Hi, good morning, everyone. Before we get started, I just wanted to get up here and say a huge congratulations to our four chiefs who did fellowship match yesterday and to a person just matched at exceptionally amazing locations. And so we are so proud of them. Congratulations to Tyler, Tony, Brandon, and Lydia. We could not be prouder of you guys. All right. Thanks, everyone, for coming to today's grand rounds. We have three residents presenting, I believe, two different cases in one potentially research project. The first case is presented by Sam Wilkinson, one of our very own PGY-2s. The title of his presentation is Unwrapping a Frosted ICL Mystery. I apologize for my voice. I am not ill. I am going to try to be the next Marlboro man, I guess. I just can't talk. So an interesting thing about Sam, for those of you who are Harry Potter fans, I chose that Sam's Patronus would be a Capybaras because it is actually the nicest animal in all of the animal kingdom. So give it up for Sam. Good morning. Thank you, Tyler. That is far too kind of an introduction. All right. And before I go any further, I just want to say huge congratulations to all of our chiefs. I really look up at each one of you and thank you so much for all your mentorship and looking forward to at least six and a half more months with all of you. So thank you for inspiring us, PGY-2s. And for those who don't know me, I am Sam, like Tyler said, PGY-2. And I'm going to be presenting a case report that I actually wrote with Dr. Ethridge and with Dr. Volershell. So let's dive into it. I have no financial disclosures, unfortunately. So our patient is a 45-year-old male who was referred to UVitis Clinic for two months of vision changes and pain. The patient underwent bilateral posterior chamber faking implantable calmer lenses three months prior to presentation. There are no complications during either surgery. He presented to his one month post-op visit complaining of floaters, blurry vision, redness, and achyocular pain in his right eye. His left eye was asymptomatic. Prior to presenting to UVitis Clinic, he was noted to have prolonged duration of post-operative cell in the anterior chamber for which he was treated with extended topical steroid. He was on pred forte four times a day and actually started on calm again twice a day due to a history of steroid response. His doctorate history was only remarkable for ICL surgeries and steroid responses previously mentioned. Medical history included depression and osteoporosis. Positive review of systems included mild headache and urinary hesitancy. His social exposures are listed here, which include minor travel exposure and living with three cats. On exam, his visual acuity was 20-30 in his right eye without improvement with pinhole, and he's 20-20 in his left eye. Intractor pressures were 16 and 17. His pupil exam was unattainable due to prior dilation, otherwise his ocular vitals were normal. On SlitLamp exam, he had fine KP and 1-plus AC cell in the right eye, and the left eye was unremarkable. His ICL was in good position, but he had white round deposits that could be seen between the ICL and the native ones. Here we have a photo, a SlitLamp photo, showing the white opacities between the ICL and the anterior chamber in this area here. Here's a second photograph that I don't think shows up very well on the projector. This third photograph also shows those white deposits. On this exam, I showed trace vitria cell. It was a little bit limited by a hazy view, but was otherwise normal. A macocet and B-scan were obtained. Here's a segment of the macocet, which was notable for mild vitria cell, but no cystoid macular edema. Similarly, the B-scan showed mild vitrious debris. Leading the differential diagnosis at this point is postoperative endophthalmitus. Other considerations included postoperative inflammation, or sorry, other considerations of postoperative inflammation included postoperative iritis, and less likely, if possible, delayed onset toxic anterior segment syndrome. The presentation was most concerning for chronic postoperative endophthalmitus, so a tappan inject was performed. The vitria sample was sent for aerobic and anaerobic culture. Empiric treatment was initiated with intravitual vancomycin and seftazidine. He's continued on predforte and vermontidine, and cyclopenylate was added to his drop regimen. He presented the clinic one week later. His exam showed decreased ac cell, but the white deposits on the lens actually increased. The gram scene was negative, however, culture results were still pending. Patient was switched from predforte to durazole, and he was added Valtrex to his medications. At the two-week follow-up, his exam was stable. Fortunately, his culture has added a little bit more clarity to the patient's diagnosis. His culture, his anaerobic culture grew C-acnees, which is formerly called P-acnees. He's given intravitual and intracameral vancomycin, as well as intravitual dexamethasone. The patient was counseled that he would likely need removal of the ICL for definitive treatment, however, he was hesitant to undergo any further surgery at this time. Over the following months, the patient continued to have trace ac cell and worsening white deposits despite repeat intravitual and intracameral injections of vancomycin. The team continued to discuss need for removal of the ICL. Here's a slit lamp photo showing the progression of the white deposits along the ICL surface and the anterior capsule. You can see the deposits take up nearly 50% of the ICL surface. Here's a photo from the initial presentation or comparison, so you can see significant progression between the two photos despite the intravitual vancomycin. Throughout this treatment of the intravitual and intracameral injections of vancomycin, his vision remained 20-30 or sometimes even better. Unfortunately, because of the persistent white deposits and recurring AC inflammation, he had to undergo an ICL extraction and AC washout. The extraction took place about six months after the original ICL surgery. Unfortunately, even after that intervention, the tail continued. Despite removal of the ICL, his inflammation recurred and persistence of the plaque now on the anterior capsule or surface of the native ones as seen in this photograph here. The ICL has been removed, but those deposits are on the anterior capsule. Here's retroalumination showing those deposits as well.  An IOL was not placed at this time. He also had a paris point of atrectomy and intravitual injection of vancomycin. Eventually, he did receive a secondary sulcus IOL, which led to uncorrected visual acuity of 2020. Our patient's case has come to a happy ending, but I want to take a few minutes to discuss endophthalminus and ICL. This is the long journey that he went through after his ICL surgery. Exogenous endophthalmitus is much more common than its counterpart endogenous endophthalmitus. Typically, we see postoperative endophthalmitus cases presenting one to two weeks post-op. Presentation more than six weeks post-op or classified as chronic. Chronic postoperative endophthalmitus typically presents with better visual acuity and is less likely to cause hypopion than its acute counterpart. The incidence of postoperative endophthalmitus is between 0.01% and 0.367% of all postoperative patients. Acute endophthalmitus occurs about two to five times as often as chronic endophthalmitus. Acute endophthalmitus is also typically caused by coag negative staff in about 70% of cases. As you know, coag negative staff is normal skin flora. Most cases of chronic endophthalmitus are caused by C. acnes as seen in this patient. Some studies estimate 41 to 63% of chronic endophthalmitus is caused by C. acnes. Fungal species are the second most common causal pathogen. C. acnes is a gram-positive bacillus that is normal flora for the contactiva and skin and is responsible for active vulgaris. C. acnes is resistant to monocyte and neutrophil mechanisms for killing, which help the organism survive. It can also form biofilm on intraocular implants, which is one of the reasons why exploitation of intraocular implants is so important. C. acnes has been reported in numerous chronic endophthalmitus cases, but this case report is actually only the second report of C. acnes causing an ICL infection. Of note, the first case also required ICL removal and lensectomy. C. acnes and endophthalmitus is often misdiagnosed as iritis. The delayed course and relatively subtle presentation can fool providers. Patients presented with decreased vision and eye pain. They typically have ac cell and white plaques on ocular implants. Diagnosis is made by vitreous culture or PCR. The first case reported actually had a positive PCR in a negative culture, so kind of the opposite of what we had in our case. C. acnes and endophthalmitus is typically treated with vancomycin. Because of the propensity for C. acnes to form a biofilm, the removal of implants is often necessary. And in many reports of post-cataract cases with C. acnes, a capselectomy has been required as well. And we'll just briefly talk about ICLs. Implanable calmer lenses or ICLs can be used to treat refractive error. In the US, they're currently only used for myopia. They're often for patients who cannot tolerate contact lenses or glasses or who have a profession requiring uncorrected 2020 vision. And ICL is often considered alongside other refractive interventions such as LASIK or PRK. The ICL is placed anterior to the capsule without disturbing the crystalline lens. None complications include infection, inflammation, glaucoma, anophilia cell loss, iris abnormalities, and retinal detachment. The rates of ICL and ophthalmitus are fairly comparable to cataract surgery. And I'd like to thank Dr. Law Rochelle, Dr. Monson, and Dr. Etheridge for their work on this case report. We published it in the American Journal of Ophthalmology case reports. Here are my references, and we'll open up for questions. So, the comment is that refractive surgery is not safe, so informed consent is really important. And then another just comment about ICLs, it would be perhaps presenting earlier, but it's important to look at the vault of the ICL. Pigment dispersion rubbing on the back of the iris can create kind of an UGG syndrome. The other thing you would need to think about in a case like this, if it were earlier, is possible bleeding from PI, creating fibrin. So, great workup. It's amazing how strong sea acne is and how resistant it is to killing, but cool case. Congrats. Thank you. Hey, is it too late to make a comment on this case, Sam? Please do. Okay, this is interesting, because as you said, this is only the second case of the PI acne or sea acne's ever reported in an ICL. The thing that makes us so difficult is when you get a nitus of this bacteria growing in there, you just can't get rid of it. And it just requires removal of pretty much everything. And when we get the pseudo-fakic ones where you've got the PI acne's in the capsular bag, we've done capsulatomies, we've done removal of the IOLs, we've done vitrectomies, and these are just incredibly difficult to get rid of. You really have to remove the entire capsular bag and pretty much everything in there, because these will just keep coming back. What's fascinating about this is it wasn't inside the capsular bag, it was actually on top of the capsular bag. So this is, as your case report said, exceedingly unusual. But again, you have to remove everything once this bug takes hold in order to get rid of it. And that's exactly what you guys did here. Unfortunately, the patient did well and you were eventually able to put in an IOL. But this is fascinating. I've never seen one of these before. Thank you, Dr. Manlis. All right. And thank you, Sam. Our next case is presented by Aisha Patil. Excuse me. Entitled A Curious Case of Vision Loss in a 10-Year-Old Girl. Her Patronus is a Kwaku. It's believed to be the happiest animal and the most selfieable, if that is a word. All right. I'm excited today to present a case that I saw as a very early PGY2 on the consult service back in July. This is a case of a 10-year-old girl presenting with vision loss. So initially, she actually presented with a drooping of her right upper eyelid. She presented outpatient to her hematology office, said she was having intermittent drooping of the right upper eyelid for three days, kind of waxing and waning. And she has a complex past medical history, so they were worried about a stroke and sent her to the ER. I saw her while she was inpatient. Very cute girl was having kind of this droopy right eyelid that she was noticing was getting worse when she was tired. And then on further probing, she said, oh, I also see double vision if I look up. And I also see it when I'm looking to the side. And if I'm just looking with my right eye only, the image is kind of smeared off to the side. So a lot of different symptoms, I really only came with asking her a lot more questions. Otherwise, she had had a recent fall. She was playing with her sibling, so hit her head, no loss of consciousness. She wasn't having any eye pain or significant headache as well. And then in looking at her history, she actually was recently diagnosed with lupus, just a week prior, immune thrombocytic perpura, a few months prior, and then chogren syndrome. So she had a pretty strong autoimmune history. She hadn't started any treatments yet for these conditions, but she was taking several naturopathic supplements. The list was at least 10 different medications. Most of them I hadn't heard of yet. For her ocular history, she had a high astigmatic refractive error. And then she also had this history of a left droopy eyelid, which was her currently unaffected eye. This was confirmed in photos seen when she was like two or three years old, but they never really made much of it. And her left droopy eye wasn't actually present when I saw her in the hospital. Otherwise, she has a strong family history of autoimmune conditions. She has Crohn's disease in her family as well as lupus, von Willebrands, psoriasis, RA, and Hashimoto's. So just about everything there. And for her social history, she had several siblings. I think it was five or six and was currently homeschooled. On her initial presentation, she was 2100 in that right eye, her affected eye. 2020 in her left, her pressures were normal. She didn't have an APD. Interestingly, on Ishihara plate, she was nine and a half out of 13. She had a lot of difficulty seeing the second letter in that right eye. And she also had some red ball desaturation in her super temporal field, as well as infrared temporal field. Visual field was also a deficit to counting fingers temporarily. Otherwise, her EOMs were full and she was ortho and primary gaze and something that was kind of picked up in her later exams, but she did have abnormal saccades when she was looking to the right. More of her first exam, she had normal optic nerves, a normal dilated fundus exam, a very normal cranial nerve exam, and neurology was also consulting for her. They did a very thorough full neurologic exam that was also normal. So naturally, with her visual field concerns, we ended up doing a bedside visual field. And to my surprise, this was her virtual visual field. It's not the most reliable. She did have quite a bit of false negatives, especially in the left eye, more than the right. But grossly, you can appreciate she does have a right homonymous hemianopia here. So with that, and that virtual visual field and field loss presentation, kind of the working differential for her was something left-sided retrochiasmal potentially, like a structural CNS lesion, ischemia, something thrombotic related to her lupus, potentially complex migraine, functional seizure presentation, a few other more rare things as well that we had considered for her given her history. So she ended up getting a pretty extensive inpatient workup. We did an ultrasound with Dr. Harry, also quite interesting, relatively normal, but she had this little echo density there that you can see with the arrow, which you can see with the CRA, but you can also see as just a calcium deposit, kind of a benign calcium deposit. So she ended up getting more workup. She had MRI brain orbits with, without, that was normal, reviewed with radiology to make sure there wasn't anything there. She had a lumbar puncture done with the neurology service that interestingly showed seven oligoclonal bands. So it was slightly positive there. And she had a normal MG panel and thyroid panel. So after her inpatient visit, there wasn't anything acute that we could really see on her workup. And so she was discharged after starting steroids for her lupus treatment. And we saw her in neurology clinic, or neuro ophthalmology clinic six weeks later, and did a more formal visual field. And again, she has this persistent right-sided haemonomous hemianopia. This also is maybe not the most reliable test. She has high false negatives in the left eye with 36% and 18% in the right, but it's still very much noticeable on this test. She also had a MAC OCT that was done and an OCT RNFL. Both of those were normal. We also did an OCT MAC GCL. This is a normal result. This was kind of interesting while I was learning about this. This is something that if you don't see like an obvious optic neuropathy and looking at the nerve, you can get an OCT MAC GCL scan that could show you some more objective differences and potentially reflect visual field changes. So hers was overall very normal though. She also had a multifocal visual evoked potential test done. And this was also normal in both eyes. This looks at electric potential evoked by visual stimulus. So this is a great test if you're not sure what's going on, but someone is having subjective vision loss. And then more of her outpatient visits with neuro-ophthalmology, we were able to do a Goldman visual field for her. Here you can see she still has a constricted visual field, not seeing the right side as well, does better with a size five stimulus, the black outline and pretty constricted with the red. She also had a normal EEG with neurology during this time, making seizures a little bit less likely for her presentation. So this is kind of the timeline that she underwent as far as this extensive workup. She was seen several times in neuro-optimology following this visual field loss, as well as getting other scans to see what could be correlating with that. And really most notably, I think we kind of went through her tests that were all normal. And by September 28th, this is when she kind of came into clinic. She's still having some cognitive issues that her mom is noticing. She's saying, oh, I can't really write with my right hand anymore. I'm going to switch to my left, kind of random things that are popping up.  So that had kind of come up at that visit. And it wasn't until recently on the November 21st visit when she was seen outpatient did that visual field loss finally resolve on confrontation. It was entirely full. Her mood was getting better. She was seeing occupational therapy. So these were some of the changes. So you can probably tell what the diagnosis is going to be at this point. But she ended up presenting with what seems to be functional vision loss after a very extensive and normal workup. So this is kind of an unusual presentation for vision loss in the kid because it's kind of difficult to, I guess, fake a right homonymous hemianopia. And she otherwise had a very interesting medical history with several autoimmune conditions. So I think it was warranted the extensive workup that she had for this and overall reassuring that it has since resolved. So I'm going to spend a little bit of time talking about what functional vision loss looks like in children and different tools that we can use to decipher whether it's functional or true organic vision loss. So functional vision loss is now the preferred term over nonorganic and nonphysiologic. It was earliest reported back in 1938 as hysteric blindness. And when that would come up, people would get admitted to psychiatric institutions for this. So it's interesting to see how far it's come at this point. And it does require an exclusion of structural pathology that could explain those symptoms. So typically you'll have ophthalmic testing that will demonstrate inconsistency in the symptoms that they're discussing or that this patient is presenting with. Otherwise, I came across an interesting study that looked at children with functional vision loss from 2007 to 2014 in the UK. They had a total of 85 patients and they were trying to characterize what they're most presenting with. And it seems like children typically present with bilateral visual acuity, decrease in visual acuity bilaterally, or a visual field defect unilaterally, or sometimes a headache. They also saw sudden scotoma, dyschromatopsia, amaurosis, or double vision. I'm just going to go through three quick cases to show different tools that we can also use to decipher functional vision loss. So the first case is a 14-year-old girl, history of a headache, myopia, and her migraine headaches were accompanied by transient visual loss. She lives in Idaho. Initially, she was presenting with left eye vision loss. And a month later in a separate headache, she started having right eye vision loss, which she describes as complete darkness. So she ended up getting admitted to the hospital there where she had several MRI brains with and without done that were normal, an MRI head and neck that was normal as well, and an LP and a neuro ophthalmology person who saw her there who just wasn't sure what was going on. She then presented to us and she was subjectively NLP, couldn't see anything. But what they were able to do in clinic was test her with an OKN drum bilaterally. So there's a picture of an OKN drum. For a classic drum, if you're 2400 or better, you should be able to see it. So you should see a smooth pursuit followed by a reflexive saccades. And so they were able to say, you know, you at least have 2400 vision with that test. You can also use the eye handbook, which I think you have to be like 2200 or better to see that one. So that's a great test. I wish I knew about that when I was a young PGY2 on consult, but I hadn't learned about it yet. But otherwise, she had a normal workup. She is now doing cognitive behavioral therapy and low vision therapy services. Unfortunately, not all patients recover their vision with these services and she is still not able to see. So she's enrolling in a school for the deaf and blind. A second case here, this is another one that I saw while on call. This is a nine year old girl presenting with the history of headaches and decreased vision in her right eye after hitting her right cheekbone on the monkey bars while at school a year ago. So when I saw her in the ER, she was initially counting fingers.  We weren't quite sure what was going on, but she was then seen really closely in neurophthalmology and they were able to do what's called a fogging test. And this can be done with a foreopter or trial glasses.  And at the same time, while distracting them, you can put a high power lens in front of their good eye, fogging it. And then you're essentially checking their affected or bad eye and having them read the chart. So she ended up actually being 20, 25 for treatment. A big component of her presentation was actually bullying at school and difficult family dynamics. So she was referred to child psychology. And for the last case, this is an interesting one as well. This was an 18 year old male unexplained vision loss in his right eye after being poked twice in the eye while wrestling in high school. It sounds like he was diagnosed as a traumatic optic neuropathy for at least a couple of years before he was referred to our neurophthalmology department. He ended up having normal MRI brain, normal looking optic nerves and a retinal exam. He presented to us initially with hand motion vision in that right eye that then corrected to 20, 20 vision or 20, 30 vision, sorry, after checking with vertical prism test. So this is one example of that kind of test, but essentially you can have a base down prism in front of the unaffected eye while you're occluding the bad eye. And that should induce a vertical double vision for that one eye. And then you remove the occluder and drop the prism. And if they do have binocular vision, they will have binocular depopia is my understanding. So that's one of the tests that they did in clinic. They also check stereocuty, which is a great test to check because you do need good binocular vision for that. And treatment, he was overall reassured that this was not an organic cause of vision loss for him and just in was doing cognitive behavioral therapy as well. So a few other tests for a Goldman visual field or sorry for a Humphrey visual field, maybe a less reliable test, you may see like a clover leaf pattern, but this can also be seen in functional vision loss. You could also do a Goldman visual field at several different distances from the patients. And for physiologic vision, you should have an increase in that visual field degree. So it should go out as more of a tunnel and you should have a larger area that they're able to see. But for maybe some functional patients, it'll continue to be more of a constricted tunnel vision. This could be another test you could try is the proprioception test with both eyes open, checking to see if they can touch their fingers, which you should be able to do regardless of your vision loss. And then including their good eye, and they may demonstrate that they're no longer able to do it. And that's a good way to see that this is actually functional as well. I came across the mirror test as well, which I haven't seen this used in clinic, but if someone is having profound vision loss, LP versus NLP, you could theoretically take a mirror while you're distracting them kind of close by and walking across and seeing if they're following the mirror, because people have a hard time kind of not looking at themselves. And as you bring it in, you may also see some meiosis or accommodation in those eyes as well, suggesting that they're actually seeing that mirror. A manifestation of that. And we need to get to the bottom of those things. We have an obligation to the patient. Most of the time, though, simply telling the child who wants glasses that when we're done, I sit them down and look at them and say, you know, I can't explain why you're not seeing whether that I today, but I think it's going to get better. And if it doesn't, we're going to come back and we're going to do bright light therapy again with the indirect and see if we can help it get better. And I also think it's very important to talk to the parents and say, you want a healthy, happy child.  Don't take them out in the hall and smack them, because you've spent time taking time off work and that that's counterproductive. You have to give them room to get better. But most of the time, they don't need counseling. I could count on one hand, the kids I've sent to counseling in the last 40 years. And almost always with helpful suggestions and parents acting as a team with us, we're able to get to the bottom of things. This is a great presentation. It's an important topic and thanks for bringing it up. Thank you. I said, this was a really interesting case for us on the consult service because this patient had a homonymous hemianopsia. And of course we saw the field and we were like, this kid needs an MRI. So I'm curious if you know if there's ever been any case reports of a homonymous defect in cases of functional vision loss. And I'm curious to see what the neuro-ophthalmology department has to say about the classic visual field defects that we see. Every kind of visual field defect that there can be, it can be non-functional, non-physiologic. And I'll bring to your attention a poster that was at Nanos, actually when it was up at Snowbird, where they had a bunch of students, medical students and pre-med students in quotes fake a visual field and you can do any visual field defect you want. Anything you care to do, you can do. It is a subjective test. So number one, number two, absolutely non-physiologic visual field defects of all types have been described. I'm just going to point out a few things. You can't really test a golden visual field at different distances. You can do that at the tangent screen. The other thing you can do at the tangent screen is have the patient maintain fixation on central fixation, but have them turn their head. And you can do that at the chair side, but you can also do it at the tangent screen. And if the visual field expands where it shouldn't, that can be useful. A hemianopic visual field defect, of course, will not expand along the vertical if it's real. So when you're looking at a constricted visual field, it should expand as you move away from the tangent screen. But a physiologic hemianopia, the vertical meridian will stay right on that vertical. So it doesn't give you information one way or another. Megan and I were conferring, I have never done the base down prism test by occluding the other eye. I think that gives way too much information to the patient that you don't want to share. And so I just hold the prism down over the good eye and just ask them if they see double and if they can read both the lines, because that can give you the acuity. I would never draw attention to the affected eye by covering it and then uncovering it. And that's just giving them too much information. And the final thing, and I think that Dr. Katz really emphasizes this, you have to have a couple of tests in your wheelhouse that you can do quickly and easily. You can't be like reading the instruction book as you're doing it, because otherwise it gives the patient way too much time to kind of ponder the possibilities. I think that functional neurologic disorders writ large have actually really come in a huge move forward in the last couple of years. And non physiologic visual loss definitely fits in that category for children and adults. And our understanding of it is becoming much more sophisticated than in quotes, they're faking it because a lot of times this is a non not a conscious event, especially with children. And I agree with Bob about warning parents about bad stuff that could be happening in this child's life that they may not be aware of. And children are sometimes very overt about the stressors in their lives. And sometimes they are terrified of revealing anything. So it's your job to tell the parents, then it's the parent's job to be the parent and to try and figure out what might be going on. I have had a patient who was in their 30s or 40s. And I always ask, have you ever had anything wrong with your eyes, eye injuries, surgery, cross eye glasses? And like, no, but when I was, you know, 12 years old, I wanted glasses because my friend had them and I fake vision loss and he got me glasses, but I didn't really need them. So, you know, insight can happen with age as well. There were a few comments in the chat too. And I think Dr. Bigunta took care of a lot of them. But there was a question about the ptosis as well. And how can we explain alternating ptosis? This was a little bit less clear. Her ptosis since also resolved, she did have a negative myocenia panel as well. So not totally clear there. Okay. Thanks, everyone. All right. Great presentation. So the next presenter, McKenna Coletti. She's actually a neurology resident here at the University of Utah on her neuro ophthalmology. We're recently completed her neuro ophthalmology component of her training. Her presentation is entitled outcomes of children diagnosed prenatally with midline brain defects. And for her Patronus, I chose an axolotl because it's the most mysterious animal in all of the animal kingdom. And I don't know her very well. So maybe we can get to know each other better. And we'll be so mysterious. You might have to help me get out of here though. Sorry. Okay. Hi. Like he mentioned, my name's McKenna Coletti. I'm one of the PGY4 pediatric neurology residents. And I wanted to present some research that I've been doing that's been really multidisciplinary. So I'm excited to share it. I have no disclosures. Here's an outline of what we're going to talk about. So we'll talk a little bit about midline brain defects in general. And then we're going to talk about one of the cases and then kind of what we've discovered with the cohort that we found. So we have a 20 year old G2P0010 currently estimated to be at 31. Oh no, I think I went too far. Oh no, we're good. Currently at 31 weeks referred to the Utah fetal clinic for an ultrasound at 28 weeks with an absence septum polysidum. Prenatal history is pretty unremarkable. This is her first pregnancy. She has asthma, frequent UTIs. She has albuterol and fovent for her medications, and she has an maternal nephew with autism. So in cases like this, we get a prenatal MRI. So this was her MRI or rather the baby's brain MRI at 31 weeks. Again, we noticed absence septum polysidum. The corpus callosum for this child is a little bit hard to see, but thin, but present, and ocular globes, nares, dental buds, palate are all normal. So other midline features looking pretty good. So the question I mean remains is when we have this information, what can we tell families? Like what are the doctors in the fetal clinic telling families about what they can expect for their children long term? Will they need more subspecialties? Do they need to deliver at a hospital that has pediatric subspecialty care? And then what are the risks associated with these defects?  But the reason why we see these midline brain defects is really a difficulty with the proencephalic maturation after primary neuralation. So this is a seroive inductive elements that result in the formation of the brain and spinal cord. And so these structures really start to form at 10 weeks to 17 weeks. And so the 20 week anatomy scan is a great time to pick up these defects. And really there, the structures are fundamental to the development of the corpus callosum, the chathum septum polysidum, and the optic nerve chiasm. So midline brain defects, they're a common type of prenatally diagnosed central nervous system malformation. And in fact, like for our fetal clinic, they're a third of all our referrals. So when we think about midline brain defects, we really think of them on a spectrum from absence septum polysidum. So just missing that thin membrane to whole prosencephaly. And we see it really like the whole prosencephaly spectrum is more common seen in about two out of 16,000 live births, where kind of the most more mild end is less common about like two to three and 100,000. So back to our case. So during the prenatal consultation, we do have some things to say and have had for a while. So the thing that we worry the most about is septic optic dysplasia. And so this is a condition that's made when children have two of the three triad of optic nerve hypoplasia, pituitary hormone abnormalities and a midline brain defect. These midline brain defects are also associated with the developmental delay epilepsy, and they're increasingly recognized on a prenatal ultrasound. And so we're getting more and more referrals for these folks yearly. But we don't have a lot of data. We have really small patient cohorts, not a lot of places in the country are doing prenatal MRIs. And we have had like struggles in the past with when these kids are born somewhere else, whether they get like quick subspecialty care, often they're presenting really late. Risk factors for these kids genetically for septic optic dysplasia are the HES-X1, SOX2, SOX3, OTC, X2, and then other risk factors like medications, as you would expect. However, most of the time we don't have a good reason why the midline defects are happening. So kind of seeing this need for a more streamlined approach to assessing midline brain defects and subspecialty care. Primary children's now has, we have a algorithm that we've been using for the last couple of years. This was put in place by some of the pediatric endocrinologists as well as some of our ophthalmology colleagues. And then Betsy Ostrander kind of on the pediatric neurology side spearheaded this. And so this is kind of lets us have, we've distributed this across the valley as well. So when a child is born, kind of what are the first steps that should be happening? So very soon after we should get a neurology consult, we need endocrine labs on day three of life as to not be seeing some of mother's effects in the endocrinology labs. And then from an ophthalmology perspective, it's kind of been a mystery to us when we should check these kids for optic nerve hypoplasia. But we started this protocol with saying that like in the first six weeks should be good, should be an appropriate time. So the study objectives are to evaluate how affected our standard approaches at identifying complications of fila-identified MBDs to support multidisciplinary care. We want to identify the clinical outcomes of these patients and then evaluate the timing of diagnosis because that still is kind of a big black box about when we should be checking these labs and when they should be seeing. So we did a retrospective review of neonates that were referred to the Tafino Center with midline brain defects. We have a cohort of 100 patients, which is much larger than any of the cohorts that I've been able to find. And then we use the standardized pathway. Kind of just quickly, this is the like some of the results that we have in terms of demographics, but this is kind of the meat and potatoes of what we found. So when you're kind of looking at this graph, each of the different colors is representing kids on the spectrum of midline brain defects. The purple is isolated CSP or absent septum polysidum. And then dark blue would be complicated absent septum polysidum. So if you have an absent septum polysidum, but you have a cyst or some other malformation that's not associated with the midline. And then the dark green is absent or dysmorphic corpus callosum, which in definition is also an absent septum polysidum. So moving along that pathway, and then we kind of we grouped all of the mild ventricular meglis in this non-complicated group, because it's quite common. And it's also often just called ventricular meglis, because we don't have the septum polysidum. It's difficult to measure the ventricle size. And then dark gray, we're seeing complicated or dysmorphic corpus callosums. And then the light green is holoprosencephaly spectrum. So I mean, broadly looking at this, you can see bad things happen to the light green, which is the holoprosencephaly spectrum at a rate much higher than anything else. And so that's, I mean, I think that is pretty intuitive. But patients with absent corpus callosum with mild ventricular meglis or patients with absent isolated septum polysidum really had the fewest complications and were by far the most likely to be alive without complications at six months. And complications include any of these diagnoses that we're talking about. And, but we, I think what's really important to notice is just the risks of endocrine, ophthalmologic, and neurologic complications are present across this entire spectrum. So I think there's some tendency to want to say, if you just have an absent septum polysidum, maybe they don't need to do all these things. But we certainly are seeing that these kids are at high risk of, yeah, developmental delays. Thank you. Yeah, Dee Dee's still onto delays. Thank you. Okay, so I think this is kind of showing us that we really want to know about these complications really early in the children's lives, so that we can help support them with therapies and treatment. So I zoomed in for you guys. This is the ad problems that these kids are having. But I really just wanted to point out that I think, especially with ophthalmologic abnormalities, it's pretty actually consistent across what we're finding across these brain midline brain defects of their, the problems associated and getting close to 50%. Obviously, it looks like the absent disc or corpus callosums with mild ventricular migraine have a lower rate of ophthalmic diagnosis is. And so 22 of our 100 children in this cohort had at least one diagnosis related to a midline defect. So the three really that we had though were bilateral nerve hypoplasia, unilateral nerve hypoplasia or an ophthalmia. I think another really interesting thing is we've kind of looked at the timing that these diagnoses were made. So excuse this tiny plot, but for the first two, for the isolated septum plucidum and complicated absence septum plucidum, these kids were diagnosed with these complications in the hospital. So we had someone came over saw optic nerve hypoplasia. And it's usually diagnosable very early, but the other types of visual impairment are going to become evident later. And so we had several cases in this cohort of optic nerve hypoplasia diagnose and then maybe retracted later in the clinical course, probably likely related to the size of neonates optic nerves. And then there's this question of MRI versus clinical diagnosis of optic nerve hypoplasia. So the MRI concern for optic nerve hypoplasia is not diagnostic, but is at very high risk of subsequent ophthalo diagnosis is and should be followed closely. So if we go back to our case, this is our the brain MRI of our infant that was born. This is him on day of life 20. So again, he's noted to have an absence septum plucidum. He's got flat roof rentacles and inferiorly pointed frontal horns. This is something that we kind of usually just incidentally when you don't have a septum plucidum. He has a partially empty cell in pituitary infendibulum. And then he has hypoplastic optic nerves. So the question here is at this point, does he meet criteria for septic optic dysplasia? And I'm going to answer it for you that we still don't know the answer because this is a clinical diagnosis. So we know that he has midline brain defects, so he's meeting one. And we cannot say about his ophthalmologic status without an exam. And then we don't know we although we know he has that risk for having endocrine abnormalities giving his empty cell. We don't know that yet. So we get a lot of reads on the MRIs, like especially prenatally that people have septic optic dysplasia, but it's not something that you can diagnose radiographically unless they have no optic nerves, I suppose. So he was evaluated by ophthalmology at three months of life, found to have hypoplastic nerves. He doesn't fix but function unsure. He's a young guy at eight months of life. He was seen again. He has hypoplastic nerves. He fixes briefly and then he responds to some lights and bright colors. And then at 15 months he has hypoplastic nerves. He's reacting to light and he's briefly fixating. And then at three years of age, he had exam under sedation, were we able to get this image? And so he had bilateral, moderate optic nerve, hypoplasia.  He wasn't a very cooperative guy earlier than that. So for our case, he was evaluated by endocrinology at day three of life. He had a little bit of a low random cortisol, but everything else looked okay. But then at one month of life, he represented to the hospital with failure to thrive and hyponatremia and was diagnosed with diabetes and sepidus. And in terms of his neurologic outcomes, he was diagnosed with infotaspasms at four months. He needs globally developmentally delayed, but he needs to make progress. So if we're looking, thinking about our patient on this graph, he kind of fits really, his prenatal diagnosis was isolated septum plucidum. So he's a purple fellow, but he did have a diagnosis of developmental delay, septic doctor's dysplasia at this point, optic nerve hypoplasia and endocrine abnormalities. So given that 60% of patients with similar prenatal imaging did not have this diagnosis, he's probably more on the more severe end for kids that were had isolated absent septum plucidum in utero. So some of the things to think about is optic nerve hypoplasia is diagnosed early, but other types of visual impairment may become later. And then kind of the things that we've talked about a little bit before MRI is helpful to know if we should continue to screen, but it's not a clinical diagnosis. Some of the future, I'm sorry, the conclusions, all again, things that we kind of mentioned before, but the farther you are on the midline brain defect spectrum, the worst things are going to be for you. And then multidisciplinary coordination of care is essential for these patients to have the most robust outcomes. Things we'd like to look for tons of them, but some specifically up to the ophthalmology related things is what's the visual function in these patients. So we have like diagnosis of optic nerve hypoplasia, but our cohort is quite young, most being under a year of age. So it's kind of, it's difficult to know at this time. And then what's the rate of optic nerve hypoplasia seen pre and postnatal MRIs and the coordination with the clinical exam? And then should we be considering other ophthalmologic diagnosis in our study? And what if we just took babies that had optic nerve hypoplasia on their prenatal MRI and then worked backwards to see like, did they have other endocrinoptermalities? We haven't done that. Special thanks to these folks that helped to write this paper. We're working to write this paper. And then Dr. Vodugunta for her feedback. Regarding timing of diagnosis of general hypoplasia, I find that it can be variable and I feel confident in saying patients sometimes I come in premature babies and sometimes they're not kind of, they're six months or older. Yeah. So it's, she says she feels sometimes she feels confident and less confident in premature babies and sometimes not confident until they're about six months of age of older. And curious to see what other peds ophthalmologists think about that as well. And I think that that was, I saw this in some of the cases and I really appreciated folks just following and tell me we kind of have a good idea of how that's going to shake out. But does anyone else have a comment on that? One place you could look at those optic nerves in neonates, we take photos on all the kids we do retinopathy prematurity screening exams. It is very common for neonates, healthy normal neonates with healthy nerves, no bad IVH or none at all to have what looks like a double ring sign. And that is what gives many folks the sense that a lot of these neonates have optic nerve hypoplasia where they clearly do not. In terms of the vision outcomes with optic nerve hypoplasia, having followed many, many kids and most of them here primary that vary in terms of the holopros and cephalos spectrum, I've been really unimpressed with my ability to prognosticate when we're seeing an infant with optic nerve hypoplasia. I mean, the kids who have basically a little nubbin of a nerve, you know, they're going to be visually impaired, but I give parents hope that things may be better than we think. And we start all of them on, you know, intensive vision support services early on, because early intervention has been shown to make the biggest difference in vision impairments when you do it early. So those kids need to be in there, but I'm not sure we have the answer yet to prognosticate in terms of vision outcome. This is a good thing you're doing. Thanks for the presentation. Looking at it from the complete other end of things, when we see adults who are incidentally found to have big visual field defects, they had no knowledge of. Obviously, many of them have, you know, gone completely through their childhood and young adulthood and they're bearing children. And the question for us is always, well, how much do you have to do as far as trying to look at their pituitary access? Because they seem to be okay. They survived their childhood. And so we have some suggested testing that we also do for them. It's obviously, as you go along, the chances that they've survived their childhood is a good indicator that they're okay. If you're interested in looking at more from that side, there's a huge cohort of kids with optic nerve hypoplasia being followed by Mark Bortchart in LA. Enormous numbers of children and most of them don't have any of the other issues. So it's kind of looking at the same ophthalmic outcome from totally different sides. And many of them are very visually impaired, but their pituitary is fine. The rest of their brain is okay, nothing major. He's got an enormous cohort there that they're studying, trying to figure out what it is that's causing it, not just diabetic mothers. Yeah. And Dr. Hoffman, you follow a lot of the patients that I've been looking at, so I appreciate you for that. Yeah. Thank you. Thank you. Appreciate it.


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So let us start. Today, I would like to show you this example of taking off concerning the non-uniqueness of solutions to ut minus the plus of u equal to 0. So we are now in n equal 1 space dimension. So we work here in r. And then u of 0 equals some function equal 0 at time 0. And you also, so it is clear that the 0 solution is a solution of this problem, because at time 0. So u equal 0 is a solution, obviously. So the point is to construct another solution. The point of the example is to construct another non-zero solution to 1 so that, at the end, there cannot be uniqueness of solutions to this problem. Remember that, on the other hand, on bounded domains, yesterday we saw that in, say, 0 capital T times omega, omega bounded, the solution is unique. Remember, in the solution, if it exists, it's unique. So the point here is that the domain is unbounded. OK, so the idea of T-conoff is to first reason formally, look for a solution for u of Tx of the form some coefficients depending on T x to the j for any Tx in 0 plus infinity times r. So we try to find the solution of this form. Of course, we will have to see that the series is converging suitably strongly. But for the moment, let us forget for the moment the problem of convergence. So let us reason formally. And assume that we also require, so this is r, x, this is T. We also require, look for such that on this yellow half line, u is equal to some given g given to be chosen on x equals 0. It is on this line, x equals 0 on, say, 0 plus infinity product 0. That is on this half line. And ux equal to 0 on same line. So this seems to be too much. But if one is able to construct a solution with the following properties, this will be a solution. So now you will see why we are asking that the derivative in this direction along this line is 0 on this line, and u is g. So I mean, if one is not able to do, then one cannot construct anything. Otherwise, one can try to look for solution with these properties. And of course, one has also to impose this at the end. So let us see what happens if we formally compute ut. So assuming convergence sufficiently strong, I differentiate under the design of series. So formally, we have g prime j of t xj formally. ux formally is equal to sum from 1 to plus infinity jj of t x to the j minus 1. And uxx is equal to sum from 2 to plus infinity jj minus 1 gj of t x to the j minus 2. OK? This is formal. In principle, one cannot do this. But at the end, assume that I am able to construct coefficients. Suitable coefficients. Assume that I am able to construct coefficients gj, depending on g, so that this converges uniformly, for instance. And all derivatives converge uniformly, for instance, on, say, for bounded sets, at least in space. Then it will turn out that this can be done. I can differentiate under the derivative of the series. It is actually the series of the derivative. OK? So for the moment, I have this and I have this. So what happens if I impose such a new to be a solution of this? What happens? Well, let me rewrite this as the sum from 0 to plus infinity of j plus 2, j plus 1, gj plus 2 of t x to the j. This is a simple change of variables, OK? I change variable. I call j minus 2. I call it k. OK? This is a change of variable. OK, so if I impose u to be a solution, I am imposing that this is equal to this. This must be equal to this. OK, now let's see what happens if I require u equal to g on this line. So u of t, 0, is actually equal to what? u of any time, 0 is 0 of t. Hence, I want my first requirement is on the coefficients. Of course, I'm looking for the coefficients. So I require the first coefficient to be equal to g that I still have to choose. So this condition is due to this requirement. Now, I am assuming also a derivative in this direction along this line. I'm looking for a solution having 0 derivative here, so some sort of even solution. So ux, ux, so this says that g1, you see, so ux at t, 0, ux at t, 0, I see it from here. It is just g1 of t. And therefore, on the second unknown coefficient, I require g1 equal to 0. So let me write now what happens. So for the moment, I have imposed this condition and this condition. From this condition, I deduce this. From this condition, ux on this half line, I deduce g1 equal to 0. Now I have to impose the validity of the equation. The validity of the equation says, you see, this is a series. This is another series. I want this to be equal. And therefore, I require that all coefficients of the same xj coincide. So I require that gj prime of t must be equal to j plus 2, j plus 1, gj plus 2, gj plus 2. That is, I require the following condition. gj plus 2 of t to be equal to gj prime of t divided by j plus 2 times j plus 1 for any j bigger or equal than 0. So I have a condition on the first coefficient. Condition that the 0th coefficient must be g. The first coefficient must be 0. And therefore, I see what, and then this. So what I deduce from this? So g0 equal to g, let me erase this. I deduce that, first of all, on all odd indices, this must be equal to 0. Is it OK? Because you see, we have that the first odd coefficient is 0. And then, for instance, g3 for j equal to 1 is equal to g1 divided by 0. Therefore, g3 is 0. And therefore, g5 is 0, g7 is 0, g9 is 0. Yeah, but g1 is identically 0. And therefore, g1 prime is identically 0, and so on. This is always the case when you have a sort of problem and you are looking for a solution written as a series. And then you have that one coefficient is 0. And then you have a relation between not 1 and the previous 1. But this differs by 2, j plus 2 and j. Then you have this. On the other hand, so what happens to g2, for instance, which is the first even number index, j equal to 0? This says that this is equal to g0 prime divided by 2, OK? But g0 prime is equal to g. And what happens to g4? What happens to g4? g4 must be equal to g2 prime divided by 4 times 3, which is equal to g. So g2 prime is actually this. And therefore, I have g2 divided by 4 factorial, OK? And in this way, it's not, it is. 4 times 3 times 2, OK? So by induction, one shows, therefore, that g for any. So g2k of t is equal to gk divided by 2k factorial, OK? So now, we keep only, therefore, this information on the blackboard. So we have this, and then we have this. So what happens to our series? Our series, our function u, we are looking for. You see now, what is this? So this is j from 0 to plus infinity gj of t x to the j, which is equal now to what? It is equal to k equal 0 plus infinity gk 2k factorial x to the 2k, gk of t. Please check this, because this is actually, why is it true? Because this sum is actually only on the even indices, because the odd indices, the coefficients are 0. So g2k, so this is simply sum over k. j is equal to 2k, and so it is this, divided by this. This seems to be correct, OK? Now, you see, what happens, what does it mean? So our u, now, our candidate, because for the moment, everything is formal. There is no proof. Everything is formal. But our candidate at the moment has coefficients, which depend only on one function g, because this is the k derivative of only one function g. So now, we have to choose g. And we hope, we choose g, and we hope that with this choice, this object satisfies all required properties. Is it clear the line of reasoning? So now, choose g. Now, choose g, so that this sum satisfies hopefully all required properties. What does it mean? Satisfies all required properties. Well, choose g, so that this sum is converging. All derivatives in time and space are converging. This is smooth enough. And by construction, it will satisfy, so this object automatically, by construction, satisfies u equal g on the yellow half line, and ux equals 0 on the yellow half line. So what we have, so this is all required, namely, the sum is converging sufficiently well, strongly, sufficiently strongly. And u is smooth enough. And u at time 0 is equal to 0, or required properties so that I can differentiate under the sum and in these holes. OK, so let now, so up to now, it is clear, otherwise, please ask, is it OK? So I keep the expression of the function u on the blackboard. So let right now, the expression just only here, u of tx, sum of, now we have gk of t, then we have 2k factorial x equal to k. Which equality? This, yes, yes, OK, yes, it's OK. Sufficiently strongly. Uniformly, we'll see this at the end, OK. So this is what we have proven up to now. If such a solution exists, it must have the following form. Now, choice of g define g of t equal to e to the minus 1 over t squared for t bigger than 0 and 0 for t equal to 0, OK. Which is a c-infinity function, OK. Now the point, you see what happens also. If everything is smooth with this choice, if everything is smooth formally with this choice, I have that this is gk of 0. And gk of 0 is 0. So if you want extend it, if you like, but this is not important, but you can extend it like this. And then this is equal to 0 formally. By the way, notice also that this function g is now c-infinity, say, but of course, is not analytic. It is not real. It is a standard example of a c-infinity function which is not analytic. All derivatives in 0 are 0. So if there is a hope that this is the correct object, so there is a hope because this function g satisfies, for instance, this property. Now the point is to study the convergence of this series. So now we are trying to estimate gk of t to see the behavior of gk of t. Study gk of t. Estimate, say. Now consider, so we have for positivity. So now we have positivity. Consider now an extension of g as follows. g of z is equal to, say, e to the minus 1 over z square for any complex z different from the origin and 0. So now g is g is holomorphic out of the origin. So we can now compute, in particular, if t is positive. If t is positive, I am out of the origin. And I compute this quantity here. Let me call the notation that I'm using. gk, let me use omega. Omega minus t to the k plus 1. So this is omega, complex number. k plus 1 in the omega. What is this? OK, now take gamma. Gamma is a parametrization of the circle centered at t. This is the Cauchy formula for holomorphic functions. I mean, the Cauchy formula usually is for k equal to 0. But then we know that it extends to all derivatives. So gamma is a parametrization of the circle centered at t0, the complex number t0 of radius, small enough such that we don't touch the origin, because in the origin, there is a problem. Actually, it's not analytic. It's not homomorphic. Say, for instance, radius t over 2. So I have t0. And then I have this circle. And then I have by Cauchy formula for holomorphic functions. So in this way, I will try. So remember that I want to estimate this. So why I'm doing this? Because I want to estimate this. This is the goal. I want to see that the series converges. So I have this. Hence, I can say that gk of t is less than or equal to k factorial over 2 pi, the integral over gamma of g of omega divided by omega. These are complex numbers. So this is k plus 1. And then I have d omega. So this is a circle centered at t. So let me define this circle. So the circle is the image of the map gamma. That is the circle is the set of all z such that z is equal to t plus 1 of t. Namely, z minus this real number is a circle of radius t over 2. So this is the circle centered. So z minus t. So any point of the circle centered, so this is t over 2. This is t0. Any point of this circle can be written as t plus t half e exponential. So this is the circle. So therefore, this is what? k factorial over 2 pi, the integral over gamma. Now, omega minus. So omega is on this circle. So omega minus t is equal to 1 half, omega minus t is equal to t over 2 times this unit vector. So this is just t over 2 to the k plus 1. Then I have 1 half t theta. And then I have e to the minus 1 over omega square. Is it OK? So what do I have now? This t over 2 cancels with this, with one of these. Therefore, what remains here is equal to. So let me check. We have to be careful about the numbers, of course. So this is 2 over t to the k, the integral of gamma. So now, e to the minus 1 over omega square.  And therefore, this is equal to this, which is equal to the product of the two exponential. And 1 is 1. So it is just e to the minus real part for 1 omega square. Is it OK for the moment? Now, remark. So if z, so this is t, then I have the circle of radius t over 2. Now, let me make a larger picture. This is t. This is t over 2. And this is 3 half t. And then I have this circle here, which is the image of gamma. And I am integrated on this set. But I am integrating something which depends on 1 over z. So this is z. And then I have to see where is 1 over z. Because now I am integrating 1 over omega z or omega square, whatever you want. OK, the claim is now that if z belongs to the circle, the circle centered at t of radius t over 2, then 1 over z. So assume that t is very small. So 2 over t is very large. So I have this 2 over t here, which is 1 over this. Then I have this is 3 half t. So this point through the map 1 over z goes to 2 over t. So the claim is that all points then lie on a circle. Lie belongs to the circle centered at 1 over t. Not sorry, not 1 over t. Centered at 2 over t minus this divided by 2. So centered at 2 plus. Yes. So what is this difference? So this difference is 2 over t minus. So it is 4. This difference is this. And therefore, I think this difference is this. So 1 half of this plus 2. So centered at 4 over 3 t and radius 2 over 3 t. Radius, sorry. Radius, yes, 2 over 3 t. This is the remark. Well, the remark should be proven. So 1 is to prove that if z belongs to this, then 1 over z belongs to the w complex such that w is equal to the center 4 3 t plus the radius. So assume for the moment that we have proven this remark. Maybe we can leave this remark as an exercise. So for the moment, assume this remark. Assume. OK, so now 1 over omega here. Now omega travels along this circle. So 1 over omega travels by the remark along this new circle. And therefore, 1 over omega is of this form. So it is of this form. And therefore, it is of the form 4 over 3 t. Multiply it by 1 plus 2 e to the phi if omega belongs to gamma. Then 1 over omega by, sorry for the notation. Let me call this z again, z maybe. Then if omega travels on this circle, then by this remark that we leave as an exercise, 1 over omega lives on this circle. So now I put this in front of the parenthesis, so this is twice. Now 1 over omega square is equal to 16 over 90 square 1 plus 4 e to the 2 phi plus 4 e to the phi plus 4 e to the phi 1 over omega square. 16 over 90 square. OK, there is some number that now I don't find, unfortunately. Is it OK? This z square? It should be 1 up. There is 1 up, where? 1 over omega is 4 over 3 t plus this. OK? Sorry, I have made a mistake here. Thank you. Thank you. I have 1 over 2. This is the claim, sorry. And therefore, now we have this is, thank you, is 1 over 4, and this is 1, maybe. Hopefully that now the constants are correct. OK? So now the real part, this is the problem of complex integrals. By the way, so the real part of this is equal on this circle. The real part of this is, therefore, t is real. So 90 square times, now we have 1, 1 4 cosine of 2 phi plus cosine of phi for any phi in 0 to phi. So now I want something like this, right? Therefore, I need that this bigger or equal than something. OK? So let me try to find the minimum of this function. So we minimize now the own parenthesis, minimize now min. So this is, so let me consider now the function say f of phi equal 1 plus 1 4 cos 2 phi plus cos phi on its domain. So let me minimize it. So f prime of phi is equal to minus 1 half sin of 2 phi minus sin of phi, which is equal to minus sin of phi cos phi minus sin of phi. OK? So, hm? So f of 0 is equal to 1 plus 1 4 plus 1, which is equal to f of 2 phi. Now let me see what happens at the interior critical points. So we have sin of phi must be equal to minus sin of phi sin of phi. So solutions to, so f prime equal to 0 in the interior. Then solutions are sin of phi equal to 0, which means phi equal to, so we have already checked what happens at the boundary. Now, phi equal to phi is a solution, phi. And on this, f of phi, f of phi is equal to 1. Then the cosine is equal to minus 1, I think. It's 1 4? F of phi is 1 over 4, like this? OK. So if sin of phi is equal to 0, and then otherwise we have cosine of phi, so sin of phi difference from 0. The other solution is cosine of phi equal to minus 1. Sin of phi equal to minus 1. And so we are, again, phi equal to pi. So the minimum point is phi equal to pi, where the value is 1 over 4. OK, so what I'm saying is simply that phi, f of phi, is always larger or equal than 1 over 4 for any phi. Then it is also probably less than or equal than 9 over 4. But this is, for the moment, I mean, we also have this. So this is larger. So hence, this object here, can I erase this part? OK. So this is larger than or equal to this I keep. It is positive. And then 1 over 4. 4 over 90 squared. Now, we can continue this inequality, getting that gk of t is less than or equal to what? k factorial divided by 2 pi. So then I have 2 over t to the k e to the minus 490 squared times 2 pi. Is it OK for the moment? OK, this is, I mean, there is an exponential, which does not depend on theta. So it comes out of the integral. Then I add the length of this, which is 2 pi k factorial. So let me check if I'm correct or not. I don't know. Apparently, it seems to be correct. OK. Therefore, this is the estimate that we were looking for on the coefficients. So now u of tx, therefore, is less than or equal to what? k from 0 to plus infinity. Now we have this 2k factorial x to the 2k. And then I have this e to the minus 4 over 90 squared outside. Then I have 2 to the k, k factorial here. And then I have t to the k. Do you agree? Is it OK? Of course, if I made some mistake, please let me know. Otherwise, we do not converge to the solution. Now, claim for any k to k factorial over 2k factorial is small. Actually, smaller is small. By the way, you see 2k factorial is very large in comparison to this and this. But let us try to see whether this is true for any k. Do you immediately see that this is true? How can you see it?  Is it so immediate? Why? We open this 2k factorial. Yes. And 2 for until 2k, it goes with 2k. So you open this, and so you cancel the last k factorial with this? No. We cancel even numbers with 2 degree k multiplied by k. Yeah. Yes. If you don't see it, maybe we can prove it by induction. Just check it by induction. k equal to 1, k equal to 0 is OK. k equal to 1 is OK.  And maybe you can use induction so that this is less than or equal to what do you have? You have a 2 here, then you have 2k factorial. Because 2k, 2 to the k, k factorial, k factorial is less than this 2k factorial. What remains is this 2 and k plus 1 squared. And so you have to show that this is less than this. So you have to show that this is less than this. And to show this, you simply divide it by 2k factorial. So it is enough to show that 2k plus 1 squared is less than or equal to 2k plus 2 times 2k plus 1, which is obviously true. Because this is equivalent to say that k plus 1 is less than or equal to 2k plus 1. So it's true. I mean maybe there is a quicker way. Just induction works very well. OK, so this is true. So we can continue our inequality by putting here e to the. So now I rewrite this. And then I have here k from 0 to plus infinity 1 over k factorial x squared over 2 to the k. Now you see why we like 1 over k factorial, because you can recognize that this is what is the exponential. So this is equal to the e x squared over t minus 4 over 90 squared. That is, if you want, e to the 1 over t x squared minus 4. So actually, our series is converging even uniformly. We have a bound e 1 over t x squared minus 4. We have this bound, if I'm not wrong. So you see, so the series, the sum, the infinite sum, is converging uniformly for bounded x for any positive t. Hence, u is continuous. For any non-negative t, u is continuous. And u, therefore, u of 0 is equal to 0. Fine, so what we have proven up to now is that u defines a continuous function, which is 0 and 0. Next, what remains is to show that similar properties are true for the ut, ux, and uxx. So there are similar kind of computations for the series. The infinite sums given by g prime of t and uxx. So at the end, and now I don't repeat the computations, maybe. So at the end, u turns out to be even more c infinity r. By construction, once you have this regularity, you can differentiate under the infinite sum.  So u, sums, OK. So this way, Iconov found another solution in an infinite half space, non-zero solution of the heat equation. So this is an example that shows that uniqueness can be very tricky. So what we have not proven, but maybe I can leave it as a homework, we have not proven the claim that if z is in that circle, then one over z is in the other circle. I leave it. OK, so now I would like to start a discussion. I don't know if I'm able to conclude today. Last discussion on parabolic equations. So on y, the notation for the heat kernel was phi, this fundamental solution of the heat equation. It was phi. Y phi of tx is so useful. So remember that we have some constant here, what is important, n over 2. And then, minus x squared over 4t. And this is defined, say, for any tx in 0 plus infinity times rn. And if you want, we can extend it at time 0 with the value 0, but not at the origin. Remember that the origin in time space 0, 0 is a singularity. So why this is so important? So this will conclude our discussion on parabolic equations. Then we will say something on elliptic equations the next lectures. And then we will start functional analysis. So there will be a big parenthesis on functional analysis. And maybe at the end of the course, we will be able to come back to PDs as in applications of functional analysis. This is the scheme of the course. For the moment, we have not used any functional analysis. We have worked in smooth space continuous c1, c2, whatever. No theorem seems to abstract the linear function. OK, so now, yes, we have the following theorem. So let u 0 bar be a function in this class, a bounded function and contiguous. The function u of tx defined as follows. u of tx equal to the integral t y minus x u 0 bar of y dy for any tx 0 plus infinity times rn. Satisfy the following properties. 1 u is c infinity. ut minus Laplace of u is equal to 0 times rn and 3 limit. As tx goes to 0x of u of tx is equal to u by x. For any x, you see? This is very interesting result, which says that if you want to solve for initial condition, proper initial u 0 bar, you want to solve the heat equation with this initial condition. One solution is given by this expression here. This is called, do you know what is the convolution? OK, so this is the convolution in space of the heat kernel at time t together with the initial condition. So we can produce a smooth solution taking the convolution. Remark.  Maybe it is also the reason why phi is called fundamental solution. You can construct using phi, you can construct solutions to this kind of problem by convolution. Remark, before proving this, assume, in addition, u 0 bar is not identically 0, but is, say, bigger or equal than 0. Then, u of tx is bigger than 0. What does it mean? So assume that your initial condition, Professor, what if u is less than 0? You use 0 bar? Yes, u 0 bar is 0 in measures. u 0 bar is this. So the striking fact. So this is x, and this is time, and this is u 0. So u 0 is identically 0 up to a small bump, very small, somewhere. Of course, u 0 is the bump is such that u 0 is non-negative. Then the striking fact of this solution, which makes me make it. Well, the striking fact is that now take any positive time. Any positive time. Then you look to this object here, and it happens that the solution. So let me do it a little bit larger. It's impossible to define a small bump, maybe very far. Maybe very, very far. And then you take any positive time. Well, at any positive time, your solution will be something like, but positive here. Positive, positive, not 0, strictly positive. So this means that if somewhere very far, the solution is positive, then instantly, and the initial condition, sorry, is positive, then instantly your solution will be positive here, almost at minus infinity. So this says that this comes from this expression. This says that there is this fact that the so-called infinite speed of propagation of the signal. Something that happens very far initially at time 0, then produces a disturbance here very far for any time, even for t equal 10 to the minus 1 million. So this is a property of parabolic equations, like the heat equation. And of course, it's not anymore true for the wave equation. We have seen, with example, remember, we have studied this initial condition. And we have seen that there were large regions where the solution remains 0. We have two waves interacting, but at the end, the solution was 0 in a large region. So the compactness of the support here was preserved. On the other hand, here, the compactness of the support is not anymore preserved. So this is maybe one of the most important differences between parabolic and hyperbolic second-order equations. In some sense, but maybe this is questionable, say, parabolic equations are not so physical, maybe, because it is difficult to imagine a disturbance that propagates with infinite speed instantly in real physics. This is maybe questionable. Of course, for the wave equation, we know that we have a finite speed of propagation. This is not anymore true in the case of parabolic. So this is really very, very qualitative important fact concerning the difference between parabolic and hyperbolic. And this is immediate from the fact that this kernel here is strictly positive everywhere. Of course, it goes to 0 very quickly and exponentially, but it is always strictly positive. And this fact has this implication. So we have this. So we can now prove the theorem. No, we cannot, because there is no time. So maybe tomorrow, so please, for tomorrow, try to do the part of the claim on the circles concerning the tickle of example. And then tomorrow, we'll prove this, which will be the last theorem on parabolic equations.


what's up guys we're back with another ghost hunters video and today we have three people joined in the ghost hunter squad we've got ex gamer hose and marcus we'll be opening some ghost packs in an attempt to find a ghost rare so we have 20 ancient prophecy and 20 crimson crisis it's the same 40 packs that we used last time so it's going to be pretty easy but these are very epic packs last time we did not find the ghost in our hunt but i have a good feeling we're gonna get it this time but before that we have a giveaway i'll be giving away these three cards that is a dualist pack kibus super rare acetrap hole from dds and a super rare thousand is your strip from dl one all you gotta do like this video be subscribed let me know do you believe we're gonna pull a ghost share which ghost rare do you want to see and what ghost share packs do you want to see in the future all right our first ghost hunter marcus is gonna open up 10 of these packs so here we go let's see if we can pull the red dragon archfiend assault mode let's see if we can do it this is gonna be pretty epic also the ultimate rare would be cool as well but right now we're ghost hunters so here we go we got nightwing sorcerer we got the metaphysical regeneration vengeful servant desert protector and flying fortress sky fire cross sword beetle and more tronic and dupe frog by the way if you guys want to have some ghost hunter videos i will have ghost share packs available on my website check out ruxen34.com i have even more variety than the three sets i have right now so there's more coming it's gonna be pretty awesome next pack from marcus let's see if we can pull a ghost rare we are hunting the ghosts here we go we've got trojan gladiator beast we've got the assault overload nightwing sorceress metaphysical regeneration and wall of thorns okay submarine frog psychic sword and we're trying map i can't really remember what do we get out of the crimson crisis packs last time i can't really remember it's been a couple weeks since i recorded that last one i think maybe maybe a week and a half i don't i can't really remember guys well we got verdant sanctuary just got to use terraforming we picked that up i'm just getting all right trap reactor wi-fi we've got the e-bond era flip-flop frog hypersyncron and a black wing bore of the spear common very cool to see that as a common okay here we go what can we pull ivy shackles we've got the verdant sanctuary again we've got the borden descending lost star star blast okay crossword beetle and solar nutrient i think isn't verdant sanctuary i think actually is worth something i can't really remember i'll have the price pop up for it just so you guys can know if i'm right or if i'm wrong i can just prove myself right or wrong right now okay let's see right now we have not hit anything yet we got a prevention star we got a dark tinker dark tinker okay that looks very uh ghosty so that's interesting trogan gladiator bees we got the assault overload bs big core and seed cannon oh my goodness slow start slow start come on let's get some good guys let's get some epic can we get an ulcer rare secret rare something big we've got an e-bond aero trap reactor wi-fi salt revival totem dragon there's our first foil a super rare totem dragon i think last time i saw it this guy was like three bucks which is not bad for a super unlimited like that's actually pretty good and uh a thing to keep in mind about a lot of these packs um not so much the g the five d's but especially the originals i've seen like when we open l or uh mrl and at psv people are like wow the value is so low and it's like well it's i'm taking tcg player low on those and that's like not graded that's just regular near mint potential reprint stuff like that so you have to keep that in mind all right we've got a nightwing sorceress metaphysical regeneration eventual circuit and the shining silver force wow look at all those traps that's four out of five right there that's weird okay three packs left can we get some big can we get some big out of these packs ghost rare come on metaphysical regeneration we got the verdant sanctuary trap reactor wi-fi ebon arrow my cue the magical mist very weird to see that printed in a five d's pack but i guess it was crimson crisis up next here we go come on let's get some good vengeful servant desert protector assault counter blackling soroko the dawn and i'm another my cue okay that's weird and psychic sword come on come on crimson crisis pull us something good here trap reactor wi-fi we get a salt revival trap eater violet witch didn't we pull this last time i feel like we pulled this last time looks pretty minty which is nice ultra rare centerings pretty good i don't know if this is one you'd want to grade but you never know maybe you do there's an ultra i'm pretty sure we pulled that last time that's very weird and see cannon all right the next ghost hunter we're gonna go with ex gamers 20 ancient prophecy so he bought ancient prophecy last time and pull decent but we didn't get the ghost let's try and get that also we're gonna book in with crimson crisis we're gonna have 10 at the end for crimson crisis let's see can we pull the ancient fairy ghost rare here we go sort of sparkles cyborg doctor reinforced truth all right advanced strong blackwing fame steel chain is back fame the steel chain gotta love it one of my favorites in this set i just love the fame the fame the steel chain very good uh nice little ring to it here we go we got the brain research lab we got a battle teleportation aqua bizarre ancient forest super rare we got the arcane and knight jokers in here the original brand i think okay ancient forest super rare that's a good start one for two arcane and knight joker advanced draw blackwing fame steel chain two for two baby let's go all right here we go ancient prophecy 20 packs of it can we get some epic pulls like a ghost we have a shark cruiser at one with the sword we gotta love that card ancient crimson eight iron core luster and wow look how work that is arcane and knight joker blackling fame the steel chain three for three and we got the scary moth wow this is epic this is really epic right now all right so uh we pulled blackling fame the steel chain in every single pack so far here we go can we get the sunny pixie yes we just got bolder i don't know why i'm saying that ultimate rare earthbound immortal i don't know if that's how you say that but it feels like that's right i don't know why it feels right check that out earthbound immortal look at that ultimate rare it's a whale it's a flying whale that is an epic pull these ancient prophecy are paying off this time that is pretty good very nice and no blackwing fame the steel chain this time okay 15 more packs we've already gotten old to we've already gotten a super we've already gotten a play set of fame the steel chain can we possibly even beat that a major upset fish fork blaster we have the psychic soul fossil dig magic hat okay and death amulet all righty all righty keep it up let's stack these up like this there we go all right here we go the legend ex gamer he's been on the channel many times and live breaks he's he's got another uh pack so it's been pretty cool hopefully we can get him some more good stuff and another good on you like that's the third one we pull from ancient prophecy i think we was it him that that got two of these i know he got it i think he got at least one of them yeah that's crazy blackwing fame the steel chain that guy is in every single pack and i love it here we go say it with me blackwing fame the steel chain here we go here we go we got the infernity beast we got the imperial custom let's get the ghost for a sunny pixie quacky mirror hydro barrier hydro barrier and core reinforcement okay the water hazard cool all right next pack ancient prophecy here we go here we go we've got a fishborg blaster psychic soul fossil dig flame vell counter curry bond all right i used to think that this was say with me blackwing fame the steel chain i thought the curry bond was originally worse something but it's not it's not worth anything i said that in the wrong order but it's not worth anything now i thought it was but it's not spirit burner all right depth amulet advance draw blackwing fame the steel chain how many is that let's get a blackwing fame the steel chain counter up that's what we're gonna need because that is a lot of blackwing fame the steel chain okay the ghost hunters are gonna become the fame the steel chain hunters because we just pull this guy every time all right rekindling morphotronic magnum bar we've got a spirit burner discord join the discord youtube members check out the membership down below click it says become a member you can become a member you can get access to discord badges emotes if you're an eight dollar member you can get discounts you can get early access to the box breaks any sort of sales i'm doing boom that's what discord reminds me okay there we go check it out become a youtube member today all right here we go let's keep going you can get that epic pecanium off next to your name genetic woman sharp cruiser get the hydropressor cannon flambeville counter sauropod brachian okay the brachian very nice where is the ancient fairy dragon ghost rare come on ages of the ocean dragon lord fish fork blaster fossil dig arm c hunter xx saber full helm night okay advance draw blackwing fame the steel chain we are gonna hit at least 10 of these i think we're gonna hit 10 of these guys like how many is that like we're at like eight nine now something like that you still got several packs left it's like more packs than not i've had that card we have released restraint wave dark sea float slip of fortune scary moth and an upside down falcon b all right and cyber doctor we didn't have the steel chain it's the first time ever we have not had fame the steel chain all right i can't pick that pack up here we go can we pull something big all right i'm dropping stuff everywhere emergency assistance rekindling oh my gosh magnum bar spirit burner psychic emperor all right sword of sparkles at one with the sword six packs left in the ancient prophecy here we go let's pull the ghost rare ghosty we must find it ghost hunters slip of fortune dark sea float scary moth flameville fire dog all right armored axon kicker and sharp cruiser okay we're 75 of the way through the epic packs of ancient prophecy we have pulled an ultimate rare which is pretty awesome but can we pull the ghost scary moth and curry bomb again cyborg doctor all right come on guys you must believe in the ghost rare here we go water hazard we got ancient leaf armored axon kicker brain research lab and the core reinforcement dark sea float okay okay all right that's fine we're ghost hunters hunting ghosts ain't easy guys it ain't easy but we're gonna do it we're gonna find something we got a sunlight unicorn sunlight unicorn damage translation dark sea float silver wing and a gap chops a gap divine soldier all right genetic woman two packs left in this ancient prophecy then we have 10 more of the crimson crisis here we go fairy wind we've got spirit burner silver wing emergency assistance jester lord okay all right reigagura and the final pack haven't seen famous still chained a lot took a little took a little break okay took a little break huh ancient crimson a last pack here genetic woman sharp cruiser flame though commando ultimate rare in the last pack two ultimates out of 20 packs check that out we're gonna have to start calling ourselves the ulti hunters because that was a good one all right here we go sunny pixie that was pretty nice we got two supers we got the discord make sure you become a youtube member we got the glungulate and two ultimate rares of 20 packs that's really good all right guys in the last ghost hunter of the day hosea s has 10 packs of crimson crisis hosea s has gotten many things on the channel before so hopefully we can pull in something epic if you guys want to be included in the ghost hunter series check out the website you guys can get those packs you don't have to have them open you can also ghost hunt on your own time so just buy them and have them shipped to you you guys can go ahead and ghost hunt so check out ruckson34.com there will be um some packs up right now and more in the future are coming okay morphotronic board and we got the descending lost star hydra viper morphotronic slingon and the pollutant virus mysterious triangle eight more packs left in the opening building 40 packs in today's video so it might be a little bit longer i don't think it's gonna be too crazy long because last time i did this it wasn't that i edited it down pretty pretty short so trap eater royal swamp have i ever seen this card i probably have but i don't remember it this card is used as synchro material all other synchros become fish type that's interesting all right that's a weird one okay and we got uh submarine frog at the end okay can we get something big out of crimson crisis our angel prophecy were pretty good our crimson crisis that we pulled a few foils but nothing too crazy so let's see if we can verton sanctuary boarding ivy shackles totem dragon again what in the world two totem dragon now all right super rare let's check out the back it looks pretty nice okay i'm a little bit paranoid every time i pull like repeat foils i just want to check all right let's make sure that we're all good all the foils are in good condition all that stuff after that last craziness okay here we go trap reactor wi-fi we got the ebon arrow flip flop rock metaphysical regeneration and assault armor we're trying map bore of the spear all right a few more ghost hunter packs here we go ghost hunter will be back though it will be back telekinetic power well descending lost star mercenary blackwing gale the whirlwind that's a good one you're the whirlwind has a rare what do we got left four packs is that four packs four remaining here we go telekinetic power well descending lost star soul mercenary royal swamp eel see what in the world we are pulling a lot of similar foils that is so super weird but pretty uh i don't know good that they're in good shape though super rare's royal swamp eel three foils left our packs hope they're foils three packs with foils left here we go assault overload fake explosion they're all gonna be ghosts here we go telekinetic power roll we got the bdb dragon i still couldn't say it even after all that all right two packs the dragon two packs remain ghost wherever the flip flop rock metaphysical verdant boring and the end may do finally made an appearance all right and one more pack one more chance at the first of open 40 packs and now 80 we're gonna have a ghost counter i think ghost our Ghost Hunter videos were 0 for 79 so far, Vernon Sanctuary, Morph Tronic Borden, and Wall of Thorns, oh my goodness. All right, so we're 0 for 80 on the Ghost Hunter series. How long will it take us to pull one? That is the question in the Ghost Hunter series. All right, so that was a lot of fun, guys. The Ghost Hunter series will continue. Let me know if you enjoy these Ghost Hunter videos. Like the video so I know, leave a comment down below letting me know that you wanna see more Ghost Hunters. You wanna see a variety of sets. We could even do like Ghost from the past. We could do other 5Ds. We could do some GX potentially, stuff like that. It's pretty fun for me and I think it'll be interesting if we keep a running count. How many packs have we opened since Ghost Hunter started until we pull it go? So I think that it could be cool. So let me know what you think about that. That's it for this video, guys. I'll see you guys later. Peace. And this day, shining at this. Ooh, the revival jam. Oh, and oh!