An organ deformation model using Bayesian inference to combine population and patient-specic data

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An organ deformation model using Bayesian
inference to combine population and
patient-specific data
Øyvind Lunde Rørtveit∗1,2, Liv Bolstad Hysing1,2, Andreas Størksen
Stordal3,2, and Sara Pilskog1,2
1Haukeland University Hospital, Bergen, Norway
2University of Bergen, Norway
3NORCE Norwegian Research Centre, Bergen, Norway
November 18, 2022
Abstract
Objective: Organ deformation models have the potential to improve de-
livery and reduce toxicity of radiotherapy, but existing data-driven motion
models are based on either patient-specific or population data. We propose
to combine population and patient-specific data using a Bayesian framework.
Our goal is to accurately predict individual motion patterns while using fewer
scans than previous models.
Approach: We have derived and evaluated two Bayesian deformation
models. The models were applied retrospectively to the rectal wall from a co-
hort of prostate cancer patients. These patients had repeat CT scans evenly
acquired throughout radiotherapy. Each model was used to create coverage
probability matrices (CPMs). The spatial correlations between these CPMs
and “true” CPMs, derived from independent scans of the same patient, were
calculated.
Main results: Spatial correlation with ground truth were significantly
higher for the Bayesian deformation models than both patient-specific and
population-derived models with 1, 2 or 3 patient-specific scans as input. Sta-
tistical motion simulations indicate that this result will also hold for more
than 3 scans.
Significance: The improvement over known models means that fewer
scans per patient are needed to achieve accurate deformation predictions.
The models have applications in robust radiotherapy planning and evalua-
tion, among others.
∗Corresponding author. oyvind.rortveit@uib.no
1
arXiv:2210.15296v2 [physics.med-ph] 17 Nov 2022
1 Introduction
In radiotherapy (RT), the dose is carefully shaped to the patient anatomy as seen in
the CT acquired before start of treatment (plan CT), to achieve a good compromise
between disease control and risk of inducing complications. Since the variability of
the organ positions and deformations is unknown before start of treatment, different
measures have been adopted to safeguard against motion uncertainties through
planning margins (Stroom et al.,1999;van Herk et al.,2000), robust optimization
(Unkelbach et al.,2018) and/or treatment plan adaptation (Yan et al.,1997).
A statistical model for the deformation of organs of individual patients using
principal component analysis (PCA) of the organ’s surface shape vectors was first
proposed by S¨ohn et al. (2005). The main drawback of the patient-specific model
is that the number of data samples (in the form of organ contours derived from 3D
images) per patient is often low, which limits the robustness of the motion estimates
(Th¨ornqvist et al.,2013b).
Budiarto et al. (2011) proposed a population based statistical model, under the
assumption that, although the size, shape and position of organs differ greatly be-
tween patients, the patterns of deformation are generally the same. The advantage
is that an estimate of a patient’s deformation patterns exists even when only a
single observation is available. When applied to prostate target deformation, they
showed that about 50% of the variation could be explained by 15 population de-
formation modes (i.e. principal components). Subsequent uses of the population
model include Bondar et al. (2014), who used it to create margins for rectal cancer
patients, Rios et al. (2017), who modeled bladder deformation for prostate cancer
RT, Szeto et al. (2017) who modeled daily variations in the thorax, and Magallon-
Baro et al. (2019), who modeled deformation in the stomach, duodenum and bowel
for pancreatic cancer RT. A weakness of the population model is its inability to
model patient-specific deformation patterns, even when multiple scans are available
for the patient in question. The aim of the current work is to combine the strengths
of the population and patient-specific models by introducing Bayesian models that
take in to account both the population deformation patterns (in terms of a prior
distribution) and patient-specific measurements, forming an individualized poste-
rior distribution. Bayesian models have previously been applied to the problem
rigid shifts of the patient, termed setup errors (Lam et al.,2005;Herschtal et al.,
2012).
In this paper, we introduce two Bayesian models, which differ in their choice
of priors. The choice of model to use will be a trade-off between accuracy and
simplicity. We derive necessary algorithms to efficiently calculate the approximate
posterior distributions in high dimensions. We apply the introduced models to a
realistic example with complex motion, in terms of the rectal wall of prostate cancer
patients. We use the models to estimate coverage probability matrices (CPMs), i.e.
3D-arrays of voxels where the value in each voxel is the probability that the voxel
will be covered by the rectal wall at any given time. We compare the accuracy
of CPMs estimated using the two Bayesian methods, the patient-specific model by
S¨ohn et al. (2005) and the population model by Budiarto et al. (2011).
In addition to the presentation of new models, this is to our knowledge the first
2
comparison between these two previous models, as well as the first time such an
organ deformation model has been applied to the rectum.
2 Methods
In the class of deformation models that we study, an organ shape is represented by
a set of points on the organ surface, as illustrated in figure 1. These representations
are derived from organ contours segmented from 3D images; for simplicity we refer
to one organ shape of this kind as a “scan”. The x,yand zcoordinates of all P
points are gathered into a shape vector s:
s= [x1, y1, z1, x2, y2, z2, . . . , xP, yP, zP]T.(1)
With this representation, we can use standard multivariate statistical distributions.
To compare organs across scans, we need corresponding points between all
shapes in the data set. This correspondence is found using deformable and rigid
contour registration both within and between patients. Details are beyond the scope
of the current work, but can be found in Rørtveit et al. (2021) .
An assumption for all the following methods is that, for a specific patient, the
shape coordinates follow a multivariate Gaussian distribution:
s∼ N(µ, R).(2)
The mean shape vector µrepresents the patient’s mean organ shape, and the
Figure 1: A rectum shape represented by a set of organ surface points.
covariance matrix Rdescribes the variance of the coordinates as well as the covari-
ance between each pair of coordinates. When µand Rare given, we can use the
distribution to draw new random organ shapes for the patient.
The difference between the methods is how µand Rare estimated. In the
Bayesian methods introduced in section 2.3 ,µand Rare considered random sam-
ples from specific prior distributions, whose parameters are calculated from the
training data. Point estimates of µand Rare derived from the posterior distribu-
tions.
3
Due to the high dimensions of the shape vectors, all covariance matrices are
parametrized using principal component analysis (PCA), see e.g. Fujikoshi et al.
(2010, chapter 10). Under PCA, a covariance matrix is represented by a few eigen-
vectors and corresponding eigenvalues. These are usually found through singular
value decomposition (SVD) of a data matrix D, whose columns are normalized
mean-subtracted samples, such that R=DDT.
2.1 Patient-specific model
In the patient-specific model introduced by S¨ohn et al. (2005), only data from the
patient under consideration is used. The mean shape is thus the average of the
available shapes s1, s2, . . . , sJfor that patient;
ˆµ= ¯s=1
J
J
X
j=1
sj,(3)
while Ris the sample covariance matrix:
ˆ
Rps =1
J−1
J
X
j=1
(sj−ˆµ)(sj−ˆµ)T.(4)
2.2 Population model
The population model introduced by Budiarto et al. (2011) rests on the assumption
that the covariance matrix is the same for all patients, and only the mean differs.
The mean is calculated as the mean shape vector for the individual patient as in
(3), while the covariance matrix is the average of the sample covariance matrices
for each patient in the training set:
ˆ
Rpop =1
M
M
X
i=1
ˆ
Ri=1
M
M
X
i=1
1
Ji−1
Ji
X
j=1
(si,j −¯si)(si,j −¯si)T.(5)
2.3 Bayesian models
In Bayesian inference, new data is combined with prior knowledge (such as pop-
ulation statistics) to update probability distributions. In the following, the mean
and covariance matrix for a given patient are considered random parameters that
vary across the population according to a prior distribution, f(µ, R). When data
for a new patient is available, we can compute the posterior distribution f(µ, R|s),
where s={s1, s2, . . . , sJ}, through Bayes theorem:
f(µ, R|s) = f(s|µ, R)f(µ, R)
f(s)(6)
Bayes theorem gives us a distribution of the possible values of µand R, as opposed
to single values. Nevertheless, due to the complexity of the posterior distributions
4
in our subject matter, we shall resort to looking at point estimates of µand R, such
as the expected value or mode of the posterior.
The Bayesian models we present differ in the selection of the prior density.
We resort to priors that result in computationally feasible posterior distributions,
since Markov Chain-Monte Carlo methods are computationally expensive in high
dimensions. In the following sections, we present two priors which each represent a
Bayesian model.
2.3.1 Normal-Inverse-Wishart prior
A natural way of combining the estimates of the covariance matrix from the patient-
specific model and population model is to use a simple weighted average:
ˆ
R=λˆ
Rpop + (1 −λ)ˆ
Rps (7)
The weighting should be proportional to the number of samples used to compute
the estimates. By setting λ=ν
ν+Jfor some ν, we obtain
ˆ
R=1
ν+J(νˆ
Rpop +Jˆ
Rps),(8)
which is the same result as obtained from assuming an inverse Wishart (IW) prior
for Rand using a specific point estimate for ˆ
Rfor the posterior, as shown below.
IW is a matrix distribution, and a conjugate prior to the multivariate Gaussian
likelihood with known mean and unknown covariance matrix. This means that
the posterior distribution for Ris also IW, and the parameters are obtained from
equations involving the prior parameters and the data. The parameters of the IW
are the scale matrix Ψ and the degrees of freedom ν. Formally, if µis given, and
the prior for Ris IW,
R∼ IW(Ψ, ν),(9)
and the likelihood is Gaussian,
s|R∼ N(µ, R),(10)
then the posterior R|s, where s={s1, s2, . . . , sJ}is also IW,
R|s∼ IW(Ψ0, ν0),(11)
with posterior parameters
Ψ0= Ψ +
J
X
j=1
(sj−µ)(sj−µ)T(12)
ν0=ν+J. (13)
We define Ψ = νˆ
Rpop and use the point estimate ˆ
R=1
ν0Ψ0for the posterior.
Inserting both these expressions into (12), we get
ˆ
R=1
ν+J νˆ
Rpop +
J
X
j=1
(sj−µ)(sj−µ)T!(14)
5
摘要：

AnorgandeformationmodelusingBayesianinferencetocombinepopulationandpatient-speci cdatayvindLundeRrtveit*1,2,LivBolstadHysing1,2,AndreasStrksenStordal3,2,andSaraPilskog1,21HaukelandUniversityHospital,Bergen,Norway2UniversityofBergen,Norway3NORCENorwegianResearchCentre,Bergen,NorwayNovember18,2022A...

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