Improve monoexponential curve fitting in the initial portion I've calculated the Tissue Volume and Tissue Mass from a Computed Tomography image of the lung of a given patient.
The y-axis of the attached figure is Volume as a percentage of the total lung capacity, so it goes from 0 to 100%. The x-axis is the patient's lung cumulative Mass in grams, so it goes from 0 (or close to  0) to 475g (total lung mass). I would like to investigate the behavior of this curve in some patients with different lung diseases. Some of these diseases destroy the lung and consequently air volume will increase and mass content will reduce. My idea was to fit the Volume x Mass curve with a monoexponential equation (Equation is shown below) and use the K3 ("Time constant") coefficient to quantify how fast is the curve response. Depending on the severity of the disease, the curve can increase faster, and as a consequence, it will have a smaller K3 compared with a healthy subject.
The problem is: the monoexponential seems to fit this type of curve poorly, mostly in the beginning portion of the curve, in special between 0-100 grams. 
Some persons told me to use weighted regression to overcome this limitation. But I am not familiar with this technique. Can anyone help me to implement this regression in Python, R or Matlab?  Here is the link of a GitHub repository with the code and an example of the Volume x Mass curve and its fitted curve.
Apart from the weighted regression, does anyone knows any other equation that could fit the Volume x Mass curve with better results and still give me the "Time/Mass constant" or some sort of coefficient that tells me how fast is the curve response? Maybe some sort of biexponential equation.
The monoexponential Equation that I used:
 $$V_{(\%)} = K1 + K2*e^{(-M/K3)}$$
where K1, K2, and K3 are the coefficients to be estimated and M is the lung tissue mass;
At the picture below, the BLUE line is the Observed and the RED is Predicted.
Thanks in advance
 A: This is not a full answer, but only an explanation about the difficulty that you mentioned :

The problem is: the monoexponential seems to fit this type of curve poorly, mostly in the beginning portion of the curve, in special between 0-100 grams. Some persons told me to use weighted regression to overcome this limitation.

If the observed curve (blue) is drawn as $V^2$ as a function of $M$ (curve below) one can see that it becomes quite linear in the beginning portion of the curve:

This means that, for lowest $M$ values, the function behaves roughly as:
$$V\simeq C\,\sqrt{M}\tag 1$$
where $C$ is a constant.
Of course this oversimplified relationship is not valid for $M$ large.
Now, consider the supposed relationship :
$$V_{(\%)} = K_1 + K_2*e^{(-M/K_3)}$$
For low values of $M$ the main term of series expansion is :
$$V\simeq K_1 + K_2*(1-\frac{M}{K_3})$$
And $V(0)=0$ implies $K_2=-K_1$
$$V\simeq K_1 - K_1*(1-\frac{M}{K_3})$$
$$V\simeq\frac{K_1}{K_3}M \tag 2$$
Comparing Eqs.$(1)$ and $(2)$ shows that the supposed  monoexponential Equation behaves like $M$ while the observed phenomena behaves like $\sqrt{M}$.
This appears the cause of discrepancy that you observe: The monoexponential Equation is not a correct equation to model the phenomena on the low range of $M$.
So, I doubt that a weighted regression will solve the problem. Of course, the fitting can be improved on the low range of $M$ but the fitting will become bad for the large values of $M$.
I think that the global improvement cannot come from playing around with the regression itself, but should come from a more advanced study of the phenomena and a better mathematical model than the simple monoexponential Equation.
For example with the combination of the function $a\:\sqrt{M}$ and a function of logistic kind :

Note that the numerical values of the parameters obtained from non-linear regression are probably not accurate because the data isn't the correct data but comes from scanning the figure published in the question. Scanning a figure is not an accurate method.
A: I don't know if you found an answer, but for me the model is not correct with the problem. Thus, if the expected behavior of the problem is essentially exponential, it appears to be multiexponential. We have some processing algorithms using the Tikhonov regularization that can be applied to fit a multi-exponential recovery or decay curve. Very nice the proposal.
