ARMA Modeling for Estimation of Permeability from Perfusion MRI Mohammad Mehdi Khalighi 1,2 , Hamid Soltanian-Zadeh 1,2 , James R. Ewing 3 1 Electrical and Computer Engineering Department, University of Tehran, Tehran, Iran 2 Radiology Image Analysis Lab, Henry Ford Health System, Detroit, MI 48202 3 Neurology Department, Henry Ford Health System, Detroit, MI 48202 ABSTRACT We develop noninvasive MRI techniques that quantify the permeability of the Blood-Brain Barrier (BBB). Using such gadolinium compounds as Gd-DTPA and gadomer- 17, changes in R1 (R1 = 1/T1) can be measured and used as estimates of tissue concentration versus time, thus permitting estimates of BBB permeability parameters. Estimating BBB permeability parameters requires deconvolution in a linear system, a task that has been accomplished heretofore by nonlinear least-squares procedures[1-3], although these techniques tend to instability in the presence of noise. We introduce a method using Z-transform and Autoregressive Moving Average (ARMA) modeling to perform this deconvolution. This method has the advantage that it linearizes the optimization procedure by which parametric estimates are formed, and it stabilizes the deconvolution in the presence of noise. Keywords: auto regressive moving average, ARMA, Z- transform, permeability, perfusion, magnetic resonance imaging, MRI 1. INTRODUCTION Most proposals to quantitate BBB permeability have used a compound of gadolinium as the indicator, and some T1- related magnetic resonance imaging (MRI) parameter to judge tissue concentration of the gadolinium. Early proposals for the quantitation of BBB permeability have emerged from Tofts and Kermode [2] Larsson et. al. [4], and Brix et. al. [5]. Tofts, in reviews [1,6], has noted the differences between these approaches, and resolved differences in notation. Common to these approaches is the assumption that contrast changes due to magnetic resonance contrast agent (MRCA) in the vascular volume are negligible. This assumption is challenged by Henderson et al [3]; these authors present the full model and demonstrate maps of BBB permeability parameters using fast T1-weighted imaging procedures. We have adopted their model, but used slower imaging techniques to estimate changes in R1, which have been shown to be proportional to tissue concentration of Gd [2]. In order to optimize our parametric estimation, we have sought stable approaches to deconvolution in the presence of noise. 2. METHODS 2.1 Theory The goal is to estimate the quantity of indicator inside and outside the capillary bed. This is approached via a single capillary model [7,8]. After the distribution of indicator in the capillary is considered, the concentration of indicator in the tissue can be inferred. We will use the so-called adiabetic approximation [9] and consider that the concentration in the tissue changes very slowly when compared to the concentration in the capillary. So the concentration of agent in the tissue can be written as [3]: ) ( ) ( ) ( ' ' t C r e FE t rC t C a t V FE a tis ees α + ⊗ = − (1) where a C is the plasma concentration of indicator at the entrance to the capillary, F is the capillary flow (ml/min) ees V is the fraction tissue volume (ml/gr) occupied by the extracellular indicator, ' E is the extraction fraction in where there is no extravascular concentration of indicator, i.e., the portion of indicator that leaks out of the capillary in a single pass, r is the constant of proportion relating arterial concentration of indicator to capillary concentration and α is the vascular fraction in tissue (ml/g). In a real experiment we always have noise and so we can re-write equation (1) as: ) ( ) ( ) ( ) ( ' ' t e t C r e FE t rC t C a t V FE a tis ees + + ⊗ = − α (2) where ) (t e is random white noise. This equation is in continuous time domain and if we sample ) (t C tis and ) (t C a each T seconds, then in discrete time domain we have: 1 ,..., 1 , 0 ) ( ) ( ) ( ) ( ' ' − = + + ⊗ = − N n n e n C r e FE n rC n C a n V T FE a tis ees α (3)