Computational and Mathematical Methods in Medicine

Volume 2015, Article ID 425475, 11 pages

http://dx.doi.org/10.1155/2015/425475

## A Sensitivity Analysis of fMRI Balloon Model

Computer, Electrical and Mathematical Sciences and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia

Received 3 March 2015; Accepted 22 April 2015

Academic Editor: Nelson J. Trujillo-Barreto

Copyright © 2015 Chadia Zayane and Taous Meriem Laleg-Kirati. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Functional magnetic resonance imaging (fMRI) allows the mapping of the brain activation through measurements of the Blood Oxygenation Level Dependent (BOLD) contrast. The characterization of the pathway from the input stimulus to the output BOLD signal requires the selection of an adequate hemodynamic model and the satisfaction of some specific conditions while conducting the experiment and calibrating the model. This paper, focuses on the identifiability of the Balloon hemodynamic model. By identifiability, we mean the ability to estimate accurately the model parameters given the input and the output measurement. Previous studies of the Balloon model have somehow added knowledge either by choosing prior distributions for the parameters, freezing some of them, or looking for the solution as a projection on a natural basis of some vector space. In these studies, the identification was generally assessed using event-related paradigms. This paper justifies the reasons behind the need of adding knowledge, choosing certain paradigms, and completing the few existing identifiability studies through a global sensitivity analysis of the Balloon model in the case of blocked design experiment.

#### 1. Introduction

Functional Magnetic Resonance Imaging (fMRI) technique has been the area of great interest since it measures the activity of the brain, which gives much insight on brain functions. The contrast agent in fMRI is based on the rate of oxygen consumption in the brain through the Blood Oxygenation Level Dependent (BOLD) signal. Whether the aim is to characterize a region of the brain or to compare different regions and/or subjects, it is necessary to link the BOLD indirect measure to the stimulus.

Several studies have considered modeling the hemodynamic response of the brain. These models fall in two categories: general linear modeling-based detection methods, which assume a known shape of the hemodynamic response function (HRF) and estimation ones which track the most appropriate HRF form in regions eliciting evoked activity in response to stimulus presentation. The first class of models uses statistical tools (discrepancy) to fit some basis functions (Poisson function [1], Gaussian function [2], Gamma function [3], inverse Logit function [4]) to the BOLD for the aim of activation detection. The second is more concerned with the physiological aspects that underlie the BOLD transients. It is natural at this point to think that understanding the brain functions requires a much more elaborated framework to map the neural activity to changes in the measured BOLD signal than a convolution by some predefined hemodynamic response functions. Furthermore, it has been shown [5–7] that to be more consistent, this mapping has to encompass nonlinear effects to describe more accurately the pathway from neural activation to the BOLD.

Among the nonlinear hemodynamic models, linking changes in the physiological variables, namely, the blood flow, the volume, and the oxygenation level to the neural activity in a localized area of the brain, is the Balloon model. This model has been first proposed by Buxton et al. [6]. Several variants of the Balloon model have been developed such as Friston et al.’s [8] variant considered in this paper. The main feature of the Balloon model is that it describes the BOLD signal as a nonlinear combination of the blood volume and deoxyhemoglobin concentration, which are expressed as nonlinear functions of the neural activity and the blood flow.

Models in fMRI, in particular the Balloon model, have been used in many studies to estimate underlying parameters. Indeed, these models become criticized because (1) our comprehension of the mechanisms underlying the hemodynamic response is evolving and (2) they are often used without previous evaluation of their identifiability. Therefore, this paper is relevant to this second point. Indeed, estimating the state, the input, and parameters for the Balloon model using BOLD measurements is ill-posed. The ill-posedness, also seen as an identifiability issue when the input signal is given, consists in having a small change in the BOLD signal while observing large variations of parameters and thus poor accuracy of the parameter estimates. Different strategies have been used to add prior knowledge that makes the problem well-posed. Riera et al. [9] used radial basis functions to express the input while Havlicek et al. [10] supposed Gaussian priors for the parameters. Small prior variances are sought to overcome the ill-posedness of some parameters. In the same field of Kalman filtering techniques, Hu et al. [11] used a square root unscented Kalman (SR-UKF) filter to estimate the states and some parameters with Gaussian priors.

Besides, even if it were possible to develop theoretical tools to get an analytical expression for the parameters from the nonlinear equations describing the dynamic behavior of the Balloon model, the input signal has to be “rich enough” so that the parameters can be estimated accurately. The typical example for that is when one parameter quantifies the output behavior with respect to the derivative of the input, and if the input is constant, then the parameter cannot be estimated efficiently. Thus, the input sequence has to be well designed to capture the effect of each of the unknown parameters on the output signal.

In practice, usually there is unfortunately a trade-off between the good contrast-to-noise ratio (due to slow or static behavior) and the estimation efficiency (often assessed for fast time scales). In general, the challenge is attributed to experimental limitations such as possible damage of the studied system (due to high switchings in frequencies, e.g.) and poor signals quality (low signal to noise ratio SNR) when dealing with high frequencies. This question has been raised for the fMRI experimental paradigms by Liu et al. [12] and a theoretical model for the quantification of the so-called detection power/estimation efficiency duality. The approach was developed within the hemodynamic general linear modeling framework.

In this study, we will focus on the identifiability study of the Balloon model parameters which represents an attempt to answer the question: *Given the output measurement ** (BOLD) and the input ** (Neural activity), can ** (Balloon model parameters) be determined uniquely?*

The paper is organized as follows. In the second section, we introduce the nonlinear hemodynamic Balloon model and present the commonly used fMRI paradigms. The third section reviews the existing sensitivity approaches of the Balloon model and completes them by a global sensitivity study when the stimulus follows block design experiment. Finally some conclusions are presented in the fourth section.

#### 2. Problem Statement

##### 2.1. The Balloon Model

The dependence of the BOLD signal upon the variations of the normalized blood flow , volume , and deoxyhemoglobin content has been proposed by Buxton et al. [6] and completed by Mandeville et al. [13]. The model presents the venous compartment’s structure as a Balloon where a stimulus resulting in a local neuronal activity leads to an increase in the blood flow as illustrated in Figure 1.