FalconBC: Flow matching for Amortized inference of Latent-CONditioned physiologic Boundary Conditions

Chloe H. Choi; Alison L. Marsden; Daniele E. Schiavazzi

cs.LGMar 18, 2026

FalconBC: Flow matching for Amortized inference of Latent-CONditioned physiologic Boundary Conditions

Authors: Chloe H. Choi, Alison L. Marsden, Daniele E. Schiavazzi

Organizations: Department of Mechanical Engineering, Stanford University, Stanford, CA. USA · Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA, USA. · Bioengineering, Stanford University, Stanford, CA, USA. · Pediatric Cardiology, Stanford University, Stanford, CA, USA. ·

Abstract

Boundary condition tuning is a fundamental step in patient-specific cardiovascular modeling. Despite an increase in offline training cost, recent methods in data-driven variational inference can efficiently estimate the joint posterior distribution of boundary conditions, with amortization of training efforts over clinical targets. However, even the most modern approaches fall short in two important scenarios: open-loop models with known mean flow and assumed waveform shapes, and anatomies affected by vascular lesions where segmentation influences the reachability of pressure or flow split targets. In both cases, boundary conditions cannot be tuned in isolation. We introduce a general amortized inference framework based on probabilistic flow that treats clinical targets, inflow features, and point cloud embeddings of patient-specific anatomies as either conditioning variables or quantities to be jointly estimated. We demonstrate the approach on two patient-specific models: an aorto-iliac bifurcation with varying stenosis locations and severity, and a coronary arterial tree.

Figures & tables

Panel of Figure 2 — Figure 2: Collection of diseased models with left and right iliac artery stenoses.

Panel of Figure 3 — Figure 3: Zero-dimensional model components for the selected aorto-iliac bifurcation model.

Figure 4: Encoder/decoder diagram . Encoder produces latent vector

\bm{z}

; template points (below

\bm{z}

) and Fourier features (above

\bm{z}

) feed into decoder via a concatenation block. Encoder and decoder are highlighted with dashed boxes.

Figure 5: Posterior distribution from CFM over time $t\in[0,1]$ .

Figure 6: Samples from the DREAM and CFM posterior distributions for the aorto-iliac bifurcation example.

Figure 7: Single RC BCs: Results of posteriors for five different offline validation points (each row corresponds to a unique observation) subsampled from 10 offline testing data points. In the rightmost column, the true observations and the associated

\pm 3

standard deviations are displayed as a black star and a black bar, respectively.

$N=100$	$P_{\text{dia}}$ (mmHg, %)	$P_{\text{sys}}$ (mmHg, %)
mean (abs/sign)	4.93 / 2.25 (6.34 / 3.26)	6.59 / -1.58 (5.34 / -1.08)
std (abs/sign)	3.64 / 4.96	4.84 / 7.11
$N=500$	$P_{\text{dia}}$ (mmHg, %)	$P_{\text{sys}}$ (mmHg, %)
mean (abs/sign)	4.95 / 1.00 (6.48 / 2.01)	6.52 / -0.67 (5.50 / -0.43)
std (abs/sign)	3.50 / 4.86	4.74 / 7.02
$N=1000$	$P_{\text{dia}}$ (mmHg, %)	$P_{\text{sys}}$ (mmHg, %)

Table 1: Single RC BC: Summary statistics of reconstruction error for

N=100,500,1000

. Absolute and signed errors are reported in mmHg. Relative errors are shown in parentheses (%).

Vessel	$R_{\text{tot}}$ [ $\text{dynes}\cdot\text{s}/\text{cm}^{5}$ ]	$C$ [cm 5 /dyne]
Right iliac	3228	5.949 $\times$ 10 -4
Left iliac	4399	3.678 $\times$ 10 -4

Table 2: Nominal values for the two RCR BCs with fixed ratios (four dimensions, two per branch outlet) for aorto-iliac model.

Vessel	$R_{p}$ [ $\text{dynes}\cdot\text{s}/\text{cm}^{5}$ ]	$C$ [cm 5 /dyne]	$R_{d}$ [ $\text{dynes}\cdot\text{s}/\text{cm}^{5}$ ]
Right iliac	233.2	3.161 $\times$ 10 -4	3126
Left iliac	281.2	1.775 $\times$ 10 -4	4308

Table 3: Nominal values for aorto-iliac model with two RCR BCs (6 dimensions)

$N=100$	$P_{\text{dia}}$ (mmHg, %)	$P_{\text{sys}}$ (mmHg, %)	Flow split (% flow, % rel)
mean (abs/sign)	5.31 / -1.59 (6.92 / -1.99)	7.52 / 0.80 (6.47 / 1.08)	3.98 / -0.24 (9.38 / 0.19)
std (abs/sign)	3.89 / 6.00	5.30 / 7.34	2.92 / 4.36
$N=500$	$P_{\text{dia}}$ (mmHg, %)	$P_{\text{sys}}$ (mmHg, %)	Flow split (% flow, % rel)
mean (abs/sign)	4.25 / 1.07 (5.69 / 1.68)	6.88 / -0.97 (5.76 / -0.42)	3.67 / 0.02 (8.79 / 0.53)
std (abs/sign)	3.16 / 4.81	4.75 / 6.72	2.73 / 4.19
$N=1000$	$P_{\text{dia}}$ (mmHg, %)	$P_{\text{sys}}$ (mmHg, %)	Flow split (% flow, % rel)

Table 4: Two RCR BCs: Summary statistics for

N=100,500,1000

data. Absolute reconstruction errors are shown. Relative errors are reported in parentheses (%).

Figure 8: Two RCR BCs (6 dimensions): Results of marginal posteriors for five offline validation points subsampled from 100 offline testing data points. In the rightmost column, true observations and corresponding

\pm 3

standard deviations are illustrated with a black star and a black bar, respectively.

Figure 9: Marginal distributions of RCR BCs (6 dimensions, 3 per outlet), conditioned on parametric inflow, target pressures and flow split. Results of marginal posteriors for five offline validation samples (one per row). In the rightmost column, the true pressures and flow split are illustrated with a black star, with

\pm 3

standard deviations in black bars.

$N=100$	$P_{\text{dia}}$ (mmHg, %)	$P_{\text{sys}}$ (mmHg, %)	Flow split (% flow, % rel)
mean (abs/sign)	9.08 / 1.23 (13.23 / 3.82)	8.48 / 0.93 (8.65 / 2.11)	7.26 / 1.38 (19.14 / 6.79)
std (abs/sign)	5.11 / 6.33	5.66 / 7.22	4.37 / 5.63
$N=500$	$P_{\text{dia}}$ (mmHg, %)	$P_{\text{sys}}$ (mmHg, %)	Flow split (% flow, % rel)
mean (abs/sign)	4.47 / 0.89 (6.90 / 2.08)	6.78 / -0.99 (6.63 / -0.42)	4.35 / -0.14 (10.51 / 0.98)
std (abs/sign)	3.23 / 4.72	4.46 / 6.10	3.01 / 4.24
$N=1000$	$P_{\text{dia}}$ (mmHg, %)	$P_{\text{sys}}$ (mmHg, %)	Flow split (% flow, % rel)

Table 5: Amortized inference of two RCR BCs (one per outlet) conditioned on parametric inflow. Absolute errors statistics for training dataset size

N=100,500,1000

. Relative errors are reported in parentheses (%).

Figure 10: Two RCR BCs and Fourier coefficients: Results of 1000 estimated Fourier curves subsampled from the posterior distribution. True curve is in black.

Figure 11: Marginal posteriors and corresponding true boundary conditions at testing. In the rightmost column, true observations and corresponding

\pm 3

standard deviations are illustrated with black stars and black bars, respectively.

Figure 12: Posterior predictive time traces of inlet pressure and outlet flows. Shaded regions indicate

\pm

one standard deviation.

Figure 13: Encoding-decoding anatomical point clouds. Marginal posterior distributions at testing. In the rightmost column, true underlying systolic, diastolic pressures and flow split and corresponding

\pm 3

standard deviations are illustrated with a black star and a black bar, respectively.

Figure 14: Encoding-decoding anatomical point clouds. Posterior predictive time traces of inlet pressure and outlet flows. Shaded regions indicate

\pm

one standard deviation.

Figure 15: Reconstructed geometries using the complete encoder-decoder architecture in Figure 4 .

Figure 16: Joint estimation of anatomical point clouds and boundary conditions from CFM. The point clouds are obtained by decoding the marginal posterior samples for the anatomical latent variables, representing stenosis location and severity.

Panel of Figure 17 — Figure 17: Coronary model and closed-loop boundary conditions.

Figure 18: Marginal posterior distributions for the resistance scaling factors at the coronary outlets.

Figure 19: Distributions of simulated (hexbin) versus measured (yellow ellipse) univariate and bivariate marginal posterior predictive samples for branch-specific coronary flows computed by FalconBC.

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Hyperparameter	Range
Hidden layers $\ell$	{1, …, 4}
Neurons per hidden layer $n$	{1, …, 128}
Learning rate $h$	[1e-4, 1e-2]
Mini-batch size $s$	{8, …, 64} for $N=100$
	{16, …, 128} for $N=500,1000$

Appendix

Table 6: Hyperparameters and corresponding admissible ranges tuned using Optuna.

Example	$\ell$	$n$	$h$	$s$
Validation	2	102	0.000431	32
Single RC $(N=100)$	2	7	0.0004859	16
Single RC $(N=500)$	1	20	0.0001535	16
Single RC $(N=1000)$	2	102	0.000431	32
Two RCRs with fixed $R_{p}/R_{d}$ $(N=100)$	1	106	0.0002250	16
Two RCRs with fixed $R_{p}/R_{d}$ $(N=500)$	1	30	0.0012486	16

Appendix

Table 7: FalconBC hyperparameters selected via Optuna. The parameters are: number of hidden layers

\ell

, number of neurons per hidden layer

n

, learning rate

h

and mini-batch size

s

Vessel	$R_{\text{tot}}$ [ $\text{dynes}\cdot\text{s}/\text{cm}^{5}$ ]	$C$ [cm 5 /dyne]
Right iliac	3228	5.949 $\times 10^{-4}$
Left iliac	4399	3.678 $\times 10^{-4}$

Appendix

Table 8: Nominal values for the two RCR BCs with fixed ratios (four dimensions, two per branch outlet) for aorto-iliac model.

Figure 20: Two RC BCs: Results of marginal posteriors for five offline validation points subsampled from 50 offline testing data points. In the rightmost column, true observations and corresponding

\pm 3

standard deviations are illustrated with a black star and a black bar, respectively.

$N=100$	$P_{\text{dia}}$ (mmHg, %)	$P_{\text{sys}}$ (mmHg, %)	Flow split (% flow, % rel)
mean (abs/sign)	5.50 / 2.97 (7.53 / 4.49)	6.64 / 0.41 (5.71 / 0.63)	3.71 / -0.54 (8.31 / -0.77)
std (abs/sign)	3.53 / 4.76	4.72 / 6.88	2.77 / 4.21
$N=500$	$P_{\text{dia}}$ (mmHg, %)	$P_{\text{sys}}$ (mmHg, %)	Flow split (% flow, % rel)
mean (abs/sign)	4.59 / 1.10 (5.95 / 1.70)	7.01 / 0.92 (5.90 / 1.09)	3.86 / 0.04 (9.22 / 0.91)
std (abs/sign)	3.35 / 4.89	4.94 / 6.98	2.87 / 4.26
$N=1000$	$P_{\text{dia}}$ (mmHg, %)	$P_{\text{sys}}$ (mmHg, %)	Flow split (% flow, % rel)

Appendix

Table 9: Two RC BCs: Summary statistics for training dataset size

N=100,500,1000

. Absolute errors are reported in mmHg. Relative errors are shown in parentheses (%).

Figure 21: Joint estimation of two RCR BCs and inflow Fourier features: Results of marginal posteriors for five offline validation points subsampled from 50 offline testing data points. In the rightmost column, true observations and corresponding

\pm 3

standard deviations are illustrated with a black star and a black bar, respectively.

$N=100$	$P_{\text{dia}}$ (mmHg, %)	$P_{\text{sys}}$ (mmHg, %)	Flow split (% flow, % rel)
mean (abs/sign)	6.65 / -1.16 (9.85 / -0.95)	8.46 / -1.85 (8.17 / -0.84)	3.85 / -0.16 (9.16 / 0.19)
std (abs/sign)	4.54 / 6.41	5.43 / 7.39	2.91 / 4.45
$N=500$	$P_{\text{dia}}$ (mmHg, %)	$P_{\text{sys}}$ (mmHg, %)	Flow split (% flow, % rel)
mean (abs/sign)	4.62 / 0.32 (6.75 / 0.92)	6.00 / 0.46 (6.06 / 0.83)	3.81 / 0.18 (9.45 / 1.25)
std (abs/sign)	3.27 / 4.65	4.25 / 6.08	2.82 / 4.18
$N=1000$	$P_{\text{dia}}$ (mmHg, %)	$P_{\text{sys}}$ (mmHg, %)	Flow split (% flow, % rel)

Appendix

Table 10: Joint estimation of two RCR BCs and inflow Fourier features: Summary statistics for training dataset size

N=100,500,1000

. Absolute errors are reported in mmHg. Relative errors are shown in parentheses (%).

Figure 22: Marginal posteriors and corresponding true boundary conditions at one testing sample. In the rightmost column, true observations are illustrated with black stars, with

\pm 3

standard deviations in black bars.

Figure 23: Reconstructed geometries from the estimated latents, which are then input to the decoder.

Figure 24: Flow profiles for different patients used for training data

Figure 25: Inflow sampled from a flow matching algorithm, given the training dataset in Figure 24 . These generated inflows are then used as training data for FalconBC.

Figure 26: 3D versus 0D results, where flow split indicates that going through the left iliac artery.

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