Hello, this is Tom Stavros, and today we're going to talk about ultrasound features of breast masses as prognostic biomarkers. My disclosure is that the study from which the data comes that I'm going to present today in all their extensive data was enrolled and funded by Cento-Medical Incorp. However, there will be no mention of Cento-Medical or its products during this talk. Now, when we talk about ultrasound features of breast masses, we have to question whether we ask enough of ultrasound, and I think we aren't. We could ask more of it. We're going to look at ultrasound features from a scoring system that was designed for a different purpose. And we decided to use these features to assess them as prognostic biomarkers after the fact, after it was designed, after the original data was evaluated. We'll talk more about the original purpose in the talk that follows this. And I'm going to show you in this talk that there are numerous ultrasound features that are significant prognostic biomarkers, but they are so numerous and they interact in such complex ways that it's difficult to put them all together without the help of artificial intelligence. Now, why we want to use ultrasound features as biomarkers is to get closer to personalized medicine. We want to get away from the one-size-fits-all medicine to precision or personalized medicine. Now, the types of biomarkers, imaging biomarkers that can be used are diagnostic, monitoring, predictive, and prognostic. This comes from a very good article by Olina Weaver and Jessica Leong, that is my kind of go-to article that I rely on for biomarkers. And today we're going to talk primarily about prognostic biomarkers and primarily how they correlate with tumor grade, receptor status, and of course, receptor status is used as a surrogate for molecular subtypes. Now, we do know that some other type of breast imaging features can be used as prognostic biomarkers. From the biography, we know that triple negatives tend to be round and circumscribed or indistinct, and that the aeropositive and luminal molecular subtype cancers had to be speculated. And we know that segmentally distributed fine linear and branching microcalcifications are most likely to occur.  We know from MRI that ring enhancements suggest triple negative or high grade, and that clustered rings of non-mass type enhancement on MR tends to be caused by grade 3 DCIS or what Laszlo would call ductal adenocarcinoma and Cy2DAB. And we know that on mammography and MRI, multi-focality is most common in her two type and luminal B types, less common in luminal A, or triple negative, which are mostly unifocal. And that large bulky lymph nodes tend to favor her two or luminal B cancers. The triple negatives often skip the lymph nodes and go hematogenously. Now, in the distant past, all the way back in the 1990s and early 1990s, I noted the correlation between sound transmission and histologic grade. I also noticed the taller than wide tended to be low grade. And that the size of enlarged ducts correlated with grade, big, big large involved ducts in the surrounding tissues, or what the associated findings in byrads tend to go with grade 3 and small ducts with grade 1 and medium size with grade 2. And that the relative thickness of the ecogenic rim and the length of the spicules tend to go with low grade. So these are all things I noticed anecdotally, but really didn't have a lot of data to support except for sound transmission. Now, here's this, here's data that we accumulated way back in the 1990s. We had no idea what molecular subtypes were, whether people had positive ER or PR or R2s or KI-67s, because we didn't do any of that stuff back then. But I could clearly show that acoustic shadowing reduced the chances of being grade 3 to about 0.4 of what the overall population had, and enhanced sound transmission increased the risk that it was grade 3 by almost 2.4. So these were relative risks, not odds ratios, but clearly there was a correlation in sound transmission and histologic grade. In fact, we came up with differential diagnoses for the tuber type based on sound transmission or what we now call posterior features.  Enhanced sound transmission tended to go with grade 3 invasive ductal cancer, especially triple-negative with some larger mucinous carcinomas, with some grade 3 carcinoma in situs that didn't have a lot of calcations with medullary carcinoma and invasive papillary carcinoma. Now, more recently, some articles have come out showing some sort of correlation of ultrasound features with prognostic biomarkers. Unfortunately, these are both from outside the United States. This one is from Toronto from a friend of mine, Fred Au, and they looked at 60-sum cancers and found that speculated margin was associated with low grade and ER and PR status and shadowing was associated with positive progesterone receptor. On the other hand, they showed that microlobulated margins and angular margins tended to go with ER negative status and enhancement with grade 3. Another recent article out of China showed that luminal A cancers tend to have speculated margins with posterior acoustic shadowing and that luminal Bs tended to have indistinct margins and triple negatives. We're more likely to have circumscribed and microlobulated margins lack a thick echogenic rim and to be markedly hypochoic and have posterior sound enhancement. So, you know, the findings on sound transmission or posterior features are holding up. I now have a larger database of 653 invasive malignant breast cancers to which I have scored all the ultrasound features and which represent the basis for the data that I'm going to show you. Now, if we use ultrasound features as prognostic biomarkers, we need something to compare to. So, receptor data turns out to be a good thing to use in addition to histologic grading. Estrogen receptor, progesterone receptor, HER2 receptor status, and KI67 are the things that are used as surrogates to decide molecular subtypes. So, first we're going to look at the individual receptor correlations, then we'll put them together to come up with surrogate molecular subtypes, and we'll look at a ultrasound correlation with the molecular subtypes based on this. It's also important the latest ICCN guidelines for TNM staging use ER and PR status to either upgrade or downgrade the staging of tumors. So, it's become more important in staging to have an estimate of what the receptor data are. Now, if we look at sound transmission, which is what we noticed way back in the 1990s, this is an extremely powerful biomarker for estrogen receptor status, progesterone receptor status, HER2 status, and KI67. And if we look at these p-values, some of them are ridiculous. We can see that the correlation with ER and PR status is 10 to the minus 15th and 10 to the minus 16th. We can even distinguish ER positive, PR positive from ER positive, PR negative, which has a worse prognosis than when they're both positive. There is a correlation with HER2 status with a p-value less than 0.5, 0.5, but it's not as strong as for some of the other features. We also have an extremely powerful correlation of sound transmission or posterior features with KI67. And we can even look at the primary breast lesion and look at some of its old sound features, especially sound transmission, and get some information about the likelihood of positivity of axillary lymph nodes. Again, the p-value is less than 0.05, but not better than 0.01. So I've color-coded these. If the p-value is less than 0.001, I've color-coded a green. If it's better than 0.05, but not as good as 0.001, then I've color-coded an aqua. But you can see that all of these are statistically significant. Now if we specifically look at sound transmission versus ER status, we've got a p-value less than 10 to the minus 16th, or p-value equal to 9.8713 times 10 to the minus 16th. We can see that if we have enhancement, the relative risk of it being grade 3, or ER negative, I'm sorry, is 3.8 times more than it is to be ER positive. On the other hand, if we have acoustic shadowing, it's 3.6 times more likely to be ER positive. And so if we look at this case, clearly there's acoustic shadowing on the left, there's enhanced through transmission on the right. The likelihood is that the shadowing lesion is ER positive, and the lesion with enhanced sound transmission is ER negative. If we look at the correlation of PR status, it's just a teeny bit less strong, but it's still p-value equal to 3.6 times 10 to the minus 15th. Again, enhancement increases the likelihood of being PR negative by 3.8 times, and shadowing increases the chance of it being PR positive by 2.3 times. And again, if we look at these two lesions, the one on the left has enhanced through transmission, the one on the right has weak shadowing. It's more likely that the shadowing lesion is PR positive, and that the lesion with enhanced sound transmission is PR negative. If we look at a combination of ER and PR, we can see that enhancement makes it 2.7 times more likely than its ER positive PR negative, whereas shadowing makes it 1.4 times more likely that both ER and PR are positive. Now if we look at HER2 status and sound transmission, we do have a significant p-value, but it's nowhere near as powerful a correlation as there is with ER and PR. But if there is a partial enhancement, we found that it's 2 times more likely to be HER2 positive, and if they're shadowing it's 1.8 times more likely to be PR negative. So what's interesting about these HER2 lesions is we found that they're quite wide, and that the sound transmission behind them tends to vary, and so we found out that if you have a mixture of shadowing and enhancement, it about doubled the chances that it was HER2 positive, especially if it was a lesion that was much wider than tall.  The HER2s would be these 2s and 3s, a mixture of enhancement with either normal and or partial shadowing. And so these are two HER2 lesions. In fact, these are some of the reference keys, the images that we use. Here's partial enhancement, partial normal, and here's partial enhancement with an area of shadowing right next to it. Both of these were proven HER2 positive cancers. Now, Ki67, again, these are color coded, so that if it's aqua, it's a p-value of less than 0.05, and if it's green, it's a p-value of less than 0.001. So we can see that just about everything correlates with Ki67. Now, if I say negative, that means that the lower the score, the more likely it is to lead to a high Ki67, and the higher the score, it leads to a lower or correlates with a lower Ki67. We can see that sound transmission is the most powerful, it has the best p-value, but there are a lot of other tremendously significant findings on Ki67. So again, in our sound transmission scoring system, enhanced through transmission is a very low score, it's a zero, whereas partial shadowing or mixture of shadowing enhanced through transmission, as we're seeing here, would be a four. So basically, the left one is likely to have a very high Ki67, this one in fact had a Ki67 of 60, and the right where they're shadowing is more likely to be a lower Ki67, and in fact this one had a Ki67 of 5. Now, what's interesting is we can look at the index lesion in the breast without even looking at the lymph nodes, and ascertain something about the likelihood of positivity of lymph nodes. And so what I did is when we looked at the lymph nodes, we tried to make it correlate with the Z11 study. So we had either negative lymph nodes, one or two positive lymph nodes, or three or more positive lymph nodes. And this kind of perplexed me initially because the things that would normally correlate with high grade, such as enhanced through transmission, did not necessarily correlate with positivity of lymph nodes. But if you think about it, it kind of makes sense, because we know that triple negatives tend to skip the lymph nodes and go straight hematogenously. And so since enhanced through transmission goes with grade three, triple negative, it's possible that those are so aggressive they go straight hematogenously without going through the lymph nodes. Anyway, shadowing increases the risk of positivity of the lymph nodes, whereas sound transmission decreases the chances of three or more positive lymph nodes. So you can see that there's about a two and a half fold increased risk of three or more positive lymph nodes when there's acoustic shadowing versus enhanced sound transmission. Now, we talked about histologic grade, and that was the first thing I noticed way back in the 1990s. So if we look at sound transmission, or in the current version, edition five or by reds, we'd call it posterior features, we can see that all of these have a very strong correlation with sound transmission versus grade. So distinction grade three from other grades, 10 to the minus 17th, distinction grade two from other grades 0.00011, and distinguishing grade one from other grades 0.000196. So we can see that if we look at enhancement, the relative risk of it being grade one or grade three, I'm sorry, is 3.5 times more likely than grade one. So enhancement goes with the increased risk of grade three with a relative risk of 3.5 times. On the other hand, acoustic shadowing makes it 2.9 times more likely to be grade one than grade three, and it makes grade 3.35 times as likely as grade one. Notice that there's a much bigger difference between grade three and grade two than there is between grade two and grade one. Grade one and grade two are more similar, and I'm going to show why at the very end of this talk, but this does make sense when we look at the genetic assays that are done in grade two cancers. If we look at the feature scores for grade one and grade two with 95% confidence intervals, we can see that indeed grade three is way, way, way lower scores, and remember lower scores go with better sound transmission, higher scores go with more shadowing, so we can see that grade one and grade two shadow more, and there's a higher likelihood of enhanced through transmission and very low scores with grade three. So we're looking at these two lesions. The left shadow is intensely. The right has enhanced through transmission. The right is likely to be the grade three. The left is likely to be the grade one. Shadowing goes with grade one, enhanced through transmission with grade three. Now what about molecular subtypes? Well, this is the ultimate in personalized medicine because they respond differently to treatments, and treatments can be customized for the molecular subtypes. Now the best way to define molecular subtypes is by genetic assays, but many times people use surrogates, and so we've defined surrogate definitions of the molecular subtypes using ER, PR, HER2, and KI67. We've already shown that sound transmission correlates with all four of these things extremely strongly, so it should not be surprising that if these are the four components for molecular subtypes, sound transmission is going to work to some extent, at least, for molecular subtypes. So if we talk about the subtypes, there's four. There's luminal A, which is defined as ER and or PR positive and HER2 negative with a KI67 of less than 13, or ER and or PR positive with a HER2 positive. Now, I'm sorry, that's a little bit. The ER2 will always be negative. In luminal B, you've got ER and or PR positive and HER2 negative, but a KI67 greater than 13, or ER and PR positive with a positive HER2. Now, I will say something about the cutoff value of KI67. In Europe, they're using St. Cal in 2013, which uses a cutoff of 20, and ASCO and CAP, which is the Pathologic Society, are using 10, so there are different cutoffs that people use to distinguish A from B. Now, some people are also using grade, but the best way is to use the KI67. The HER2 positive group is ER and PR negative, but HER2 positive, and the HER2 positive is defined as either an IHC immunohistochemistry 3+, or an IHC 2+, with a re-amplified HER2 fish, either one of those counts. And then a triple negative, obviously, is ER and PR negative, or HER2 negative. Now, we kind of talk as if triple negative is one thing. It's now been shown from genetic assays and prognoses that there are really multiple different subtypes of triple negatives. If we look at the survival curves, luminal A has a much better survival curve than any of the others. You can see that the others have much worse survival curves, although I will say that HER2 is the manifestation of personalized medicine. Without the HER2 blocking drugs, the prognosis of HER2 is terrible, but with the right combination of HER2 antibodies, the prognosis of HER2s has gotten much, much better. In general, the luminal A's make about more than a third, almost 40%. Luminal B's are similar. Triple negatives on HER2s are usually 10% to 15%. Now, luminal A's are the most common subtype. They arise from the epithelial cells of the assymus of the Tdlu. They're unifocal about three-quarters of the time, multifocal about a quarter of the time. They're very positive ER and PR and HER2 negative. They have a low Ki67 of less than 14 or less than or equal to 13, and they respond to tamoxifen and aromatase inhibitors. And they have the best long-term survival, but they do tend to recur late, so you'd have to follow them for a long time. And when there are metastases, they tend toward bone metastases. A typical path report for a luminal A would be grade 1, irregular shape speculated. ER 100% positive, PR 100% positive, HER2 negative, 0 or 1 plus, and then Ki67 to 5. Luminal B's make up a slightly smaller percentage than luminal A's. They also arise from the epithelial cells of the assymus of the Tdlu, and therefore they're AADs, assynure adenocarcinones of the breast. They're more typically, they're more commonly multifocal in about half of the cases. The ER is positive, but more weakly so. The PR is positive, but more weakly so. The HER2 is negative and 70% positive and 30%. The Ki67 is going to be over. It's going to be 14 or higher. There is a higher chance of P53 mutation rate, and P53 is a tumor suppression gene. So when the tumor suppression gene is faulty, the normal tumor repressive mechanisms don't work. They respond to tamoxifen, aromatase inhibitors, but these people usually also get chemotherapy. Luminal A usually gets hormone therapy only, whereas luminal B's get a combination of hormone therapy and chemo. Their survival rate is worse. It's about 40% of 5 years, and like luminal A's, they tend to metastasize the bone. A typical path report for a luminal B would be a grade 2 or 3, irregular shape, indistinct margins, ER 30%, PR 10%, HER2 1+, or 2+, with non-amplified fish, and a Ki67 say of 25%. The HER2 molecular subtype is only about 10% to 15% of all cancers. These are thought to arise from the luminal epithelial cells of the duct. So they're DABs, and they may undergo epithelial or mesenchymal transformation. These are multifocal in the highest percentage of cases, 65%. And it's very common to see almost a few lesions with lots of small foci that involve two, sometimes even three quadrants. So the largest focus in my experience might be 12 millimeters, but you might have 10 or 15 foci. These have the highest percentage of positive lymph nodes, and I've noted, personally, I've not seen this literature, but when I see a lesion where the lymph nodes are so bulky that they have a larger volume than the breast lesion, I think HER2 first. Some of these may have a little bit of HER2, a VR and PR expression. They tend to have pretty high Ki67s. I find it 50 or 60 often. These have a higher P3 gene mutation rate, up to 75%. They more often have extensive DCIS, or what Laszlo calls DAB components, usually grade three, and they are often associated with fine linear branching calcifications. So you may see an evasive mass together with extensive calcified DAB components. These respond to the trastuzumide, pertuzumab, targeted antiher2 drugs that are antibodies or kinase inhibitors. They have a 31% survival of five years without treatment and a high recurrence rate, but they do respond very well to antiher2 drugs. Interestingly, they tend to have visceral myths and to metastasize towards brain. And brain is very important. A lot of these insurance companies are not approving PET CT scans that include the brain. It's a big mistake if there's a HER2 positive case, not to include the brain on your PET CTs, because about a third of these patients get brain myths. And the reason for that is that trastuzumide is a large antibody, too large to cross the blood vein barrier, so it will prevent disease below the blood vein barrier, but not above the blood vein barrier. So a typical report here would be grade two or three, irregular shape, indistinct margins, nuclear grade three with comedone necrosis on the DAB components. ER put negative PR negative HER2 three plus or two plus with an amplified fish and KI67 at 60%. Triple negatives is again 10 to 15%. They are thought to rise from the myoepithelial cells of the assymus, so they are AABs, but they are thought to rise from myoepithelial cells. They're usually unifocal. They're more common with BRCA1 mutations. They're more common in African-Americans. They have the highest P3 gene mutation of all lower lymph node metastasis rate. They tend to go hematogenously more early. They're the highest chance of presenting as an interval cancer between screening mammograms. They tend to have less associated DCIS or DAB components, but they have a higher occurrence rate, usually within one to four years. So if you can get to five years with a triple negative, you might be cured. They also tend to have metastasis to visceral liver lung brain. Typical report here again would be grade two or three, usually three. Irregular in shape or indistinct margins. ER negative, KR negative, HER2 negative, and KI67 might be 90. Now let's look at how sound transmission works on these molecular subtypes. As you would expect, we can distinguish triple negative from other types really well with a p-value of 10 to the minus 13. Distinguishance luminal A from other molecular subtypes, again, 10 to the minus 14. Not quite as good with HER2, but it's a significant p-value of less than 0.05. So if we actually look at the graphs of what percentage of cases have what kind of sound transmission, we can see that triple negative by far has the highest enhanced sound transmission rate, and it makes it 5.5 times more likely that it's triple negative than luminal A if there's enhanced sound transmission. On the other hand, if there's acoustic shadow, we can see that luminal A has the highest rate there, and if there's acoustic shadowing, it's 6.8 times more likely to be luminal A than triple negative. So again, if we look at these two lesions, the one on the left has enhanced sound transmission, the one on the right shadowing, it's taller than wide with a thick halo. The one on the right is going to be the luminal A, the one on the left is going to be the triple negative. Now, the sound transmission with molecular subtype and histologic grade, you know, it's all interrelated. So the, you know, molecular subtype and histologic grade are not independent of each other. So if we look at our 524 invasive breast cancers where we have all the receptor data, we can see that luminal A's are about half and half grade one and grade two. Luminal B's in the other hand have many fewer grade ones, and mostly grade twos and quite a few grade threes. Her twos have virtually no grade ones. They're all grade two or grade three, mostly grade three, and then triple negatives even more. But, you know, we can say that it's not surprising that grade and molecular subtype correlate similarly with enhanced through transmission because, again, we can see that sound transmission, enhanced sound transmission goes with grade three and with triple negative, but grade three also goes with triple negative. So you can see the interrelationship, and by the same token, if we look at shadowing, shadowing means it's more likely to be grade one or luminal A, but we can see why because most luminal A's are grade one. So we can see the interrelationship. You know, what I'm saying is the histologic grade and molecular subtype are not independent of each other because they're related. Now, if we look at the actual scores for molecular subtypes with a 90% confidence intervals with luminal A and the left luminal B and then her two and the triple negative, we can see that triple negative and her twos have much lower sound transmission scores than luminal A or luminal B, but we can see that there's even, there's not even any overlap between luminal A and luminal B. So we can't really tell her two from triple negative, but we can tell her two and triple negative from luminal A and from luminal B, and we can also tell luminal A from luminal B. So again, if we look at this sound transmission, molecular subtype and histologic grade, the lesion on the left is taller than wide with the thick halo and speculations and acoustic shadowing. It's going to be a luminal A grade one, and the one on the right is circumscribed or indistinct with enhanced through transmission, not much of a thick echo than a halo. We're going to think triple negative in grade three. Now, what about taller than wide? Well, when we talked about an atom, we talked about how the TDLUs are located in the periphery and that AABs arise from the TDLUs and that, you know, enter and posterior TDLUs tend to be oriented not parallel to the skin, whereas terminal TDLUs tend to be parallel, but the cancers that arise from the enter and posterior TDLUs will be taller than wide. So AABs, at least some percentage of AABs is going to be taller than wide. We talked about how DABs arise in the center of the mammary zone and the ducts, and the ducts are horizontally oriented. So the DABs are always going to be wider than tall. So what this means is that if we see taller than wide, it ought to be an AAB and probably a grade one AAB or a luminal A AAB. So again, we showed how a small AAB from an anterior TDLU looks like a tennis rack with head up to handle down lies in the superficial half of the breast. From posterior TDLU looks like an upside down tennis rack with the handle up in the head down in the deep half of the breast, but some terminal TDLUs are going to be wider than tall. They'll never be taller than wide. We can see that as a lesion gets larger, its chances of being taller than wide falls off because, you know, larger lesions represent either DABs that were never taller than wide or they represent combination AABs and the DABs where the DAB component is making the lesion wider than tall. So how does the data look on that? Well, we can see that grade threes do have a lower percentage of non-parallel orientation in grades one or grade two, but it was not statistically significant. The P value was .09. Triple negatives do have a lower percentage of non-parallel orientation of luminal A subtypes. And so that did have a significant P value of .002. One thing important here is that in the byrads, addition five, round is considered not parallel. And so many triple negatives are not parallel by virtue of being round. So that sort of the definition that you should define a round lesion is not parallel is the correct thing to do, but it makes this correlation weaker, I think. So if we look at a non-parallel orientation versus grade, we can see that indeed grade one, which goes with luminal As, is 1.7 times more likely than grade three. And if we look at the molecular subtypes, the one thing that sticks out is that the HER2 is only one-third as likely as luminal A to be taller than wide. And that makes sense because we know the HER2s often have a large DAB component or maybe entirely DAB, and they correlate with linear casting calcifications, which are in a horizontally oriented duct, so it does make sense. So here we're seeing a taller than wide lesion that should be an AAB. It looks like it's arising from an anterior TdLU. It has normal sound transmissions, probably going to be a grade two AAB, luminal A. And then the one on the right is round and has advanced through transverse mass triple negative. Here we have a taller than wide lesion, same lesion I showed you previously. Again, that's going to be a lumalase single TdLU cancer arising from an anterior TdLU. And here we have multifocal disease with a very wide distribution, much wider than tall. I mean, this is a very classical pattern for multifocal HER2. Now, what about the size of ducts? So in Byred's edition five, this would be considered part of associated findings. We noticed in the past that larger ducts tend to correlate with grade three. A DCIS components are in site two components, and smaller with grade two and intermediate size with grade, smaller with grade one and intermediate size with grade two. We know that the mechanisms of duct enlargement in DAB or DCIS are four components. You have tumor cells within duct lumen. You either have secretions or necrosis within the lumen. You can have variable amounts of perioductal lymphocytic reaction and perioductal desmophlegia. And we know that all four of those tend to be most common in grade three DABs, less common in grade two, and certainly grade ones may have only tumor cells with or without secretions, but generally known necrosis. So if we look at these two ducts side by side, on the left is a normal duct with some minimal duct hyperplasia. On the right is a DAB grade three, and what I want to focus on is the widest part of the duct just to the right edge of the histologic slide and the outlined rectangle shown by the arrow. And at the point where the arrow is, is the widest point of the duct. There we can see purple stuff in the lumen. That's the micropapillary tumor cells. The white stuff is the secretions that are secreted by the micropapillary DAB. The pink stuff is the perioductal desmophlegia on the walls of the ducts. And then the purple stuff outside of the, is the, is the perioductal lymphocytic reaction. But the reason the duct is so large there is that all four components are contributing. So we would think that when you get large ducts in the surrounding tissue, it's going to suggest more likelihood of all four components contributing and more likelihood that it's grade three and more likelihood that it's going to be her two. Now, how good are we as seeing the large ducts? Well, if it's grade three, we found that you can have four, six, eight, 10, even 12 times normal size ducts. So it's not difficult to detect and large ducts caused by grade three. Nuclear grade two, you may get two to four times normal size. So if it's two times normal size, it could be tough to pick up. But at four size, four, you can probably pick it up. So we're intermediate in our ability to pick up grade two. We're pretty good at grade three. Grade one frequently does not enlarge the duct enough to efface the surrounding loose stromal tissues within the surrounding the ducts. And so we often can't even detect grade one.  Intermediate size ducts, as we're showing in the middle, would tend to go with grade two. And small ducts that are hard to tell from curved speculations tend to go with grade one. Same with microlabulations. We know that microlabulations are often clusters of enlarged ducts. So when we have large microlabulations, up to say two, three millimeters, this tends to go with grade three. Very small ones tend to go with grade one and intermediate sized ones with grade two. So if you look at large ducts in the surrounding tissue and their correlation, you can see that there are multiple correlations. Now what's interesting is that HER2 should be the best correlation, but we didn't get statistical significance. I think there's an artifact of small numbers. Unfortunately, during the study that was funded by a company, the research coordinators at the sites did not realize that a 2 plus HER2 IHC required a fish. And so the study got closed before it was evident that there was missing data on the HER2s. So almost all the, I shouldn't say almost all, but a large percentage of the 2 plus IHC HER2s did not have the data for the fish entered. They got the fish. Somebody knows it at the site, but it never got put into the central database. So I really think the HER2 should be the best correlation, but it isn't because of small numbers.  Now, if we look at actual numbers, even though this wasn't statistically significant, if you had enlarged ducks in the surrounding tissues, it made it just about two times more likely that it was HER2 positive than HER2 negative. So it's kind of what we predict, just not significant p-values due to two small numbers. Now, if we look at grade again, we said that if we see enlarged ducks, it ought to be more likely grade three. You can see that that's true. If there are enlarged ducks, it makes it 2.4 times more likely to be grade three than grade one. Now, I also mentioned when we talked about molecular subtypes that the types that were often associated with extensive DCIS or DAB components are limelabie and HER2. You can see that indeed, if we have enlarged ducks, it makes it 2.9 times more likely to be HER2 than triple negative, and 2.3 times more likely than Illuminal A. So indeed, enlarged ducks tend to go with grade three, and they tend to go with Illuminal A over two. So on the left, we have some moderately enlarged ducks surrounding an invasive cancer that's multifocal. That turned out to be Illuminal B, with grade two DAB components. On the right was a mass with much larger ducks in the surrounding tissues. This turned out to be a HER2 grade three. Now, here's a case where we have lots of very prominent ducks in the surrounding tissues, lots of small lesions, and lots of calcations in the ducks. This, the calcations in the ducks, definitely tends to go more with HER2. And so this, in fact, was a grade three HER2 lesion. Now, what about the relative thickness of the echrogenic rim? I showed you some recent articles showing the thickness of the rim, and speculations tended to go with Illuminal A in grade one. So what evidence do we have to support this? Well, on the left, we have a small taller than wide lesion with a very thick echrogenic halo, and very long spicules. This is a 38 millimeter probe, and the spicules are essentially 3.5 centimeters wide. Even though the width of the lesion is about 4 millimeters. That's very typical of Illuminal A, or grade one, and or grade one. Now, the right is a roughly round lesion with enhanced through transmission, and it has an echrogenic rim, but it's relatively less thick, and the spicules present are less prominent. So if we look at the relative thickness of the echrogenic rim to the hypococentral nitus, you can see a big difference, and the same relative length of the spicules versus the width of the internal rim, much less with Illuminal A. I mean with triple negative, sorry. So here I'm just showing you ratios. So the white measurement would be the width of the hypococentral nitus. The pink would represent the width of the lesion, including its echrogenic rim, and if I were taking a ratio of pink divided by white, you can see that the pink divided by white is much bigger for the grade one Illuminal A on the left than it is for the grade three triple negative on the right. So when we plot this out, and we look at the relative thickness of the boundary zone to the hypococentral nitus, you can see that indeed it's greater for grade one than grade two than grade three. Unfortunately, there's a little bit of overlap. You can see that the 95th percentile, or the upper confidence interval for grade three just very slightly overlaps that for grade one, and greatly overlaps grade two. If we look at the echrogenic rim thickness versus molecular subtype, we can see that the confidence interval of triple negative overlaps Illuminal B on her two, but does not overlap Illuminal A. So we can tell Illuminal A from triple negative, but not necessarily Illuminal A from Illuminal B, Illuminal A from her two, or triple negative from her two, or Illuminal B. So we can tell the opposite ends of the spectrum from each other, but not the fine gradations in the middle. Now if we look at the length of the spicules versus the hypococentral nitus, again, we can see that grade three has significantly shorter spicules compared to the hypococentral nitus compared to grade one and grade two, and in fact, doesn't overlap the confidence interval for either grade one or grade two. So you can tell grade three from grade one and grade two, but you can't tell grade one from grade two. Now, if you look at molecular subtypes, again, triple negative can easily be distinguished from Illuminal A and Illuminal B. Illuminal A and Illuminal B can even be distinguished from each other, but you can't tell triple negative from her two, and you can't tell Illuminal B from her two. But what this data is showing is that indeed, the Illuminal A is most likely to be speculated and have a thick hypoallergenic halo that's very thick compared to the hypococentral nitus. The spicules are very long compared to the hypococentral nitus, whereas triple negative is very likely to have a thin or absent hypoallergenic rim with absent or short spicules in the surrounding tissue. So basically, we're showing that both boundary zone and peripheral zone features can distinguish Illuminal A from triple negative, not necessarily from Illuminal B or her two. Now, am I troubled that we do a better job of telling grade one from grade three than we do from grade one from grade two? Not at all. There are several studies now published where they've done genetic assays and showed that, and this is very interesting. I'm going to tell a little side story. A friend, a pathologist mind, told me that there is no such thing as grade two. There's just grade one and grade three, and he calls it grade two, and he can't decide whether it's grade one or grade three. So he meant that as a joke. He wasn't being serious, but in effect, there's some truth to that. So these are genetic assays, and what have been shown is that grade one and grade three are markedly different from each other genetically. But there are really two types of grade two. There's what they call a G21 and a G22, or G23, I'm sorry. So about two-thirds of grade twos genetically are similar to grade one, and about one-third of grade twos are genetically similar to grade three. Now, what that means is that whatever imaging biomarkers we see for grade two should be more similar to grade one than they are to grade three, because a larger percentage of grade twos are more genetically similar to grade one. They've shown the exact same thing with proteomics, and the exact same thing with non-coding RNAs, which are the key things that determine epigenetics. So genetically and epigenetically, grade twos are a mixture of lesions, about two-thirds of which are similar to grade one, and one-third of which are similar to grade three. So I think the data here show this. You can see that in terms of percentages and actual scores, grade one and grade two are more similar to each other, and then grade two is to grade three. Now, this is all very complicated. I've shown you innumerable features that correlate. So here I'm showing various grades versus each other, and again, we've used the same color coding. If I give it an aqua color, it's better than 0.05. P value less than 0.05. And if it's green, the P value is less than 0.001. So you can see there are lots of correlations. And then if we look at luminal A and luminal B and triple negative and heart two, so if we look at my leg, our subtypes again, lots of different correlations. And what this means is it's very complicated. How's our mind going to figure all this stuff out? Now if we look at a luminal A and what its most common feature score is for the various categories, and the five categories are shape, echo texture, sound, rend, mission, margin, boundary zone, and peripheral zone, what I've shown in yellow is different between the two. So really the only thing that's similar between triple negative and luminal A is the presence of an echogenic rim or echogenic halo. And yet it's not like it's 100% and 0%. And we've got 51% versus 31%. 43% versus 52, 65 versus 57. So these percentages are not staggering and be large. So how's our mind going to deal with this? Well, it's very difficult. Now if we look at less significant features, what I'm looking at is various shape scores, echo texture scores, so they're color coded by the five different categories. And if I made it bold in yellow, it means that there's a 3 to 1 difference in the percentage of cases that are triple negative versus luminal A. So if you look at oval shaped, triple negatives have 8.9% or oval shaped, where it's only half a percent of luminal A. So that's an 18 to 1 ratio. If you look at oval or round shape, it's 11.4 versus 2.1, a 5 to 1 ratio. But these are small percentages. It's not like 100% of triple negatives are that and 0% of little ways are that. So again, even though these ratios are greater than 3 to 1 differences, it involves only a subset of the cases. And so it's difficult for our mind to put all this together. So here's this taller than wide lesion with shadowing at thick halo and very long specules. This should be luminal A. And this is a oval shaped or circumscribed lesion with very thin equigenic rim compared to the central nitrous and very short spicules. This is triple negatives got enhanced through transmission. I mean, these are classics.  And a lot of the things in the middle, the herd twos and the little bees are tough to pick out. So we could use some help there. So we're going to talk about this in the next talk. What I'd like to do is get machine learning to help us. AI machine learning. So what we could do is put in many of the things we know correlate with prognostic features such as age, race, BR status, mammogram features, MR feature size, universal multifocalody, large lymph nodes, the ultrasound feature scores, and just have the machine figure out what the likelihood of each malignant subtype is. So we're going to talk more about that in the next talk. But in summary, what I've shown you is that we can and we probably should ask more of our salt cell. And if we want to really deliver personalized medicine, there's a tremendous amount of information on ultrasound that we're just throwing away and not using. There are numerous ultrasound features of breast masses that are significant prognostic biomarkers, but it's really difficult for our puny human brains to put them all together without the help of AI. So in the next talk, I'm going to show you how I would visualize this could work in the real world. Thank you.


This video is the third part of the Wienberg lecture on Wienberg's algorithm and Katz-Mudin algebras. It might not make a whole lot of sense unless you've seen the first two parts of the lecture, so I'll put links to them in the description of the video below. Now if I think of it, it might not make a lot of sense even if you have seen the first two parts of the lecture. What we're going to discuss in this lecture is whether you can assign nicely algebras to Wienberg's group. So to recall what we did in the first two parts of this lecture, Wienberg looked at the lattices i,n,1. So this is the odd unimodular Laurentian that is consisting of ornums m1 up to mn, mn plus 1, with all the mi in z, and m1 squared plus plus mn squared minus mn plus 1 squared is the norm of this vector, the square of its length. Notice there's a minus sign here. And Wienberg calculated the Dinken diagrams of the reflection groups, and you remember he found these rather intriguing diagrams that two of them looked like this, for example. Now, there's one obvious way to get Dinken diagrams, which is that if you take a Lie algebra, the Lie algebra, you might want it to be finite dimensional and semi-simple, then this gives you a Dinken diagram. For example, if we take the Lie algebra of all n by n matrices over the reals, then this gives us a Dinken diagram, which looks like this, with n minus 1 points. So this is the a n minus 1 Dinken diagram. And we can ask, do Wienberg's Dinken diagrams also correspond to various Lie algebras? So what happens is we can look at spherical reflection groups, and these correspond to the usual finite dimensional Lie algebras. We also looked at Euclidean reflection groups. And these give the so-called affine Lie algebras, more or less. So a typical example of this would be the group of n by n matrices with coefficients that are Laurent polynomials in the reals. Actually, that's not quite true. There's a little extra fuss you have to do about the center of the Lie algebra, but I won't worry about this. And finally, you can ask, what about hyperbolic reflection groups? So for example, we can look at Wienberg's examples of the lattices i, n1 and the even lattice i, i, n1. And we can ask, what's the algebras to these give rise to? And I'll be discussing this lecture, what we want to put here. So Katz and Moody gave one answer to this. What they found was a way of associating any to any Dinken diagram, some sort of Lie algebra. This is called a Katz Moody algebra. And so for the finite dimension Euclidean Dinken diagrams, this gives the finite dimension and Euclidean Lie algebras. So what we need to do is to see, how do you get from a Dinken diagram to Lie algebra? So what does a Dinken diagram look like? Well, a Dinken diagram has a bunch of points and the points have various numbers of lines between them. The way you get from a Dinken diagram to Lie algebra is as follows. First of all, for any point, you get a copy of the Lie algebra SL2. So that's just little two by two matrices A, B, C, D with A plus D is zero under the usual Lie bracket. And the lines between the points tell you how these two little copies of SL2 interact. For example, if you have two points like this, this means that the two corresponding copies of SL2 commute with each other. And if you have two copies of SL2 and a single line between them, then what happens? This means the two copies of SL2 don't quite commute. They sort of interact in the way that, well, there are two obvious copies of SL2. There's one copy here and there's another copy there. And you see they sort of overlap in some sense. And a line between two copies of SL2 means they overlap in this sense. And if you assign more lines, then more complicated things happen. Well, that's a sort of rather informal way of describing the algebra of a Dinken diagram. More precisely, you can write down the Harish-Chandra-Sare relations. So what we do is we take numbers E, I, F, I and H, I for each point. So these are going to be generators of the Lie algebra. And now we want some relations telling you how these generators fit together. So the relations look like this. First of all, H, I, H, J commute with each other. This means the H, I... If you take all the H, I's, they form an abelian Lie algebra called the Cartan algebra, Cartan sub-algebra, Lie algebra. The Cartan sub-algebra sort of behaves a bit like the diagonal matrices of the N by N matrices. Next, we introduce relations saying what the EIs and the HIs interact, how they interact with each other. So we have H, I, E, J equals A, I, J, E, J and H, I, F, J is minus A, I, J, F, J. These numbers A, I, J are mysterious constants making up something called the Cartan matrix. And the Cartan matrix is determined by the number of lines between points in the Lie algebra. So what's going on here is this says the numbers E, E, J and H, J, sorry, E, J and F, J are eigenvectors of the Cartan sub-algebra H. So let's call capital H for the abelian Lie algebra generated by this. And this just means they're eigenvectors. In the case of the general linear group, the EIs and the FJs are things that live just above or just below the diagonal. So you can think of the EIs would have a non-zero entry in one of these positions and the FIs would have a non-zero entry in one of these positions and be the zero somewhere else. So the EIs and the HIs sort of give you the diagonal entries and the EIs and the FIs give you the entries that are just above or just below the diagonal. Next we have some more relations saying E, I, F, J equals naught for I, not equal to J and H, I, if I is equal to J. And what this relation says is that the generators E, I, F, I and H, I form a little sub-algebra that looks like SL2 of the real. So it's the three-dimensional, the algebra that everybody knows about. And in matrix form these are just the little 2 by 2 blocks on the diagonal. So a typical one of these, the algebras might look like a 2 by 2 block there. So these are the relations that are easy to understand. We also have some rather more complicated relations, one of which says that EI, EI, SON, EI, EJ is equal to zero and the same for the FIs. So what do these say? Well, first of all you have to say how many EIs you put there but it turns out it doesn't really matter how many EIs you put there. I mean, people have some complicated formula depending on these numbers A, I, J but in fact it turns out that provide you've got a sufficiently large number of EIs that all these relations are equivalent. And what these say is that the operations taking a Libra with EI or with FI are locally nilpotent. What this means is they're not quite nilpotent but they're nilpotent if you sort of apply them to any finite dimensional subspace. In the finite dimensional case, this says they are actually nilpotent. This is very nice because it means that when you act with SL2R on the cat's moody algebra, the cat's moody algebra splits up as a sum of finite dimensional representations of SL2R which is good because we know all about the finite dimensional representations of SL2R. So these relations sort of say the cat's moody algebra behaves very nicely as a representation of all these SL2Rs. So cat's moody pointed out you would take any dink and diagram whatsoever just write down these relations and this gives you the algebra. And as I said for the finite dimensional reflection groups, this gives you the finite dimensional of the algebras and for affine reflection groups, this gives you the affinely algebras. So this gives us one answer to our question. If we've got any hyperbolic reflection group, we get a dink and diagram and from the dink and diagram we get a cat's moody algebra. So, well, that seems to answer the question but it's an entirely satisfactory answer because it turns out the cat's moody algebras you get are kind of a bit wild and difficult to understand and what we really like is a nice lean algebra here. And the question is what do you mean by a nicely algebra? Well, everybody agrees the finite dimensional the algebras and the affinely algebras are nice. So let's see what's nice about them. Well, what's nice about them is we can write down an explicit character formula which was discovered by Hermann Weier and it's called the Weier character formula. The finite dimensional one as well. Cat's noted that it could be extended to all cat's moody algebras to the Weier cat's formula. It's not completely straightforward extending it because if you look at all the proofs of the Weier character formula and finite dimensions none of them actually work if the le algebra is infinite dimensional but cat's discovered that if you take one of these proofs and kind of stretch it a little bit you can get it to work for infinite dimensions. The main problem cat's had to solve is this proof used the Casimir operator which doesn't quite make sense for infinite dimensional le algebras but cat's discovered a clever way to regularize it so it did make sense and was then able to push through a proof. Now I'm going to be most interested in the case of the character formula for the one dimensional trivial representation. It sounds kind of stupid writing down the character formula for the one dimensional representation because we know the character of the one dimensional representation. But the Weier cat's character formula says that the character of a representation is equal to a certain sum over a certain product and this is rather nice because if we know the character is one then we're going to get a nice identity saying the sum, a certain sum is equal to a certain product. So we get an identity saying the sum is a product which is the Weier denominator formula and the Weier denominator formula looks like this and what we do is we take a sum over something called the Weier group and this is equal to a product over something where these are something called a positive roots and row that appears all over the place here and here is something called the Weier vector. This is just a sign plus or minus what we're depending on the determinant of the element of the Weier group. So I just give a couple of examples of the Weier denominator formula for finite dimensional and affine algebras. So for finite dimensions the Weier denominator formula is just the Vandermonde identity at least for the le algebras a and minus one. So you remember the Vandermonde identity says if you've got a matrix 1, 1, 1 x0 x1 x2 y0 x0 squared x1 squared x2 squared and you want to know it's determinant. This is the product over i less than j of x i minus x j up to a sign which I always get wrong and you see what we have here is a product over i less than j and you can see that i less than j corresponds to three upper triangle entries here which are the three positive roots of a little 2 by 2 matrix and the sum is over if you expand out this determinant you get six terms which correspond to the six entries of the symmetric group S3 on three points and this is just the Weier group of the le algebra of 3 by 3 matrices. So here this is really a sum over the Weier group and this is a product over positive roots. So the usual Vandermonde identity is a typical example of the Weier denominator formula in finite dimensions. The simplest example of the Weier denominator formula for affine algebras is the famous Jacobi triple product identity and to get the Jacobi triple product identity what you do is you draw this picture here and what this is is it's really the root system of it's the root system of SL2 are with coefficients this and the wrong polynomials so here what we have is the root system of SL2 and it sort of gets shifted to the right or the left whenever you sort of multiply by Z and what I'm going to do is I'm going to take the positive roots to be these points here and to each positive root I'm going to assign a factor as follows so this is going to be here I'm going to put 1-q squared, 1-q to the 4 and so on here I'm going to put 1-qz here I'm going to put 1-qz to the minus 1 here I'm going to put 1-q cubed z to the minus 1 here 1-q cubed z and so on so what we get is a big product that looks like a product over n greater than 0 of 1-q to the 2n 1-q to the 2n-1z 1-q to the 2n-1z to the minus 1 and if you expand out this product what you would mostly expect to get is some sort of horrible complicated mess it's fairly obvious that all the terms of this product when you expand it out are going to lie inside a certain parabola and normally you'd expect to have something very complicated on the inside but it turns out that all terms inside the parabola vanish what's happening is there's sort of massive amounts of cancellation going on so what we're left with is the terms that are exactly on the parabola which are easy to work out it's just sum over n of minus 1 to the n q to the n squared z to the n so it's very easy to remember this identity if you remember this picture of the root lattice of obviously algebra here then the product is just the positive roots and the sum is just the sum over the boundary of the obvious region containing the terms of this product and Mcdonald found such identities or all the other affine root systems they called them Mcdonald identities many of them were sort of found by Dyson who sort of overlooked the connection with the algebras which he was very frustrated by Mcdonald came up with these identities slightly before affine-cats moody algebras were well known and cat sort of said that when he looked at the Mcdonald identities and looked at affine-cats algebras it was just sort of immediately obvious that the Mcdonald identities were the denominator formula for the affine-cats moody algebras so that was a very nice explanation the Mcdonald identities were originally a bit puzzling because the sum was, the product was not only over the roots of the affine reflection group which were these terms and these terms but there were some extra terms here which were a bit puzzling but these turned out to be the norm zero roots of the corresponding affine algebra so the reason why we get nice identities is we know, first of all we know the simple roots now knowing the simple roots tells us what the vial group is just the reflection group of the simple group and this tells us what the sum is in the vial denominator formula we need to know the root multiplicities I didn't quite make these obvious but if you go back to the vial denominator formula we should really take several factors if the root alpha has multiplicity greater than one so there should really be a factor here where you put in the multiplicity of alpha so we also know the root multiplicities in both the finite dimensional and affine cases and this tells us what the product is in the vial denominator formula and now the problem is that if we write down a Dinkindragon we certainly know what the simple roots are but the root multiplicities seem to be very complicated in general now for finite dimensional algebras they're easy they're just one but affine algebras they're much more complicated they're usually either one or the rank or something something a bit related they're easy to describe so what do the root multiplicities of these more general cats moody algebras look like well people did a lot of computer calculations and there are some examples in cats' books so here's the root multiplicities of a little rank two if you look at them the multiplicities it's rather hard to say anything much about them let me just magnify a bit so you can see so if I zoom in a bit so you can see the multiplicities I mean they're not too difficult to calculate by computer but it's hard to say anything very interesting about them and you can write a product formula for them to some extent a more interesting case is it was done by Frenkel and Feingold who looked at this rank 3-le algebra this rank 3-le algebra this turns up as the reflection group of the modular group so its vial group is essentially just gl2 over the integers and if you look at the multiplicities something quite interesting has happened first of all you see that various numbers are repeating quite often so we've got a number 30 appearing 3 times so 3 different orbits of roots and then we get numbers 42 appearing 3 times and 56 and so on and if you look at the multiplicities well there's a sequence 2 3 5 7 11 and every number theorist will immediately recognize these numbers as the values of the partition function they're also primes for the first few but that's just a coincidence they're really values of the partition function so what's the partition function well the partition function is just the number of ways of writing the number as a sum of positive integers so the partition numbers look like this and for example the fourth entry is 5 and that's because we can write 4 is equal to 4 or 3 plus 1 or 2 plus 2 or 2 plus 1 plus 1 or 1 plus 1 plus 1 plus 1 and these numbers are usually noted by pn and Euler found this very nice formula for p of n sum of pn times q to the n is just equal to product over n greater than 0 of 1 over 1 minus q to the n so it's 1 over 1 minus q 1 over 1 minus q squared and so on so that looks very exciting we've got a lot of numerical evidence that we're getting the partition function appearing here except we don't if you go on further you know you go up to here for example again you can see one of the coefficients is 627 but the other is only 626 so we're getting the partition function for quite a long way except it suddenly starts breaking down and as you as you go further it breaks down more and more so we've got something rather puzzling going on here it almost looks as if we've got a nice formula for multiplicity but then it goes wrong so I will explain later on why this the algebra almost has root multiplicities that are given by the partition function well the most striking hyperbolic reflection group was Conway's reflection group so you remember Conway calculated the reflection group of the 26th dimensional even Lorenzian lattice and found that it's dink and diagram is just the leach lattice for details see the second part of this lecture so the obvious thing to do is to look at the what are the multiplicities of the corresponding Katz-Mudi algebra so we're taking the whole leach lattice as a dink and diagram but so it's got an infinite number of vertices and it's sort of very complicated and looking at Katz-Mudi algebra and Igor Frenkor made this absolutely amazing discovery he showed that multiplicities of a root alpha is at most p24 of 1 minus alpha squared over 2 so what's p24 well the sum of p24n q to the n is the product of n greater than 0 of 1 over 1 minus q to the n to the 24 so it's like the partition function except you've got this 24th power here so these series numbers get big quite quickly so it's 1 plus 24q plus 324q squared and so on and furthermore he showed that there are lots of vectors that have exactly this multiplicity so we get equality if alpha beta equals 1 where beta squared is equal to 0 and beta does not correspond to the leach lattice so you remember there are 23 orbits of vectors of norm 0 correspond to the 23 neomy lattices and if a root is in a product 1 with at least one of those then the multiplicity is exactly 24 if beta does correspond to the leach lattice the multiplicity actually becomes less than this number here and the way Igor Frenkor proved this is he used string theory more precisely he used the no ghost theorem from string theory which predicts the dimension of some sort of space of strings to be given by this number here in 26 dimensions so the you know string theory has this problem that predicts space time is 26 dimensions which is a real headache for physicists but is really great for mathematicians because it's actually the right dimension for this lattice here as Igor Frenkor discovered so we can also ask how good is Igor Frenkor's upper bound for other vectors well let's take a look at some vectors alpha in the 26th dimensional lattice and see what the multiplicity is so if alpha squared is equal to 2 and then there's one orbit and the multiplicity is just one so it's exact if alpha squared equals 0 then there are 24 times infinity orbits that's because there are 24 orbits of primitive norm 0 vectors and you can multiply any of them by a positive integer and for 23 of these the multiplicity is equal to 24 and for 1 the multiplicity is equal to 0 that's the leaf lattice causing trouble alpha squared is equal to minus 2 and things start getting really complicated there are 121 orbits and you can work out the multiplicity for each of these and it turns out 119 have multiplicity 324 exactly equal to Frenkor's upper bound and two of them have multiplicity slightly less and for alpha squared is minus 4 it's similar so there are 665 orbits and most of them have multiplicity 3200 which is the next coefficient and there's a small hand for which have multiplicity slightly less and you see this is suspiciously similar to what was happening with the example calculated by Feingold and Frenkor that many of the roots have multiplicity is given by some nice coefficient but eventually the roots start having multiplicity a little bit less than that so something is almost working but starts to break down after a while and so how can we explain this well this gives a strong hint that is a slightly larger algebra with multiplicity exactly p24 of 1 minus alpha squared over 2 and what can this algebra be well it can't be a cat's moody algebra because that doesn't quite work but it turns out you can generalise the notion of the cat's moody algebra so this is a generalisation of cat's moody algebras and the generalisation is quite simple so the cat's moody algebra depends on these numbers aij and it always assumes that aii is equal to whenever ii is whenever 2 of these entries is the same then the diagonal entry is 2 and this corresponds to the fact that all simple roots have norm greater than 0 the norm of the simple root is closer to aij the exact way depends on how you normalise things well what we should do is we can also allow aii to be less than or equal to 0 and most of the theory of cat's moody algebras goes through with some slight changes so these correspond to so-called imaginary simple roots where the norm is less than or equal to 0 the term imaginary is a bit unfortunate because it conflicts with another use for the term imaginary in the theory of Li algebras here the term imaginary means that the norm which is roughly the square root of aii is an imaginary number rather than a real number and we need to modify the relation slightly so if aii is less than or equal to 0 we discard the relations that say ei, ei and so on ej is equal to 0 and of course we discard the ones and so on and we just keep a tiny piece of them we keep the relation ei fj equal to 0 if aij is 0 so other than that the relations are very similar to the relations for cat's moody algebras and again the whole theory of cat's moody algebras extends the slightly more general case in particular we get a character formula and a denominator formula which is a little bit more complicated because the imaginary simple roots give some extra correction terms so another way of thinking of the difference is that these simple roots of aii was 2 all correspond to copies of SL2 of r as we saw earlier the simple roots of aii or 0 correspond to copies of the Heisenberg algebra which is another three-dimensional of the algebra it's a sort of degeneration of SL2 in some sense the ones with aii are less than 0 also correspond to copies of SL2 of r except that I mentioned earlier that the cat's moody algebras is a sum of finite dimensional representations of this SL2 r well if aii is less than 0 this is no longer true and the cat's moody algebra is in general some infinite dimensional discrete series representations of SL2 so there are some differences but most of the theory goes through and so we can ask can we find the algebra one of these more general forms of cat's moody algebra we can so the algebra can be described as follows so this is going to be the root lattice and for each of these vectors we need to describe the multiplicity and the multiplicity of alpha is going to be p of 1 minus p so p24 of 1 minus alpha squared over 2 for alpha not 0 and 26 of alpha is equal to 0 so this is a really hugely algebra if we sort of draw the light cone then here it's got a lot of vectors of norm 2 each of which has multiplicity 1 and then it's got a bunch of vectors of norm 0 on the light cone each with multiplicity 24 and this light cone is living in 26 dimensional rents in space and then it's got some more vectors lying on the surface of vectors with alpha squared equals minus 2 and these vectors all the vectors lying on here have multiplicity 24 and then there are more vectors lying on the next surface above it or with multiplicity 3200 and then as you go further up the multiplicity you get bigger and bigger and bigger so this light algebra has a very narrow waist in some sense but it gets huge up here and it gets huge down there and we can ask what is its denominator formula and you can write down its denominator formula explicitly it looks like this first of all we take a sum of the vial group and we take the sign and then we take omega of sum over tau n of e to the n row and this is equal to e to the row times product over alpha greater than 0 1 minus e to the alpha the malt of alpha and this multiplicity is just given by this expression here and what's going on is this looks just like the denominator formula or finite dimensional affine the algebra except we've got these extra correction terms these come from the imaginary simple roots here tau of n is Romanogen's tau function you remember it's defined like this so sum of tau n q to the n is just q times product over 1 minus q to the n 24 which is you know it's 2 minus 24 q squared and so on so so this counts as a nicely algebra because we have a simple expression for both the simple roots that are just given by points of each lattice and the root multiplicities they're just given by this values of p24 now we can ask can we find similarly algebras for the other reflection groups and this turns out to be rather difficult there's one case in which you can do this suppose we take an automorphism sigma of the leach lattice and actually strictly speaking we don't take an automorphism of the leach lattice we take an automorphism of a certain double cover of a leach lattice but I'm not going to worry about that too much and we can look at the fixed point lattice lambda sigma where we just take all points of the leach lattice fixed by sigma then this lattice behaves in a way that's very similar to the leach lattice in particular we can find reflection groups whose dink and diagram is not quite this lattice but something similar so what we can do is we can take things like lambda sigma plus a little two-dimensional Laurentian lattice plus a little two-dimensional lattice with some other number there and this is similar to the 26-dimensional Laurentian lattice in particular you can define its reflection group and the dink and diagram turns out to be describing terms of several cosets of this fixed sub lattice and Peter Nieman looked at this for certain sigma of prime order and worked out the corresponding denominator formulas you get and when sigma has ordered 23 you find something really quite interesting the fixed sub lattice lambda sigma is dimension equal to 2 it has determinant 23 and I don't know what that's doing there and the corresponding Laurentian lattice contains the lattice of 2 minus 2 minus 2 2 minus 1 minus 1 which was studied by Brenkel and Feingold so just recall that Brenkel and Feingold were studying the lattice corresponding to this cot and matrix or this lattice and that was the one whose root most species were given by values of the partition function so here this lattice is now going to be four-dimensional but it's containing three-dimensional lattice and what Peter Nieman found was that the Lie algebra corresponding to this two-dimensional fixed sub lattice of lambda has root multiplicities related to the coefficients of 1 over eta of tau what's eta of tau well eta of tau is equal to q to the 1 over 24 times product of 1 minus q to the n to the 1 where n is equal to 0 where q is equal to e to the 2 pi i tau now you notice the coefficients of this are going to be very similar to the coefficients of the partition function so this is going to give you q to the minus 1 over 24 times 1 plus q plus 2 q squared plus 3 q cubed and so on where these are the values of the partition function and then we need to multiply it by eta of 23 tau which is going to give us 1 plus q to the 23 plus 2 q to the 46 and so on yes we should have a q to the minus 23 over 24 and what now you see but what's going on here is the coefficients so the first 23 coefficients are given by values of the partition function but after that they start getting a little bit bigger than the partition function and the root multiplicities aren't always given exactly by the coefficients of this function it's a little bit more complicated I won't go into exact details so many of the coefficients are given by the coefficients of this function so this sort of explains the root multiplicities found by Bencore and Feingold it turns out that these numbers here aren't really values of the partition function they're really coefficients of this function here and the Lie algebra we should be looking at is not quite this one but a certain rank for Lie algebra that contains this as a very large subspace I should say by the way that Franco doesn't agree with me that this explains his results he wants to see a Lie algebra whose coefficients are given exactly by the partition function not by this slightly modified form Neil Scheithau is systematically looked at all the automorphisms of the Lie lattice with a non-trivial fixed point sub lattice and describe the Lie algebras you get from all of these so we get a whole family of Lie algebras whose root multiplicities and simple roots are known explicitly and parameterised by certain elements of Conway's group of symmetries of the Lie lattice so this gives you a family of reflection groups related to similar to Conway's reflection group all of which are related to very nicely algebras well next you can ask what about Wienberg's other reflection groups so Wienberg also found reflection groups of say the E10 Dink and Diagram for the 10 dimensional even Lorenz and Lattice and you can ask do these correspond to nicely algebras and the answer is yes and I will be discussing this in the final part of this lecture in the next video and the key point here is that to construct these Lie algebras we need to use automorphic forms so I'll be explaining the relation of Wienberg's reflection groups to certain automorphic forms


Our first speaker is Stacey Lindow and Stacey is an associate professor of obstetrics gynecology and medicine geriatrics. She received her amenity degree from Brown, did a residency at Northwestern, and got a master's here at the Harris School of Public Policy Studies where she was also a Robert Wood Johnson clinical scholar. And Stacey is one of these rare people that is a national expert in two quite different areas. Initially she was an expert on the area of sexuality in older persons and really bringing to light a taboo subject where the quality of life of sexuality in older people is a big deal. And Stacey was one of the first to really describe what is the issues and the scripted statistics regarding this and how can we start trying to improve sexual quality of life among older people. And our next area which she'll talk about in this talk today really is thinking about how do you mobilize the assets of the community to improve community health. And really Richard Kohler and I were talking about this a little bit earlier that Stacey is really sort of at the front edge of a cutting wave where with the increased emphasis on population health and as funding improves for looking at population health what Stacey will talk about will really be sort of at the forefront. Stacey also is a favorite of my medical students here as a role model for someone who is an outstanding clinical investigator and someone who really thinks about how can we improve the care for some of our most vulnerable populations. Stacey will talk about community RX connecting health care to self care. It was a story because I can't help but remember when I see Mark I was two weeks postpartum with our third son and I got a call that I needed to come in to help Mark with a meeting with a donor. It was very important and I felt very special that I've been called to this task. So I stuffed myself literally into a dress and came to campus to have a conversation with a donor about why they might think about supporting not only the ethics center but something beyond that.  And when the firecracker goes off these sparkles land all over the globe in the form of these ethics fellows who then build new flames all over the earth. And then I think I almost started to cry and then I thought, well, you know, I've rest my case. I'm going to go home now and nurse the baby. But I think we were actually successful ultimately in that bid. And I'm continually inspired by you, Mark, and I know everyone in this room is. Thank you and Marshall, thank you for your mentorship and for that generous introduction. So I want to start by acknowledging a broad range of partners who have worked with me to develop this program called Community Rx. And I think this gets to the recent question, actually, which is how do we ensure that the citizenry understands the research we're doing? And I think one path to that is to engage with the citizenry and identifying the questions we're asking and shaping the ways we do the research in ensuring that the data we collect is good for the next New England Journal paper but also is useful to the people who are asking the questions and whose lives are touched. And the work this work Community Rx is very much an outgrowth. I think of a very concerted effort to work closely with a broad range of community stakeholders to solve, I think, a very basic but recalcitrant problem in health. You saw that first slide up there about, you know, don't copy this. And this work is funded by primarily a grant from the Centers for Medicare and Medicaid Services. I had to submit my slides for review. They're sort of very close collaboration with our funders around this work. And any ignorance is mine alone and not that of the federal government. The third bullet point actually also raises some interesting ethical issues that we could talk about maybe as part of the question and answer. But I am expected as part of this funding to create a sustainable business model for the innovation we've developed. And what this means is that a new entity has been founded, an LLC entity. I'm the founder and the manager. And whether or not it will become the sustainability solution for Community Rx remains to be seen. But it does present a potential actual or perceived conflict of interest in presenting the study findings. So intersectoral health is a mouthful and not a term that many of our stakeholders really resonate with, but it's an idea that the World Health Organization has been talking about for four or five decades and that the Institute of Medicine has recently signed on to. And the idea is that if we're going to move from a healthcare system to a system that promotes health or a health system, that there is this collective influence and responsibility that all sectors have for creating and sustaining the conditions necessary for health. And on the one hand this is an inclusive statement bringing in entities, community-based service providers, social service providers who maybe have been left out of the healthcare economy for forever. On the other hand, this could be seen as sort of distributing or redistributing responsibility for the health of the population. So we're going to move in this direction and the Robert Wood Johnson Foundation, for example, would call this a culture of health. We hear the phrase health in all policies. All of these ideas relate to this basic concept of multiple sectors coming together to take on responsibility for the health of the population. If we're going to realize this idea, we need to create some infrastructure to support it because right now the practice of medicine is largely a siloed from the practice of all the other things that happen in the community, in education, in housing, in the justice system. Those systems are largely disconnected day to day from our operations of clinical care. So here's the ethical problem as I see it. And the caption says it's the only treatment option that we'll say she has under her current health plan. And so I'm a gynecologist and I'm the cave lady up there standing on the cliff sort of looking like what just happened. And my patient actually came to me on solid ground.  And I've been able to diagnose her problem and I've been able to give her a prescription, let's say, for vaginal dilators, which my patients who've had pelvic radiation need to recover their sexual function. And then I say to her, based on our conversation, there are some other things you might need. Taking care of your weight is important for survivorship after cancer. Eating healthy, exercise. Maybe swimming would be good because it's a low intensity physical exercise. Where is she going to go to get these services? I really have no idea. My patients come from all over. I can't possibly manage all the information about all those other things that she needs to keep herself healthy as a cancer survivor. And in fact, just a sort of a funny anecdote of the problem that comes from this, we do prescribe these dilators and they are medical grade, there's a medical grade option for these dilators that costs about $150 and I have to write a prescription for it. The alternative would be for the patient to go to a local shop like the Pleasure Chest, which unfortunately when I mentioned this to a patient I called it the lion's den and then we both realized that was actually a strip club. So I mean, this is the problem with not having good quality information about where are the resources of the community. She looked at me, she was sort of like, I know what you mean, but I think that's a strip club. And at the same time I'm like, oh my God, don't go to lion's den for that. But, you know, why do we tolerate such poor quality and really the absence, the total absence of reliable information about all those other resources we need to take care of our health to manage with disease, to take care of a loved one every day. And so to me the ethical problem we're talking about here is that throwing people off the cliff. They're gripping on to that prescription for their anti-hypertensive or for their dilators. And we're yelling after them, but there are all these other things you need to do and I have no idea where you can go and I don't know if you can afford it or if you're eligible for that service. And so this is the problem we're trying to address. I think it's largely a justice problem, but there's not much beneficence or even autonomy in this picture either. So the Robert Wood Johnson Foundation did a survey of 1,000 primary care physicians and learned that physicians wish they could write prescriptions to help patients with their social needs. They learned that four in five physicians say their patient's social needs are as important as their medical conditions. And they learned that four in five physicians are not confident in their capacity to address their patient's social needs. So let's just look for a minute at electronic prescribing of drugs. This is the trend. What technology has been adopted this quickly? And we know that electronic prescribing of drugs in many ways has been part of the incentive to adopt electronic medical records, which has been challenging at best. But e-prescribing of drugs went from zero to 20% to ubiquitous in the last six years. E-prescribing of drugs makes it really easy to connect our patients to the medications they need. It requires good quality information about the medicine and all the doses and all the various formulation. And all of that information is built right into our workflow. What if the same quality information were available for the resources the community people need to take care of themselves? And we could inject the doctor-patient encounter, the nurse-patient encounter, with that information real time so that it was as easy to prescribe a drug as it was to not prescribe the lion's den or to prescribe swimming or a cancer wellness support center. This is what the architecture looks like for e-prescribing of drugs. It's complicated. And I think the most important point for this slide is to ask how do we know if the patient ever took the drug? We ask them the next time they came back. And the same would be true if we had as elegant an architecture and an underlying information system to connect people to the resources of the community. So the Health Care Innovation Award from the Centers for Medicare and Medicaid became available as an opportunity in 2012. There were 900 million taxpayer dollars appropriated to the Centers for Medicare and Medicaid for this new innovation center. And 107 of these were awarded in the first couple of rounds nationally. They were looking for innovations that would improve health, improve healthcare, reduce costs, promote the workforce of the future. The recession was hitting hard around the country at this point. And that would define and test a clear pathway to sustainability. That's not typical for NIH funding. It's becoming more common, I think, that taxpayers are looking for return on their investment even for grant-type funding. So we created an idea called HealthyRx, the connection between healthcare and self-care, or what I would say to people who understand e-prescribing, e-prescribing of everything else. What we had when we applied for this funding was a program called MAPS Core. This program grew out of relationships with community members that had been cultivated over several years. And when I say community, I mean anyone who cared about health and vitality on the south side of Chicago and who heard about what we were doing and came to the table. That's the definition of community. And community members said if we're going to work together with you researchers at the University of Chicago on issues related to health or urban health, we have three stipulations. One is figure out a way to engage youth, employ youth, give the youth of our communities a reason to come to the University of Chicago, expose them to science. Another was take a broad definition of health. It's not just the absence of disease or infirmity, it's all the things we do as a society to be healthy and its economic vitality. And third it was we better collect data that we can use and you better be able to deliver it to us fast. One of the core questions we were all struggling with are what are the assets of our community that we can build toward better health? There wasn't a good answer to that. And so we started employing youth to walk up and down every block of every street with smart phones and start cataloging all the resources of the community. And over time we just completed our sixth year with this. This became known as MAPS Core and it became a data engine for this e-prescribing system. So when we applied to CMS, this is what we had. We had a really unique workforce solution. I don't think anyone else applied with a youth-based workforce in the future and we had data about the resources of our community. And the idea was that we would connect up this resource database that the kids are creating to the electronic medical record workflows in clinical practices in our communities. So these are the communities. This is the city of Chicago, O'Hare. For those of you who have been stuck there, it's all the way up here. Over the last six years we've mapped all of this geography. It's about 90 square miles. And we have resources, thousands of community resources identified. Why did we have to do that? Well, the best available data, the data behind Google, the data behind Yellow Pages, the data behind Yelp is not good enough to be able to refer a patient to a place for services. It would be the equivalent of, here's a prescription for an antihypertensive. I'm not sure which one the pharmacist is going to give you. Or here's a prescription for your antihypertensive and I'm only 60% sure the pharmacy is actually there. Okay, so that's not tolerable. In the health, human services sectors had the poorest quality information in these mainstream data sets. And our hypothesis is that the poorer the community, the more invisible are the resources of the community. In connecting up the community resource database to the electronic medical record workflows, we needed to create logic so they could talk to each other and we call these ontologies. The ontologies are driven from fields in the electronic medical record that the doctor or nurse has to populate and we know the age, the gender of the patient, what language the patient speaks hopefully, we know what problems the patient has today, we know the address of the patient. Those are data that power this system. And the software pulls on those data fields that we have to populate anyhow and when we close the chart to move on to the next patient, sends a signal to this database that began with the youth data collection and sends back what we call a healthy RX prescription. I'll show you the prescription in a minute but over the demonstration period, this is the geography where we're deployed. Our first site was the Komet Holman Health Center, a really innovative, federally qualified health center of the Near North Health Services Corporation located at 43rd and Berkeley here on the south side of Chicago. And we've now spread to 21 sites across five healthcare organizations and integrated the technology with three electronic medical record platforms. It's been hard work but honestly, it's not that hard to solve this problem. It's just that we hadn't thought we really hadn't taken ownership of the problem before. This is what a healthy prescription looks like. It prints out with the after-visit summary for those of you who practice medical care. This is a summary that has to be printed in order there's an incentive around printing this summary in order to meet certain kinds of regulatory requirements that Medicare and Medicaid require. So a sample healthy RX prints out and what it does is connect a patient to a specialized type of community health worker. It has the lady Pamela sayings at the top of the page, and then provides in different categories the two closest programs and services to the patient's home address that are designed to address the problems in her problem list. This isn't all the information she could use. She's directed to the community health worker if she wants more information, if she wants to give feedback about her experience, et cetera. There's a public website so that even if a patient isn't coming to a clinical site where we're operational, their own healthy RX are created for somebody they care about. Now, the only resources that are in this database are the ones now in the 16 zip code area where we're conducting our work. And since April of last year, we've generated more than 125,000 of these healthy prescriptions and served close to 60,000 unique individuals. So you saw those data from the very beginning about provider confidence in their ability to meet their patient's unmet needs. These are preliminary data. We're monitoring ourselves, and we're also being evaluated by an external evaluator for our productivity and our performance on this project. These are serial cross-sectional surveys of providers who've been using the healthy RX system in a whole variety of sites over time. So at time three, we had 43 providers who had been using the system for at least 18 months. And what we see, this is just one of many metrics, is a much higher proportion of providers that are saying they feel confident or very confident in their ability to meet their patients unmet social needs. Is that because we're injecting every single patient visit with this information, this colorful information about the resources of the patient's community? We don't know cause and effect, but this is a promising sign. I want to end with a quote. These are pictures of our MAPS Core Mappers at the upper picture, is a picture of them at the Scientific Symposium at the end of the summer, where they collected about the assets of the community and presented back to community stakeholders, family members, and others. And C.K. Prahalad, who was a business professor at the University of Michigan. Unfortunately, I read his book for the first time a few weeks after I understand he passed away because I had intended to drive to Ann Arbor to meet him. He said, if we stop thinking of the poor as victims or as a burden and start recognizing them as resilient and creative entrepreneurs and value conscious consumers, a whole new world of opportunity can open up. Now, I don't entirely agree maybe with the diction of how he presented his idea. I tend not to call people the poor. But I think his point was, again, salient to some of the earlier questions we've had today. And that is, if we work with the people who have been resilient in solving their problems, if we work with the businesses and organizations who have been resilient in sustaining themselves even under very difficult conditions and we combine the talents and skills of scientists and physicians with the talents and skills of community practitioners and residents that we can find creative and I believe sustainable solutions to very basic problems. Thank you very much. I'm Joe Katva from Elkhart, Indiana. And I don't know if I have a question or just an observation. I'm incredibly excited about this. I didn't know about it at all and you just made my entire day worthwhile. I'm a little struck by your cartoon where you suggested that this is a justice issue. And you made the offhand comment that I don't think there's much autonomy here. I think the difference, though, is that you're actually conceiving of autonomy differently than most of the presentations in the earlier part of the day. We have a tendency, I think, to think of autonomy as isolated individuals. So whether I as myself have, you know, all these powers and rights. But when your model actually understands autonomy as a collective endeavor, right? So when you talk about these community assets and using youth to discover what the community assets are, you're talking about empowering autonomy but it's a social endeavor to do that. At least it seems to me. My offhand comment didn't contemplate your perspective and I'm very grateful that you shared it. I'll use it next time. And what I meant was that something, I think, more simple-minded, which is without basic information, patients, neither patients nor their healthcare providers can make an informed decision.   Thank you. I'm really inspired by your work, too. I just had two quick questions. You didn't show any data about utilization or uptake on the patient's side. Yes. And second, what do you do with the one quarter of all patients who are probably functionally illiterate? Thank you. On the first point, we're early in understanding whether patients actually go to the places listed on the HealthyRx. What we were funded to do was to implement a call-in survey. So every one of those HealthyRx prescriptions, all 125,000 has in the upper right-hand corner, call in for $25. You can help us make HealthyRx better. It's more market research than I'm used to but this is what we were funded to do. And we've had hundreds of people call in. In fact, every month more people call than we can afford to pay to participate. And what we see now is that 15%, one-five of the people who call in, report going to at least one place listed on the HealthyRx that was a place they had never been to before. In my mind, that's a fairly optimistic view of things. Even more interesting, I think, is that 50% of the people who call in, it's a convenient sample, but 50% tell us they've told somebody else about the HealthyRx information. And 100%, which you have to be skeptical of in science, 100% say they said something positive. And then people go on to tell us what they told the other person. So we now have funding for a gold standard study, an NIH-funded R01, you know, Prospective Randomized Trial, where we will follow people, very meticulously, intervention in a controlled group to see what they do with this information, how it flows and spreads. That'll take five years. So that's my best answer. With respect to literacy, I'm concerned about that, and it's actually my interest in research started in part with studying health literacy. At this stage, even low literacy people can see a person's face and can read a phone number. And in the very brief training we're giving to providers, and we've trained more than 1,000 providers to do this, we're asking them to focus on Ms. Sayings, the community health worker, and her phone number. And most people with low literacy are able to make that connection. Thank you. Very quick question, quick answer. Okay, yes. Beautiful work. Congratulations. Thank you. Do you think people who are participating in this are looking at themselves rather than having an illness now thinking about health and healthy lifestyle in a much more transformative and positive way? Gosh, I hope so. Do you feel that way from hearing what I've talked about? Yeah, I feel like they're owning health.  They don't have hypertension. They have a strategy of dealing with a condition that actually may go away if they lose weight, exercise, change. Yes. I mean, it's a completely different way of looking at what you have versus how you manage it in a... Yes. You know, the most common ontology that drives the Health Air X is wellness. So if you don't have one of those conditions, you default to a wellness Health Air X. And we have them tailored to different age groups and genders. But I think one of the... This is anecdotal. But one of the anecdotal experiences is both for patients and providers is the revelation of how much is actually here. And this other idea that we can bring the... raise the economy up by bringing local businesses and organizations into the fold of health and wellness. Thank you. Thank you.