Quantitative Photoacoustic Tomography (qPAT) L1-504

Medical ImagingQuantitative multi-spectral chromophore recovery (multi-physics joint inverse)δ=5 · advancedL_DAG = 8.5📋 Stub — not mineable

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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.

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Forward model E

P hi (r, λ) = in t e g r a l G_{o} pt i c a l (r, r^{'}; m u_{a}, m u_{s}^{'}, λ) \cdot S (r^{'}, λ) d r^{'}; H (r, λ) = m u_{a} (r, λ) \cdot P hi (r, λ); p_{0} (r) = G amma \cdot λ \sum H (r, λ); p (r_{d}, t) = in t e g r a l G_{a} co u s t i c (r_{d}, r^{'}; c) \cdot p_{0} (r^{'}) d r^{'}; j o in tl y in v er tp (r_{d}, t) \to m u_{a} (r, λ) g i v e nm u_{s}^{'}, G amma, c, i l l u mina t i o n S

Quantitative Photoacoustic Tomography (qPAT): joint multi-physics forward couples optical radiative transfer / diffusion (light fluence Phi(r, lambda) under tissue scattering and absorption), Grueneisen-coupled thermoelastic source generation (initial pressure p_0 = Gamma * mu_a * Phi), and acoustic wave propagation (p(r, t) from p_0(r) under tissue speed-of-sound c(r)). The forward DAG has 8 primitives with two coupling constraints (n_c = 2): (i) light-fluence to energy-absorption multiplicative coupling H = mu_a * Phi, position-dependent and saturating at high absorption; (ii) thermoelastic-source to acoustic-wave causal coupling p_0 -> p, governed by the homogeneous wave equation in tissue. Recovery is posed as the joint inverse problem that recovers mu_a(r, lambda) (and optionally mu_s'(r) via multi-illumination) from measured p(r, t) across N_wavelengths. Multi-spectral acquisition disambiguates chromophore concentrations (oxy-hemoglobin, deoxy-hemoglobin, melanin, lipid, water, contrast agents). Difficulty tier delta = 5 with raw condition number kappa ~ 300 (limited by acoustic bandwidth and high-absorption non-uniqueness) and effective kappa_eff ~ 40 after model-based two-step or one-step Bal-Uhlmann reconstruction. Mismatch parameters: mu_s_prime_uncertainty, gruneisen_uncertainty, acoustic_speed_heterogeneity, transducer_position_error, light_fluence_inhomogeneity, background_absorption_drift. Additive Gaussian thermal noise sets the data-fidelity floor. See forward_model field for the closed-form joint imaging equation.

L-DAG

L.optical_source -> L.optical_diffusion -> L.beer_lambert -> L.thermoelastic_grueneisen -> L.acoustic_wave_propagation -> L.transducer_detection -> int.temporal -> int.spectral

L.optical_sourceL.optical_diffusionL.beer_lambertL.thermoelastic_grueneisenL.acoustic_wave_propagationL.transducer_detectionint.temporalint.spectral

Well-posedness W

Existence:: true
Uniqueness:: conditional
Stability:: conditional
κ:: 300

Existence of recovered 3D multi-spectral absorption coefficient mu_a(r, lambda) is guaranteed within the declared Omega bounds. Uniqueness holds under multi-illumination or multi-wavelength acquisition (Bal-Uhlmann 2010; Bal-Ren 2011); single-illumination single-wavelength qPAT is non-unique due to the multiplicative H = mu_a * Phi coupling and is excluded from the spec range. Stability is moderately conditioned (kappa_eff ~ 40 after model-based reconstruction) — acoustic bandwidth dominates spatial resolution; mu_s_prime_uncertainty dominates absorption-coefficient bias; gruneisen_uncertainty contributes a scaling factor. Joint Hadamard well-posedness for the coupled optical-thermoelastic-acoustic forward is established by Cox-Arridge-Beard (2009), Bal-Uhlmann (2010), Bal-Ren (2011), Tarvainen et al. (2013), and Cox-Tarvainen-Arridge (2014). See joint_well_posedness_references.

Solvability C

Solver class:: linear-operator + convex optimisation [two-step: UBP + GN-IR for mu_a; one-step: Bal-Uhlmann joint inversion] | gradient-based MCMC [Bayesian qPAT, Tarvainen]| linear-operator + deep neural prior [DAS-Net, J-Net]
Convergence rate q:: 2
Complexity:: O(H * W * Z * N_wavelengths * (transducer_count * time_samples)^(2/3)) per iteration via k-Wave time-domain forward / adjoint; frequency-domain variants O(H W Z N_wavelengths log(H W Z)); learned variants O(H W Z N_wavelengths * F_theta_cost) per forward pass

Specs (0)

No L2 specs registered yet for this principle.