Arterial Spin Labeling (ASL) techniques potentially allow the absolute, non-invasive quantification of brain perfusion. This can be achieved by fitting a kinetic ...
Bayesian estimation of perfusion using PASL-MRI Juliane Farthouat1,2, João Sanches1,2, Patrícia Figueiredo1 1 Instituto Superior Técnico, Lisboa, Portugal; 2ISR, Lisboa. Abstract Arterial Spin Labeling (ASL) techniques potentially allow the absolute, non-invasive quantification of brain perfusion. This can be achieved by fitting a kinetic model to the data acquired at a number of inversion times (TI). Here, we propose a Bayesian framework for ASL model estimation based on the maximum a posteriori (MAP) criterion. A priori information concerning the physiological variation of the model parameters is used to guide the solution. Monte Carlo simulations showed improved performance when compared with a standard Least Squares (LS) approach.
In this study, we aimed to implement and validate a Bayesian estimation algorithm, incorporating a priori information about the physiological variation of the model parameters. M
t
1. Introduction Perfusion describes the distribution of nutrients to the tissues by blood flow through the capillary bed and is defined as volume of blood per unit time and per unit volume of tissue. Arterial Spin Labeling (ASL) magnetic resonance imaging (MRI) techniques offer a noninvasive way of generating perfusion images that are potentially quantitative [1]. They consist on magnetically labeling the water molecules in the arterial blood and then measuring the magnetization of the tissues after a certain time interval, the inversion time ( TI ). The magnetization difference M measured as a function of TI in pulsed ASL (PASL) experiments can be described by a standard kinetic model [2] (Fig. 1): 0 TI t Af r1bTI k (TI t ) M (TI , ) 1) e (e t TI t k r1bTI k (TI t ) e k (TI t ) ) (e TI t e where f , t , , T1b , T1 is the vector of parameters,
including perfusion f ( s 1 ) and the transit time t (s ) . In principle, the magnetization collected at a single TI point is sufficient to obtain a perfusion estimate, provided that the values of the other model parameters are available or can be assumed. However, this is not always the case, particularly in pathological conditions such as cerebrovascular disease. In these cases, it would be possible to estimate perfusion, as well as other unknown parameters, by fitting the PASL model to M data collected at multiple TI points. Because of the intrinsically low Signal-to-Noise Ratio (SNR) of PASL data, substantial signal averaging is usually performed, which is reflected in proportionally increased acquisition times. This limitation is especially critical when multiple TI points should be sampled in order to perform quantitative parameter estimation. In this case, a trade-off exists between the number of TI points sampled and the number of averages collected at each point. For these reasons, PASL model estimation becomes a very difficult problem.
TI
Figure 1: M as a function of TI in PASL (Eq. 1).
2. Problem formulation
Let Y y1 , y2 ,..., y N be a set of observations:
yi M (ti , ) i where i ~ N (0, y2 )
is assumed to be Additive White
Gaussian Noise (AWGN) and M (TI , ) is the magnetization difference predicted by the PASL model (Eq. 1). A common approach to estimate the parameter vector is the Least Squares (LS) method, formulated as the following optimization task:
ˆ arg min Y ΔM(θ)
2
Because of the non-linearity of the model, this is an illposed problem that needs appropriate techniques to be solved. Here, a Bayesian framework is proposed whereby the model parameters are estimated using the maximum a posteriori (MAP) criterion [3]. Prior information about the physiological distributions of the parameter values, obtained from the literature, is then used in order to obtain more accurate and stable solutions [4]. The proposed algorithm is formulated as the following optimization task:
ˆ arg min E (Y , )
where E (Y , ) is the energy function composed by two terms:
E (Y , ) Y ΔM( ) y2 (θ θ 0 )T C 1 (θ θ 0 ) 2 2
Data fidelity term
Pr ior term
with C diag ( 1 , 2 ,..., P ) containing the known uncertainty of the model parameters. The optimization is accomplished by using the Newton method:
n1 n H 1 ( n ) E (Y , n )
where H E (Y , ) and E (Y , ) are the Hessian 2
i j
matrix and the gradient of E (Y , ) with respect to , respectively. In this work, using synthetic data, the parameters i y2 / 2 are computed and used in the
4. Results
i
estimation of . When using real data, an accurate estimation of the amount of noise corrupting the data, y2 , is needed [5].
Figure 3. SNR of the estimated curves as a function of the SNR of the synthetic data, obtained using both the standard LS ( ) and Bayesian approach ( ).
5. Discussion
Figure 2. Mean (top) and standard deviation (bottom) of the absolute value of the parameter estimation error, for
f , t , using both the standard LS (
Bayesian approach (
) and
).
Monte Carlo simulations were performed by fitting the model to 1000 synthetic PASL datasets with added Gaussian noise (10, 50, 75 100, 125 and 150 % of the maximum signal). We used typical gray matter values at 3T drawn from their physiological variations as reported in the literature: f ~ N (0.012 ,0.0042 ) s 1 ; t ~ N (0.7,0.32 ) s ;
~ N (0.7,0.12 ) s ; T1 ~ N (1.3,0.12 ) s ; T1b ~ N (1.5635,0.21152 ) s . A uniform sampling strategy of 100 TI points in the interval [0, 5000] ms was employed. A 2-parameter f , t model estimation was performed using the Bayesian ) as well as a standard least squares
). approach ( The mean and standard deviation (SD) of the absolute values of the parameter errors were calculated at each noise level. Moreover, the Signal to Noise Ratio of the data ) and the estimated curves ( ( determined as:
6. References [1] Figueiredo, PM et al., 2005, Quantitative perfusion measurements using pulsed arterial spin labeling: effects of large region-of-interest analysis. JMRI, 21(6):676-82.
3. Methods
approach (
A Bayesian algorithm was implemented and validated for the estimation of a perfusion kinetic model, incorporating a priori information about the physiological variation of the parameters. Monte Carlo simulations showed improved performance when compared with a standard LS approach.
) were
[2] Buxton, RB, 1998, Quantitative perfusion measurements using pulsed arterial spin labeling: effects of large region-ofinterest analysis. MRM, 40(3),383-96. [3] Steven M. Kay, Fundamentals of statistical signal processing: estimation theory, Prentice-Hall, Inc., Upper Saddle River, NJ, 1993. [4] Sanches, J. M. Nascimento, J. C. Marques, J. S., Medical Image Noise Reduction Using the Sylvester– Lyapunov Equation, IEEE Transactions on Image Processing, Volume: 17, Issue: 9, pp.1522-1539, Sept. 2008. [5] Moon and Stirling, Mathematical Methods Algorithms for Signal Processing, Prentice Hall, 2000.
7. Acknowledgements FCT grant PTDC-SAU-BEB-65977-2006. BIAL grant Nb.16/2004.
and
