Quantitative MRI & Dynamic Models

Transcription

1 Quantitative MRI & Dynamic Models The pre-eminent role of imaging now requires a new level of metric - quantitative measurements Robert I Grossman, Radiology Dep, NYU why measuring relaxation times? T1 mapping T2/T2* mapping, diffusion models MR contrast agents DCE image analysis basic model for DCE MRI numerical techniques for qmri and DCE-MRI: fitting model to data 1

2 Applications of qmri qmri allows statistical analysis of tissue by providing scale & units detects subtle changes that may not be apparent to radiologist' eye T1: tumor assessment, cartilage damage, conversion to tracer concentration T2* iron content, tissue oxygenation problems: propagation of noise lack of standard methods hard to deliver quickly -- complicated image post-processing Damedian "Patent 832" Table 2 envisions qmri even before MRI 2

3 T1 measurement User manipulates flip angle θ TI TR Several images are acquired. System of equations (2 or more) solved for M0 and T1 Saturation recovery method Acquisitions with different TR 's For accuracy we need to know what T1 is expected (a) TR < 0.5 T1 (b) TR > 2 T1 Relatively slow 3

4 Variable flip angle gradient echo method Repeated flip angles: Two or more equations with different θ 's Ernst angle E : flip angle that gives the maximum signal (in a fixed time) For accuracy we need to know what T1 is expected (a) θ < E (b) θ > E Faster than saturation recovery Needs accurate flip angles 4

5 Inversion recovery method sequence: TI readout... TR-TI... (a more general equation exists for arbitrary flip angles) Gold standard because it gives accurate T1 Used in spectroscopy Slow, but one could use bssp for faster acquisition (Bokacheva MRM 2006) 5

6 R2, R2*, ADC measurement Functional form so can be transformed to a linear regression. Special methods DESPOT1, DESPOT2 (Deoni MRM 2003, MRM 2005) DESPOT: combined T1 and T2 mapping DESPOT1: T1 is calculated from a series of SPGR DESPOT2: T2 is calculated from a series of SSFP images Both acquired over a range of flip angles with constant TR. T1 information used in computing T2. TRITONE (Fleysher & Fleysher MRI 2008) addresses the issue of B1 inhomogeneity by collecting three conventional 3D SPGR EPI images to produce unbiased T1 maps. FireVoxel demos: (a) T1 mapping (b) ADC mapping 6

7 MRI contrast agents Most clinical MRI contrast agents are complexes of gadolinium. Gd 3+ is a highly toxic metalic ion. Chelating is needed to administer in the patient. Approved chelating agents: DTPA, DOTA & derivatives. the electron spin resonance of Gd 3+ matches the Larmor frequency & induces electron-nucleus interactions. This indirectly shortens the T1 relaxation time of hydrogen protons (water and fat) that are in close proximity of Gd 3+. 7

8 Common MRI ligands 8

9 MRI signal vs concentration R1 = 1/T1 R2 = 1/T2 units of R1,2 s -1 1/T1,2obs = 1/T1,2tissue + 1/T1,2agent 1/T1,2obs = 1/T1,2tissue + r1,2 [Gd] [Gd] = concentration in mm linear relationship between R 1 and [Gd] linear relationship between R 2 and [Gd] 9

10 Factors affecting relaxivity macromolecular content water exchange water diffusion susceptibility DCE-MRI analysis 10

11 DCE-MRI: complete workflow 1) generate time series from DICOM files 2) coregister 3) form time activity curves TAC (may need segmentation) 4) convert to concentration (may use pre-contrast T1/T2) 5) fit model to concentration curves 11

12 Signal to concentration: S(t) C(t) in two steps 1. Compute T1(t) from above eq. given: S0 = signal before contrast arrival T10 = pre-contrast T1 of tissue θ = flip angle 2. Compute tracer concentration C(t) in tissue 1/T1(t) = 1/T10 + r1 C(t) r 1 = relaxivity, or increase in R1 per unit concentration of Gd 12

13 Single compartment perfusion model A bucket or a pool of volume V (ml) well-mixed input flow = output flow = k (ml/min) A(t) = concentration in inlet tube = input function (measured) C(t) = concentration in the pool = outlet tube (measured) Conservation of mass: foe each delta t amount of tracer entering the pool : k A(t) amount of tracer exiting the pool: k C(t) amount of tracer inside the pool: V C(t) V C'(t) = k A(t) k C(t) linear 1st order differential equation with constant coefficients can be solved for k/v = flow rate in (ml blood/min)/(ml tissue) 13

14 Linear 1st order differential equation with constant coefficients a1 y'(t) + a0 y(t) = g(t) The solution is given by convolving g(t) and an exponential function: where k= a0/a1 & q(t) = (1/a1) g(t) Proof: divide by a1 to get the "standard form": y'(t) + k y(t) = q(t) y(t) = particular solution + const general solution of homogenous equ. particular general y(t) is the tissue concentration, t=0 is just before initial contrast appearance in the arterial system, so we usually want y(0)=0 const =0 14

15 V C'(t) = k A(t) k C(t) regional flow (tissue perfusion) = k / V often expressed in: ml blood per min per 100 gram of tissue 15

16 Basic DCE-MRI model of tissue concentration (Tofts model) Contrast injected into blood plasma compartment plasma ~60% of blood volume (blood cells = other 40%) plasma considered a variable fraction of tissue reversible exchange between blood and extravascular extracellular space EES EES 10-60% of tissue 16

17 vp plasma included as part of tissue volume (high in liver, kidney, low in brain) Cp(t) = arterial (plasma) input function (measured) kep = K trans /ve flow of tracer between plasma & EED = kep differences in concentrations (passive exchange) ve Ce'(t) = kep Cp(t) kep Ce (t) Tofts model: 1st term often omitted 17

18 Tofts et al JMRI 1999 Standard terminology and units 18

19 Properties of parameters ve volume of EES regions with high ve take longer to reach the peak K trans transfer rate rapid enhancement high K trans For healthy blood vessel (no Gd leakage): K trans = vessel permeability x surface area permeability-limited regime regional blood flow >> PS For leaky vessels (in many tumors) K trans = blood flow flow-limited regime PS >> regional blood flow 19

20 Two representative DCE-MRI studies Pickles et al. Breast Cancer Research & Treat 2005 Ocak et al. AJR 2007 Some clinical DCE-MRI results using Tofts model are hard to interpret. More sophisticated models are able to estimate both flow and leakage: AATH model Multi-compartment models Both require good temporal resolution for meaningful fitting. 20

21 Estimating model parameters model (green curve): data (white crosses): model fits the data compute parameters to minimize the difference (sum of squares) This problem (also many T1 mapping approaches) is an example of non-linear minimization Minimization problem = maximization problem maximize f(x 1, x 2..x n ) is equivalent to minimizing g() = -f() (we can say optimization but then there is confusion with control theory that uses analytical techniques) 21

22 Grid method Brute force, but becomes more effective as computers become faster 22

23 Random grid vs. uniform grid 23

24 A sequence of 1-D minimizations (iterative) General rule in function minimization: we don't expect good one-dimensional techniques to perform well when extended to high dimensionality 24

25 Simplex method of Nelder & Mead (amoeba) One of the most successful iterative (stepping) methods. MATLAB: fminsearch() Simplex is an n-dimensional figure specified by n + 1 points in n-space. Triangle in two dimensions, tetrahedron in 3D. At each step the program remembers the value of minimized function at n + 1 points. Starting simplex points are arbitrary (healthy tissue parameters). P H point is where the function is highest (worst) P L point is where the function is lowest (best so far) 25

26 From the original simplex, a new simplex is formed by replacing P H by a "better" point. Each search is in an `intelligent' direction, pointing from the highest value to the average of the other values P. P is the center of gravity (centroid) of n other points (after excluding the worst point) Ameoba can: reflect expand contract see video: 26

27 Reflection The first attempt to find a better point is made by reflecting P H with respect to P: P* = P + (P - P H ) Ameoba is told to reflect if P* is lower than the second worst, but not better than the best. then iterate keep P* if f(p*) < f(p 2 ) & f(p*) > f(p L ) 27

28 ... otherwise consider expansion (move down hill faster) else if f(p*) < f(p L ), a new point is tried at P** = P + 2 (P - P H )... otherwise consider contraction else if f(p*) > f(p H ), a new point is tried at P** = P (P - P H ). 28

29 The best new points replaces P H in the simplex, then we iterate. If none of reflection, expansion, contraction are applicable (all are larger than PH), a whole new simplex is formed around P L, with dimensions reduced by a factor of 0.5. Note we don't let amoeba to collapse into a hyperplane of n - 1 dimensions amoeba is designed to take as big steps as possible, and therefore amoeba is relatively insensitive to shallow local minima -- fine structures in the function caused by noisy data, rounding errors, etc. 29

30 Analytic methods Many require that we compute partial derivatives. Steepest descent -- series of minimizations along the direction of local gradient. Unfortunate property: successive searches are in orthogonal directions. Newton's method (requires the matrix of 2nd derivatives or Hessian matrix) Conjugate gradient method. Variable metric method of Fletcher and Powell. Least squares fit: Levenberg-Marquard algorithm. MATLAB: lsqnonlin() Interval arithmetic & Global optimization FireVoxel demo: prostate perfusion 30

31 To make a good model (not just DCE-MRI) 1. Ensure estimated parameters represent known physiology (validation). This work leads to successive revisions. 2. Make sure that changes in model parameter lead to changes in measured data. This could lead to a revised, simpler model. 3. Ensure precision of estimated parameters using phantoms, repeated imaging, Monte- Carlo simulations. This also often leads to a revised, simpler model. Model parameter becomes a biomarker if sensitive to physiological changes -- responds to progression of disease, responds to successful treatment reliable (good precision) agrees with well established reference measures (accurate) 31