Wikipedia:Reference desk/Archives/Science/2018 June 17

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June 17

Bateman equation

Suppose A decays to B via first order rate constant k1 and B decays to C via first order rate constant k2. All the treatments of the Bateman equation in radioactive decay I've found assume that k1 is never close in magnitude to k2. This is really frustrating because I want to use it to approximately model to my pharmacological diester duration of action problem. If k1 = k2 (you would think this is a simple situation!!) then you get division of zero over zero and l'Hôpital's rule doesn't solve the problem. (I've even tried using the ratio k2/k1). Help? If k1=k2=0.086 (corresponding to half lives of 8 days each), then how much time does it take for half of A to break down to C? Is it more like 16 days or 64 days? Yanping Nora Soong (talk) 01:43, 17 June 2018 (UTC)

I get about 18 days, just using a secret engineer's method (a spreadsheet model, renders most calculus redundant). The mass of B is a maximum at about day 11. Greglocock (talk) 06:01, 17 June 2018 (UTC)

In your problem when $k_{1}=k_{2}=k$ and $A(0)=1$, $B(0)=0$ and $C(0)=0$

$C(t)=1-e^{-kt}(kt+1)$.

which, when $C=1/2$, leads to $kt\approx 1.65$ and $t\approx 1.65/0.086=19.2$ days. Ruslik_Zero 10:19, 17 June 2018 (UTC)

Thank you!!!!! I am also going to write a script for a Monte Carlo method later. Now I suspect because of increased steric bulk that k2 is slightly bigger than k1 (but around the same order of magnitude), but the problem k1=k2 was bugging me because I was worried that the half-life was unbounded. It turns out that HRT users report that a med that includes this diester (Climacteron) is injected every 30 days (although manufacturer monograph says 4-8 weeks). Yanping Nora Soong (talk) 17:21, 17 June 2018 (UTC)

Your problem is the repeated eigenvalue issue in linear differential equations. It’s usually as far as i know solved by multiplying C1e^kt by t and then adding C2e^kt, which is the same thing really as what the engineer said above. Something similar in decaying harmonic motion is called “criticallydamped.” — Preceding unsigned comment added by 67.128.146.22 (talk) 18:10, 17 June 2018 (UTC)

Hmmmm... per my logic in the last question section I would think that if the half-life of each of two bonds is 8 days, then after 19.2 days, each bond is only unbroken 19% of the time, so the amount with both broken at that point is 81%*81% = 66%. I think the difference in the math above is that you suppose the half-life of A is the same as the half-life of B -- but in the previous question A has *two* ester bonds either of which can be broken while B just has one, so we'd expect a different rate constant. Again, I don't really know the stability of A and it doesn't have to be half of B or any other particular value, but I just thought I should point this out. Wnt (talk) 21:42, 17 June 2018 (UTC)

I've realized this -- there are two intermediates -- but I made that approximation to try to obtain a value that made sense. I saw a "complex" chain treatment that looked like my problem, and I'll try a Monte Carlo method later. Thank you so much!!! Yanping Nora Soong (talk) 02:04, 18 June 2018 (UTC)

• For the complete math: the original problem is a "simple" matrix differential equation ${\begin{bmatrix}{\dot {A}}\\{\dot {B}}\end{bmatrix}}={\begin{bmatrix}-k_{1}&0\\k_{1}&-k_{2}\end{bmatrix}}{\begin{bmatrix}A\\B\end{bmatrix}}$ (add a column/row if you want C), which does have an analytical solution in the general case. Tigraan^{Click here to contact me} 11:47, 18 June 2018 (UTC)

Dividing line between angina and heart attack

Doesn't angina always involve at least some loss of oxygen to the heart, and therefore a possibility of damage? So if a self-interested hospital wants to increase revenue, what’s to stop the hospital from calling a case of angina a heart attack? Is there news or opinions in media about this as a conflict of interest? Thanks 67.128.146.22 (talk) 10:01, 17 June 2018 (UTC)

There are tons of literature about differential diagnosis of heart attack and angina pectoris. You can search yourself in Google. Ruslik_Zero 12:23, 17 June 2018 (UTC)

”you can search yourself in Google” is a weird response for a volunteer at reference desk.67.128.146.22 (talk) 17:57, 17 June 2018 (UTC)

At an academic reference desk, the answer is sometimes "here are the search terms you need; look through them, and feel free to come back if you want further assistance". Nyttend (talk) 01:10, 18 June 2018 (UTC)

(I am assuming, based on your IP address, that you're interested in the U.S. "healthcare system".) Most U.S. hospitals bill on a fee-for-service basis; that is, the charges are based on the services and procedures performed, not on the specific diagnosis attached to a particular patient. If a patient undergoes cardiac catheterization, the bill is the same whether the diagnosis is "angina" or "infarct". (That oversimplifies a bit; in practice the billing goes into rather a lot of arcane detail, with separate charges for staff, operating room time, drugs, instruments, tests, etc.)

Now, if a hospital makes a habit of over-diagnosing and thereby performing unnecessary tests and procedures, pushback happens in a number of ways and places. Insurance companies tend to notice when one hospital seems to have unusual patterns of diagnosis and treatment, and will start to deny reimbursements that aren't accompanied by sufficient documentary evidence. Overdiagnosis to increase billings is a gross breach of medical ethics, and can lead to both civil and criminal penalties for the parties involved.

(More common and more difficult to deal with is over-testing and over-treatment done in good faith. For physicians, it's hard – and sometimes legally risky – to tell a patient that expensive tests aren't necessary; it's hard to tell a patient that the best way to treat their condition is sometimes to do nothing. This New Yorker article is an accessible overview.) TenOfAllTrades(talk) 15:00, 17 June 2018 (UTC::thanks good response67.128.146.22 (talk) 17:57, 17 June 2018 (UTC)