Regression when $X$ is random and unobservable

I am trying to fit a curve to empirical data where I know that uncertainty is in the predictors ($X$) (there could be also an uncertainty in $Y$, but let’s keep things simple for the start). In other words, I know only the expected values, but not the true values of $X$.

My model is somewhat complicated, but I’ll try to use a simplification as an example. Say I want to determine the rate of the decay b of a compound. Based on my domain knowledge I assume the following model:

$$
N_t = N cdot e^{-bt}
$$

I set up the experiment in which I start with theoretically always the same amount N0 of the compound, wait some pre-defined time t0, which I vary over the repeated runs of the experiment, and measure the actual remaining amount Nt of the compound. However, I have errors both in N0 and in t0, so that the actual (but unobserved!) starting amount N and the waiting time t are random variables, say, normally distributed around N0 and t0, respectively.

In code:

    import numpy as np
    import matplotlib.pyplot as plt

    np.random.seed(1)
    b = .2345
    N0 = 10
    N  = N0 + np.random.randn(10)
    t0 = np.linspace(0, 10, 10)
    t  = t0 + np.random.randn(10)
    Nt = N * np.exp(-b * t)
    plt.plot(t0, Nt, '.', label='empirical')
    plt.plot(t0, N0 * np.exp(-b * t0), label='theoretical')
    plt.legend(loc='upper right')
    plt.gca().set_xlabel("t")
    plt.gca().set_ylabel("N(t)")

How would I determine b in a maximum likelihood manner?

I have read
Simple linear regression model with random x,
Regression with random X,
Find P(Y=y | X=x) when X is a continuous random variable, and
What are the Differences in Linear Regression of Y vs X when both Y,X are Random and Regressing Y vs X when X is Mathematical,
but they don’t seem to answer my specific question (or in an abstract way which I was unable to comprehend).