The problem below is data science that I nee to solve using python programming and regression. Can any one solve it

Question

The mathematical model of tumor growth  

4.  Now our goal is to fit the time evolution of the growth of this tumor, using a mathematical model. To do so, we need to find the best-fit parameters of the model. The mathematical model we will use here is called the Gompertzian growth model. Here, we will use a slightly modified form of the Gompertzian function of the following form,

N(t,λ,c)=N0×exp(λ [1−exp(−ct)]) ,

where N(t,λ,c) is the predicted number of tumor cells at time t, N0 is the initial number of tumor cells at time t=0 days, λ is the growth rate parameter of the model, and c is just another parameter of the model. We already know the initial value of the number of tumor cells, N0=100,000±10,000. Therefore, we can fix N0 to 100,000 in the equation of the model given above.

However, we don’t know the values of the parameters λ and c. Thus, we would like to find their best values given the input tumor cell data using some Python optimization algorithm.

This Gompertzian growth model is called our physical model for this problem, because it describes the physics of our problem (The physics/biology of the tumor growth).

Combining the physical model with a regression model  

Now, if our physical model was ideally perfect in describing the data, the curve of the model prediction would pass through all the points in the growth curve plot of the above figure, thus providing a perfect description of data. This is, however, never the case, as it is famously said all models are wrong, but some are useful. In other words, the model prediction never matches observation perfectly.

Therefore, we have to seek for the parameter values that can bring the model prediction us as close as possible to data. To do so, we define a statistical model in addition to the physical model described above. In other words, we have to define a statistical regression model (the renowned least-squares method) that gives us the probability π(logNobs|logN(t)) of observing individual data points at each of the given times,

π(logNobs|logN(t,λ,c),σ)=1σ2π‾‾‾√exp(−[logNobs(t)−logN(t,λ,c)]22σ2) ,

Note that our statistical model given above is a Normal probability density function, with its mean parameter represented by the log of the output of our physical model, logN(t,λ,c), and its standard deviation represented by σ, which is unknown, and we seek to find it. The symbol π, whenever it appears with parentheses, like π(), it means the probability of the entity inside the parentheses. However, whenever it appears alone, it means the famous number PI, π≈3.1415.

Why do we use the logarithm of the number of cells instead of using the number of cells directly?The reason behind it is slightly complicated. A simple (but not entirely correct argument) is the following: We do so, because the tumor cell counts at later times become extremely large numbers, on the order of several million cells (For example, look at the number of cells in the late stages of the tumor growth, around t=20 days). Therefore, to make sure that we don’t hit any numerical precision limits of the computer when dealing with such huge numbers, we work with the logarithm of the number of tumor cells instead of their true non-logarithmic values.

We have seven data points, so the overall probability of observing all of data  together (the time vector and the logarithm of the number of cells at different times) given the parameters of the model, (|λ,c,σ), is the product of their individual probabilities of observations given by the above equation,

(|λ,c,σ)=∏i=1n=8π(logNobs(ti)|logN(ti,λ,c),σ)=∏i=1n=81σ2π‾‾‾√exp(−[logNobs(ti)−logN(ti,λ,c)]22σ2) .

Frequently, however, you would want to work with log instead of . This is again because the numbers involved are extremely small often below the precision limits of the computer. So, by taking the logarithm of the numbers, we work instead with the number’s exponent, which looks just like a normal number (not so big, not so small). So, by taking the log, the above equation becomes,

log(|λ,c,σ)=∑i=1n=8logπ(logNobs(ti)|logN(ti,λ,c),σ)=∑i=1n=8log[1σ2π‾‾‾√exp(−[logNobs(ti)−logN(ti,λ,c)]22σ2)] .

Answer 1

.... and the data?