Pharmacokinetic Dynamic PET (PK-PET) L1-505
Unclaimed Principle — open for contribution
This Principle is declared in the catalog but has no reference solver, no pinned dataset, and is not registered on-chain. There is no reward pool. Submitting a cert against this Principle today will record the cert for reproducibility but pay zero PWM.
To claim it as a Bounty #7 contribution: open a PR adding (1) a reference solver, (2) ≥1 dataset pinned to IPFS, (3) updates to the L3 manifest with dataset CIDs. After verifier-agent triple-review, the founders' 3-of-5 multisig signs PWMRegistry.register() and the Principle becomes mineable.
Forward model E
dC_T/dt = K_1\cdot C_p(t) - (k_2 + k_3)\cdot C_T + k_4\cdot C_T_bound; dC_T_bound/dt = k_3\cdot C_T - k_4\cdot C_T_bound; A(r, t) = alpha(r)\cdot (C_T(r, t) + C_T_bound(r, t)); y(s, t) = integral A(r, t)\cdot h_PET(s; r) dr + n_Poisson(s, t); jointly invert y(s, t) \to (K_1, k_2, k_3, k_4)(r) given C_p(t), h_PET, alphaPharmacokinetic Dynamic PET (PK-PET): joint multi-physics forward couples (i) compartmental ODE tracer kinetics describing time-evolution of free and bound tracer concentrations C_T(r, t), C_T_bound(r, t) under plasma input C_p(t) and rate constants (K_1, k_2, k_3, k_4)(r); (ii) tissue activity distribution A(r, t) = alpha(r) * [C_T(r, t) + C_T_bound(r, t)] coupling chemistry to photon physics; (iii) PET projection forward y(s, t) = integral A(r, t) * h_PET(s; r) dr where h_PET is the line-of-response system matrix including attenuation, scatter, normalization, and detector response. The forward DAG has 8 primitives with two coupling constraints (n_c = 2): (i) ODE kinetics state -> tissue activity (deterministic, multi-compartment); (ii) time-resolved activity -> time-resolved projection (linear in activity, time-coupled across frames). Recovery is posed as the joint 4D inverse problem that recovers (K_1, k_2, k_3, k_4)(r) directly from y(s, t) (direct kinetic reconstruction) or via a two-step pipeline (4D-EM-ML for activity then voxelwise kinetic fitting). Standard tracers: FDG (irreversible 2-tissue with k_4=0 for metabolic rate K_i), FLT (proliferation), 11C-PiB (amyloid), 18F-FDOPA (dopamine synthesis), [18F]-Florbetapir (amyloid). Difficulty tier delta = 5 with raw condition number kappa ~ 200 (limited by count statistics and identifiability of multi-compartment model) and effective kappa_eff ~ 25 after temporal regularization and direct kinetic ML reconstruction. Mismatch parameters: input_function_uncertainty, attenuation_correction_error, scatter_correction_error, motion_during_scan, partial_volume_effect, count_statistics_dropoff. Poisson noise dominates count statistics; Gaussian thermal/electronic noise sets a smaller residual floor. See forward_model field for the closed-form joint imaging equation.
L-DAG
Well-posedness W
- Existence:
- true
- Uniqueness:
- conditional
- Stability:
- conditional
- κ:
- 200
Existence of recovered 4D kinetic parameter maps (K_1, k_2, k_3, k_4)(r) is guaranteed within the declared Omega bounds. Uniqueness holds for irreversible 2-tissue compartmental models (FDG-like, k_4 = 0) with sufficient temporal sampling and known C_p(t); reversible 2-tissue models (4 free parameters) require either reference-region constraint or sufficient SNR for full identifiability. Stability is moderately conditioned (kappa_eff ~ 25 after 4D-EM-ML or direct kinetic reconstruction) — input_function_uncertainty dominates K_1 bias; partial_volume_effect dominates small-structure quantitation; count_statistics_dropoff dominates late-frame variance. Joint Hadamard well-posedness for the coupled compartmental-PET forward is established by Carson (1996, 2003), Gunn-Gunn-Cunningham (2001), Patlak-Blasberg-Fenstermacher (1983 Patlak plot), Logan et al. (1990 Logan plot), Wang-Qi (2013 direct kinetic estimation), and Reader-Verhaeghe (2014 4D image reconstruction).
Solvability C
- Solver class:
- linear-operator + convex optimisation [direct kinetic ML, 4D-EM-ML, NEGML; nested ML for kinetics + activity] | graphical analysis [Patlak plot for irreversible tracers, Logan plot for reversible tracers] | linear-operator + deep neural prior [DeepKinet, DeepPET]
- Convergence rate q:
- 2
- Complexity:
- O(H * W * Z * N_frames * (LOR_count)^(2/3)) per iteration via OS-EM with kinetic basis; direct kinetic ML O(H W Z N_frames N_compartment_params) per iteration; learned variants O(H W Z N_frames * F_theta_cost) per forward pass