Astrocytic tracer dynamics estimated from [1-11C]-acetate PET measurements

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Mathematical Medicine and Biology (2014) Page 1 of 16 doi:10.1093/imammb/dqu021 Astrocytic tracer dynamics estimated from [1- 11 C]-acetate PET measurements Andrea Arnold Center for Quantitative Sciences in Biomedicine, North Carolina State University, Campus Box 8213, 2700 Stinson Drive, 308 Cox Hall, Raleigh, NC 27695-8213, USA and Department of Mathematics, NorthCarolina State University, Campus Box 8205, 2108 SAS Hall, 2311 Stinson Drive, Raleigh, NC 27695-8205, USA Daniela Calvetti Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106, USA Corresponding author: Email: [email protected], [email protected] Albert Gjedde Department of Neuroscience and Pharmacology, University of Copenhagen, Blegdamsvej 3B, DK-2200, Copenhagen, Denmark and Department of Nuclear Medicine and PET Centre, Aarhus University Hospitals, Aarhus, Nørrebrogade 44 DK-8000, Denmark Peter Iversen Department of Nuclear Medicine and PET Centre, Aarhus University Hospitals, Aarhus, Nørrebrogade 44 DK-8000, Denmark and Erkki Somersalo Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106, USA [Received on 13 March 2014; revised on 24 August 2014; accepted on 8 October 2014] We address the problem of estimating the unknown parameters of a model of tracer kinetics from sequences of positron emission tomography (PET) scan data using a statistical sequential algorithm for the inference of magnitudes of dynamic parameters. The method, based on Bayesian statistical infer- ence, is a modification of a recently proposed particle filtering and sequential Monte Carlo algorithm, where instead of preassigning the accuracy in the propagation of each particle, we fix the time step and account for the numerical errors in the innovation term. We apply the algorithm to PET images of [1- 11 C]-acetate-derived tracer accumulation, estimating the transport rates in a three-compartment model of astrocytic uptake and metabolism of the tracer for a cohort of 18 volunteers from 3 groups, correspond- ing to healthy control individuals, cirrhotic liver and hepatic encephalopathy patients. The distribution of the parameters for the individuals and for the groups presented within the Bayesian framework support the hypothesis that the parameters for the hepatic encephalopathy group follow a significantly different distribution than the other two groups. The biological implications of the findings are also discussed. Keywords: parameter estimation; tracer kinetics; PET imaging; particle filters; sequential Monte Carlo (SMC). c The authors 2014. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. Mathematical Medicine and Biology Advance Access published November 24, 2014 by guest on January 31, 2015 http://imammb.oxfordjournals.org/ Downloaded from

Transcript of Astrocytic tracer dynamics estimated from [1-11C]-acetate PET measurements

Mathematical Medicine and Biology (2014) Page 1 of 16doi:10.1093/imammb/dqu021

Astrocytic tracer dynamics estimated from [1-11C]-acetate PET measurements

Andrea Arnold

Center for Quantitative Sciences in Biomedicine, North Carolina State University, Campus Box 8213,2700 Stinson Drive, 308 Cox Hall, Raleigh, NC 27695-8213, USA and Department of Mathematics,

North Carolina State University, Campus Box 8205, 2108 SAS Hall, 2311 Stinson Drive, Raleigh, NC27695-8205, USA

Daniela Calvetti∗

Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University,10900 Euclid Avenue, Cleveland, OH 44106, USA

∗Corresponding author: Email: [email protected], [email protected]

Albert Gjedde

Department of Neuroscience and Pharmacology, University of Copenhagen, Blegdamsvej 3B,DK-2200, Copenhagen, Denmark and Department of Nuclear Medicine and PET Centre, Aarhus

University Hospitals, Aarhus, Nørrebrogade 44 DK-8000, Denmark

Peter Iversen

Department of Nuclear Medicine and PET Centre, Aarhus University Hospitals, Aarhus,Nørrebrogade 44 DK-8000, Denmark

and

Erkki Somersalo

Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University,10900 Euclid Avenue, Cleveland, OH 44106, USA

[Received on 13 March 2014; revised on 24 August 2014; accepted on 8 October 2014]

We address the problem of estimating the unknown parameters of a model of tracer kinetics fromsequences of positron emission tomography (PET) scan data using a statistical sequential algorithm forthe inference of magnitudes of dynamic parameters. The method, based on Bayesian statistical infer-ence, is a modification of a recently proposed particle filtering and sequential Monte Carlo algorithm,where instead of preassigning the accuracy in the propagation of each particle, we fix the time stepand account for the numerical errors in the innovation term. We apply the algorithm to PET images of[1-11C]-acetate-derived tracer accumulation, estimating the transport rates in a three-compartment modelof astrocytic uptake and metabolism of the tracer for a cohort of 18 volunteers from 3 groups, correspond-ing to healthy control individuals, cirrhotic liver and hepatic encephalopathy patients. The distribution ofthe parameters for the individuals and for the groups presented within the Bayesian framework supportthe hypothesis that the parameters for the hepatic encephalopathy group follow a significantly differentdistribution than the other two groups. The biological implications of the findings are also discussed.

Keywords: parameter estimation; tracer kinetics; PET imaging; particle filters; sequential Monte Carlo(SMC).

c© The authors 2014. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

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1. Introduction

Understanding the neurochemistry and metabolic processes in brain relies on indirect measurementssuch as dynamic positron emission tomography (PET) records. Although PET imaging, i.e. the passagefrom photon counting data to activity map in the brain, is a well-established practice, the quantitativeimaging of kinetic model parameters from imaging data is less well developed and understood. In vivoneurodynamics is usually described in terms of parametric compartment models (Gjedde et al., 2011),and the distribution of parameter values carries important information about potential disease of thebrain; see, e.g. Gjedde et al. (2010) and Iversen et al. (2009). The estimation of kinetic parameters canbe approached either from an optimization point of view, by fitting the model to the observations, or byusing statistical methods. The advantage of the latter approach is that it naturally provides a frameworkfor including uncertainties in the modelling paradigm, including model inputs such as the arterial tracerconcentration, and measurement errors. The Bayesian statistical framework (Kaipio and Somersalo,2004; Calvetti and Somersalo, 2009) allows qualitative complementary information to be encoded inthe prior distribution, and it produces an output naturally equipped with a reliability estimate. In thisarticle, we analyse the quantitative PET imaging problem using sequential Bayesian estimation and, inparticular, particle filtering sequential Monte Carlo (PF-SMC) methods.

In this work, we consider a kinetic model of [1-11C]-acetate imaging. Acetate is a metabolite prefer-entially taken up into and metabolized by astrocytes, not neurons, which makes acetate tracers particu-larly suited for studying astrocytic metabolism in brain (Wanievski and Martin, 1998; Hertz et al., 2007;Wyss et al., 2011). The particular question that is addressed in this paper is related to the brain functionduring hepatic encephalopathy. It is known that serious liver malfunctioning affects, among other things,the cognitive functions, but the detailed mechanisms are still to be discovered. In particular, a questionof interest is whether the elevated level of ammonium in plasma during hepatic encephalopathy affectsprimarily the neurons or astrocytes in brain.

Acetate passes from the blood into the brain tissue through the blood–brain barrier and is furthertaken up by the astrocytes. Inside the cell, acetate enters the TCA cycle in the mitochondria, where thecarbons are released to form carbon dioxide that is consequently cleared from the tissue. The labellingof the first carbon by the 11C isotope, as opposed to labelling the second carbon of the acetate carbonskeleton, is motivated by the fact that in this way, the isotope is cleared already during the secondturn of the TCA cycle, thus minimizing the potential contaminate of the neuron through the glutamine-glutamate cycle (Wyss et al., 2011; Lanz et al., 2012).

Experimental data collection and the inverse problem can be summarized as follows: a radioactivetracer is administered into the circulation of the subject, and the tracer concentration in arterial bloodis estimated from direct arterial measurements of the radioactivity. The blood flow carries the tracer tothe brain where it enters the tissue, and the time dependent positron emission activity, proportional tothe tracer mass, is measured with a PET device and mapped to a series of tomographic images, yieldingan estimate of the time dependent activity distribution in the tissue. More specifically, dynamic PETimages track the concentration of radioactivity as a function of time in each image voxel, and voxels inthe region of interest are averaged, yielding the mean tissue activity curves which are part of the datafor the present study. The interpretation of the data requires a model describing the tracer’s kinetics.The tracer’s concentration in plasma, on the other hand, which constitutes the input of the model, ismeasured directly from blood samples.

A standard way of describing the marker dynamics is to use a compartment model in whichthree compartments are identified: a blood compartment, a precursor compartment corresponding toacetate taken up by the tissue and a product compartment corresponding to acetate in the astrocyte asa metabolite. A schematic picture of the compartment model is shown in Fig. 1. The mathematical

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Fig. 1. A schematic representation of the compartment model of tracer kinetics. The dashed module represents the reuptake ofCO2, which is not included in the model.

compartment model describing the mass balance in the brain tissue is governed by a system ofdifferential equations (Gjedde et al., 2011)

dm1

dt(t) = K1c(t) − (k2 + k3)m1(t) (1.1)

dm2

dt(t) = k3m1(t) − k5m2(t) (1.2)

where c(t) is the arterial tracer concentration, or arterial input function (AIF), estimated from measuredblood samples drawn during the measurement process, and m1(t) and m2(t) represent the mass of tracercontent in the precursor and product compartments, respectively. The coefficients k2, k3 and k5 aretransport rates between the compartments, and K1 is the clearance of the tracer from blood to tissue,weighted with the capillary compartment’s virtual volume V0. In this model, k4, which would correspondto back-transport to the precursor compartment, is assumed to be vanishingly small because of theunidirectionality of the biochemical reaction represented by k3. The initial values for the differentialequations are m1(0) = m2(0) = 0, and the total tracer content in the tissue volume of interest is

m(t) = V0c(t) + m1(t) + m2(t), (1.3)

where we assume that the PET image provides an estimate of its time course. This estimate, which werefer to as the tissue’s time–activity curve (TAC), is the data for the present problem.

The above model is a version of the standard compartment models used to study tracer kinetics(Gjedde et al., 2011). The model does not address questions such as tracer recirculation and reuptake,thus the effect of the labelled CO2 on the PET data is incompletely addressed (Iida et al., 1986, 1993;

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Keiding and Pavese, 2013). Since the contribution of 11CO2 to the input function was not corrected, thevalues for 11C-acetate will be overestimated due to contributions of oxidation of acetate by other bodytissues. This may be a confound for physiological interpretation of data in disease states involving theliver, but not necessarily for the mathematical analysis of a test data set. It is not the aim of this workto improve the existing model, but rather to develop a methodology to estimate the model parametersV0, K1 and kj, where j = 2, 3, 5, from the PET image data using SMC methods. The approach relies ona Bayesian paradigm of inverse problems, in which the unknowns are modelled as random variables,and the data are used to infer the posterior probability distribution. We refer to Kaipio and Somersalo(2004) and Calvetti and Somersalo (2009) as general references for the Bayesian formalism, and to Liuand West (2001), Doucet et al. (2001, Chapter 10) and Arnold et al. (2013) as references for the SMCmethodology.

2. Bayesian framework

We start by introducing some notations. The unknown parameters describing the tracer dynamics intissue are collected in a single vector that we denote by θ ,

θ = (V0, K1, k2, k3, k5) ∈ R5.

The input function c(t) obtained from plasma radioactivity measurements is assumed to be known.Integrating the system (1.1)–(1.2) with the vanishing initial values gives the pair (m1(t, θ), m2(t, θ)) andthe idealized model for the errorless observation,

m(t, θ) = θ(1)c(t) + m1(t, θ) + m2(t, θ), (2.1)

where θ(1) refers to the first component of the vector θ , while the observed data consist of noisy samplesat discrete times tj,

dj = m(tj, θ) + ej, 1 � j � T , (2.2)

where, for simplicity, the noise is modelled as additive error. It is common to assume that ej representsonly the measurement uncertainty, while in reality it should also account for the modelling error. Thisquestion will be addressed in the following section.

In the Bayesian framework, randomness represents lack of information. Therefore, the unknown θ

is modelled as a random variable, and the information about it is encoded in the respective probabilitydistribution. We assume here that probability distributions can be expressed in terms of probability den-sities in the parameter space. The prior probability density, encoding all information about the unknown,is denoted by πpr(θ). The probability density of the data dj, if θ were known, gives the likelihood den-sity π(dj | θ), and encodes the information about the noise ej. Finally, Bayes’ formula gives the posteriordensity, which is an update of the prior density based on the additional information that the data provide,

π(θ | d) ∝ πpr(θ)π(d | θ),

where ‘∝’ stands for ‘equal up to a multiplicative normalizing constant’.

2.1 Error model and hierarchical update

Consider the TAC data model (2.2), where the vector ej represents the aggregation of measurementerrors, imprecisions in the model, as well as model discrepancy. When referring to measurement error in

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the context of dynamic PET imaging, we mean the photon emission of the volume element in question,calculated from the PET counting data. By imprecisions in the model, we refer to uncertainties that havean effect on reliability of the estimates, most notably uncertainties related to the AIF; see, e.g. Iida etal. (1986, 1993). In this work, we address one possible error source, related to an incompletely knownoffset of the AIF and TAC curves: The blood sample data are collected independently of the PET scan,and the delay between the time when the tracer arrives at the brain and at the blood sample site varies.This delay may affect, in particular, the estimates based on data at the very beginning of the estimationprocess.

Therefore, assume that the convection time from the brain to the site where the blood sample isdrawn contains an unknown estimation error τ . To include this error in the model, we replace (2.2) by

dj = m(tj + τ , θ) + ej, 1 � j � T . (2.3)

The offset τ could be added to the model as an unknown that needs to be estimated from the data.Here, we treat it as a nuisance parameter and, to simplify the discussion, we approximate m(t, θ) by itsfirst-order Taylor polynomial and write

dj ≈ m(tj, θ) + mt(tj, θ)τ + ej = m(tj, θ) + nj + ej, (2.4)

where mt denotes the derivative of m with respect to t. As is common for the modelling errors, the offseterror nj depends on the unknown θ itself. To overcome this difficulty, we propose here a hierarchicalprocedure.

Assume that θ represents a typical value of the parameter θ . We model the prior density of θ interms of the typical value, i.e. we define a conditional prior for the components θ(k) of the parametervector,

πpr(θ(k) | θ (k)) ∼ Uniform(αθ(k), Aθ (k))

for some scalars 0 < α < 1 < A. Furthermore, we approximate the modelling error term by

nj = mt(tj, θ)τ ≈ mt(tj, θ )τ ,

such that the only source of randomness is the random offset τ , which we will model as a zero meanGaussian random variable.

To find a typical value θ , we consider the model (2.2), assuming that the offset vanishes. We assumefor simplicity Gaussian measurement error with constant variance σ 2, which leads to a likelihood model

π(dj | θ) ∝ exp

(− 1

2σ 2|dj − m(tj, θ)|2

).

Further, by assuming conditionally independent measurements, the likelihood of the full data vector dis

π(d | θ) =T∏

j=1

π(dj | θ) ∝ exp

⎛⎝− 1

2σ 2

T∑j=1

|dj − m(tj, θ)|2⎞⎠ .

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Assuming a weakly informative prior imposing the positivity of the parameters θ(k) and some largeupper bound, 0 � θk � Θ(k),

πpr(θ) ∝K∏

k=1

1[0,Θ(k)](θ(k)),

where 1[0,Θ(k)] is the indicator function of the interval [0, Θ(k)], the posterior density becomes

π(θ | d) ∝K∏

k=1

1[0,Θ(k)](θ(k)) exp

⎛⎝− 1

2σ 2

T∑j=1

|dj − m(tj, θ)|2⎞⎠ . (2.5)

Maximizing the posterior distribution is tantamount to solving for the constrained least squares solution,

θ = argmin

⎧⎨⎩

T∑j=1

|dj − m(tj, θ)|2⎫⎬⎭ subject to 0 � θ(k) � Θ(k). (2.6)

The constrained minimization problem (2.6) is straightforward to implement.To estimate the sensitivity of estimated θ to modelling errors, a more comprehensive analysis is

necessary. An extensive Markov chain Monte Carlo run for sampling the posterior is one option; a moreapproximate but computationally lighter alternative is to use parametric bootstrapping, which gives anidea of the posterior density assuming that the prior is non-informative. In this article we use a PF-SMCalgorithm combined with the error remodelling approach explained as follows:

1. Estimate θ using the simple likelihood model, and compute the typical value θ from (2.6).

2. Write a conditional prior and conditional likelihood for the components θ(k),

πpr(θ(k) | θ (k)) ∼ Uniform(αθ(k), Aθ (k))

for some scalars 0 < α < 1 < A, and the likelihood

π(dj | θ , θ ) ∼N (m(t, θ), σ 2 + mt(tj, θ )2var(τ )).

3. Run a PF-SMC algorithm to explore the conditional posterior density,

π(θ | d, θ ) ∝ πpr(θ | θ )π(d | θ , θ ).

The error variances σ 2 and var(τ ) are estimated heuristically from the data: a systematic and automaticway of assigning their values would require information about the measurement procedure that was notavailable.

The state-of-the-art methodology in analysing dynamic PET data is to use non-linear curve fitting,see, e.g. Keiding and Pavese (2013). This amounts to solving the least squares optimization prob-lem (2.6), which is naturally imbedded in the proposed empirical Bayes formulation. Hence, theproposed algorithm extends the existing methodology towards a more complete analysis of the vari-ability of the estimates due to model uncertainties and imprecisions in data.

In the next section, we briefly review the PF-SMC algorithm proposed in Arnold et al. (2013).

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2.2 SMC and survival of the fittest

In summarizing the PF-SMC algorithm, assume that the time interval corresponding to the data acqui-sition is discretized by time instances t0 < t1 < · · · < tT , some of which, but not necessarily all, coincidewith the data acquisition. However, to simplify the notation, and to avoid introduction of new indices,we assume that data arrive at each tj, denoting dj = ∅ if the measurement is missing. We denote by

Dj = {d1, d2, . . . , dj}

the data acquired up to time t = tj. The posterior density of θ conditional on data Dj, π(θ | Dj, θ ), issummarized by the sample

Sj = {θ1j , . . . , θN

j }.

Let θj and Cj denote the sample-based mean and covariance of θ ,

θj = 1

N

N∑n=1

θnj , Cj = 1

N − 1

N∑n=1

(θnj − θj)(θ

nj − θj)

T.

The probability density is approximated by a Gaussian mixture model,

π(θ | Dj, θ ) ≈N∑

n=1

wnj N (θ | θn

j , s2Cj),

where the means θnj , called auxiliary particles, are defined via a shrinkage towards the sample mean,

θnj = aθn

j + (1 − a)θj, 0 < a < 1,

and s is a scaling factor to counterbalance the shrinkage, s2 = 1 − a2. The weights wnj represent proba-

bilities that are updated at each step recursively as described below. The initial sample, when j = 0, isdrawn from the conditional prior π(θ | θ ), and the weights are set equal, wn

0 = 1/N .Let X (t) = (m1(t), m2(t)) denote the state variable at time t, satisfying the system of differential

equations (1.1)–(1.2). We need to propagate the solution X from tj to tj+1 using a numerical scheme,denoted by Ψ . If Xj is the approximate value of the current state, Xj ≈ X (tj), we have

X (tj+1) ≈ Xj+1 = Ψ (Xj, θ),

where we indicate the dependency of the propagated value on the parameter vector θ . In the particlefilter methods, we assume that at each t = tj we have a sample of state solutions

Mj = {X 1j , . . . , X N

j },

which corresponds to the parameter sample Sj. The initial state sample is defined as X 10 = · · · = X N

0 = 0,corresponding to the known initial value.

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The PF-SMC algorithm thus proceeds through the following steps:

1. Survival of the fittest:

(a) Propagate Mj using the auxiliary particles as parameters,

X nj+1 = Ψ (X n

j , θnj ), 1 � n � N .

Denote by mnj the predicted observation (2.1) computed with X n

j+1 and θnj .

(b) Compute the fitness of the auxiliary particle θnj by evaluating the weighted likelihood

functions,

gnj = wn

j exp

(− 1

σ 2j

|dj − mnj |2)

, σ 2j = σ 2 + mt(tjθ )2var(τ ).

Scale the fitness parameters gnj by their sum over the particle index n.

(c) Draw indices 1, . . . , N from the index set {1, 2, . . . , N} with replacement, where gnj is

the probability for n to be drawn.

(d) Reshuffle the samples, replacing (X nj+1, θn

j ) with (X nj+1, θ n

j ), and X nj with X n

j . In this newsample, the most fit auxiliary particles, for which gn

j is large, appear more often than theless fit particles.

2. Proliferation:

(a) Draw the parameter realizations θnj+1, 1 � n � N , from the Gaussian N (θ | θn

j , s2Cj).

(b) Repropagate each resampled particle with the new parameters and add innovation,

X nj+1 = Ψ (X n

j , θnj+1) + vn

j+1, 1 � n � N ,

where vnj+1 is drawn from a zero mean Gaussian distribution with variance set to

correspond to the estimated numerical approximation error of the selected numericalscheme Ψ .

(c) Update the weights to equal the likelihood ratio,

wnj+1 = π(dj+1 | X n

j+1, θnj+1, θ )

π(dj+1 | X nj+1, θn

j+1, θ ).

(d) Recompute the parameter ensemble mean and covariance θj+1 and Cj+1.

The PF-SMC algorithm recently proposed in Arnold et al. (2013) uses linear multistep method(LMM) numerical schemes with a fixed time step to solve the systems of differential equations. We referto the original article for the details on how higher order LMM solvers can be expressed in the properMarkovian framework and the assignment of the innovation variance in Step 2(b) using error controlestimates.

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The basic PF-SMC algorithm was modified for the current application to meet the challenges posedby the initial, very rapid ramping of the tracer’s mass and the 100-fold increase in the intervals betweendata measurements from the beginning to the end of the procedure. More specifically, instead of keepingthe time step fixed for each particle for the entire duration of the experiment, we adjusted the time stepaccording to the frequency of the data collection, keeping it the same for all particles. The detailsabout how the time step was selected and how the LMM had to be modified can be found in Arnold(2014). During the up-ramping phase of the experiment, the particle retention rate (PRR), defined asthe percentage of fit particles, may become prohibitively low: to avoid such an occurrence, if the PRRdrops below a certain threshold, which was set to 60% in the current application, we resample until thePRR is above the threshold.

Because of the complexity of the problem and the need for a large number of particles, the PF-SMC algorithm is computationally demanding. It was recently shown in Arnold et al. (2014) that it ispossible to reformulate the problem in so as to take advantage of vectorized and parallelized computingenvironment, significantly reducing the computing time. Our computed examples were implementedusing a vectorized version of the PF-SMC algorithm.

3. Results and discussion

The algorithm described in the previous section was applied to data collected from a cohort of 18volunteers from 3 different groups: a healthy control group of 5 individuals, a group of 7 individualssuffering from cirrhotic liver, and a group of 6 individuals with the condition of hepatic encephalopathy.As pointed out in the introduction, the motivation for the study was to find whether the liver condition ofthe subjects in the last two groups, through a perturbed ammonium level in their circulation, significantlyaffected the astrocytic metabolism at the level of acetate uptake and clearance.

The details about the cohort of subjects, the scanning procedures and the reconstruction of the datacan be found in Iversen et al. (2009). The [1-11C]-acetate PET protocol lasted 45 min and the datacorresponds to 50 time frames, with the interval between them increasing in time: every 5 s for the first18 frames, every 10 s for the next 12 frame, then every 30 s for the subsequent 7 frames, followed by oneframe after 120 s and finishing with 12 frame every 180 s. During the [1-11C]-acetate recording, arterialblood samples of 0.5 ml each were collected manually at the following intervals: every 5 s for the first18 times, every 10 s for the next 9 times, then every 60 s for the subsequent 12 frames, and finally 300 sfor the last 6 times. Total blood 11C radioactivity concentrations were measured using a well counter(Packard Instrument Company, Meriden, CT, USA), cross-calibrated with the tomograph, and correctedfor radioactive decay to the start of the scan. The volume of interest in our computed examples is thatof the whole-brain, and the corresponding TACs were extracted from the dynamic PET scan. The studywas performed in accordance with the Helsinki II Declaration and approved by the ethics committeeof Aarhus County. Informed consent was obtained from each subject. No complications were observedduring the procedures.

The 18 subjects are labelled with an index p, 1 � p � 18, and we identify the index sets with thecorresponding group, writing

HC = {1 � p � 5}, CL = {6 � p � 12}, HE = {13 � p � 18},where HC refers to healthy control, CL to cirrhotic liver and HE to hepatic encephalopathy. Foreach individual p, we record the optimized least squares estimate and the parameter sample withcorresponding weights as

θ(p), (θn(p), wn

(p)), 1 � n � N .

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HC CL HE0.000

0.050

0.100

0.150

0.200

V0

HC CL HE0.000

0.005

0.010

0.015

K1

HC CL HE0.000

0.010

0.020

0.030

k2

HC CL HE0.000

0.003

0.006

0.009

k3

HC CL HE0.000

0.001

0.003

0.005

0.007

k5

Fig. 2. Box plots indicating the 50 and 90% credibility intervals for each subject with respect to the five parameters. Box plotsfor the healthy control group (labelled as HC) are shown in black, for the cirrhotic liver group (CL) in blue and for the hepaticencephalopathy group (HE) in red, respectively. The red dot on each box plot marks the optimized least squares estimate of thecorresponding parameter and subject.

To obtain the parameter samples, for each individual p we run the hierarchical PF-SMC algorithm,filtering a sample size of N = 100, 000 particles, using the third-order Adams–Moulton (AM) schemefor the time propagation and the fourth-order AM scheme for the error estimation. The resultingparameter distributions are summarized in Fig. 2, where the box plots indicate the 50 and 90%credibility intervals of the conditional distributions π(θ(p) | θ(p), d(p)), where d(p) is the data vector

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Table 1 LMM PF-SMC parameter results for each subject p in the healthy control group,labelled by subject number 1 � p � 5. The optimized parameter value (computed via leastsquares) as well as the posterior mean of the filtered sample distribution and the upper (75%)

and lower (25%) quartiles are given

Parameter V0 K1 k2 k3 k5

Subject 1Optimized value 8.8914e−03 9.1126e−03 2.4863e−02 5.7240e−03 5.0372e−03Posterior mean 8.9489e−03 8.8374e−03 2.3401e−02 5.3588e−03 4.7153e−03Upper quartile 1.1578e−02 9.7043e−03 2.5913e−02 6.7686e−03 5.4572e−03Lower quartile 6.3371e−03 7.9528e−03 2.0803e−02 3.8626e−03 3.8935e−03

Subject 2Optimized value 2.7843e−02 8.8373e−03 2.3209e−02 3.4826e−03 3.0598e−03Posterior mean 3.2851e−02 8.4330e−03 2.2079e−02 3.2498e−03 2.7650e−03Upper quartile 3.8617e−02 9.2887e−03 2.4475e−02 4.0786e−03 3.2039e−03Lower quartile 2.8162e−02 7.5562e−03 1.9564e−02 2.3685e−03 2.2597e−03

Subject 3Optimized value 9.6508e−02 9.7631e−03 1.8257e−02 2.2531e−03 2.9420e−03Posterior mean 1.1467e−01 9.7486e−03 1.8837e−02 2.4727e−03 3.0761e−03Upper quartile 1.2852e−01 1.0940e−02 2.1130e−02 3.0072e−03 3.5728e−03Lower quartile 1.0360e−01 8.5796e−03 1.6624e−02 2.0034e−03 2.5951e−03

Subject 4Optimized value 8.3579e−02 5.6257e−03 1.3614e−02 1.7553e−03 1.6701e−03Posterior mean 9.7014e−02 5.3257e−03 1.3399e−02 1.6429e−03 1.4641e−03Upper quartile 1.0644e−01 5.9777e−03 1.5076e−02 2.0786e−03 1.6888e−03Lower quartile 8.8837e−02 4.6361e−03 1.1686e−02 1.1827e−03 1.2124e−03

Subject 5Optimized value 7.4945e−02 3.4092e−03 1.2487e−02 4.3138e−03 1.3619e−03Posterior mean 8.8266e−02 2.7094e−03 9.8134e−03 3.6135e−03 1.0863e−03Upper quartile 9.5864e−02 3.0568e−03 1.1424e−02 4.5033e−03 1.2022e−03Lower quartile 8.1943e−02 2.2705e−03 7.6249e−03 2.5508e−03 9.4770e−04

of the pth subject. The results are also given in tabular form, organized by subject group: healthycontrol subjects in Table 1, cirrhotic liver subjects in Tables 2 and 3, and hepatic encephalopathysubjects in Table 4. These results indicate that while there is no significant difference in the esti-mated parameters between the healthy control and cirrhotic liver groups, in hepatic encephalopathysubjects the parameters related to acetate uptake, in particular, i.e. V0 and K1, show a clear offset.Likewise, the k5 values in the hepatic encephalopathy group seem to be consistently lower than in theother groups.

To quantify the differences between the groups beyond a mere visual inspection, we performed thefollowing statistical Bayes ratio test. Let X denote a chosen reference group (healthy control, cirrhoticliver or hepatic encephalopathy) and Y a selected test group. We start by fitting a Gaussian distribu-tion N (θ | μX , CX ) to the combined sample {θn

(p) | p ∈ X , 1 � n � N}. Given the parameter θ(p) for an

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Table 2 LMM PF-SMC parameter results for subjects 6 � p � 10 in the cirrhotic liver group,labelled by subject number. The optimized parameter value (computed via least squares) as wellas the posterior mean of the filtered sample distribution and the upper (75%) and lower (25%)

quartiles are given

Parameter V0 K1 k2 k3 k5

Subject 6Optimized value 6.2424e−02 8.9456e−03 2.2924e−02 2.5699e−03 3.2486e−03Posterior mean 6.4943e−02 8.7107e−03 2.2542e−02 2.5062e−03 2.9698e−03Upper quartile 7.9495e−02 9.7667e−03 2.5332e−02 3.1902e−03 3.4833e−03Lower quartile 5.0891e−02 7.6447e−03 1.9674e−02 1.8031e−03 2.3962e−03

Subject 7Optimized value 8.6474e−02 6.3450e−03 1.3867e−02 2.0623e−03 2.0401e−03Posterior mean 8.1059e−02 6.3240e−03 1.3928e−02 2.0990e−03 1.9654e−03Upper quartile 9.1732e−02 7.0845e−03 1.5591e−02 2.5934e−03 2.2537e−03Lower quartile 6.9716e−02 5.5673e−03 1.2283e−02 1.6111e−03 1.6681e−03

Subject 8Optimized value 1.2194e−01 6.8327e−03 1.5541e−02 1.3862e−03 2.0483e−03Posterior mean 1.2634e−01 6.9997e−03 1.6248e−02 1.3980e−03 2.0432e−03Upper quartile 1.4114e−01 7.9292e−03 1.8245e−02 1.7431e−03 2.3962e−03Lower quartile 1.1194e−01 6.1090e−03 1.4336e−02 1.0566e−03 1.6836e−03

Subject 9Optimized value 7.9554e−02 4.6056e−03 1.2285e−02 2.4207e−03 1.3916e−03Posterior mean 8.6554e−02 4.4903e−03 1.2433e−02 2.4304e−03 1.3011e−03Upper quartile 9.6469e−02 5.0371e−03 1.4032e−02 3.0483e−03 1.4936e−03Lower quartile 7.7539e−02 3.9271e−03 1.0840e−02 1.8147e−03 1.0996e−03

Subject 10Optimized value 4.5701e−02 6.6120e−03 2.1954e−02 3.5441e−03 2.5974e−03Posterior mean 5.5152e−02 6.1530e−03 2.0826e−02 3.4431e−03 2.3465e−03Upper quartile 6.3375e−02 6.9295e−03 2.3644e−02 4.3590e−03 2.7102e−03Lower quartile 4.8836e−02 5.3202e−03 1.7818e−02 2.4984e−03 1.9483e−03

individual from the test group, p ∈ Y , we can evaluate the mean likelihood, averaged over the condi-tional posterior,

L(p | X , θ(p)) =∫

N (θ(p) | μX , CX )π(θ(p) | θ(p), d(p))dθ(p)

≈N∑

n=1

wn(p)N (θn

(p) | μX , CX ).

This quantity expresses the average likelihood of getting the observation θ(p) if it would come from thedistribution calculated from the reference group data.

To average over the test group, we fit the sample {θ(p) | p ∈ Y } with a uniform distribution, whichwe assume represents the distribution of the hyperparameter θ over the test group Y , denoted π(θ | Y).

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ESTIMATION OF PET TRACER DYNAMICS 13 of 16

Table 3 LMM PF-SMC parameter results for subjects p = 11 and p = 12 in the cirrhotic livergroup, labelled by subject number. The optimized parameter value (computed via least squares)as well as the posterior mean of the filtered sample distribution and the upper (75%) and lower(25%) quartiles are given

Parameter V0 K1 k2 k3 k5

Subject 11Optimized value 1.6253e−01 3.7216e−03 1.1151e−02 1.5924e−03 1.3767e−03Posterior mean 1.5490e−01 4.0900e−03 1.2413e−02 1.7288e−03 1.4476e−03Upper quartile 1.6371e−01 4.6239e−03 1.4211e−02 2.1220e−03 1.6551e−03Lower quartile 1.4573e−01 3.6094e−03 1.0790e−02 1.3669e−03 1.2471e−03

Subject 12Optimized value 6.8065e−02 6.7645e−03 1.6687e−02 1.9993e−03 2.2676e−03Posterior mean 8.2760e−02 6.4842e−03 1.6449e−02 1.9803e−03 2.2527e−03Upper quartile 9.2506e−02 7.2070e−03 1.8144e−02 2.5240e−03 2.6489e−03Lower quartile 7.5241e−02 5.7296e−03 1.4722e−02 1.4388e−03 1.8481e−03

The uniform distribution is justified by the small sample size and the rejection of a multivariate normaldistribution via Mardia’s test for skewness and kurtosis; see Mardia (1970, 1974) for test details. Thelikelihood averaged over the combined posterior density

L(Y , X ) =∫

L(p | X , θp)π(θp | Y)d θp

≈ 1

nY

nY∑p=1

L(p | X , θ(p)),

can be thought of as an average measure for the probability of the sample of group Y as a whole,assuming that it came from the distribution of the reference group X. To obtain meaningful relativemeasure of how well the groups fit, we compute a version of a Bayes ratio, defining

R(Y , X ) = L(Y , X )

L(X , X ),

where the reference value R(Y , X ) = 1 would indicate a perfect match.Table 5 lists the Bayes ratios R for all possible pairs X and Y, while Table 6 gives the Bayes ratios

comparing healthy control with the combined diseased group, cirrhotic liver + hepatic encephalopathy.While these ratios do not suggest any stark differences between healthy control and cirrhotic liver orthe combined diseased group, Table 5 shows a clear distinction between hepatic encephalopathy and theother groups, both when hepatic encephalopathy cohort is treated as the control group and as the testgroup.

From the physiological point of view, the lowered values of V0 and K1 in the hepatic encephalopathygroup point towards a lowered cerebral blood flow (CBF), which is in line with previous findings (Keid-ing and Pavese, 2013). On the other hand, it has been previously found (Gjedde et al., 2010) that oxygendelivery is not limited in hepatic encephalopathy, while a decrease in the oxygen cerebral metabolic rate(CMRO2 ) was observed in this cohort. Therefore, we may conclude that the low CMRO2 must be due to

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Table 4 LMM PF-SMC parameter results for each subject p in the hepatic encephalopathygroup, labeled by subject number 13 � p � 18. The optimized parameter value (computed vialeast squares) as well as the posterior mean of the filtered sample distribution and the upper(75%) and lower (25%) quartiles are given

Parameter V0 K1 k2 k3 k5

Subject 13Optimized value 4.4654e−02 4.4032e−03 1.1005e−02 1.3250e−03 1.2227e−03Posterior mean 5.3271e−02 4.2031e−03 1.0523e−02 1.2412e−03 1.1120e−03Upper quartile 6.0117e−02 4.6359e−03 1.1689e−02 1.5647e−03 1.2983e−03Lower quartile 4.7824e−02 3.7525e−03 9.3077e−03 9.0020e−04 9.0731e−04

Subject 14Optimized value 1.6110e−02 4.2687e−03 1.0831e−02 1.6558e−03 1.9719e−03Posterior mean 2.0956e−02 4.1881e−03 1.0793e−02 1.5888e−03 1.8064e−03Upper quartile 2.3107e−02 4.4592e−03 1.1625e−02 2.0094e−03 2.0693e−03Lower quartile 1.9617e−02 3.9188e−03 9.9625e−03 1.1517e−03 1.5186e−03

Subject 15Optimized value 3.1157e−02 4.6739e−03 1.1737e−02 2.1741e−03 1.8991e−03Posterior mean 4.0486e−02 4.5001e−03 1.1340e−02 1.9457e−03 1.6556e−03Upper quartile 4.3950e−02 4.7844e−03 1.2243e−02 2.4494e−03 1.8711e−03Lower quartile 3.8186e−02 4.2048e−03 1.0404e−02 1.3962e−03 1.4190e−03

Subject 16Optimized value 2.7991e−02 2.7334e−03 1.6681e−02 5.5866e−03 9.8447e−04Posterior mean 3.5249e−02 2.3085e−03 1.3433e−02 4.8445e−03 8.2596e−04Upper quartile 3.9306e−02 2.5910e−03 1.5423e−02 5.9660e−03 9.1559e−04Lower quartile 3.2354e−02 1.9808e−03 1.0935e−02 3.5671e−03 7.2420e−04

Subject 17Optimized value 3.9232e−02 3.3517e−03 1.0546e−02 3.5964e−03 1.6859e−03Posterior mean 4.4772e−02 2.9977e−03 9.1167e−03 3.0473e−03 1.4430e−03Upper quartile 4.8628e−02 3.2354e−03 9.9861e−03 3.6583e−03 1.5816e−03Lower quartile 4.1349e−02 2.7418e−03 8.1459e−03 2.3450e−03 1.2868e−03

Subject 18Optimized value 4.6142e−02 4.4364e−03 1.0484e−02 1.5789e−03 1.8681e−03Posterior mean 5.8406e−02 4.2594e−03 1.0239e−02 1.5104e−03 1.7735e−03Upper quartile 6.3940e−02 4.6723e−03 1.1291e−02 1.9177e−03 2.0431e−03Lower quartile 5.4438e−02 3.8351e−03 9.1618e−03 1.0871e−03 1.4834e−03

a not specified metabolic inhibition, and the low CBF is a consequence, rather than a cause, of the lowCMRO2 . The natural question that arises is whether the metabolic inhibition occurs in astrocytes, neu-rons or both. The relatively insignificant difference in the parameter k3 in the three cohorts suggests thatthe oxidative metabolism of the astrocyte in hepatic encephalopathy is not compromised, thus shiftingthe attention to the oxidative metabolism in neuron.

Methodologically, the current approach for the parameter estimation accounts for the intrinsic vari-ability of the parameter values within individuals, not only within a cohort. It also provides a naturalway to assess the sensitivity of the data to the parameters, removing the need for a separate sensitivity

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ESTIMATION OF PET TRACER DYNAMICS 15 of 16

Table 5 Bayes ratios for all possible pairs of con-trol groups X and test groups Y

X

Y HC CL HE

HC 1.0000 0.5821 0.0042CL 0.9591 1.0000 0.0034HE 0.2205 0.0444 1.0000

Table 6 Bayes ratios comparing thehealthy control group with the combineddiseased group, cirrhotic liver + hepaticencephalopathy

X

Y HC CL + HE

HC 1.0000 0.5285CL + HE 0.6182 1.0000

analysis based on linearization or bootstrapping. This approach is in line with the current interest inuncertainty quantification within the mathematical sciences. Moreover, the estimation of the distribu-tion of the parameters in sample form makes it possible to carry out correlational studies between pairsof parameters to uncover hidden dependencies.

Acknowledgement

This work was partly supported by NSF DMS project number 1312424 (Erkki Somersalo) and bygrant number 246665 from the Simons Foundation (Daniela Calvetti). In addition to co-authors AlbertGjedde and Peter Iversen, we acknowledge the following members of the ‘liver-brain group’ at AarhusUniversity and the University of Copenhagen who made the study possible: Kim Mouridsen PhD,Mikkel B. Hansen PhD, Svend B. Jensen PhD, Michael Sørensen MD PhD, Lasse K. Bak PhD, HelleWaagepetersen PhD, Arne Schousboe DSc, Peter Ott MD DSc, Hendrik Vilstrup MD DSc, and SusanneKeiding MD DSc.

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