Comparing rate constants for 2 exponentials given 3 semi-observable data points per curve

Question

I believe that a certain measure for a reaction, over time, follows an exponential curve of the form: $$ f(t) = ae^{-kt} + b $$ where $t$ corresponds to time. All three parameters, $a$, $k$ and $b$, are positive.

From one experiment to the next, all three parameters, $a$, $k$ and $b$, can change. I am interested in how quickly a reaction converges compared to another: what is the relationship between the constants $k_1$ and $k_2$, for two separate experiments/runs. I don't particularly care about the values of $a_1$, $a_2$, $b_1$ and $b_2$.

Unfortunately, $f(t)$ cannot be directly observed. The only reliable information I have is the time, $t$, when an experiment reaches certain milestone values of $f(t)$. I have access to 3 such milestones, which I will call $u$, $v$ and $w$. In other words, I can tell when $f(t) = u$, or $v$ or $w$. I don't know what the actual values of $u$, $v$ and $w$ are, but I know that $u > v > w$.

Let's call $t_{u,i}$ the time at which experiment $i$ reaches milestone $u$: $f_i(t_{u,i}) = a_ie^{-k_it_{u,i}} + b_i = u$. We can do the same with $t_{v,i}$ and $t_{w,i}$.

In other words, given two experiments, 1 and 2, we have:

$$ u = f_1(t_{u,1}) = f_2(t_{u,2}) $$ $$ v = f_1(t_{v,1}) = f_2(t_{v,2}) $$ $$ w = f_1(t_{w,1}) = f_2(t_{w,2}) $$

I'm looking to express $k_2$, the rate constant for the second experiment, in terms of $k_1$. Ideally, I would like to find the ratio between the two (how much faster one experiment converged than another experiment).

So far, I have found that if $b_1 = b_2$, the solution is simple: $\frac{k1}{k2} = \frac{t_{v,2} - t_{u,2}}{t_{v,1} - t_{u,1}}$. The ratio between the two values of $k$ is simply the ratio between the time lag between two corresponding events. Also, if $u$, $v$ and $w$ are equally spaced in time, the problem is not that hard. I think, actually, that if this holds for both experiments then the asymptotes have to be equal. However, if $b_1 \ne b_2$ and the milestones are not equally spaced in time, the problem seems to be much more complicated.

Here's what else I have found so far:

• I think the ratio between $t_{v,i} - t_{u,i}$ and $t_{w,i} - t_{v,i}$, the time lag between consecutive events, might be key: if it takes a very short time to go from the first to the second milestone, and a much longer time from the second to the third, then the asymptote is probably nearby.
• There seems to be only one exponential going through a set of 3 different points $\{(t_{u,i}, u), (t_{v,i}, v), (t_{w,i}, w)\}$, which makes me think that there are unique values $k_1$ and $k_2$. I wonder if the ratio between the two depends on the actual values of $u$, $v$ and $w$ or if it can be obtained independently of these values.
• I have identified a few extreme cases: when $t_{v,i} - t_{u,i}$ approaches 0, $k_i$ goes to infinity; when $\frac{t_{w,i} - t_{u,i}}{t_{w,i} - t_{v,i}}$ approaches $\frac{w-u}{v-u}$, $k_i$ goes to 0; and when $\frac{t_{w,i} - t_{u,i}}{t_{w,i} - t_{v,i}}$ is constant across two experiments, the asymptote $b$ seems to also be constant and the ratio between $k_2$ and $k_1$ is easy to compute.
• At some point this morning, I managed to express the solution as a polynomial by scaling the x axis such that $t_{u,1} = 0$, $t_{v,1} = 1$, and $t_{v,2} = s$, scaling the y axis such that $b_0 = 0$ and $u = 1$, and shifting the experiment 2 curve horizontally so that $t_{u,2} = t_{u,1} = 0$. If $r = e^{-k_1}$ and $a = k_2/k_1$ then I had the following equation: $r^{at_{v,2}}[1 - r^s] - r^{at_{w,2}}[1 - r] + r^s - r = 0$ but I have no idea if that equation can be solved for $a$.

If there is no analytical solution, then I'd be open to suggestions for numerical methods to find the ratio.

I've been at this for days and have a lot of trouble keeping my mind off of this problem, which is key to a technological problem I am trying to solve. I would extremely appreciate any help you might have!