An algorithm to sample an anatomy with uncertainty

Cesare Corrado1, Steven Williams1, Iain Sim1, Sam Coveney2, Mark O’Neill1, Richard Wilkinson2,Jeremy Oakley2, Richard Clayton2, Steven Niederer1

1 King’s College London, London, United Kingdom2 University of Sheffield, Sheffield, United Kingdom

Abstract

Image artefact, resolution and contrast all impact ourability to quantify anatomy. In the context of patient spe-cific simulations this uncertainty in shape can lead to un-certainty in model predictions. We propose and apply amethod for quantifying uncertainty in shape and demon-strate the impact of uncertain shape in activation time pre-dictions in a model of cardiac electrophysiology activa-tion.

1. Introduction

Patient-specific models of the heart are gaining impor-tance in the treatment of the heart diseases as they help topredict procedure outcomes and guide ablation targets, [1].Recently, we developed and validated a novel method togenerate patient-specific models of left atrial electrophys-iology from clinical measurements [2] and tested its abil-ity to differentiate aberrant activation patterns [3]. Clini-cal measurements, however, are affected by noise; hence,quantifying how measurement uncertainty affects modelpredictions represents an important step in communicat-ing the confidence of model predictions to cardiologists.An important building block in this process is to considerthe impact of uncertainties in the measurements of atrialgeometry that are used to generate the mesh on which themodel is solved. In this work, we propose a method toquantify the uncertainty on the anatomy of the left atrium.We then show the effect of these anatomical uncertaintieson simulations of local activation times.

2. Method

In the next subsections, we introduce a method to char-acterize the uncertainty about the true anatomy X givenan observations Xobs, i.e., find p(X|Xobs). Dealing withthe uncertainty in the high-dimensional space where X islocated is computationally expensive and often this com-plexity furnishes information about the uncertainty that isnegligible. Hence, we characterise the uncertainty through

its principal components, [6].

2.1. Anatomies

We recorded a training set of M=17 left atrial anatomyusing an electro-anatomical mapping system (EAM,StJude Velocity) and a test set of 4 left atrial anatomy us-ing a different EAM (Biosense Carto). On each of the21 anatomies, we manually clipped the pulmonary veins(PV) and the left atrial appendage (LAA). We thus ob-tained anatomies described by 7021 ± 1661 points (train-ing set) and by 9209 ± 2368 (test set) in an irregular tri-angle shell mesh. Next, we aligned each anatomy to a ref-erence atlas [4] formed by Np = 6132 vertices. We thenregistered the atlas to each anatomy, by minimising the dis-tance between the two manifolds using Deformetrica [5].Each anatomy could then be described by the same fixednumber of points.

We denote the vector ([x1, y1, z1, ..., xNp , yNp , zNp ]T ) of

the x,y and z point coordinates of the deformed atlas ob-tained by unrolling the point coordinates with Xobs ∈R3Np ; with X ∈ R3Np the anatomy in the absence of un-certainty and with e ∈ R3Np a random vector representingthe uncertainty in observations of the anatomy, such that:Xobs = X + e. We further assume a normally distributeduncertainty: e ∼ N (0,ΣX), with mean of zero and a co-variance matrix ΣX, with the correlation depending on therelative spatial distance.

2.2. Principal component analysis (PCA)

We reduced the number of parameters representing un-certainty in the anatomy by describing the anatomy as alinear combination of a reference mean value µ and ofNmodes � 3Np modes:

Xobs ' µ+ Uλ

We determined the modes that capture the largest vari-ability on the atrial anatomies with the principal compo-nent analysis (PCA) [6] on a training set, reducing the di-mensionality of the problem to Nmodes = 16 independentparameters.

We first evaluated the empirical mean µ:

µ =1

M

M∑k=1

Xkobs

Next, we build the matrix Aobs ∈ R3Np×M where the k-thcolumn is obtained as: Aobs∗,k = Xk

obs − µ and we evalu-ated the PCA loadings [6] with the singular value decom-position (SVD):

Aobs = UΘV T

Here U ∈ R3Np×(M-1) is the matrix of the left sin-gular vectors and coincides with the loadings, and Θ ∈R(M-1)×(M-1) is the diagonal matrix of the singular valuesΘi. The variances σ2

i captured by each mode are obtainedas follows:

σ2i =

Θ2i

M− 1

and depicted in figure 1.

Figure 1. Plot of the variances σ2i ; of each mode

2.3. Error propagation

We decompose a new observation X∗obs as the sum ofthe reconstruction from the principal component represen-tation, the truncation error and the uncertainty:

X∗obs = X∗ + e = µ+ Uλ∗ + e⊥ + e

here the vector λ∗ ∈ RNmodes represents the factor load-ings for X∗, and e⊥ represents the truncation error in theabsence of uncertainty. Even though deterministic, thetruncation error e⊥ is unknown and hence modelled witha probability distribution: e⊥ ∼ N (0,Σe⊥). Finally, wedefine the global error as follows: eTOT = e+e⊥, with dis-tribution: eTOT ∼ N (0,ΣeTOT) where ΣeTOT = ΣX + Σe⊥and we introduce a model for ΣeTOT .

2.4. Prior and posterior distribution on λ∗

λ∗ prior distribution We assume the prior distributionλ∗ ' N (0,Σλ) and Σλ = ΘTα2Θ. Using the samplesfrom the training set, we then evaluate α2 and prove theexistence of variables λi ∼ N

(0, α2I

), such that λi =

Θλi. For each sample, we have:

λi = Θ−1UT(Xi

obs − µ), i = 1 . . .M

If the null hypothesis holds, λij , i = 1 . . .M, j =1 . . . Nmodes represent Ndof = Nmodes ×M realizations ofN(0, α2I

). Next, we approximate α2 with the sampling

covariance, s2:

s2 =1

Ndof − 1

M∑i=1

Nmodes∑j=1

(λij − µ

)2, µ =

1

Ndof

M∑i=1

Nmodes∑j=1

λij

Finally, we performed a Kolmogorov Smirnov test tocheck for normality, and found no evidence to reject thenull hypothesis (p=0.56). The QQ plot showed excel-lent agreement with a standard normal distribution (notshown).λ∗ posterior distribution Being the problem conjugate

we obtain a normal posterior distributionN (µλ,post,Σλ,post),and Bayes formula p(λ∗|X∗obs) ∝ p(X∗obs|λ∗)p(λ∗)yields:

p(λ∗) ∼ N (0,Σλ)

p(X∗obs|λ∗) ∼ N (µ+ Uλ∗,ΣeTOT)

p(λ∗|X∗obs) ∝ exp

(−λ∗

T

Σ−1λ,postλ∗ − 2λ∗

T

Σ−1λ,postµλ,post + a

2

)

and we thus obtain the posterior distribution for λ∗:

p(λ∗|X∗obs) ∼ N (µλ,post,Σλ,post) (1)

Σλ,post =(Σ−1λ + UT Σ−1eTOT

U)−1

µλ,post = Σλ,postUT Σ−1eTOT

(X∗obs − µ)

2.5. Sampling

Once a new anatomy X∗obs is measured with EAM, wedraw Nsamples i.i.d. samples from p(λ∗|X∗obs), obtainedby applying (1); next, we obtain the anatomy samples(X1

obs, ..,XNsamplesobs ) with the formula: Xk

obs = µ+ Uλk

3. Results

We sample from the uncertainty distribution of each ofthe 4 anatomies depicted in figure 2. We assumed the fol-

Figure 2. Anatomies

lowing form for ΣeTOT :

Σixk ,jxleTOT = ν2 exp

(−(dij

l

))2

δxk,xl

Here ν2 represents the characteristic scale of the covari-ance, l represents the characteristic length of the spacecorrelation, dij represents the geodesic distance betweenpoints i and j, δxk,xl represent the Dirac delta between thespace components xk and xl and index ixk represents theentry in ΣeTOT corresponding to node i and component xk.

As the typical discrepancy between anatomies obtainedwith EAM and anatomies obtained with MRI scan is' 5mm [7], in this work we considered the followingcharacteristic values: ν2 = [25, 49, 100, 225, 784] mm2,while we arbitrarily chose a width l equal to 10 mm.

Next, for each sample, we computed LATs with agraph-based eikonal model [8] and the modified Mitchell-Schaeffer (mMS) ionic model [9]. For the model parame-ter values we selected τin = 0.1, τout = 2.5 and vgate = 0.1and we chose an isotropic diffusivity of σm = 1.0cm2/sm,consistent with [2, 10]. To take into account the effect oflow mesh resolution on the propagation through a graph,we set the corrective coefficient to δ = 0.92. This valuefurnished the minimum RMS between the LATs evalu-ated with the eikonal model and those computed solv-ing the monodomain equations with the finite elementsmethod (FEM) on a discretized domain characteristic sizeh ' 300µm.

Next, we evaluated the expected value and the stan-dard deviation of LATs with the Monte Carlo [11] for-mula. For each anatomy and for each value of ν2, wedraw Nsamples = 12, 000 i.i.d. anatomy samples; overall,the procedure required ∼ 70 minutes for a single case andfor a single value of ν2. The l2 norm of the differencebetween the expected value evaluated with 12k samplesand the expected values evaluated with 10k samples was2.6ms± 0.58ms, or∼ 1% of the total activation time. Fig-ure 3 shows the distribution of expected value for LAT, andthe standard deviation for each of the 4 patients and whenν = 28 mm.

Figure 4 shows the maximum value of LAT standard de-viation as a function of the uncertainty standard deviationν; each line represents a patient.

Figure 4. Maximum value of the standard deviation onLATs as a function of the standard deviation of the uncer-tainty ν; each line represents a patient.

4. Discussion

In this work, we have presented a method to evaluatethe statistics of a space-dependent variable and to infer ashape when the data quality is not high. We parameterisedthe uncertainty on the domain using PCA. When the uncer-tainty affects the computational domain, a widely used ap-proach is represented by stochastic collocation techniques,[12, 13]. This approach, however, does not take into ac-count any spatial correlation on the domain uncertainty.Indeed, this uncertainty is described by parameterising theuncertain domain through a linear composition of shapefunctions, weighted with Karhunen-Loeve (KL) approxi-mation. This is not the case with the approach we pre-sented: first, the spatial correlation is implicitly describedby PC loadings; second, among all the possible orthogonalshape functions that represent space uncertainty, the PCloadings are those that minimise the truncation error; third,the coefficient weighting the PC loadings are mutually in-dependent and ordered: we can thus capture the largestvariability (in the statistical sense) keeping the same accu-racy using a smaller number of parameters.

Patient 1 Patient 2 Patient 3 Patient 4

Figure 3. Top: Distribution of LATs expected values for patients 1,2,3,4. Bottom:Distribution of LATs standard deviationfor patients 1,2,3,4.

5. Conclusions

We have developed a method to sample from ananatomy probability distribution and to infer model uncer-tainty. We took advantage of a PCA decomposition for theanatomy, obtaining a method computationally efficient.

Acknowledgements

This work was supported by the EPSRC (EP/P101268X/1), the Wellcome centre and the Department of Health viathe National Institute for Health Research (NIHR) compre-hensive Biomedical Research Centre award to Guy’s & StThomas’ NHS Foundation Trust in partnership with King’sCollege London and King’s College Hospital NHS Foun-dation Trust.

References

[1] Colli Franzone P, Pavarino L, Savare G. Computationalelectrocardiology: mathematical and numerical modelling.Complex systems in Biomedicine Springler, 2006:187-241.

[2] Corrado C, et al. A work flow to build and validate patientspecific left atrium electrophysiology models from cathetermeasurements. Medical Image Analysis 2018; 47:153-163.

[3] Corrado C, et al. A predictive personalised model for theleft atrium. 2017 Computing in Cardiology (CinC) 2017.

[4] Tobon-Gomez C, et al. Left Atrial Segmentation Chal-lenge: A Unified Benchmarking Framework. Statistical At-lases and Computational Models of the Heart. Imagingand Modelling Challenges: 4th International Workshop,STACOM 2013, Held in Conjunction with MICCAI 2013,Nagoya, Japan, September 26, 2013. Revised Selected Pa-pers Springer Berlin Heidelberg, 2014; 1-13.

[5] www.deformetrica.org[6] Hotelling H. Analysis of a Complex of Statistical Variables

with Principal Components. Journal of Educational Psy-chology 1993; 24:417-441.

[7] Nguyen, U C, et al. A novel approach for left ven-tricular lead placement in cardiac resynchronization ther-apy: Intraprocedural integration of coronary venous elec-troanatomic mapping with delayed enhancement car-diac magnetic resonance imaging. Heart Rhythm 2017;14(1):110-119.

[8] Corrado C, Zemzemi N. A conduction velocity adaptedeikonal model for electrophysiology problems with re-excitability evaluation. Medical Image Analysis 2018;43:186-197.

[9] Corrado C, Niederer S. A two-variable model robust topacemaker behaviour for the dynamics of the cardiac actionpotential. Mathematical Biosciences 2016; 281:46-54.

[10] Corrado C, et al. Predicting spiral wave stability by person-alized electrophysiology models. 2016 Computing in Car-diology (CinC) 2016.

[11] Kroese, D, et al. Why the Monte Carlo method is so impor-tant today. Wiley Interdisciplinary Reviews: ComputationalStatistics John Wiley & Sons, Inc, 2014:386-392.

[12] Castrillon-Candas J, Nobile F, Tempone R. Analytic regu-larity and collocation approximation for elliptic PDEs withrandom domain deformations. Computers & Mathematicswith Applications 2016; 71(6):1173-1197

[13] Sankaran, S and Marsden, AL. A Stochastic CollocationMethod for Uncertainty Quantification and Propagation inCardiovascular Simulations. ASME. J Biomech Eng 2011;33(3):031001-031001-12

Address for correspondence:

Cesare CorradoDep. of Biomedical Engineering; St Thomas’ Hospital, SE17EH,London (UK)cesare.corrado@kcl.ac.uk