please make a report of this paper: show the strong point and the weak points



Latent 3D Brain MRI Counterfactual
Wei Peng1, Tian Xia3, Fabio De Sousa Ribeiro3, Tomas Bosschieter1
Ehsan Adeli1, Qingyu Zhao2, Ben Glocker3, and Kilian M. Pohl1
1 Stanford University, Stanford, CA 94305
2 Weill Cornell Medicine, New York, NY 10065
3 Imperial College London, South Kensington Campus London SW7 2AZ, UK
Abstract. The number of samples in structural brain MRI studies is
often too small to properly train deep learning models. Generative mod-
els show promise in addressing this issue by effectively learning the data
distribution and generating high-fidelity MRI. However, they struggle to
produce diverse, high-quality data outside the distribution defined by
the training data. One way to address the issue is using causal models
developed for 3D volume counterfactuals. However, accurately modeling
causality in high-dimensional spaces is a challenge so that these mod-
els generally generate 3D brain MRIS of lower quality. To address these
challenges, we propose a two-stage method that constructs a Structural
Causal Model (SCM) within the latent space. In the first stage, we em-
ploy a VQ-VAE to learn a compact embedding of the MRI volume. Subse-
quently, we integrate our causal model into this latent space and execute
a three-step counterfactual procedure using a closed-form Generalized
Linear Model (GLM). Our experiments conducted on real-world high-
resolution MRI data (1mm) demonstrate that our method can generate
high-quality 3D MRI counterfactuals.
1 Introduction
Generative AI models have demonstrated great potential to advance numerous
medical fields including neuroimaging with 3D brain MRIs [18,22]. The synthesis
of medical images is particularly promising for tasks like improving image qual-
ity, imputing missing modalities [32], and modeling disease progression [31,7,8].
Recent advances in diffusion probabilistic models (DPMs) [6,24] have played a
significant role in this development as they are able to accurately capture the
underlying data distributions. This results in synthesized medical images con-
taining a great amount of detail [28,11] that also differ from the training sam-
ples [9,30]. Nevertheless, a key challenge remains generating realistic samples
that are outside the distribution defined by the training dataset [10]. Overcom-
ing this challenge is essential to the generalizability of large-scale models [30].
One way to address this shortcoming is by incorporating causality between fac-
tors during the generation progress [17]. For example, aging causes a thinning of
the cortex that is accelerated in those with alcohol use disorder or Alzheimer’s
disease, which provides a causal interaction not taken into account by statisti-
cal models. Here, we propose to integrate causality in the synthesis of 3D brain
arXiv:2409.05585v1 [cs.CV] 9 Sep 2024
2 Peng et al.
MRIs so that their generation can be guided by (interventions on) factors such
as age, brain regions, and diagnosis.
Specifically we rely on probabilistic Structural Causal Model (SCM) [17,19],
which can incorporate the relationships between variables, directed from cause to
effect. It indicates that intervening on a cause should lead to changes in the effect,
rather than the other way around. This forces the model to consider our premises
and assumptions about causality [3]. Contrary to previous SCMs [15,29], we focus
on the synthesis of 3D high-fidelity image counterfactuals of real-world data.
However, building an SCM in a high-dimensional space presents considerable
computational challenges, particularly when dealing with voluminous 3D MRIs.
To that end, we build a counterfactual model that can learn the causality and
perform counterfactual generation in a latent space, which allows us to produce
3D brain MRIs of higher quality than other counterfactual models.
Our model first employs a VQ-VAE [14] to encode the high-dimensional 3D
brain MRI into a low-dimensional latent space. We then integrate a causal graph-
ical model [3,16] into this latent space such that interventions are performed in
the latent space rather than the observation space (which contains MRIs). The
three-step counterfactual procedure, i.e., abduction, action, and prediction, is
subsequently achieved by a Generalized linear Model (GLM) [13] with a closed-
form solution.
To the best of our knowledge, this is the first work that can perform high-
fidelity counterfactual generation of 3D MRIs. Our counterfactual generative
model does not only improve interpretability and generative performance but can
also diversify (MRI) datasets. Additionally, counterfactual explanations could be
key for preventive purposes, as it is inherently capable of showing how one’s brain
might change under certain ‘conditions’ such as prolonged substance use. Our
main contributions can be summarised as follows:
– We present a novel causal generative modeling framework in a latent space
for producing high-fidelity 3D MRI counterfactuals with Markovian proba-
bilistic causal models;
– Our model fulfills all three rungs of Pearl’s ladder of causation and the three-
step procedure of the counterfactual generation can be realized efficiently by
a novel GLM method;
– We show that our model can perform high fidelity 3D counterfactual gener-
ation and also demonstrate its axiomatic soundness of our counterfactuals
by evaluating its context-level generation.
2 Methodology
Our method comprises a two-stage process (outlined in Fig. 1) that performs
counterfactual generation of MRIs given attributes, e.g., age and brain regions
of interest (ROIs). Given an intervention on (at least) one of these attributes, a
Deep Structural Causal Model (DSCM) deploys a causal graph to compute the
counterfactual of the attributes. For example, consider a brain MRI scan of an 80-
year-old female, from which we extract the attributes age=80, sex=female, and
Latent 3D Brain MRI counterfactual 3
Fig. 1: Architecture of the LSCM for image: Stage I (black box) consists of a
VQ-VAE to encode the real 3D MRI to a quantized vector representation. Based
on this, in stage II (blue boxes), a Latent SCM is constructed. The three-step
procedure of counterfactual inference is achieved by a efficient GLM.
the two ROI scores ROI_1_score=0.3 and ROI_2_score=0.7. Now to generate
the brain MRI at age 50 years old, we intervene and set age=50, after which the
DSCM computes the counterfactual attribute values, e.g., age=50, sex=male,
ROI_1_score=0.4 and ROI_2_score=0.6. These counterfactual attributes are
then used to compute the counterfactual feature embeddings, which are fed into
a VQ-VAE decoder [14] in order to generate the corresponding MRI. These two
stages are discussed below.
2.1 Deep Structural Causal Models
A structural causal model (SCM) [17] is a triple M := ⟨U, V, F ⟩ consisting of
two sets of variables: (i) exogenous variables U = {u1, . . . , uN }; (ii) endogenous
variables V = {v1, . . . , vN }, and a set of functions known as causal mechanisms
F = {f1, . . . , fN }, which determine the values of the endogenous variables: vk :=
fk(pak, uk), where pak are the direct causes (parents) of vk. It is possible to
perform an intervention on any vk by, e.g., setting it to a constant do(vk := c)
and thereby disconnect vk from its causal parents. SCMs enable the estimation of
counterfactuals (hypothetical scenarios) via a three-step process: (i) abduction:
infer the posterior distribution P (U | X), which represents the current state of
the world; (ii) action: perform one or more interventions to obtain a modified
model; (iii) prediction: infer counterfactual values of the endogenous variables
using the modified model and the posterior P (U | X).
In this work, we use a deep structural causal model (DSCM) [15] to model
the causal relationships between patient attributes such as age, brain regions of
interest, and diagnosis. Following [4], a DSCM uses a conditional normalizing
flow [23] as the mechanism for each endogenous variable vk := fk(uk; pak), to en-
able tractable and explicit abduction of the exogenous noise uk = f −1
k (vk; pak).
4 Peng et al.
DSCMs can thereby tractably estimate counterfactuals given observed attributes
and intervention(s). The counterfactual of any vk is given by
bvk := fk(f −1
k (vk; pak); cpak), where cpak are the counterfactual parents of vk
resulting from an upstream intervention.
However, it has been shown that directly applying DSCMs [15] to 3D brain
MRIs is impractical due to the complexity of constructing and optimizing invert-
ible neural networks [27,23]. Therefore, we separate the MRI variables x from
V and learn its counterfactual in a low dimensional latent space. For this pur-
pose we adapt the Latent Structural Causal Model (LSCM) to 3D brain MRIs.
We achieve this by using a generalized linear model (GLM) [13] in the latent
space of a VQ-VAE model. After estimation of all counterfactual attributes, we
obtain the counterfactuals of image latent features z’s parents (see next section
for details of z), i.e. cpaz, which will be used to generate bz. The resulting coun-
terfactual feature encoding is converted into an MRI using the decoder of the
VQ-VAE model.
2.2 Latent SCM for 3D Counterfactual MRIs
To fully leverage the latent SCM described in Section 2.1, a high-dimensional
MRI x first has to be encoded to a corresponding latent feature z, for which
we use a VQ-VAE. Then, we obtain its counterfactual bz based on cpaz, which
is finally decoded to the MRI counterfactual bx. We now discuss the process
of obtaining these latent features z and estimating the counterfactuals bz, after
which decoding is trivial given a trained VQ-VAE.
Computing latent feature z. To obtain a latent feature z from a 3D MRI
x, we employ a Variational Autoencoder (VQ-VAE) [14]. VQ-VAEs comprise
an encoder E and decoder D, as well as a code book C that performs vector
quantisation (VQ). That is, code book C empowers mapping a 3D MRI x to a
quantized encoding.
The first step in the process of computing the latent feature z is using the
encoder E to map the 3D MRI x ∈ RD×H×W to its lower-dimensional fea-
ture representation z := E(x) ∈ Rd×h×w×nD . Correspondingly, the decoder D
aims to reconstruct a 3D MRI ˆx = D(z) from a compressed embedding z. The
training objective of the model is to minimize the reconstruction error. Once
latent feature z is obtained, we apply vector quantization, which discretizes the
continuous latent space using a set of embedding units C := {(k, e(k)) : k =
1, 2, . . . , NC and e(k) ∈ RnD }. This allows an MRI to be represented very effi-
ciently by a set of indices (a.k.a. code) as defined with respect to the code book.
Notably, this process also serves a regularization purpose to avoid overfitting the
model during training.
After the encoding and vector quantization steps are finished, the latent rep-
resentation z is described by d × h × w vectors of nD dimensions each. This is
achieved by searching the nearest neighbor of the embedding units in codebook
C for each feature vector in zc. In other words, each of the d × h × w vectors
zc ∈ RnD from z is paired with its closest representative e(k) from the codebook
Latent 3D Brain MRI counterfactual 5
such that the corresponding code (a.k.a. index) is I(zc) := arg mink(zc − e(k)).
Subsequently, e(I(zc)) is the representative such that the quantization of the la-
tent encoding may be formulated as Q(zc) = [e(I(zc))]z⊣ z, where [·]zc⊣ z denotes
the vectors zc defined according to z. Building on this foundation of vector quan-
tization, we propose a fine-grained quantization process that acquires multiple
quantizations per vector, namely the vector’s own quantization as well as the
quantization of its residual. However, we first reshape the MRI’s latent encoding
z ∈ Rd×h×w×nD to R2×d×h×w× nD
2 so that z consists of twice as many vectors
zh ∈ R nD
2 with half the size; in other words, each zc ∈ RnD gets split into two
vectors zh ∈ R nD
2 . For each of the vectors zh, we determine the code I(zh) as
done above (with e(k) ∈ R nD
2 also being half in size), and then compute the
code corresponding to the residual, i.e., I(zh − e(I(zh))), so that the quantized
encoding can be written as
Q(zh) := e(I(zh)) + e(I(zh − e(I(zh))). (1)
For every zc comprised of two zh, we stack the two quantized encodings Q(zh)
to reconstruct a vector of the original size, thus obtain a quantized vector rep-
resentation of z, which is subsequently fed into the decoder.
Estimate latent counterfactual bz and bx. Given feature z = Q(E(x)), we
will build a closed-form mechanism f for z, such that its exogenous variable is
given by uz = f −1
θ (paz, z). We achieve this by using the GLM. Specifically, we
first flatten the feature as a vector with dimension K = d × h × w × nD . Then,
a feature matrix Z ∈ RN ×K is built for all the N samples in the dataset. At the
same time, the paz for all training samples is accumulated to construct a matrix
P ∈ RN ×m, where m is the number of variables in paz. Then, an ordinary least
square estimator [13] is applied to solve the GLM’s normal equations so that the
linear parameters B can be represented by the closed-form solution
B = (PT P)−1PT Z.
Subsequently, the Abduction step of the counterfactual generation can be rep-
resented as the computation of the residual component UZ ∈ RN ×K acting as
the exogenous variable for features Z, which is given by
UZ = Z − PB = (I − (PT P)−1PT )Z.
Given that we construct P explicitly, we find that the solution for B, and thus
for UZ, is of closed-form. This is computationally feasible as we perform the
GLM in the latent space, not in the high-dimensional MRI space. Our method
is scalable and easily lends itself to cases containing many more samples. Like
metadata normalization [13], we can accumulate the B with momentum.
The Action step of the counterfactual generation can be realized by using
the causal graph to produce counterfactual parents cpaz. Once we have the cpaz,
the Prediction step of counterfactual generation can be achieved by adding
factor-specific embeddings back to the exogenous variable as follows:
bZ = UZ + bPB.
6 Peng et al.
Fig. 2: Counterfactual cases. Left: the counterfactual, the differences with origi-
nal input, and the uncertainty for each intervention. Right: The assumed causal
graph is based on [25]. Variables in the graph are: age (a), diagnosis (d), Parietal
(p), Frontal (f ), Insula (i), and latent features of MRI (z).
By recovering its spatial resolution and decoding using the generator, we get
the MRI counterfactual in the observation/MRI space, i.e., x = D(ˆz), ˆz ∈ bZ.
3 Experiments
We train our VQ-VAE model using 8566 t1-weighted brain MRIs pooled from
two datasets: the Alzheimer’s Disease Neuroimaging Initiative [20] (ADNI (1511
subjects)) and the National Consortium on Alcohol and Neurodevelopment in
Adolescence [1] (NCANDA (808 subjects)). Besides, 400 subjects are left for
testing. Processing includes denoising, bias field correction, skull stripping, affine
registration to a template (which correctes for difference in head size and thus sex
and race), and normalizing intensity values between 0 and 1. The voxel resolution
is 1mm and we pad each MRI to spatial resolution 144 × 176 × 144. Based on
the latent space of VQ-VAE, the DSCM model is trained on an in-house MRI
dataset (PIs Drs. Pfefferbaum and Sullivan) with 826 samples from 400 subjects
consisting of individual diagnosed with alcohol use disorder (AUD) and healthy
controls. Compared to controls, the brain regions frontal, insula, and parietal
lopes are smaller in those with AUD [25]. Therefore, we build our casual graph,
as in Fig. 2, using these five variables, i.e., age, diagnosis, frontal, insula, and
parietal. After the training, we evaluate the model with respect to its ability to
create counterfactuals and the anatomical plausibility of the synthetic 3D MRIs.
3.1 Counterfactual Generation
This part demonstrates that our model can perform different interventions to
the generation process and output high-fidelity 3D counterfactuals. Given a MRI
example, we apply an intervention do() to age and diagnosis in our causal graph
Latent 3D Brain MRI counterfactual 7
Fig. 3: Show cases. 2 views of real MRI vs synthetic MRIs generated by 4 models.
Our model can produce visually similar MRI to others but we can do counter-
factual generation.
and generate the 3D counterfactual accordingly. As shown in Fig. 2, we show
MRI counterfactual from different views. Our model demonstrates a strong coun-
terfactual generation ability with high fidelity. The diagnosis intervention shows
a global but subtle changes on the brain. For the age intervention, we can see
clear changes in the Ventricle, a region known to be significantly influenced by
the aging process.
3.2 Counterfactual Evaluation
By preforming random intervention, 400 counterfactuals are synthesized. We
then compare with synthetic MRIs from recent generative models and also eval-
uate it anatomical plausibility to qualify its value for neuroimaging studies. The
comparison methods include GAN models like CCE-GAN [30], VAE-GAN [12],
α-WGAN [10], and HA-GAN [26]; and diffusion models like latent diffusion
model (LDM) [21] and cDPM [18]. Qualitative results of all 7 models are shown
in Appendix while only the best four are shown in Fig. 3 due to space limita-
tions. Of those four methods, HA-GAN was the only method that was not able
to produced the MRIs that looked like those of healthy controls. On the other
side LDM produces the noisiest MRI and the MRI of cDPM has clear slice arte-
facts. In contrast, the MRI of our model shows clear gray-matter boundaries and
looks most similar to the real MRI. Finally, our model is much faster than the
diffusion-based method as their is no need for the multi-step diffusion process.
We evaluate its anatomical plausibility by comparing its subcortical similarity
with the real data. To this end, we run Freesurfer pipeline [5] for both of the
400 real samples and the synthetic ones to conduct the brain pacellation and
compute automatic segmentation volume (aseg). We report the effect size (using
Cohen’s d coefficient [2], d) to 24 sub-cortical differences with the real brain
MRIs. As shown in Table 1, 50% of the sub-corticals have a |d| < 0.2, which
means these brain regions are very similar to the real ones. More than 90% of
the 24 brain regions have an effect size that is less than 0.4, which suggests
8 Peng et al.
Table 1: Subcortical Measurement. Cohen’s d is reported here as the effect size
(Effect). Most of the ROIs show strong correlations with the real data as they
have a small value.
ROI vessel
volume
5th
ventricle
optic
chiasm
accumbens
area
choroid
plexus
cc_mid
posterior
cc
central
cc_mid
anterior
Effect 0.0065 -0.0547 -0.4014 -0.0028 0.2400 0.2540 -0.3216 -0.1132
ROI cc
anterior
cc
posterior
3rd
ventricle amygdala 4th
ventricle pallidum WM
hypo. caudate
Effect 0.3581 0.3939 -0.1804 -0.1686 -0.3383 0.5588 -0.0417 -0.1362
ROI hippo
campus putamen ventraldc thalamus lateral
ventricle
brain
stem
cerebellum
WM
cerebellum
cortex
Effect 0.2833 -0.1373 0.6070 0.3978 0.1306 0.2529 0.1690 -0.0748
that the observed difference between the real and synthetic is relatively minor
or subtle.
4 Discussion
While our work proposes novel methodology for embedding counterfactual rea-
soning into generative AI models to enhance interpretability and generative
performance, and outperforms state-of-the-art models in generating 3D MRIs,
several challenges remain. Firstly, building a DSCM is challenging on high-
dimensional data [3] without sacrificing the quality of generated counterfactuals.
This can be further hindered if the size of the dataset is small. To mitigate this
issue as much as possible, we compute counterfactuals in the lower-dimensional
latent space, although this requires us to control information at a scalar-level,
challenging the decoder to generate high-dimensional 3D MRIs.
Secondly, for disease prevention purposes in a clinical setting, it is non-trivial
to comprehensively assess the anatomical plausibility of the counterfactuals and
generated MRIs when we intervene on a diagnosis, as these present novel findings.
5 Conclusion
We propose a novel brain counterfactual for 3D MRI generation, which fully fills
three rungs of Pearl’s ladder of causation. The model adeptly performs causal
modeling while capable of generating high-quality 3D brain volumes. This is
achieved by our latent structural counterfactual model, which contains a deep
SCM constructed in a latent space. We then run a GLM on this low dimensional
latent space to synthesize counterfactuals, to mitigate significant computational
challenges posed by performing interventions directly on the 3D brain MRIs.
The generated MRIs show high anatomical resolution, while the counterfactual
generation contains an efficient intervention strategy to generate high-quality 3D
counterfactuals.
Latent 3D Brain MRI counterfactual 9
6 Acknowledgement
Collection and distribution of the NCANDA data were supported by NIH fund-
ing AA021681, AA021690, AA021691, AA021692, AA021695, AA021696, AA021697.
Access to NCANDA data is available via the NIAAA Data Archive, collection
4513 (https://nda.nih.gov/edit_collection.html?id=4513).
This work was partly supported by funding from the National Institute of
Health (MH113406, DA057567, AG084471, AA021697, AA017347, AA010723,
AA005965, and AA028840), the DGIST R&D program of the Ministry of Sci-
ence and ICT of KOREA (22-KUJoint-02), Stanford School of Medicine Depart-
ment of Psychiatry and Behavioral Sciences Faculty Development and Leader-
ship Award, the Stanford HAI-Google Research Collaboration, the Stanford HAI
Google Cloud Credits, the 2024 Stanford HAI Hoffman-Yee Grant, and the UK’s
Engineering and Physical Sciences Research Council (EPSRC) Hub for Causality
in Healthcare AI (grant number EP/Y028856/1).
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