Generative Method to Discover Genetically Driven Image Biomarkers

Abstract

We present a generative probabilistic approach to discovery of disease subtypes determined by the genetic variants. In many diseases, multiple types of pathology may present simultaneously in a patient, making quantification of the disease challenging. Our method seeks common co-occurring image and genetic patterns in a population as a way to model these two different data types jointly. We assume that each patient is a mixture of multiple disease subtypes and use the joint generative model of image and genetic markers to identify disease subtypes guided by known genetic influences. Our model is based on a variant of the so-called topic models that uncover the latent structure in a collection of data. We derive an efficient variational inference algorithm to extract patterns of co-occurrence and to quantify the presence of heterogeneous disease processes in each patient. We evaluate the method on simulated data and illustrate its use in the context of Chronic Obstructive Pulmonary Disease (COPD) to characterize the relationship between image and genetic signatures of COPD subtypes in a large patient cohort.

N.K. Batmanghelich and A. Saeedi—equal contribution.

Appendix: Variational Bayes Inference Procedure

Combining all components of the model defined in Sect. 2, we construct the joint distribution of all variables in the model (Fig. 6):

$$\begin{aligned}&p(\mathcal {D}, \mathcal {S}, \mathcal {P}) = \underbrace{\prod _{k=1}^{K}{ p(\mu _k,\varSigma _k;\eta ^I)\, p(\beta _k;\eta ^G)\, p(v_k;\omega )}}_{ \text {population-level topics}} \times \\&\prod _{s=1}^{S}\prod _{t=1}^{T}{ \underbrace{p(c_{st}|v_k) p(\pi _{st};\alpha )}_{\text {topics for subject { s}}} } \prod _{n=1}^{N}{ \underbrace{p(z_{sn}^I | \pi _{st})}_{\text {image topic}} \underbrace{p(I_{sn} | z_{sn}^I, c_{st}, \{ \mu _k,\varSigma _k \} )}_{\text {image likelihood}} }\\&\prod _{m=1}^{M}{ \underbrace{p(z_{sm}^G | \pi _{st})}_{\text {genetic topic}} \underbrace{p(G_{sm} | z_{sm}^G, c_{st},\beta _k)}_{\text {genetic likelihood}} }, \end{aligned}$$

where N and M are the number of supervoxels and minor alleles, respectively, identified for subject s.

Fig. 6.

Left: Graphical model that represents the joint distribution. The open gray and white circles correspond to the observed and the latent random variables, respectively. The full circles represent fixed hyper-parameters. Superscript I and G denote image and genetic parts of the model respectively. Right: Update rules for the variational parameters.

We choose a factorization for the distribution q that captures most model assumptions and yet is computationally tractable:

$$\begin{aligned} q(\mathcal {S}, \mathcal {P})&= \underbrace{\prod _{k=1}^{K} \text {NIW}(\mu _k,\varSigma _k; \tilde{\eta }^I_k) \, \text {Dir}( \beta _k;\tilde{\eta }^G_k )\, \text {Beta}(v_k; \tilde{\omega }_k)}_{\text {population-level topics}} \times \\&\prod _{s=1}^{S} \prod _{t=1}^{T} \underbrace{\text {Cat}(c_{st}; {\xi }_{st}) \, \text {Beta}(\pi _{st}; \tilde{\alpha }_{st})}_{ \text {topics for subject { s}}} \prod _{n=1}^{N} \underbrace{\text {Cat}(z_{sn}^I; {\phi }^I_{sn})}_{\text {image topic}} \prod _{m=1}^{M} \underbrace{\text {Cat}(z_{sm}^G; {\phi }^G_{sm})}_{\text {genetic topic}}, \nonumber \end{aligned}$$

where we choose an appropriate approximating distribution for each latent variable and use $\tilde{}$ to denote parameters of the approximating distributions. The optimization is defined in the space of the variational parameters $\left\{ \tilde{\eta }^I,\tilde{\eta }^G,\tilde{\omega },{\xi }, \tilde{\alpha }, {\phi }^I,{\phi }^G \right\}$. We omit the derivation of the updates due to space constraints; Algorithm 1 provides pseudocode for the resulting updates. We run the algorithm five times starting from different random initializations and report the result with the highest lower bound F(q) .

Once the algorithm converges, we estimate the population-level quantities of interest as means of the corresponding approximating distributions:

$$\begin{aligned} \hat{\mu }_k = \mathbb {E} \left[ \mu _k | \mathcal {D} \right] \approx \mathbb {E}_q \left[ \mu _k; \tilde{\eta }^I_k\right]&, \quad \hat{\varSigma }_k = \mathbb {E} \left[ \varSigma _k | \mathcal {D} \right] \approx \mathbb {E}_q \left[ \varSigma _k^I; \tilde{\eta }^I_k \right] ,\\ \hat{\beta }_k = \mathbb {E}\left[ \beta _k | \mathcal {D} \right] \approx \mathbb {E}_q \left[ \beta _k^G ; \tilde{\eta }^G_k\right] . \end{aligned}$$

Each expectation above can be easily evaluated from the parameters of the corresponding distribution. In addition, we construct spatial maps that display the posterior probability of each population topic for each supervoxel in a particular subject s to visually evaluate the disease structure in that subject.