in vitro concentrations, Cint, (cell culture) and environmental chemical concentrations, Cext. Note the equation similarity between first-order input and first-order elimination used in PK/PD modeling.
This equation is coupled through Cint to two additional differential equations modeling a stochastic death hazard function, which describes the pharmacodynamics.