CEST.mat is MATLAB toolbox for simulating and fitting an arbitrary number of exchange pools in CEST MRI and NMR.

    [zSpectrum,Mstate]= CESTnPools(magField, satTime, satPower, PPM, ParamVec)

Notation

Water should always be assigned to pool A. All other pools should be assigned to other letters, from pool B, pool C, all the way to pool N. This notation will be used in the following paragraphs when constructing the five vectors needed to simulate data. The T1, T2, and pool saturation vectors are constructed in the following form:

    T1=     [T1A T1B T1C ... T1N]; 
    T2=     [T2A T2B T2C ... T2N];
    PPM=    [ppmA ppmB ppmC ... ppmN];

T1A, T2A, and ppmA are all associated with pool A (water); T1B, T2B, and ppmB are all associated with pool B; and so on. The concentrations and exchange rates vectors are constructed in another form:

    Conc=           [CB CC ... CN];
    ExchangeRates=  [kBA kCA ... kNA];

These are the exchange rates from one pool to water. Notice that the components of these two vectors begin with pool B, not pool A. Like before, CB and kBA are associated with pool B, CC and kCA are associated with pool C, and so on.

SIMULATION OF CEST MRI DATA

In order to simulate CEST MRI data using the CESTnPools function, the following parameters must be defined: Single floating point numbers

magField    The magnetic field in Tesla
satTime     The saturation time in seconds
satPower    The saturation power in micro-Tesla

Vectors

SaturationOffset    1 x K vector of saturation offsets in parts per million (ppm)
ParamVec            1 x (5n-2) vector, where n is the number of CEST pools including water

The vector is constructed by concatenating the following five vectors:

ParamVec= `[T1 T2 Conc ExchangeRates Pool_Offsets]`;

T1              1 x n vector of T1 relaxation times in seconds
T2              1 x n vector of T2 relaxation times in seconds
Conc            1 x (n-1) vector of concentrations in moles/Lt, Water is 110 Moles/Lt 
ExchangeRates   1 x (n-1) vector of exchange rates from each pool to water in Hz
Pool_Offsets    1 x n vector of pool offsets in ppm (including water)
Outputs

The CESTnPools function will solve the time-dependent Bloch equations using the parameters defined in the program.

zSpectrum   Z-spectra of length K (same as saturation offsets) 
Mstate      (3n+1) X K matrix of final magnetization components for each pool with the following format: 
Mstate=     [Mx,A ... Mx,N My,A ... My,N Mz,A ... Mz,N 1]' x K

FITTING OF CEST MRI DATA

In order to fit CEST MRI data using the CESTnPools function, some of the variables used in the simulation but me fixed using an anonymous function, otherwise the fitting algorithm will try to estiamte them. The two variables thats are always knows are:

   magField = magnetic field in Teslas
   satTime = saturatin time in seconds

If you are very confident on the saturation power applied to the sample you can also fix satPower. If you are want to estimate the actual power, just include it as one of the variables. Once you define the anonymous function, you can use it and lsqcurvefit (or other minimizer) to estimate the paramters.

Define anonymous function

CESTfunc= @(ParamVec, SaturationOffset) CESTnPools(magField, satTime, satPower, SaturationOffset, ParamVec);

Perform Non-Linear Leastsquare curve fitting_

[ParamVecPred,resnorm,residual,~,~,~,jacobian]= lsqcurvefit(CESTfunc, InitCond, SaturationOffset, yData, LowerBounds, UpperBounds);

where:

InitCond =              Initial guess for all the parameters to be estimated
SaturationOffset=       Experimental saturation offsets in ppm
yData=                  Experimental Z spectra
LowerBounds=            Lower limit for all the parameters to be estimated
UpperBounds=            Upper limit for all the parameters to be estimated

ParamVecPred contains the fitting results in the format of ParamVec. The confidence intervals (conf) can be determined using the nlparci function with the following syntax:

conf= nlparci(ParamVecPred,residual,'jacobian',jacobian);