Causal Spatio-Temporal Neural Distance Fields for Counterfactual Cardiac Anatomy

Mathias Lowes; Kurt Butler; Kristine Aavild Sørensen; Klaus F. Kofoed; Sotirios A. Tsaftaris; Rasmus R. Paulsen; Steven McDonagh

cs.CVOct 15, 2026

Causal Spatio-Temporal Neural Distance Fields for Counterfactual Cardiac Anatomy

Authors: Mathias Lowes, Kurt Butler, Kristine Aavild Sørensen, Klaus F. Kofoed, Sotirios A. Tsaftaris, Rasmus R. Paulsen, Steven McDonagh

Organizations: DTU Compute, Technical University of Denmark, Kongens Lyngby, Denmark · Causality in Healthcare AI Hub (CHAI), UK · School of Engineering, University of Edinburgh, Edinburgh, UK · Novo Nordisk A/S, Copenhagen, Denmark ·

Abstract

Counterfactual cardiac anatomy can support risk modelling and intervention planning, by predicting how a patient's heart would change under altered clinical conditions. However, current shape generative models are conditional on clinical variables and disregard causal links between these variables. Thus, they cannot answer counterfactual questions about how a given subject's anatomy would change under a different clinical profile. Modelling the heart's continuous movement over the cardiac cycle requires a representation that is itself continuous in space and time. Spatio-temporal neural distance fields provide this while separating clinical conditioning from subject-specific shape. We place a structural causal model (SCM) over the set of clinical variables (age, sex, smoking, BMI, heart rate, systolic blood pressure), before the condition encoding of the spatio-temporal neural distance field which represents the left atrial complex over a cardiac cycle by signed distance fields. An intervention of the clinical variables, propagated through a SCM before the 4D counterfactual, is constructed from the neural distance field. The generated counterfactuals follow dose-response under age interventions and transforms the male distribution to closely match the female distribution under the intervention $\mathrm{do(\mathrm{female}{=}1)}$ . By doing inverse interventions the model also approximates the original anatomy with sub millimeter precision. To our knowledge, this is the first counterfactual generation of 4D cardiac anatomy and the first integration of an SCM with an implicit neural representation as its imaging mechanism.

Figures & tables

Figure 1: (a) proposed causal graph for cardiac anatomy, where the nodes are grouped to simplify the visualization. An edge between two groups of nodes, $A\to B$ , indicates that every node in A connects to every node in B. (b) clinical variables are encoded to the latent space $\mathbf{z}_{c}=g_{\phi}(\mathbf{c})$ and concatenated with the patient latent $\mathbf{z}_{r}$ and spatio-temporal point $\mathbf{x}$ to estimate the signed distance $\hat{d}(\mathbf{x})=f_{\theta}(\mathbf{x},\mathbf{z}_{c},\mathbf{z}_{r})$ .

Figure 2: Counterfactual generation for changing age for a single patient. The original surface is represented with the black mesh and the counterfactuals with coloured surfaces. The original has $\mathrm{age}=83$ , while the counterfactuals are generated with interventions $\mathrm{do}(\mathrm{age}{=}\alpha)$ , where $\alpha\in\{40,50,60,70,80\}$ for (a) - (e), respectively. The top row is at $t=0\%$ (contraction phase) and the bottom row is at $t=35\%$ (dilation phase).

Age (y)		40	50	60	70	80
Vol (ml)	GT $\pm$ 5y	81.8	79.6	82.6	84.0	109.7
	Baseline	74.3	75.2	76.7	79.2	81.4
	SCM	76.6	78.1	78.6	83.8	91.8
FC (%)	GT $\pm$ 5y	40.1	39.1	37.0	36.2	28.3
	Baseline	39.0	37.9	36.3	34.5	32.8
	SCM	44.3	39.0	37.3	34.7	32.5

Table 1: Dose response comparisons for the interventions

\mathrm{do}(\mathrm{age}{=}\alpha)

, where

\alpha\in\{40,50,60,70,80\}

. Mean left-atrial volume at

t=0\%

(Vol) and fraction change (FC) comparing the Baseline model, the SCM model, and the test data whose age is within

\pm 5

y of the intervened age (GT

\pm

5y).

Figure 3: Dose response comparisons for the interventions

\mathrm{do}(\mathrm{age}{=}\alpha)

, where

\alpha\in\{40,50,60,70,80\}

. The crosses indicates the mean of the population within 5 years of the given age. (a) mean of volumes at

t=0\%

, (b) fractional change of volume over cardiac cycle. It should be noted that there are few subjects in limit age groups, this makes the GT mean unreliable.

Group	$n$	Mean	Std	Median	Range
GT male	85	94.76	20.78	91.48	[54.25, 160.90]
GT female	111	76.03	17.66	72.41	[45.63, 169.36]
Baseline	85	72.91	17.37	69.34	[35.65, 128.19]
SCM	85	76.87	20.83	71.39	[42.15, 164.20]

Table 2: LA-complex volume distributions at end-diastole (ml) for the two ground-truth subgroups and the two counterfactual models under

\mathrm{do}(\mathrm{female}{=}1)

. Counterfactuals are computed on the

n{=}85

GT-male test subjects. Bold indicates closest to GT female distribution between the baseline and SCM models.

Figure 4: Volume distribution comparisons for counterfactuals with

\mathrm{do}(\mathrm{female}=1)

on the male test set. (a) volume distribution of the LA complex at

t=0\%

, the black line indicates the mean of the distribution. (b) mean volume over time, shaded area indicates 95% CI for the group mean.

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Surrogate Modeling PDE Surrogate Modeling

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High-fidelity computational models of cardiac mechanics provide mechanistic insight into the heart function but are computationally prohibitive for routine clinical use. Surrogate models can accelerate simulations, but generalization across diverse anatomies is challenging, particularly in data-scarce settings. We propose a two-step framework that decouples geometric representation from learning the physics response, to enable shape-informed surrogate modeling under data-scarce conditions. First, a shape model learns a compact latent representation of left ventricular geometries. The learned latent space effectively encodes anatomies and enables synthetic geometries generation for data augmentation. Second, a neural field-based surrogate model, conditioned on this geometric encoding, is trained to predict ventricular displacement under external loading. The proposed architecture performs positional encoding by using universal ventricular coordinates, which improves generalization across diverse anatomies. Geometric variability is encoded using two alternative strategies, which are systematically compared: a PCA-based approach suitable for working with point cloud representations of geometries, and a DeepSDF-based implicit neural representation learned directly from point clouds. Overall, our results, obtained on idealized and patient-specific datasets, show that the proposed approaches allow for accurate predictions and generalization to unseen geometries, and robustness to noisy or sparsely sampled inputs.

Abstract

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