# Nodes, Schmodes

Ultimately, the game changer in prognosis and therefore management of my breast cancer vis-à-vis chemotherapy and radiation is whether the cancer has already spread to my lymph nodes.  Dr. Kavanagh said in her phone call with me (See It’s a Date post.) that my MRI changed the probability I have lymph node involvement to just 1%.  Ever inquisitive, and also hoping I remember enough basic probability theory to execute, I wanted to know whether my diagnosis probability estimates bear that out.

If you look through enough medical studies on breast cancer, you can find the following data.

1)  For patients with pure DCIS, the probability of a positive lymph node is 1-2% (Let’s say 1%.  And, yes, I realize that by definition if the cancer has spread to the lymph nodes then it can no longer be called in situ — the IS part of DCIS.).  We’ll call this P(L|DCIS) = 0.01.

2)  For patients with DCIS-MI, the probability of a positive lymph node is 3-20% (One study says 7%.  We’ll go with that.).  Thus, P(L|DCIS-MI) = 0.07.

3)  For patients with IDC and lesion <2 cm, the probability of a positive lymph node is 29% (Actually, 29% was the case for all  invasive breast carcinoma <2cm, but we’ll just close our eyes here and say it can apply to IDC in particular.).  Therefore, P(L|IDC<2cm) = 0.29.  There is an overlap between DCIS-MI and IDC<2cm, namely that DCIS-MI is also known as IDC<1mm.  To find the probability of positive lymph nodes given IDC and lesion between 1mm and 2cm, simply subtract.

P(L|1mm<IDC<2cm) = P(L|IDC<2cm) – P(L|IDC<1mm) = 0.29-0.07 = 0.22

The updated table from my Number Crunching post can be restated thusly:

P(DCIS) = 0.55; P(DCIS-MI) = 0.44; P(IDC>1mm) = 0.01.

Ideally, I would instead have P(1mm<IDC<2cm) instead of P(IDC>1mm), but we’re going to assume that P(IDC>2cm) is negligible and therefore

P(1mm<IDC<2cm) = P(IDC>1mm) – P(IDC>2cm) = P(IDC>1mm).

Recalling what I know about conditional probabilities:

The probability of my post-surgery pathology showing DCIS and cancer in my lymph nodes is now P(DCIS,L) = P(DCIS) * P(L|DCIS) = 0.55 * 0.01 = 0.0055.

Similarly, the probability of my post-surgery pathology reporting DCIS-MI and cancer in my lymph nodes is P(DCIS-MI,L) = P(DCIS-MI) * P(L|DCIS-MI) = 0.44 * 0.07 = 0.0308.

Lastly, the probability of my post-surgery pathology concluding IDC w/max. tumor between 1mm and 2cm and cancer in my lymph nodes is P(1mm<IDC<2cm,L) = P(1mm<IDC<2cm) * P(L|1mm<IDC<2cm) = 0.01 * 0.22 = 0.0022.

Add those three joint probabilities together to get my total probability of having lymph node involvement reported after surgery:

P(L) = P(DCIS,L) + P(DCIS-MI,L) + P(1mm<IDC<2cm,L) = 0.0055 + 0.0308 + 0.0022 = 0.0385, or 3.85%.

Compare this 3.85% to Dr. Kavanagh’s estimate for my lymph node involvement of 1%.  Why the discrepancy?  Here are some possible explanations.

1)  I made too many erroneous assumptions.

2)  Perhaps Dr. Kavanagh did not explicitly state but now assumes that I in fact have pure DCIS, in which case the probability of lymph node involvement is simply the 1% from medical literature.

3)  Possibly the MRI results changed P(DCIS), P(DCIS-MI), and P(IDC>1mm) in ways I am inadequately capturing.

4)  Or maybe Dr. Kavanagh’s wrong.

At any rate, props to you if you made it this far.  And thanks for indulging my inordinate need to make sense of a heretofore unimaginable predicament.