Investigating Causal Genetic Effects on Overall Survival of Glioblastoma Patients using Normalizing Flow and Structural Causal Model

Abstract

Glioblastoma (GBM) is the most common and aggressive brain tumor with short overall survival (OS) of about 15 months. Understanding the causal factors affecting the patient survival is crucial for disease prognosis and treatment planning. Although previous efforts on survival prediction using multi-omics data has yielded useful predictive models, the causation of the correlated genetic risk factors has not been addressed. Recent advances in causal deep learning models enable the study of causality from complex dataset. In this paper, we leverage the recently proposed structural causal model (SCM) with normalizing flows parameterized by deep networks to perform the counterfactual query to investigate the causal relationship between gene mutation and OS with the presence of other confounders including sex, age and radiomics features. The query amounts to the question that what the survival days will be if the gene mutation status has been changed, i.e., from mutant to non-mutant and vice versa. The trained causal model will infer the counterfactual outcome given the intervention on specific gene mutation. We apply multivariate Cox-PH model to find the genes associated with survival, and investigate the causal genetic effect by comparing the original and counterfactual survival days in a bi-directional fashion. Particularly, the following two scenarios are considered: (1) intervention on a specific gene with non-mutant status to generate the counterfactual survival days as if the gene is mutant, with which the original survival days of the subjects with that mutant gene will be compared; (2) intervention on the gene with mutant status and perform the comparison with survival days of subjects with that non-mutant gene. Our experimental results show that no causation of two correlated genes (NF1, RB1) was revealed in the cohort (n=181), while their genetic effects on OS in terms of prolonging or shortening are generally in accordance with clinical findings.

1. INTRODUCTION

Glioblastoma (GBM) is the most common and agressive malignant primary adult brain tumor with a median overall survival (OS) of 14.6 to 16.7 months.¹ Maximal safe surgical resection combined with radiotherapy and temozolomide (TMZ) chemotherapy has been the standard treatment pipeline for GBM in clinical practice.² There have been studies on using multi-omics data, i.e., clinical information, radiomics and genomics data, to predict the OS and derive radiomic signature or survival prediction index.³ As a result, these studies often present the correlation between the risk factors and the OS without addressing the causal aspect, since correlation does not imply causation. Uncovering the causal mechanism of cancer prognosis will bring benefits to the scientific research and clinical practice, e.g., causal gene of certain clinical phenotypes can serve as more precise drug targets.

Causal deep learning (DL) models integrate causality into DL research which enable the answer to interventional and counterfactual questions. This paper builds upon the deep structural causal model^4,5 to formulate the causes (gene mutation status, sex, age, imaging phenotype) and effects (survival days after surgery) via a causal graph and learn the structural assignments by using normalizing flows parameterized by deep networks. To the best of our knowledge, this work should be among the first to apply causal DL models for studying the causality for disease prognosis, and can be further extended to identifying causal relationship in other applications of biomedical domain.

2. METHODS

2.1. Study Cohort

Multi-parametric MRI (mpMRI) scans (T1, T1-Gd, T2, T2-FLAIR, DSC, DTI) of patients diagnosed with glioblastoma were retrospectively collected. Radiomics features, including histograms, morphological and textural descriptors, were derived using Cancer Imaging Phenomics Toolkit (CaPTk).⁶ We applied non-negative matrix factorization (NMF) to reduce the dimension of imaging features and used the coefficients as covariates. Genetic markers were obtained through a targeted next generation sequencing (NGS) panel on the resected tumor samples from the patients. A total number of 27 genes are included in the NGS panel from which 5 most frequent genes were selected for analysis. Clinical information include sex, age at the scan and survival days from the surgery. The final cohort consists of n = 181 IDH-wildtype patients with all radiomics, genomics, and clinical measures available. Specifically, we investigate the causal relationship of the survival days with respect to the 5 gene mutation status (EGFR, NF1, PTEN, RB1, TP53), while age, sex, 3 NMF coefficients are the other confounders.

2.2. Structural Causal Model

A structural causal model (SCM) denoted as M = (S, P(ϵ)) consists of a collection S = (f₁, …, f_K) of structural assignments x_k = f_k(ϵ_k; pa_k), where pa_k represents the set of parents (direct causes) of x_k and ϵ_k are noise variables of the unknown sources of variation for x_k. A SCM satisfies the causal Markov condition that each variable x_k is independent of its non-effects given the direct causes. Therefore the joint observational distribution of a SCM can be factorised as $P_M(\text{x})={\prod }_{k=1}^K{P}_M\left(x_k\mid p{a}_k\right)$, where each conditional distribution P_M(x_k|pa_k) is characterized by the corresponding structural assignment and noise distribution. The joint distribution of exogenous noise variables can be expressed as $P(ϵ)={\prod }_{k=1}^{K}P\left({ϵ}_k\right)$.

Given the above formulation, we perform a counterfactual query following the three-step process: abduction, action and prediction.⁷ The abduction step is the prediction of the exogenous noise ϵ given the observations x which is to infer the posterior P_M(ϵ|x). Next comes the action step where an intervention, denoted by $\text{do}\left(x_k≔{\tilde{x}}_k\right)$, is conducted according to the desired manipulation. A modified SCM $\tilde{M}=M_{\text{do}(\tilde{x})}=\left(\tilde{S},P_M(ϵ\mid x)\right)$ is obtained as a result. Finally, in the prediction step, the counterfactuals are predicted through sampling from the distribution $P_{\tilde{M}(x)}$ entailed by the modified SCM.

2.3. Normalizing Flow

Normalizing flow can model a complex probability density by applying transformation to simpler base distribution with same dimensionality. Given an observed variable x, an invertible and differentiable transformation f(·), and base variable ϵ ~ P(ϵ) such that x = f(ϵ), the output density is computed as p(x) = p(ϵ)|det∇f(ϵ)|⁻¹ where ϵ = f⁻¹(x) and ∇f(ϵ) as the Jacobian of the flow f : ϵ ↦ x. Denote a dataset with sample size of n as $D={\left\{x^i~p(x)\right\}}_{i=1}^n$, a normalizing flow can be fit by maximizing the likelihood to obtain the parameters:

$${\theta }^{*}=\text{argmax}\frac{1}{n}\sum _{i=1}^n\text{log}p\left({ϵ}^i\right)-\text{log}\left|\text{det}\nabla f_{\theta }\left({ϵ}^i\right)\right|$$

where we use a deep neural network to parameterize the normalizing flow. The above approach can be used for the conditional densities P_M(x_k|pa_k) in SCM. The flows $f_{{\theta }_k}$ will be fit to map exogenous noise ϵ to effect x_k (survival days) given parents pa_k (sex, age, NMF coefficents of radiomics features, gene mutation). The abduction step involving the flow is essentially ${ϵ}^i=f_{\theta }^{-1}\left(x^i\right)$. We can manipulate the value of a particular variable for the flow as the intervention $\text{do}\left({\tilde{x}}_k\right)$, as in our case we change the gene mutation status from 0 (non-mutant) to 1 (mutant) and vice versa. Then the modified flow ${\tilde{f}}_{\theta }$ can be conducted to sample from $P_{\tilde{M}(x)}$ to get counterfactuals (survival days w.r.t. change in gene mutation status).

3. RESULTS

The multivariate Cox-PH model and statistical analysis is performed in R. Figure 2 shows that NF1 (p-value: 0.0133 < 0.05) and RB1 (p-value: 0.0072 < 0.05) were significantly associated with survival days after surgery. The flow-based SCM is implemented with quadratic autoregressive spline⁸ by using PyTorch and Pyro.⁹ Categorical distribution is used for sex and gene mutations, and we use real-valued normalizing flow for other structural assignments. Specifically, the structural assignments for age and NMF coefficients are linear flows, while that for survival days is a conditional flow conditioned on the activations of a fully-connected network with age, sex, NMF coefficients and gene mutations as input. For training details, we use Adam as the optimizer, set learning rate as 3 × 10⁻⁴, weight decay as 10⁻⁴ and the number of epochs as 100. Figure 3 and Figure 4 show the distributions of factual and counterfactual survival days under the intervention of specific gene mutation status. No statistical significance was observed from the t-test. The genetic effects on survival days from the counterfactuals are in accordance with literature, e.g., mutation in RB1 may prolong the survival.¹⁰ These results indicate that correlation may not imply causation and it is desired to uncover the causal factors underlying the disease prognosis.

Figure 2.

Survival curves show the association of gene mutations with survival after surgery in GBM patients. NF1 and RB1 are significantly associated with survival among the 5 most frequent genes.

Figure 3.

Intervention on a specific gene to alter its status from non-mutant to mutant to observe how the counterfactual survival days would change comparing with the original survival days of subjects with non-mutant status of that gene.

Figure 4.

Intervention on a specific gene to alter its status from mutant to non-mutant to observe how the counterfactual survival days would change comparing with the original survival days of subjects with mutant status of that gene.

4. CONCLUSION

The current study demonstrated the utility of causal DL models in biomedical domain and drew the conclusion that significantly associated genes may not be the underpinning causes for influencing the survival days after surgery in GBM patients. Future directions include defining causal graphs with intermediate nodes for secondary phenotypes, i.e, taking account of the causal relationship from gene to imaging features via causal mediation, adding more genes or other risk factors into the model, and also applying the causal inference method on multi-institutional data consortium with larger sample size.

Figure 1.

Causal graph of SCM for glioblastoma where the causal effect of survival days after surgery is studied. The nodes consist of sex (s), age (a), NMF coefficients (i₁ to i₃), multiple key driver genes (g₁ to g₅) and survival days after surgery (x), with corresponding exogenous noise terms $({ϵ}_s,{ϵ}_a,{ϵ}_{i_1},\dots ,{ϵ}_{i_3},{ϵ}_{g_1},\dots ,{ϵ}_{g_5},{ϵ}_x)$. Invertible normalizing flow models are denoted by bidirectional arrows. The black dot indicates a conditional flow where x is conditioned on all the parent nodes.