A ﬂexible deep learning framework for survival

analysis with medical data

Gabriele Campanella

1,2(B)

, Ida Häggström

3,4(B)

, Lucas Kook

, Torsten

Hothorn

, and Thomas J. Fuchs

1,2

Windreich Department of AI and Human Health, Icahn School of Medicine at

Mount Sinai, USA

Hasso Plattner Institute at Mount Sinai, Icahn School of Medicine at Mount Sinai,

USA

gabriele.campanella@mssm.edu

Dept of Electrical Engineering, Chalmers University of Technology, Sweden

Institute of Clinical Sciences, Gothenburg University, Sweden

idah@chalmers.se

Institute for Statistics and Mathematics, WU Vienna, Austria

Epidemiology, Biostatistics and Prevention Institute, University of Zurich,

Switzerland

Abstract. Medical imaging data and electronic health records are an in-

tegral part of clinical routine and research for prognostication of patient

survival and thus directly inform patient management. However, stan-

dard regression models used to derive patient prognoses are ill-equipped

to handle such non-tabular data directly. Several neural network architec-

tures based on classiﬁcation or the Cox model have been proposed. Here,

we present deep conditional transformation models (DCTMs) for survival

applications with medical imaging data. DCTMs include the Cox model

as a special case, but parameterize the log cumulative baseline hazards

via Bernstein polynomials and allow the speciﬁcation of non-linear and

non-proportional hazards for both tabular and non-tabular data and ex-

tend to all types of uninformative censoring. DCTMs yield moderate to

large performance gains over state-of-the-art deep learning approaches to

survival analysis on a multitude of publicly available datasets featuring

tabular or imaging data from radiology and pathology.

Keywords: Computational pathology · deep learning · radiology · sur-

vival analysis · transformation models.

1 Introduction

Arguably one of the most important aspects of health and medical research is

being able to understand, prognosticate and predict patient survival in order

to improve patient management and ultimately extend their life span or time

in remission [9]. Survival analysis is used for these purposes to study time-to-

event information relating to for example death, response to treatment, adverse

treatment eﬀects, disease relapse, and the development of new treatments [4].

2 G. Campanella et al.

Traditional approaches, such as Cox proportional Hazards [5, CPH], relied on

tabular features and are not amenable to analyze high-dimensional non-tabular

data such as medical images. With recent advances in computer vision and deep

learning (DL), there has been increasingly more interest in performing survival

analysis directly from high-dimensional data in order to automatically learn

patterns that stratify patients based on their outcome without the need for

feature engineering [17,22,14,2].

1.1 State-of-the-art in deep learning for survival analysis

Early deep learning (DL) approaches to survival analysis [20] drew inspiration

from the Cox proportional hazards (CPH) model [5], using the partial likeli-

hood as a loss function and extending it to handle piece-wise constant haz-

ards, non-proportional hazards, and non-linear eﬀects. For instance, DeepSurv

[13] optimizes the ℓ

-regularized log partial-likelihood, while [24] introduces a

mixture of Cox models using variational inference. In contrast to these semi-

parametric methods, [23] proposes a fully parametric variational framework.

Other approaches include a generative Weibull model [25] and piece-wise ex-

ponential hazard functions within a penalized likelihood loss [6]. Additionally,

“DeepHazard” [32] uses a squared error loss based on the counting process, ac-

commodating time-varying covariates, non-linear eﬀects, and non-proportional

hazards while accounting for censoring.

Beyond Cox-inspired methods, survival analysis can also be framed as a clas-

siﬁcation problem. For example, [30] introduce “neural multi-task logistic regres-

sion” (N-MTLR), which divides the time axis into intervals and uses a softmax

ﬁnal layer to model event indicators. Similarly, [18] propose “dynamic DeepHit,”

an extension of “DeepHit” [19], which also employs a softmax ﬁnal layer and

handles competing risks and non-linear eﬀects.

Our proposed class of models belongs to the family of deep conditional trans-

formation models [26] (DCTMs), which have been studied for continuous [1]

and discrete [15] outcomes and can be understood as a conditional normalizing

ﬂow [16]. DCTMs extend the ﬂexible class of conditional transformation models

[10,11] (CTMs) with deep neural networks to handle non-tabular data, such as

images.

1.2 Our contribution

In this paper we present DCTMs for survival analysis with medical data as a

ﬂexible framework for deep learning based survival analysis rooted in statistical

modeling and including the CPH model as a special case.

– Our DCTMs allow the speciﬁcation of non-linear and non-proportional haz-

ards for both tabular and non-tabular (image or text) data.

– Our DCTMs can be understood as a ﬂexible modular survival head which

can be combined with arbitrary feature extractors and foundation models

tailored towards speciﬁc applications.

A ﬂexible deep learning framework for survival analysis with medical data 3

– DCTMs are shown to yield superior prediction and discrimination perfor-

mance in terms of a time-dependent c-index on several tabular, radiological

and histopathological datasets.

To the best of our knowledge, this paper is the ﬁrst to extend DCTMs to survival

regression and present strong empirical evidence for improved performance on

relevant medical imaging tasks over established approaches.

2 Background

Survival analysis. Let (T

∗

, C, X) ∈ R

× R

× X , where T

∗

denotes the true

event time, C denotes the censoring time and X denotes features. We observe

n i.i.d. realizations {(t

, x

, δ

)}

i=1

of (T, X, δ), where T = min{T

∗

, C} and δ =

1(T < C) is the event indicator. Survival analysis targets the distribution of the

event time T conditional on features X, on the scale of the survivor function,

T |X

(t|x) := P(T > t|X = x). One of the most popular choices is the CPH

model [5],

T |X

(t|x) = exp(−Λ(t|x)) = exp(−Λ

(t) exp(x

⊤

β)), (1)

where Λ : R

× X → R

denotes the positive, increasing cumulative hazard

function. The hazard function is assumed to be decomposable into a baseline

hazard Λ

: R

→ R

, independent of the covariates, and multiplicative feature

eﬀects exp(x

⊤

β) with coeﬃcients β representing log-hazard ratios [4]. The CPH

model is estimated by maximizing the maximum partial likelihood.

Conditional transformation models. CTMs are distributional regression

models of the form [10]

T |X

(t|x) = 1 − F

(h(t|x)), (2)

where the conditional survivor function of the response given the features T |X =

x is decomposed into an a priori chosen and parameter-free target distribution

and a conditional transformation function h, which depends on the features

X = x. In order for S

T |X

to be a valid survivor function, it is suﬃcient for h to

be continuous and monotonically increasing in t for all x ∈ X [11]. CTMs can

be estimated via maximum likelihood and allow various kinds of responses and

uninformative censoring. The model in (2) is closely connected to the CPH model

in (1): For F

(z) = 1 −exp(− exp(z)) (the minimum extreme value distribution)

and h(t|x) = log Λ(t|x), the two models coincide.

3 DCTMs for survival analysis

We propose DCTMs (Figure 1), which parameterize the transformation function

h in (2) via (deep) neural networks. For instance, let ϕ : X → R

denote a fea-

ture extractor, which maps the input x to a feature vector of dimension d. The

4 G. Campanella et al.

Fig. 1. Proposed deep conditional transformation model (DCTM) architecture. The

model processes tabular or non-tabular explanatory variables (e.g., images) for survival

data, handling exact or right-censored event times. Non-tabular inputs are mapped to

a latent feature space using a feature extractor ϕ (e.g., a neural network), followed by

the DCTM head. DCTMs can be trained by minimizing the negative log-likelihood

(NLL). The right panel shows the transformation function (h), cumulative distribution

function (CDF), probability density function (PDF), and NLL for an exact event.

extracted features are ﬁnally passed through a last layer with linear activation

and weights w. We propose diﬀerent parameterizations for the transformation

function to control the desired complexity of the model: A shift, scale and general

DCTM with a non-linear function in t using Bernstein polynomials. The com-

plexity of DCTMs can be tuned via cross-validation or on a hold-out set. For all

experiments, we choose the standard logistic cumulative distribution function

(CDF) σ(z) = (1 + exp(−z))

−1

as the target distribution. Consequently, the

transformation function can be interpreted on the log-odds scale.

Assuming T takes values in an interval [l, r] ⊂ R

, the shift DCTM, DCTM

employs Bernstein polynomials of order K and additive feature eﬀects,

h(t|x; ϕ) = b(t)

⊤

ϑ + ϕ(x)

⊤

w, where b : [l, r] → R

K+1

, w ∈ R

where b

(t) :=





(1 −

K−k

and

t =

t−l

r−l

. Ensuring ϑ

k+1

> ϑ

, k = 0, . . . , K,

for all x, is suﬃcient to ensure monotonicity of h(t|x; ϕ) in t [10]. We enforce

monotonicity via ϑ = g(γ) =



, γ

+ softplus(γ

), . . . , γ

K+1

k=2

softplus(γ

)



For the more ﬂexible shift scale DCTM, DCTM

, we use

h(t|x; ϕ) = softplus(ϕ(x)

⊤

β) · b(t)

⊤

ϑ + ϕ(x)

⊤

which allows for non-proportional hazards using a scale term. Allowing the pa-

rameters of the Bernstein polynomials to fully ﬂexibly depend on x, we arrive

at the most ﬂexible general DCTM, DCTM

h(t|x; ϑ) = b(t)

⊤

ϑ(ϕ(x)), where ϑ : R

→ R

K+1

The general DCTM

, allows a ﬂexible baseline hazard function that captures

also higher moments of S

T |X

, resulting in a distribution-free model [27,16].

A ﬂexible deep learning framework for survival analysis with medical data 5

3.1 Training and evaluating DCTMs

DCTMs are ﬁtted by minimizing the empirical NLL: A single observation (t, x, δ)

contributes

L(h; t, x, δ) =

(

σ(h(t|x))(1 − σ(h(t|x))h

′

(t|x) δ = 1,

1 − σ(h(t|x)) δ = 0.

to the NLL. The empirical NLL is then given by NLL = −

i=1

log L(h; t

, x

, δ

and minimized over the n

training samples using any deep learning optimiza-

tion routine. Interval- and left-censored observations can also be handled with

the proposed method see, e.g., [11].

After ﬁtting a DCTM, the conditional survivor function of a test observation

can be computed from the estimated parameters via

T |X

(t|x) = 1− F

(

h(t|x)).

Now, all evaluation metrics can be computed from (some form of) the predicted

conditional distribution. The time-dependent c-index [8] is deﬁned as

c(t) =

i=1

j=1

1(t

< t

) · 1(

Λ(t|x

) > Λ(t|x

)) · 1(t

< t)

i=1

j=1

1(t

< t

) · 1(t

< t)

and measures the concordance between event times and predicted hazards. The

integrated inverse probability of censoring weighted Brier score (IBS)

IBS =

max

i=1

1(t

≤ t)

T |X

(t|x

)

+ I(t

> t)(1 −

T |X

(t|x

))

G(t

)

measures quality of probabilistic predictions, where

G(t) denotes an estimate of

the unconditional probability of not being censored until time t.

4 Experiments and results

We extensively evaluate DCTMs against competing methods on several datasets

involving tabular features, computed tomography (CT) images, and histopathol-

ogy whole slide images. The experiments are carried out as ablation studies to

explore the ﬂexible DCTM framework with varying parametrization complexity

and order K of the Bernstein polynomial, denoted DCTM

{S,SS,G}

. C-indices

were calculated at the training cohort event time quantiles 25%, 50%, 75% and

100%, and IBS was integrated over time steps of size 0.01 from 0 to the largest

test set event time. All DCTMs and experiments were implemented in Pytorch,

unless indicated otherwise. The source code for the DCTM framework is avail-

able on GitHub at https://github.com/sinai-computational-pathology/DCTM.

4.1 Tabular data

We ﬁrst compare our DCTMs to state-of-the-art methods (CPH model, random

survival forest [RSF], and DeepSurv) on tabular benchmark datasets. The GBSG

6 G. Campanella et al.

Table 1. Test c-indices (CI, ↑) evaluated at diﬀerent event time quantiles and

IBS (↓) for several models on three tabular datasets. DS=DeepSurv.

Metric CPH RSF DS DCTM

DCTM

gbsg

25 0.600 0.638 0.655 0.705 0.696 0.686 0.687 0.700 0.707

50 0.605 0.619 0.650 0.691 0.684 0.676 0.682 0.684 0.691

75 0.600 0.603 0.648 0.687 0.679 0.674 0.678 0.676 0.679

100 0.600 0.603 0.648 0.687 0.679 0.674 0.678 0.676 0.679

IBS 0.125 0.129 0.180 0.110 0.115 0.116 0.116 0.115 0.115

metabric

25 0.591 0.669 0.563 0.643 0.654 0.684 0.676 0.694 0.692

50 0.601 0.605 0.567 0.648 0.652 0.659 0.645 0.666 0.653

75 0.594 0.566 0.550 0.631 0.634 0.633 0.616 0.634 0.615

100 0.589 0.545 0.548 0.629 0.631 0.620 0.576 0.615 0.605

IBS 0.215 0.251 0.203 0.227 0.226 0.220 0.229 0.224 0.230

support

25 0.569 0.642 0.540 0.586 0.603 0.602 0.619 0.602 0.612

50 0.575 0.621 0.520 0.605 0.612 0.611 0.610 0.613 0.616

75 0.577 0.601 0.517 0.609 0.614 0.614 0.611 0.616 0.616

100 0.571 0.588 0.515 0.610 0.614 0.615 0.610 0.616 0.610

IBS 0.251 0.273 0.304 0.253 0.243 0.235 0.235 0.236 0.231

dataset contains 686 breast cancer patients with 56% censoring and 7 features.

The METABRIC data contains 1980 breast cancer patients with 42% censoring

and 9 features. The SUPPORT dataset contains 9105 hospitalized adults, 32%

censoring and 14 features. We use the same train/test splits as in [13]. For

each dataset, explanatory variables were z-normalized using the training split,

while the time to event data was normalized in the range 0–1. Cox and RSF

models were trained using the survival and ranger packages in R, respectively.

For DeepSurv and DCTM, the encoding portion of the model consisted of a linear

layer followed by a ReLU non-linearity. The survival head consisted of a linear

layer projecting features to the risk score for DeepSurv, and a DCTM survival

head with diﬀerent parametrizations using K = 1 or 10 bases. Models were

trained for 15 epochs with a learning rate of 0.001 using the L-BFGS optimizer.

Results are presented in Table 1. In terms of c-index, except for the two

earlier time points in the SUPPORT data, DCTMs outperform CPH, RSF and

DeepSurv on all other datasets and for all time points. For the GBSG data, the

most complex DCTM

performs best, whereas the simpler DCTM

and shift-

scale (K = 10) perform best on the METABRIC and SUPPORT data. In terms

of probabilistic predictions (IBS), DCTMs outperform the baseline methods on

the GBSG and SUPPORT data, while DeepSurv performs best on METABRIC.

4.2 Radiology data

The radiology data comprised 3346 volume CT scans of patients with head &

neck cancer from the open RADCURE cohort with curated disease-free survival

time [29]. 780 (23%) subjects had an event and the rest were censored. The

A ﬂexible deep learning framework for survival analysis with medical data 7

Table 2. Cross-validation c-indices (CI, ↑) of the test set for the models evaluated

at diﬀerent event time quantiles of the CT-image dataset, and IBS (↓). Values

are average±std (%). DS=DeepSurv, DH=DeepHit.

Metric DS DH DCTM

DCTM

radcure

25 66.2±1.2 73.0±1.1 75.0±0.7 74.8±0.9 74.2±0.8 74.8±0.9 74.4±0.6 74.2±0.4

50 65.8±1.1 72.0±1.0 73.4±0.7 73.3±0.7 72.9±0.7 73.3±0.9 72.9±0.6 72.8±0.5

75 65.8±1.1 72.0±1.0 73.4±0.7 73.3±0.7 72.9±0.7 73.2±0.9 72.9±0.6 72.8±0.5

100 65.8±1.1 50.8±21.7 73.4±0.7 73.3±0.7 73.0±0.6 73.0±1.0 73.0±0.6 72.7±0.5

IBS 13.2±0.1 16.7±0.0 10.7±0.2 10.2±0.5 10.5±0.2 9.9±0.3 10.3±0.3 9.7±0.3

planned target volumes (i.e. tumor ROIs as RTstructs) for radiotherapy were

also available, which were converted to binary masks. All CTs and tumor masks

were resampled to an isotropic voxel size of 3 × 3 × 3 mm. Of the cohort, 750

(22%) patients were held out for testing according to RADCURE splits, and the

remaining 2596 (78%) were randomly divided for 5-fold cross-validation. The

event times were normalized to range 0–1, and the CT images were clipped

to [−1000, +1000] HU and min-max normalized to 0–1. The tumor ROI masks

were added as a second image channel. A 3d ResNet34 backbone [7] was used

for feature extraction (size 512) of the image volumes, followed by a survival

head based on DeepSurv, DeepHit or variants of our DCTM. We used 3000

time thresholds in DeepHit. The DCTM variants were trained by minimizing

NLL, DeepSurv by minimizing the log partial-likelihood [13] and DeepHit by

minimizing cross entropy (CE) [19] (no ranking loss). For DeepSurv and DeepHit

we calculated a risk score (log cumulative hazard for DeepSurv and survival

probability for DeepHit) for each patient by including an additional trainable

linear layer in the model with image features as input and output size 1 for

DeepSurv, and 3000 for DeepHit. Models were trained for 20 epochs with a

learning rate of 0.001 using the SGD optimizer. The ﬁnal results were calculated

as the average across the ﬁve cross-validation models, applied to the test set.

Results are shown in Table 2. The DCTM models outperform all competitors

at each quantile by a considerable margin. While the simplest DCTM

achieves

the best discrimination performance, the IBS indicates that the most complex

DCTM

produces the best probabilistic predictions.

4.3 Histopathology data

For the histopathology experiments we leveraged three cohorts from the TCGA

[28] dataset containing overall survival information and whole slide images:

BRCA—1023 breast cancer patients with 86% censoring, LUAD—55 lung ade-

nocarcinoma patients with 65% censoring, and UCEC—548 patients with uterine

corpus endometrial carcinoma with 83% censoring.

For the analysis of the digital slides we extracted features using the state-

of-the-art foundation model UNI [3]. The models then consisted of a gated MIL

attention model [12] to aggregate features over the slide and a survival head. We

8 G. Campanella et al.

Table 3. Cross-validation c-indices (CI, ↑) for models evaluated at diﬀerent

event time quantiles of three histopathology datasets, and IBS (↓). Values are

average±std (%). DH=DeepHit.

Metric DH [21] DCTM

DCTM

brca

25 68.9±9.1 70.5±10.7 66.4±6.0 70.0±9.4 73.3±0.9 74.2±7.4 70.3±11.2

50 64.2±8.2 69.0±8.3 65.6±9.9 67.8±7.7 68.7±3.7 71.0±5.0 66.6±9.5

75 63.3±8.1 67.1±7.5 64.6±7.8 66.6±8.8 67.6±4.0 69.7±6.0 64.7±7.8

100 62.5±7.5 66.9±6.4 64.3±6.9 65.3±5.3 66.0±6.0 67.5±6.3 63.2±8.7

IBS 22.2±4.2 10.8±0.6 10.0±1.7 9.3±1.1 9.7±2.8 9.5±1.4 9.6±1.3

luad

25 64.5±7.7 65.4±6.9 62.2±3.8 61.1±7.3 62.9±6.6 60.7±8.1 61.2±5.9

50 62.3±8.0 62.2±7.9 61.2±5.9 60.5±4.1 60.9±7.3 59.0±5.2 60.1±5.3

75 61.1±6.1 61.8±5.0 61.0±5.2 61.0±5.2 61.1±5.0 59.0±6.5 59.4±5.3

100 62.1±5.0 61.7±4.7 61.0±4.5 58.7±8.0 60.0±5.9 57.6±7.1 57.4±5.0

IBS 29.9±9.1 19.2±3.1 17.1±4.9 14.9±3.5 16.1±4.3 15.4±4.4 16.6±4.4

ucec

25 64.5±5.1 66.0±6.1 65.3±3.3 68.6±3.2 69.9±6.0 68.0±9.1 68.2±7.0

50 62.6±4.9 65.8±4.6 65.8±8.7 67.3±4.7 68.3±6.2 68.4±6.5 67.1±5.1

75 64.5±8.1 67.0±5.5 66.8±7.5 67.7±4.7 67.9±5.5 68.3±8.3 67.7±5.7

100 66.9±7.9 67.1±5.6 66.9±7.7 67.7±6.7 68.2±6.1 68.1±9.1 67.3±8.5

IBS 21.0±8.9 9.4±2.6 9.6±4.2 8.4±2.5 8.7±3.1 8.5±2.6 8.8±2.9

compared our proposed DCTM head with the DeepHit model as implemented

in a recent benchmark study [21]. We omitted DeepSurv as it is incompatible

with batch sizes of one as used in histopathology experiments due to size. The

models were trained using splits from [21] in a 5-fold cross-validation scheme for

100 epochs following a cosine annealing learning rate schedule with maximum

learning rate of 0.0002 using the AdamW optimizer.

Results are shown in Table 3. For the BRCA and UCEC data, the general

DCTM (K = 1) or shift-scale DCTM (K = 10) perform best in terms of c-index.

For the LUAD dataset, the simplest shift DCTM (K = 1) performs best at the

25th and 75th percentile in terms of c-index, while DeepHit performs best for the

50th and 100th percentile. In terms of IBS, the shift-scale DCTM with K = 1

performs best on all three datasets.

5 Discussion and conclusions

We have proposed DCTMs for survival analysis and demonstrated their eﬀec-

tiveness on a wide range of datasets featuring both tabular and imaging data, as

well as various neural network architectures, feature extractors and foundation

models. In terms of time-dependent c-index, DCTMs outperform state-of-the-

art models (DeepSurv, DeepHit, RSF, and CPH) across most data modalities

and perform on par otherwise. Our ablations show that the various degrees of

ﬂexibility of DCTMs aids in trading oﬀ ﬂexibility and prediction performance.

In particular, our results on the radiology data indicate that lower order Bern-

stein polynomials (K = 1) yield better discrimination but lower probabilistic

A ﬂexible deep learning framework for survival analysis with medical data 9

prediction performance than higher order (K = 10). Furthermore, the most ﬂex-

ible model, DCTM

, does not typically perform best in terms of discrimination.

The type of explanatory data and their relationship with the survival informa-

tion likely govern the optimal DCTM complexity. In practice, this complexity

can be chosen based on a validation split or cross-validation and thereby re-

duce the risk of overﬁtting. Lastly, our parameterization of the NLL in terms of

Bernstein polynomials and the sigmoid CDF yielded more stable training curves

- a common problem when optimizing the partial likelihood (e.g. DeepSurv) -

and avoids introducing bias as in CE losses [31] (e.g. DeepHit). Using the time-

dependent c-index for evaluation is necessary due to the non-linear nature of

some of the DCTMs and allows a more ﬁne-grained view on model performance.

Taken together, DCTM for survival analysis have been shown to be a ver-

satile neural network survival head which can be combined with various input

data modalities, feature extraction methods and foundation models, and yield

prediction and discrimination performance superior or at least similar to state-

of-the-art deep learning methods for survival outcomes.

Acknowledgments. GC, IH, and TF were supported in part through the NIH/NCI

Cancer Center Support Grant (P30 CA008748). LK was supported by Novartis Re-

search Foundation (FreeNovation 2019) and the Swiss NSF (S-86013-01-01; S-42344-04-

01). TH was supported by the Swiss NSF (200021_184603). Disclosure of Interests.

TF is a founder and equity owner of Paige.AI, and Chief AI Oﬃcer at Eli Lilly.

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