standard error of independent variables using inverse linear regression

Hi,

I have four dependent variables (y(1), y(2), y(3), y(4)) that vary as a function of two independent variables (x(1), x(2)). I performed a set of laboratory tests to determine the calibration between dependent and independent variables under controlled conditions. For all four dependent variables, the relationship is fairly described by a complex linear function:

y(1-4) : a + bx(1) + cy(2) + dx(1)^2 + ex(2)^2 + fx(1)x(2) + gx(1)^3 + hx(2)^3 + ix(1)x(2)^2 + jx(1)^2x(2)

Afterwards, I use a least square algorithm to determine "best fit" x(1) and x(2) values for a given set of y(1-4) values.

For the calculation of the variability of x(1) and x(2), I would like to consider both the measurement noise (this can be easily done by measuring y values for replicate samples) and the calibration noise (i.e. taking into account the standard errors associated to the regression coefficients a, b, c, etc.). I am struggling with this issue due to the complex linear functions. Does anyone have ideas on how to proceed?

Thanks a lot, Dries