Help: Faster way to calculate inverse for series of slightly different matrices.

I have a matrix which is 6x6 and a vector 6x1. I calculate C2 = -(A^-1)*B;

Please have a look at the structure of my matrix and vector:

A =    [[-r_2s          k_ws        (Del-Del_s) 0           0           0];...
        [k_sw           -r_2w       0           (Del)       0           0];...
        [-(Del-Del_s)   0           -r_2s       k_ws        Om1         0];...
        [0              -(Del)      k_sw        -r_2w        0          Om1];...
        [0              0           -Om1        0           -r_1s       k_ws];...
        [0              0           0           -Om1          k_sw     -r_1w]];           

B = [0, 0,  0,  0,  M_s0/T_1s,   M_w0/T_1w]';

As you can see, “A” is a function of Del which appears four time @(1,3), (2,4), (3,1), (4,2). I usually calculate C1 and C2 for a series of Del with close points.

      DelVec  = (-7:0.25:7)* 773.7;               %Convert from ppm 
      Del_s   = 2.0*773.7;                        %Convert from ppm

I am wondering if there is a way to calculate C2 faster for series Del = DelVec(i). In future, the size of matrix A would be become large, as I would start processing multiple voxel at one go.

Since the solution for two successive values of Del, say DelVec(10) and DelVec(11), would not be too different, I would like to use previous solution as a first guess for next value of “Del”.

I tried using "Conjugate gradients squared method": cgs; but, it's 5-10 times slower than if I calculate C2 = -pinv(A)*B; directly

Regards, DK