Inverse pressure gradient estimation from PC-MRI velocity - PowerPoint PPT Presentation

Inverse pressure gradient estimation from PC-MRI velocity measurements — reducing the effect of errors in segmentation David Nolte, Cristóbal Bertoglio, Axel Osses, Roel Verstappen Center for Mathematical Modeling, Universidad de Chile Computational Mechanics & Numerical Mathematics, University of Groningen

Motivation: Cardiovascular flow ∙ Arterial blood pressure very important for diagnosing cardiovascular diseases ∙ Examples: stenosis, heart valve insufficiency Stenosis of the aorta: Low pressure High pressure ∙ Surgery if pressure drop critical ∙ Pressure measurements are invasive : catheters equipped with pressure probes Stenosis ▸ non-invasive methods? Blood velocity 1 / 20

Phase-Contrast MRI PC-MRI: measure flow velocities with magnetic resonance imaging 2D slices PC-MRI measures blood velocity 3D data velocity pressure Navier-Stokes equations ∙ Estimate pressure gradient from partial measurements with PDE-constrained optimization 2 / 20

PDE-constrained Optimization Approx. 2D slice Data Assimilation 3D geometry ̃ Ω measurements NSE model: obtain 3D+time flow field ( 퐮 , 푝 ) 3 / 20

PDE-constrained Optimization Forward model: Navier-Stokes equations. Incompressible, unsteady, laminar flow; blood as Newtonian fluid 휌휕 퐮 in ̃ 휕푡 + 휌 ( 퐮 ⋅ ∇) 퐮 − 휇 Δ 퐮 + ∇ 푝 = ퟎ Ω in ̃ ∇ ⋅ 퐮 = 0 Ω + Initial and boundary conditions Solver: ∙ Convection-dominated, complex flow → Fractional step-scheme (Chorin-Temam) ∙ Semi-implicit, skew-symmetric convection term ∙ Backflow stabilization at outflow boundaries ∙ FEM with ℙ 1 ∕ ℙ 1 elements for velocity/pressure + streamline–diffusion stabilization 4 / 20

PDE-constrained Optimization ∙ Inflow and outflow BCs a priori unknown ▸ Parameters 휃 ∈ ℝ 푝 need to be estimated from measurements! ∙ Initial condition: assume 퐮 ( 푡 = 0) = 0 ∙ Minimize discrepancy between model prediction and measurements 푁 ∑ 1 1 2 2 ̂ | | 휃 − 휃 0 | | 푍 푘 − 퐻푋 푘 ( 휃 ) | | 휃 = 햺헋헀 헆헂헇 0 + | | 푃 −1 Γ −1 2 2 휃 푘 =1 – 푋 푘 ( 휃 ) =  ( 휃, 푋 푘 −1 ( 휃 푘 −1 )) ∈ ℝ 푛 : model state at time 푡 푘 – 푍 푘 ∈ ℝ 푚 : velocity measurement at time 푡 푘 – 퐻 ∈ ℝ 푛 × 푚 : observation operator – 휃 0 ∈ ℝ 푝 : initial guess of parameters – 푃 0 ∈ ℝ 푝 × 푝 : initial parameter covariance matrix – Γ ∈ ℝ 푚 × 푚 : covariance of measurements 5 / 20

PDE-constrained Optimization Method: Reduced-order Unscented Kalman Filter (Moireau & Chapelle, 2011) ∙ Sequential method For 푘 = 1 , ..., 푁 푇 : 휃 푘 = 햺헋헀 헆헂헇 퐽 푘 ( 휃 ) 휃 퐽 푘 ( 휃 ) = 1 푘 + 1 2 2 | 휃 − 휃 푘 −1 | | | | 푍 푘 − 퐻푋 푘 ( 휃 ) | | | ̂ 푃 −1 Γ −1 2 2 ̂ ∙ 푃 푘 ( Cov ) updated recursively from deterministic particles ▸ Reduced rank of state covariance: rank( ̂ 푃 푘 ) = 푝 ▸ Assume initial uncertainty only present in parameters ∙ ROUKF requires 푝 + 1 particles, i.e., 푝 + 1 forward evaluations per time step 6 / 20

Geometric Uncertainties ∙ Inverse problem is set up and solved in domain extracted from medical images Issue: ∙ Uncertainty associated to vessel wall position, at least of order of image resolution ∙ Standard no-slip boundary conditions can introduce large errors in estimated pressure drop! ▸ numerical experiments to investigate effects! 7 / 20

Numerical Experiments Geometries: ∙ 3 stenoses with 40%, 50% and 60% obstruction ∙ Reference geometry ∙ ‘approximate‘ geometries with walls shifed inward by Δ = ퟣ 헆헆 ( 10% ) and Δ = ퟤ 헆헆 ( 20% ) Figure: 60% stenosis, reference and 20% wall error Reference solution: solve Navier-Stokes on true domain with BCs: ∙ Outflow condition: homogeneous Neumann, zero-stress ∙ Inflow condition: pulsating plug flow: 퐮 = 퐔 sin( 휔푡 ) ∙ No-slip on walls 8 / 20

Numerical Experiments: Reference ∙ Reference solutions. 푅푒 = 2500 at inlet at peak time ∙ velocity magnitude (top row), pressure (botom row) 40% stenosis 50% stenosis 60% stenosis ∙ Number of elements: ∼ 3 − 4 × 10 6 9 / 20

Numerical Experiments Synthetic Measurements: ∙ Generated by interpolating reference solution to coarse measurement meshes , defined at selected planes ∙ 2 planes: (A) inlet cross-section, (B) lengthwise intersection ∙ Image resolution: 퐻 = 1 & ퟤ 헆헆 (simulation: ℎ ≈ ퟢ . ퟤퟧ 헆헆 ) ∙ Temporal resolution: Δ 푇 = ퟤퟢ 헆헌 (simulation: Δ 푡 = ퟣ 헆헌 ) 10 / 20

Numerical Experiments: No-slip results Parameter estimation: ∙ Given measurements at INLET , approximate (erroneous) geometry, estimate plug flow parameter 퐔 ∙ Pressure drop: 훿푝 푘 = | Γ 푖 | ∫ Γ 푖 푝 푘 − 1 1 | Γ 푛 | ∫ Γ 푛 푝 푘 11 / 20

Numerical Experiments: No-slip results Parameter estimation: ∙ Given measurements at INLET , approximate (erroneous) geometry, estimate plug flow parameter 퐔 ∙ Pressure drop: 훿푝 푘 = | Γ 푖 | ∫ Γ 푖 푝 푘 − 1 1 | Γ 푛 | ∫ Γ 푛 푝 푘 reference Δ = 10% Δ = 20% 40% stenosis 50% stenosis 60% stenosis pressure drop (mmHg) 10 30 100 20 5 50 10 0 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 1 0 . 2 0 . 3 0 . 4 0 . 1 0 . 2 0 . 3 0 . 4 time (s) time (s) time (s) 11 / 20

Numerical Experiments: No-slip results ∙ measurements only at inlet ∙ Velocity errors:  푘 = ‖ 퐮 푘 −  퐮 푘 푟푒푓 ‖ 퐿 2 ( ̃ Ω) Δ = 10% Δ = 20% 40% stenosis 50% stenosis 60% stenosis 퐿 2 velocity error 200 100 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 1 0 . 2 0 . 3 0 . 4 0 . 1 0 . 2 0 . 3 0 . 4 0 0 0 time (s) time (s) time (s) ∙ HUGE errors in 훿푝 ; plug flow parameter recovered with good precision, within 0.5% 12 / 20

Numerical Experiments: No-slip results ∙ Add measurements in interior! ∙ Error in 휃 : 10 − 20% and 25 − 50% Δ = 10% Δ = 20% reference 40% stenosis 50% stenosis 60% stenosis 훿푝 (mmHg) 40 6 15 4 10 20 2 5 0 0 0 −2 velocity error 200 100 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 0 . 1 0 . 2 0 . 3 0 . 4 0 0 . 1 0 . 2 0 . 3 0 . 4 time (s) time (s) time (s) 13 / 20

Slip/Transpiration BCs ∙ Instead of no-slip boundary conditions: allow for some slip and transpiration on the vessel wall ̃ Γ 푤 14 / 20

Slip/Transpiration BCs ∙ Instead of no-slip boundary conditions: allow for some slip and transpiration on the vessel wall ̃ Γ 푤 ∙ Slip & transpiration BCs : on ̃ 퐧 ⋅ [ 휇 ∇ 퐮 − 1 푝 ] ⋅ 퐧 + 훽 퐮 ⋅ 퐧 = 0 Γ 푤 , (2) 푑 −1 푑 −1 ∑ ∑ on ̃ 퐧 ⋅ [ 휇 ∇ 퐮 − 1 푝 ] ⋅ 퐭 푘 + 훾 퐮 ⋅ 퐭 푘 = 0 Γ 푤 , (3) 푘 =1 푘 =1 where 훽 , 훾 are coefficients controlling the slip and transpiration. ∙ BC is consistent with Poiseuille flow; 훽 , 훾 known explicitly 14 / 20

Slip/Transpiration BCs ∙ Instead of no-slip boundary conditions: allow for some slip and transpiration on the vessel wall ̃ Γ 푤 ∙ Slip & transpiration BCs : on ̃ 퐧 ⋅ [ 휇 ∇ 퐮 − 1 푝 ] ⋅ 퐧 + 훽 퐮 ⋅ 퐧 = 0 Γ 푤 , (2) 푑 −1 푑 −1 ∑ ∑ on ̃ 퐧 ⋅ [ 휇 ∇ 퐮 − 1 푝 ] ⋅ 퐭 푘 + 훾 퐮 ⋅ 퐭 푘 = 0 Γ 푤 , (3) 푘 =1 푘 =1 where 훽 , 훾 are coefficients controlling the slip and transpiration. ∙ BC is consistent with Poiseuille flow; 훽 , 훾 known explicitly ∙ General case: estimate parameters 14 / 20

Numerical Experiments: Slip/transpiration results ∙ Measurements: inlet and interior slice ∙ parameters : slip 훾 , transpiration 훽 , plug flow 퐔 15 / 20

Numerical Experiments: Slip/transpiration results ∙ Measurements: inlet and interior slice ∙ parameters : slip 훾 , transpiration 훽 , plug flow 퐔 slip Δ = 10% slip Δ = 20% no-slip Δ = 10% no-slip Δ = 20% reference 40% stenosis 50% stenosis 60% stenosis 40 6 훿푝 (mmHg) 15 4 10 20 2 5 0 0 0 −2 0 . 1 0 . 2 0 . 3 0 . 4 0 . 1 0 . 2 0 . 3 0 . 4 0 . 1 0 . 2 0 . 3 0 . 4 time (s) time (s) time (s) 15 / 20

Numerical Experiments: Slip/transpiration results ∙ Measurements: inlet and interior slice ∙ parameters : slip 훾 , transpiration 훽 , plug flow 퐔 slip Δ = 10% slip Δ = 20% no-slip Δ = 10% no-slip Δ = 20% reference velocity error 200 100 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 0 . 1 0 . 2 0 . 3 0 . 4 0 0 . 1 0 . 2 0 . 3 0 . 4 time (s) time (s) time (s) ∙ accuracy plug flow parameter: 1 − 10% and 10 − 25% for Δ = 10% and 20% 16 / 20

Numerical Experiments: Slip/transpiration results ∙ Accuracy with slip/transpiration much beter than no-slip model for severe stenoses ∙ Plug-flow parameter overestimated with slip/transpiration ▸ increase weight of measurements at the inlet to obtain beter plug flow estimates? 17 / 20

Numerical Experiments: Slip/transpiration results ∙ Accuracy with slip/transpiration much beter than no-slip model for severe stenoses ∙ Plug-flow parameter overestimated with slip/transpiration ▸ increase weight of measurements at the inlet to obtain beter plug flow estimates? slip Δ = 10% slip Δ = 20% weighted slip Δ = 10% weighted slip Δ = 20% reference 6 30 훿푝 (mmHg) 10 4 20 2 5 10 0 0 0 −2 0 . 1 0 . 2 0 . 3 0 . 4 0 . 1 0 . 2 0 . 3 0 . 4 0 . 1 0 . 2 0 . 3 0 . 4 time (s) time (s) time (s) 17 / 20