But how can we solve the inverse problem, where we measure an absorption spectrum but don't know what absorbers caused it? In some sense, this problem is very challenging if not impossible to solve. However, if we have some a priori information (i.e., some knowledge about the system) then this is a much easier problem to solve,
We start by assuming that the total absorption of the sample is a linear combination of all the absorbers:
We have written that the sum of all the individual absorbers (i), of which there are N total, add up to the total measured absorption. Because the absorption coefficient changes with wavelength, we have written each absorption coefficient as a function of wavelength (lambda).
Lets' simplify by assuming we have only two absorbers (N=2): oxy-hemoglobin and deoxy-hemoglobin. This is a reasonable though imperfect assumption for tissues exposed to NIR light, depending upon the tissue and the wavelengths used. We can then simplify the summed expression as:
We can translate the absorption coefficient for each absorber into the molecular concentrations:
The 2.303 factor came from the conversion between base 10 and base e logarithms. In tissue spectroscopy we measure the left-hand side of the equation (i.e., the total absorption). The molar extinction coefficients are assumed known (which is a whole different story for later ...). That leaves the concentrations as the only remaining unknown variables.
Thus, to measure the concentration of a given sample, in principle only a single wavelength is needed (unwise, but possible). But you can also see that we have a problem when N increases ... run into a math problem like 1=x+y and only x is known. Linear algebra tells us we need at least as many equations (i.e, wavelengths) as unknowns (i.e., concentrations); thus to measure the concentrations of both oxy- and deoxy-hemoglobin in this example, we need at least 2 wavelengths. If we add more absorbers, we need more wavelengths to add information content. In the previous example, we would need:
The general problem is one of linear algebra (remember those days?!?; if not check here), which is expressed in matrix form as:
There are standard mathematical methods for solving this matrix problem. We typically use a least squares fit for the matrix operations (see the function 'mldivide" in MATLAB). PS - If you don't know what matrix operations means you should probably look here, but realize that a matrix is just a long chain of equations as in the above figure. PSS - If you aren't sure what least squares fit means, you can look here for a technical discussion or here for a less-technical discussion.
The little hat denotes a vector (1-D array) and the bold type represents a matrix (multi-dimensional array). The absorption coefficient is a vector of length M x 1, with each element starring as a unique wavelength. The concentration term is also a vector, but of length N x 1. Thus, the molar extinction coefficient must be a matrix of size M x N. Each column of this matrix is a spectrum of the extinction coefficients for a given absorber in the system. Each row of this matrix provides the extinction coefficients for a given wavelength for each absorber.
A few more notes about this problem are in order.
It is important to realize that we are dealing with a data fit with experimental data so that any attempt to recover the "true" concentrations will necessarily have errors. That is why we are doing a least squares fit; the solution is the "best" fit of the data in that the differences between the recovered answer and the measurement are minimized.
Constraints can be used to help provide realistic answers. For example, we can require that all the concentrations be positive. In general we don't use many constraints (other than the positive constraint) to measure the standard NIR absorbers in DOSI. However, if you are looking for really small things like cytochrome absorption, constraints are likely necessary because their contribution to the total NIR absorption signal is usually only a few percent.
It is also important to realize that the assumption of what absorbers are present in the system, is itself a model (which we refer to as a spectral model). For example, in NIR spectroscopy of muscle tissues it is common for people to ignore the effects of myoglobin. Ignore it all you want, but it is there and obfuscates the meaning of hemoglobin changes.
On the same theme, note that if you neglect an absorber the least squares method will do its best to account for that missing absorption with known absorbers. For this reason, we have added a lot of spectral content to DOSI to allow for us to deal with other absorbers.
There are special methods for trying to make sense of absorption spectra such as principal component analysis and independent component analysis. These methods can be helpful for identyfying which features of a spectrum are important, but we will show how in some problems like cancer detection they may not be necessary.
What is DOSI? >