Latent Space Approaches to Subtyping in Oncology Trials

Michael Kane and Brian Hobbs

Outline

Motivation: the "new" way clinical oncology trials are being conducted

The patient heterogeneity problem

Automated subtyping using latent space methods

Case study: predicting patient response by subtype

Case study: diagnosing mis-dosing based on adverse events

The "old" way of conducting clinical trials

New compound is developed and is thought to deliver a (small) benefit over current therapies for a specific histology

A large number of patients are enrolled (at least hundreds)

The response rate of the treatment population is tested against a control group

Entrectinib: Patient X Before

Entrectinib: Patient X After

Entrectinib

Targeted therapy (NTRK gene rearrangement)

Very stringent inclusion/exclusion criteria

Effective for other histologies (including breast, colorectal, and neuroblastoma)

8/11 responders for lung cancer in initial study

Breakthrough Trials

"A drug that is intended to treat a serious condition AND preliminary clinical evidence indicates that the drug may demonstrate substantial improvement on a clinically significant endpoint(s) over available therapies"

Benefits:

Priority review: expedited approval process
Rolling reviews, smaller clinical trials, and alternative trial designs

Alternative Designs

Often means single arm

Smaller populations

May include multiple histologies

Still work within FDA regulation, often including "all-comers"

Biomarker	Tumor Type	Drug	N	ORR (%)	PFS (months)
BRAF V600	NSCLC (>1 line)	Dabrafenib + Trametinib	57	63	9.7
ALK fusions	NSCLC (prior criz)	Brigatinib	110	54	11.1
ALK fusions	NSCLC (prior criz	Alectinib	225	46-48	8.1-8.9
EGFR T790M	NSLCLC (prior TKI)	Osimertinib	127	61	9.6
BRCA 1/2	Ovarian (>2 prior)	Rucaparib	106	54	12.8
MSI-H/MMR-D	Solid Tumor	Pembroliumab	149	40	Not reached
BRAF V600	Erdheim Chester	Vemurafinib	22	63	Not reached

Breakthrough Therapies

The patient heterogeneity problem

The patient heterogeneity problem

Hobbs, Kane, Hong, and Landin. Statistical challenges posed by basket trials: sensitivity analysis of the Vemurafinib study. Accepted to the Annals of Oncology.

A subtype is a group of patients with similar, measurable characteristics who respond similarly to a therapy.

Why isn't supervised learning enough?

> summary(lm(y ~ x1n + x2n - 1, ts1))

Call:
lm(formula = y ~ x1n + x2n - 1, data = ts1)

Residuals:
   Min     1Q Median     3Q    Max
-7.276 -2.695  0.260  2.341  7.358

Coefficients:
    Estimate Std. Error t value Pr(>|t|)
x1n   1.1018     0.2266   4.863 4.03e-05 ***
x2n   1.2426     0.2398   5.181 1.69e-05 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.647 on 28 degrees of freedom
Multiple R-squared:  0.6383,	Adjusted R-squared:  0.6124
F-statistic:  24.7 on 2 and 28 DF,  p-value: 6.572e-07

We can't tell this

from this

The model fits relationships between the regressors and the outcome

But we may want the relationships between the regressors with respect to the outcome

More formally start with supervised learning:

Consider \( n \) training samples \( \{x_1, x_2, .., x_n\} \), each with \(p\) features.

Given a response vector \(\{y_1, y_2, ..., y_n\}\), find a function \(h : \mathcal{X} \rightarrow \mathcal{Y}\) minimizing the \(\sum_{i=1}^{n} L( h(x_i), y)\) with respect to a loss function \(L\).

Construct \(h = f \circ g_y \) such that \(g_y : \mathcal{X} \rightarrow \mathcal{X'} \) is a latent space projection of the original data, whose geometry is dictated by the response.

Note that \(f\) is not parameterized by the response.

...and construct a (supervised) latent space

Let \(X \in \mathcal{R}^{n \times p}\) be a full-rank design matrix with \(n > p\), \(X = U \Sigma V\) is the singular value decomposition of \(X\).

where \(\Gamma\) is a diagonal matrix in \(\mathcal{R}^{p \times p}\), \(\mathbf{1}\) is a column of ones in \(\mathcal{R}^n\), and \(\varepsilon\) is composed of (sufficiently) i.i.d. samples from a random variable with mean zero and standard deviation \(\sigma\).

A simple example with OLS

Y = X V \Gamma \mathbf{1} + \varepsilon

Y = X V \Gamma \mathbf{1} + \varepsilon

Under the \(\ell^2\) loss function we can find the optimal value of \(\widehat \Gamma\) among the set of all weight matrices \(\tilde \Gamma\) with

The matrix \(\widehat \Gamma\) is \(\text{diag}(\widehat \beta)\) where \(\widehat \beta = \Sigma^{-1} U^T Y\) is the slope coefficient estimates of the corresponding linear model.

A simple example with OLS

\widehat \Gamma = \underset{\tilde \Gamma}{\operatorname{argmin}} \ ||Y - X V \tilde \Gamma \mathbf{1} ||

\widehat \Gamma = \underset{\tilde \Gamma}{\operatorname{argmin}} \ ||Y - X V \tilde \Gamma \mathbf{1} ||

\(X_Y = XV \tilde \Gamma\) represent the data in the latent space

Each column whose corresponding slope coefficient is not zero, contributes equally to the estimate of \(Y\) in expectation

A simple example with OLS

If the distance metric denoted by matrix \(A \in \mathcal{R}^{p \times p}\) and the distance between any two \(1 \times p\) matrices \(x\) and \(y\) expressed by

The square euclidean distance between two samples, \(i\) and \(j\) in \(X_Y\), denoted as \(X_Y(i)\) and \(X_Y(j)\) respectively is

A metric learning connection

d_A^2(x, y) = ||x - y||_A = (x - y) A (x - y)^T

d_A^2(x, y) = ||x - y||_A = (x - y) A (x - y)^T

|| X_Y(i) - X_Y(j) ||^2 = || X(i)V - X(j)V ||_{\Gamma}^2

|| X_Y(i) - X_Y(j) ||^2 = || X(i)V - X(j)V ||_{\Gamma}^2

A bias-corrected Gramian matrix estimate

\mathbb{P} X_Y X_Y^T - \sigma^2 \text{I} = U (\Sigma \Gamma)^2 U^T

\mathbb{P} X_Y X_Y^T - \sigma^2 \text{I} = U (\Sigma \Gamma)^2 U^T

\begin{aligned} \mathbb{P} \, X_Y X_Y^T & = \mathbb{P} \, U \, \Sigma \, \widehat \Gamma^2 \, \Sigma \, U^T \\ &= \mathbb{P} \, U \Sigma (\Gamma + \sigma \Sigma^{-1} \mathbf{Z})^2 \Sigma U^T \\ & = \mathbb{P} \, U (\Sigma^2 \Gamma^2 + 2 \sigma \Gamma \Sigma \mathbf{Z} + \sigma^2 \mathbf{Z}^2) U^T \\ \end{aligned}

\begin{aligned} \mathbb{P} \, X_Y X_Y^T &amp; = \mathbb{P} \, U \, \Sigma \, \widehat \Gamma^2 \, \Sigma \, U^T \\ &amp;= \mathbb{P} \, U \Sigma (\Gamma + \sigma \Sigma^{-1} \mathbf{Z})^2 \Sigma U^T \\ &amp; = \mathbb{P} \, U (\Sigma^2 \Gamma^2 + 2 \sigma \Gamma \Sigma \mathbf{Z} + \sigma^2 \mathbf{Z}^2) U^T \\ \end{aligned}

Proof:

Let \(\mathbf{Z}\) be a diagonal matrix of standard normals

A toy example with iris

> head(iris)
  Sepal.Length Sepal.Width Petal.Length Petal.Width Species
1          5.1         3.5          1.4         0.2  setosa
2          4.9         3.0          1.4         0.2  setosa
3          4.7         3.2          1.3         0.2  setosa
4          4.6         3.1          1.5         0.2  setosa
5          5.0         3.6          1.4         0.2  setosa
6          5.4         3.9          1.7         0.4  setosa
> 
> fit <- lm(Sepal.Length ~, iris[,-5])

A toy example with iris

> mm <- model.matrix(Sepal.Length ~ ., iris[,-5])
> 
> km <- kmeans(mm, centers = 3)
> table(km$cluster, iris$Species)
   
    setosa versicolor virginica
  1     21          1         0
  2     29          2         0
  3      0         47        50
>
> # ...
> table(subgroups$membership, iris$Species)
   
    setosa versicolor virginica
  1     50          0         0
  2      0         23         0
  3      0         27        20
  4      0          0        30

Comparison with K-means

> mm <- model.matrix(Sepal.Length ~ ., iris[,-5])
> 
> km <- kmeans(mm, centers = 4)
> table(km$cluster, iris$Species)

    setosa versicolor virginica
  1      0         24         0
  2     50          0         0
  3      0          0        36
  4      0         26        14
>
> # ...
> table(subgroups$membership, iris$Species)
   
    setosa versicolor virginica
  1     50          0         0
  2      0         23         0
  3      0         27        20
  4      0          0        30

Comparison with K-means

Back to cancer

Clinical trial data is not low-dimensional

Sometimes the predictive information isn't in a linear subspace of the data

Anonymous drug 1

Received "accelerated approval"

Response rate was high, but not quite as high as expected

Subtype response based on baseline characteristics

All Trajectories

Low Risk

Intermediate Risk

High Risk

Anonymous Drug 1

Was found to be effective for a certain type of cancer.

Ran into problems with severe toxicity events (449 toxicities out of 607).

Goal was to find subtypes least (or most) likely to have toxicity events.

Variable	Description
AMD19FL	Exon 19 Del. Act. Mut. Flag
AM858FL	L858R Activating Mut. Flag
LIVERFL	Mets Disease Site Liver Flag
DISSTAG	Disease Stage at entry
NUMSITES	Num. of Mets Disease Sites
PRTK	Number of Prior TKI
PRTX	Number of Prior Therapies
WTBL	Baseline Weight
SEX

AE Heatmap

AE Similarity Network

What else can you do with subtyping?

1. Improve prediction accuracy:

Recall \(g_y : \mathcal{X} \rightarrow \mathcal{X'} \)
Combine new samples with an \(f\) parameterized by \(Y\), \(h' = f_y \circ g_y \)
E.g. Bayesian power prior

2. Construct counterfactuals and create synthetically controlled trials.

Obtain data from other trials (standard of care)
Project data into the latent space
Match
Compare results

Thanks

Latent Space Approaches to Subtyping in Oncology Trials

By Michael Kane

Latent Space Approaches to Subtyping in Oncology Trials

1,512

Michael Kane

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