Multi site chemical exchange model

Longitudinal (T1) relaxation in systems with chemical exchange is influenced by the exchange kinetics
and by the ratio equilibrium magnetizations of the exchanging species.
The theory describing the effect of chemical exchange on longitudinal relaxation can be found in:
Specer, R.G. and Fishbein, K.W., Journal of Magnetic Resonance 142, pp 120-135 (2000).
An exact description of the relaxation behavior can be given by their solutions of the modified Bloch- McConnell equations.

The effect of chemical exchange is that steady state magnetization in a short TR pulse and observe experiment may not always fit the Ernst equation.
For a numerical prediction of the saturation factors for a short TR experiment with this formalism one would have to know the kinetics and the relative equilibrium magnetizations of the exchanging species.
The relative equilibrium magnetizations are often just the thing we want to measure and kinetic parameters are usually not known either.
Simulations have shown that the Ernst model is a good approximation under certain conditions [Ouwerkerk, R., and Bottomley P.A. JMR 148, 425-435 (2001)].

These conditions can be summarized as:
- Find the optimum flip angle and TR for the best SNR efficiency (SNR per unit time) for all exchanging species.
- Then move to the most relaxed parameters that still yield about 90-95 % of the maximum efficiency for all metabolites.
At a fixed TR choose the lowest flip angle
At a fixed flip angle choose the longest TR
The areas where saturation factors are least M0 and k dependent can be seen as blue areas in the plots of the variance analysis for SF.
Ideally we get an estimate for the errors by simulating the experiment.
For these simulations I wrote a Matlab script (see below). for evaluation of saturation factors for steady state pulse experiments in systems with multi site chemical exchange.
This Matlab script is available on request. Just e-mail me

Mapping the M0 and k dependency

A big problem in systems with chemical exchange is that the apparent T1's are dependent on the exchange rates and the equilibrium magnetizations.
The equilibrium magnetization is usually just what we want to measure. The saturation factors used for correcting signals we get from the more efficient
short TR and low flip angle experiments is usually measured with a fully relaxed 90 degree experiment.
These saturation factors may be applicable only to a system with the same equilibrium magnetizations and reaction rates as in the system where we determined them.
This would be bad if the SF are always strongly dependent on k's and equilibrium magnetisations.
By mapping the M0 and k dependency of the observed (effective) saturation factors we can find experimental conditions (TR and flip angle) where these dependencies are less, whilst still maintaining a good signal to noise efficiency.
An example is shown for the following system (a model for human heart at 1.5T):

exchange rates matrix k

kaa=0	kba=0.6	kca=0
kab=0.4	kbb=0	kcb=0.25
kac=0	kbc=0.5	kcc=0

equilibrium magnetisations (M0i) and intrinsic T1's

M0A = 1.5	T1A = 6.0
M0B = 1	T1B = 2.0
M0C = 0.2	T1C = 5.5

(These example parameters mimic the PCr/ATP/Pi system in human muscle at 1.5T)

Sensitivity of SFs to ks and M0s.

The sensitivity of SF to the k's and M0's in three site exchange models, was computed by expanding the derivative of SF in terms of its partial derivatives with respect to the various independent variables, as routinely used for determining error propagation (7). The root mean square fractional uncertainty in SF for species A is:

[1]

[2]

where deltaSF, deltaM0, deltak etc (dSF, dM0 etc.) , are the errors or standard deviations (SDs) in the corresponding variables, and the capital delta SF (DeltaSF) are fractional errors in each of the composite variables as defined by the respective terms in Equation [1]. Similar expressions can be written for dSF(B)/SF(B), and dSF(C)/SF(C). dSF/SF is a measure of the dependence or sensitivity of SF(TR, q) to variations or errors in the equilibrium magnetizations and rate constants.

Figures 1,2 and 3 show colormap plots of dSF/SF as a function of TR and q, calculated for PCr, ATP and Pi using the pre-ischemic heart muscle parameters above. The fractional variation in SF(A) due to the individual components, DeltaSF(M0A), DeltaSF(M0B ), and DeltaSF(kAB ), are also plotted for a 25% change in the values of M0A, M0B , and kAB (that is, dM0,/M0, = dk/k = 0.25). The variations in DeltaSF(M0C)and DeltaSF(kBC) were much smaller. Superimposed on the colormap are contours lines indicating the operating conditions for q and TR that yield 0.85, 0.9 and 0.95 times the optimum (Ernst angle) SNR efficiency (= signal to noise per unit time = signal / sqrt(TR)).

Figure 1A Sensitivity of saturation factors for species A (PCr) to changes in equilibrium magnetizations and/or exchanges rates (click on figure to enlarge)

Figure 1B Sensitivity of saturation factors for species B (gamma ATP) to changes in equilibrium magnetizations and/or exchanges rates (click on figure to enlarge)

Figure 1C Sensitivity of saturation factors for species C (Pi) to changes in equilibrium magnetizations and/or exchanges rates (click on figure to enlarge)

Figures 1A,B,C

Sensitivity of saturation factors to changes in equilibrium magnetization or rate constants, as measured by sensitivities to individual parameters: DeltaSF(M0A) ( A), DeltaSF(M0B ) (B), DeltaSF(kAB ) (C), and the combined sensitivities for all parameters, dSF/SF, for species A (PCr) (D), and dSF/SF for species B (E) and C (F). A three site linear exchange network was modeled with the following system parameters : T1A = 6s (PCr), T1B =2s (g-ATP), T1C = 5.5s (Pi), M0A =1.5, M0B= 1, M0C= 0.2, kAB = 0.4 [s-1] kCB = 0.6 [s-1]. The partial derivatives were determined by changing each parameter by 10^(-8) of the starting value, whilst keeping other parameters constant. The relative errors, were calculated with Equations [1-2] for a 25% relative error or uncertainty in M0's and/or k's. Pulse angle q and TR were each varied in 51 steps in the ranges 0 < flip < 120º and 0 < TR < 5s. The contour lines on each of the plots delimit the operating conditions for TR and q that yield 0.85, 0.9 and 0.95 times the optimum SNR efficiency of the Ernst angle experiment .

Simulation for a dual angle T1 based saturation correction.

Example

The system in table 1 is designed to mimic the creatine kinase (CK) system in skeletal muscle, observed with ³¹P MRS at 1.5 T.

PCr <=> ATP<=> Pi

Table 1 Exchange and relaxation parameters for a simulation of a three site exchange system A <=> B <=> C.

Species M_z equilirium T₁ [s] k forward [s^-1]

A 28 6,7 kab = 0.27

B 4.5 2.3 kbc = 0.5

C 2 5.5 kca = 0

Errors in calculated saturation factors for a 35 degree flip and TR = 1s

Errors in calculated saturation factors for a 35 degree flip and TR = 1s in the "skeletal muscle" system of table 1.
using the apparent T1's as would be measured with a 15-60 degree dual angle T1 measurement

Apparent T₁, determined with the dual angle method
A :   5.800066
B :   3.925183
C :   4.385051
Relative errors in saturation factors calculated for 35 degree flip
A :   0.017095
B : -0.069131
C : -0.036102
Equilibrium magnetization calculated with single exponential model
A : 28.486988
B :   4.209024
C :   1.930313

Matlab user interface for multi site chemical exchange models

The program checks the chemical exchange network for chemical equilibrium (sum off all forward reactions equals the sum of all backward reactions).
For a valid set of exchange parameters it can produce a plot of the observed signal as a function of either TR or flip angle.

Screen shots of the user interface and of the TR and flip angle dependent output are shown below.

Steady state magnetization as a function of flip angle

Observed magnetization as a function of flip angle

Species	M_z equilirium	T₁ [s]	k forward [s^-1]
A	28	6,7	kab = 0.27
B	4.5	2.3	kbc = 0.5
C	2	5.5	kca = 0