 OK, so I'm going to talk about applications of this BED modeling clinical practice. A lot of examples, I'm just going to demonstrate how I use it, how it's been used. A lot of these are examples that I pulled from my own experience, whereas somebody has asked me, what do I have to do? And I get my calculator out, and I work it out for them using the linear quadratic model. And it seems to work pretty well. So the main applications of the BED model is to design or compare fractionation or those rate schemes. And it's also used for correction of errors. And I'm going to show that. When you make an error, can you correct it using the BED model to get the effect that you wanted in the first place? And there's been a paper published on that. So let me look at different uses of the BED model. And these are the uses I'm going to show you. First of all, I'll show a simple fractionation change, very simple application of the model. Correction for errors, as I mentioned. Conversion to two-grade profraction equivalent dose, we've heard about that quite a lot. You can convert any point dose into a two-grade equivalent dose. And this is often done in practice, and particularly in treatment planning computers, analyzing those volume histograms. Effective change of overall treatment time. Very important. Correction for rest periods. I'm always being asked, what do I have to do? The patient didn't come in for two weeks. How do I make up for that? Change in dose rate when we're talking about brachytherapy. Conversion from low dose rate brachytherapy to high dose rate brachytherapy. That's how we converted our low dose rate experience in the high dose rate experience. We did that many years ago when we first went to high dose rate. And then finally, the effect of half life on permanent implant doses. For instance, if you're going from ID 125 to Palladium 103, how do you make that correction in dose? And I mentioned yesterday that I'd done this when people first started using Palladium. And it seemed to work because we still use the same doses that I was able to predict using the BED model. So let's look at a simple change in fractionation, very simple application of this equation. So here's the problem. What dose per fraction delivered in 25 fractions would get the same probability of late normal tissue damage at 60 grade delivered in 30 fractions. So the radiation oncologist decides, I'm going to give these treatments in 25 fractions instead of 30 fractions. So what dose should I give to those patients? Well, let's think about it first of all. We're giving 60 grade in 30 fractions. Now we're going to 25 fractions. Ah, less repair. We're talking about late reactions. So the total dose is going to be less than 60. It's always good to think ahead. What do I expect? Because you work out on your calculator and up comes a number. And don't just accept it. I always think first rather than think later. Because sometimes you can be fooled because this is mathematics. It must be right. No. Think about the answer first. So there's your basic BED equation that we saw before, linear quadratic in dose per fraction. So the solution's very easy. And what do I assume now for my late reactions? Well, I'm going to assume three gray for this particular tissue. I pretty much assume three gray for all normal, late normal tissue reactions. Because we don't know any better. The Quantek program would tell us better numbers. But they haven't yet come up with anybody numbers to my knowledge. So about three gray for, ah, so the BED 60 gray, one plus the dose per fraction over three, one plus two thirds is 100. It's a nice number. I always remember 160 fractions in five fractions a week, two gray per fraction, BED, beautiful number, 100. So now a new dose per fraction, we don't know what that's going to be. In 25 fractions, we just put that equal to 100. So 25 times the unknown dose per fraction, one over the dose per fraction divided by alpha over beta. Solve that, it's a quadratic equation. D is 2.27 gray per fraction. So we're going to use 25 fractions of 2.27 gray per fraction. Very simple. I didn't calculate the total dose. I should get my calculator. But it'd be less than 60 if you worked that out. OK, what about correcting for errors? There's been a paper written by Michael Joyner, one of my colleagues, published about 10 years ago, on correcting for errors in the red journal, as we call it, International Journal of Radiation, Oncology, Biology, Physics, and Biology. And he devised a very simple method to correct for errors. So he found that if several fractions are delivered at the wrong dose per fraction, for some reason or other, they've started the treatment, and then you're doing your QA on the chart, and you say, oh, made a mistake. Patients get in the wrong dose per fraction. How can I correct that? You can derive a dose per fraction to use for the remainder of the course of therapy the result in the plan BD being delivered to all the tissues. So you'll get your plan BD to the late-reacted normal tissues, as well as the plan BD to the tumor. It's independent of the alpha over beta of the tissue. Very interesting. I was surprised when he came up with this. I was surprised I hadn't thought of it myself. And he's not a physicist. He's a biologist who's not supposed to know much about mathematics. I was really impressed how he came up with this. What he came up with depends on some definitions. So we're going to define the plan total dose as dp. That's the plan. And dose per fraction, small dp. The dose given in error would be then dE and dose per fraction dE in error. And then the dose required to complete the course will use the subscript C, dC, and small dC. So this is his equation. Very simple. This is all there is to it. The total dose he discovered remains unchanged. There's no change in the total dose you plan to give. And this is the equation for the dose per fraction corrected. And you just simply put the numbers into that equation and you get your answer. So here's the question. Let's say we plan. And let's say we're doing HDR brachytherapy. We plan to give 42-gray and 7-gray per fraction. Oh dear, we made a mistake. After two fractions, our QA shows we've only given three gray per fraction. Something went wrong in our calculation. Now what do I do? Then the dose per fraction needed to completely treatment. Just put your numbers into that equation and you get 7.67 gray. And if you do the calculations of BED for late reacting tissues and for tumor, you'll find you get what you plan to get. Very simple. And it works every time. So what's the extra dose required? You now need, remember you're giving the same total dose. So the corrected dose is 36. Hence the number of fractions required. It turns out when you do that calculation, you see it's 4.7 fractions to finish the treatment. You've already given two. You now need to give 4.7 more. But you can't give 0.7 of a fraction. So I always round out to the higher number of fractions. So instead of 4.7 fractions, let's give five fractions. Because that protects normal tissue. I always want to protect the normal tissue a little bit better if I can. So just divide the dose you need to give divided by the number of fractions that you need to give. You need to give 7.2 fractions to complete the treatment. I always round out the number of fractions up, protects the normal tissues. What Mike Joyner didn't realize, and I was able to show him, is that that whole thing is independent of any geometrical sparing of the normal tissue. So if you put geometrical sparing factor into there, you still get the same answer. Doesn't matter. Very interesting. Independent of alpha over beta of the tissue, and it's independent of geometrical sparing of normal tissues. How do anybody use this? I think most people miss that paper. They don't use it. And to be honest, I didn't use it. Because I've got a calculator. I can do the calculations. I come up with the same answer. It's a very simple equation to use in clinical practice. And we often find that dose per fraction was a little bit in error. When we started, of course, of that radiotherapy, maybe somebody did a simple calculation instead of a complex calculation, and the complex calculation shows a slightly different number. You can correct for it. I still give the planned treatment to the patient. Conversion to two-grade per fraction equivalent dose, you simply use the linear quadratic model, with the given dose per fraction on one side and two-grade per fraction on the other side. And you put them equal to get the same BED. And that's the equation you get. So this is what happens in treatment planning computer when it wants to convert every little voxel of tissue that's getting a certain dose per fraction, all different because it's a heterogeneous dose distribution. This is what it would do to each of those to convert it to equivalent two-grade per fraction. So you get an equivalent DVH, a two-grade per fraction. And some of the algorithms use this, for instance, in their optimization. Let's look at another example. What total dose given at two-grade per fraction is equivalent to 50 grade given at three-grade per fraction. And then let's do it for cancers. And let's assume it's a typical cancer with an alpha over beta of 10 grade. And let's repeat that for normal tissues with an alpha over beta of three-grade. And let's look at the results. And let's think ahead of what we're going to see here. What total dose given at two-grade per fraction is equivalent to, so you're now reducing the dose per fraction. It was three-grade per fraction to a 50 grade. Now you're reducing the grade per fraction. You should be able to go, you should have to go to a higher dose now because it's less dose per fraction. And will it make a bigger difference for the cancers or for the normal tissues? Well, if you just put in the numbers into the linear quadratic model, for tumors, you need to get 54.2 grade to get the equivalent. For normal tissues, you can go up as high as 60 grade. Of course, you're getting this window of opportunity because you're going to a lower dose per fraction, so you can go to a higher dose to the normal tissues. So actually, you'll get a better result if you do this. Why not go to 60 grade and get an even better result on your tumor, which is what I would do in this case? So I tell my radiation oncologist, well, you're going to two-grade per fraction. You might as well go to a higher dose because the normal tissues can take up to 60 grade. Might as well go up to 60 grade and give a bigger biological effective dose to the tumor. Repopulation. We haven't put any repopulation in here. What if we've got a rapidly growing cancer and we need to take into account that it's growing while we're treating it? So here's a problem. It's required to change a fractionation scheme of 60 grade and 30 fractions, a two-grade per fraction, over 42 days, six weeks of treatment, to 10 fractions delivered over 14 days. So what are we doing? Hypofractionation. We're going to do everything in 10 fractions now over two weeks. How do we solve that? Well, what dose per fraction should be used to keep the same effect on cancer cells? And then, let's answer the question using ALQ model, would a new screen have increased or decreased effect on late-reacting normal tissues? Well, we're going to 10 fractions instead of 30. It's going to have a worse effect on normal tissues. So what everybody was concerned about, when we go to hypofractionation, we're going to do worse for normal tissues, and everybody believed that. But as we're going to see in a minute, that's not always the case if you've got geometrical sparing of normal tissues. So let's do this problem with no geometrical sparing of normal tissues. So let's assume, first of all, no repopulation and no geometrical sparing, and see what we come up with. And just put the numbers into the equation for the tumor, BED of tumor, comes to 72. Then for the same BED in 10 fractions, you're going to put 72 equal to some unknown dose per fraction that you're going to need, solve for the dose per fraction, 4.85 grade. So I need 4.85 times 10, 48.5 grade, to get the same effect on tumor. OK, well, what's this going to do to the normal tissues? Going to be bad news. We know that. So assuming alpha over beta is 3 for the normal tissues, and the normal BED is 100. We remember that one. That's 60 grade, 2 grade per fraction. And the normal BED for 10 fractions of 4.85 grade, which we've just discovered we need for the tumor, put those numbers 127, oh dear. We've gone up 27% in the effective dose to the normal tissues, bad news. So 10 fractions should be far more damaging to normal tissues. But we already knew that, because we thought about this problem beforehand. Well, let's now put in a geometrical sparing. Let's assume that the EUD, for instance, equivalent uniform dose for the normal tissues for this patient, is 0.6. We've got 40% sparing of those normal tissues. Now it's going to work out a little bit better. So now we've got to put this parameter 0.6 in for the normal tissues. So instead of 4.85 grade per fraction, it's 4.85 times 0.6 grade per fraction. And the BED for the normal tissues will be, for late reacting normal tissues, will now be 50. And it will go up to 57. So it was 50 before with the 30 fractions, and it's gone up to 57 with the 10 fractions. You can learn a lot from the BED model. It's not just useful mathematically, but it's useful conceptually. Because look what happens here. We've now gone up from 50 to 57. It appears that the 10 fraction scheme, even with 0.6 geometrical sparing factor, is still somewhat more damaging, 57 to compare to 50. So you still haven't got over this problem of going to hypofractionation causing more damage to normal tissues. And then you have to think, well, why did I go to hypofractionation in the first place? Two reasons. One was it's less expensive because you only give 10 fractions, so that's good, more efficient. But two, which we haven't put in here, is the tumor's growing while you're treating it. So the quicker you can treat it, the better. So let's now throw in repopulation. And let's assume this is an average growing tumor. And if you remember, the k value, the repopulation parameter for an average growing tumor, is about 0.3 BED units per day. So let's throw that in and see what happens. So we now need to recalculate the tumor BEDs to take into account that both in 30 fractions and in 10 fractions, we get 0.3 BED per day being lost due to repopulation. So we've got to subtract that from these equations. So at 30 fractions, we subtract. So you've got a subtraction of 0.3 times 42 days. So it reduces now to 55.2 BED 30 fractions. Now put that equal to what we need in 10 fractions, so 55.2 has to be equal to. And now we're only doing 14 days of treatment. 0.3 times 14 has to be subtracted. Now we're going to get a different number, solve this quadratic equation, and you get 4.26 grade. We only need 4.26 grade, not as much as we had needed before, because we're doing better. We're treating quicker in 10 days instead of six weeks. We're doing it much quicker, so we don't need to go to such a high dose. Great. So now what about the effect on late reactions there? So do exactly as we did before, putting the values in for late reactions. But instead of a higher dose, we now need 4.26 times 0.6. 6.6 is a geometrical sparing factor. Throw all that in there. And what do you find? You planned originally, if you did 60 grade in six weeks, you're going to get a BED later 50. And now, throwing in repopulation of the tumor as well as geometrical sparing, you get a BED of 47. Low and behold, we've created a hypofractionation scheme that at least mathematically ought to be better for late reacting normal tissues. So we've learned something from the L2 model here. We've learned that decreasing the number of fractions such as in hypofractionation doesn't necessarily mean increasing the risk of normal tissue damage when keeping the effect on tumor constant. This is why we may see lots more hypofractionation. And I'll talk about that later on when we have the panel discussion. Because my own opinion is that in the next five years hypofractionation is going to take off. People are going to really start using it a lot more. It's more efficient, more cost effective. And I believe even better for the patient. We've never thought that before. The LQ model demonstrates that it can be. Yes, Jacob? No, no repopulation factor for normal tissue at all. That never came in here. No, that was because I needed to go to a lower dose perfraction, you remember? Because the tumor doesn't require such a high dose perfraction anymore, because we're repopulating and we're doing it quicker. So it's interesting. So the LQ model, I find the LQ model is very good, not only for real patients, but for generalizations like can hypofractionation actually work. Many, many years ago, I used this model to demonstrate high dose rate brachytherapy for cervix cancer. Taking all these things into account, geometrical sparing, the cancer's actually growing a bit. And at that point, I was able to demonstrate mathematically that you could go to high dose rate brachytherapy and you would actually be better off. You get better results than the conventional low dose rate brachytherapy that everybody says was the ideal way to treat patients. And I had a debate with Patricia Eiffel, who's the big guru of low dose rate brachytherapy at MD Anderson Hospital. I debated her at the Astro meeting. I think I lost, because most people in the audience were doing low dose, 99% of the audience were doing low dose rate brachytherapy. And they believed what the MD Anderson told them, the big center, that low dose rate was the way to treat cancer and you couldn't treat it any better with anything else. So I lost it. But I noticed that 10 years later, Patricia Eiffel was using high dose rate. I lost the debate, but I won the war. So your first thought was, it can't be better. It can be better. And I think you'll find that clinical trials now are showing that hypofractionation is actually turning out to be better than conventional fractionation. They're doing lots of experiments with breast, for instance, in five or 10 fractions. And finding it's actually better, amazingly. We didn't think, well, I always thought so. Most people didn't believe me. I believe these models work pretty well. What about rest periods during treatment? Often happens to our patients. Patients can't come in. A machine breaks down for two weeks. You're having it fixed. What do I have to do to get the same effect that I plan to do in the first place? So there's a patient planned to receive a BED of 100, 60 grade, 2 grade for fraction, over six weeks. And this rested for two weeks after the first 20 fractions, for some reason or other. How should the course be completed if the biological effectiveness is to be as you originally planned it? Well, let's solve this for late-reacting normative tissues. Now let's think of what's going to happen here before we do the calculations. This is bad news. And the reason it's bad news is during that two weeks of rest, the late-reacting normative tissues don't gain anything because they don't repopulate. The tumor loves it, loves to have a rest because the tumor can repopulate while the patient's resting, while you're not irradiating it. So this should be bad news for late-reacting normative tissues. Late-reacting normative tissues probably do not repopulate it during the break, so they don't benefit at all. So you have to complete the course for late-reacting normative tissues. You complete the course the way you plan to in the first place. Give 10 more fractions of 2 grade. So you get up to your 30 fractions. But what about the cancer cells? Oh, they're very happy. So let's make some assumptions. Let's assume an average growing cancer, k of 0.3 BED units per day, an alpha over beta of 10. And let's solve it. Let's put the rest period in there, subtract it from the BED, simple equation. OK? The solution is that we have to go to 2.12 fractions more. Instead of 10 fractions, you need 12 fractions of 2 grade. That's bad news. But the effect of normative tissues is then going to increase. So if you increase, if you get 12 fractions instead of your original plan, 10 fractions, bad news. So you have to compromise. If this is a real patient, you've really got to compromise. You don't want to decrease too much the effect on the normal tissues. In this case, you have to go to 12 fractions instead of 10. You don't want to compromise by only going to 10 fractions. Because you know your tumor cells are then going to be repopulating. Your probability of curing the tumor is going to go down. So I always tell my radiation oncologist, that's compromise and go halfway between the two. So let's give 11 fractions instead of 12. We originally planned 10 more fractions. We'll give 11 more fractions. And you're going to put the normative tissues at a bit more risk. But what are you going to do? And you maybe redesigned the treatment to be a little more conformal. That's another way of going around it. By the way, what's really interesting here is way back before we had any of these models, what did radiation oncologists do? Well, radiation oncologists had a rule of thumb. That for every two weeks that you rest a patient, you give one extra fraction, which is just that's that compromise. They knew it wasn't good on the normal tissues, but that was what they used to use. And I always used to tell people that two weeks of rest, one extra fraction. And it hasn't changed at all. It's still the same. Now that we've got the biological models to do the same thing. OK, let's look at brachytherapy a little bit now. And yes, if you put that into this calculation, you'd find you needed four more fractions. Compromise, give two more. Again, it fits in. And that's what it would work out to. So I don't even use the BED model for this. I just use the rule of thumb because I know, having experienced it, and I asked the radiation oncologist, do you mind a bit of a compromise here? If he said, I don't want to compromise at all, I don't want to damage the normal tissues, I say, OK, go to the dose you planned. You're going to be compromising the tumor control probability. And then he usually said, oh, I don't really want to do that. So I said, OK, give two extra fractions. And he's usually happy. None of this is real science. I mean, let's be honest, we don't know what's going on with any particular tumor. All you know is they're probably very happy you're giving them a rest, the tumor cells, so that they can repopulate. Every patient's different. Different parts of the cancer in any one patient will be different, because all the cells aren't the same in any one cancer. And we just don't know the really scientific answer to that question. We have to do a compromise and average over not only over that patient's tumor, but over everybody's tumor, because we don't know any better. Maybe sometime in the future, we can do some maybe pet imaging and demonstrate which parts of the cancer are repopulating faster than which other parts. And then change our dose distribution accordingly so we give something to that part of the tumor that's different from the other part. We don't know how to do that yet. But I can see it happening in a while. A few PhD students need to write some theses on how to do it, and then maybe we'll start doing it. So change in dose rate. So this is brachytherapy now. A radiation oncologist wants to convert 60-gray, a 0.5-gray per hour, to a higher dose rate of one gray per hour, keeping the effect on tumor the same. And this is actually something that we have done in the past. We used to leave our, for instance, we used to leave our cervix cancer patients lying in bed for a week. This is the standard Manchester system. You lie the patient in bed for a week. Big problem for lots of people, including the patient, who can't move very much because they've got an applicator inside them. In fact, we've had patients take the applicator out and throw it away in the trash. Bad news. I've had to search the sewers of New York City for some radium sources, as we were using in those days. And I found them. Because very high energy, you can go around with a Geiger counter and follow the sewer. And they sometimes get tracked somewhere. And sure enough, we pull up the manhole. And I didn't go down. I sent somebody else down with tweezers to pick up the applicator, which was taken out of the patient. And they brought this applicator up. I said, put it down fast. Radioactive. But these things happen. It's very uncomfortable for the patient. So sometimes, radiation oncologists will say to me, this patient can't lie in bed for that long. Let's do it with higher activity sources. And let's do it at about one gray per hour. OK, I could use this model to calculate what dose to give. So instead of 60 gray, you're going to give less than 60 gray because you're going to hire dose rate. So what total dose is required? Very easy. You just put the numbers into the equation. But you remember that equation? Very complicated. But we don't need to use the complicated version of that equation. You could if you wanted to. And I've done it. But it takes a while to do it. R is the dose rate. And T is the time for each fraction. If you're doing, say, if you're going to fractionate it, typically we didn't use to fractionate low dose rate for acryltherapy. We do it all in one fraction. Mu is the repair rate constant that we talked about for related to the halftime for repair. But the simplified form of the equation, if the total time is less than 100 hours, we can use a simplified form of the equation. Well, for the originally planned course, it was less than 100 hours. So we can use this simplified equation. And for T greater than 100 hours, we can even use a more simplified equation. So I'm going to use the very most simplified equation, very simple linear quadratic model, and plug in some numbers. I need to assume some parameters, put those parameters in. And the approximate answer is going to be that we need 73. So the BED for the tumor planned was 73. So the BED at one grade per fraction has to be 73. So we put 73 equal to. And we put those numbers with one grade per fraction in instead of half a grade per fraction, all the same numbers. And what did we get? We get a time of 51 hours. So for this patient, we need to leave the patient in only for 51 hours. Which is great. We've reduced the time considerably. I cheated a little bit there because I used the simplified equation at one grade per hour. When it turns out the actual time is less than 100 days. I did it because I happen to know that if I've gone through all the rigmarole of that big equation and putting all the numbers in there, I've come out with about the same number. So in fact, 51 grays with the simplified equation of one grade per hour. And actually, I shouldn't have done that. I should have used a big equation. And it turns out 51.3 extra grays. So there's hardly any difference. It's well within the reliability of the mathematics because it's an approximate model, again, remember. By the way, it took me a long time to do that calculation on my pocket calculator. I remember it took me about an hour. I could have had it all planned in the computer, and it would have been a lot quicker. But it took me about an hour to come up with this answer. It wasn't an easy thing to do. But I know by experience that the simplified form works out well enough. So 51 grade that we need to give. OK. What about converting from low-dose rate to high-dose rate? I remember I mentioned that I did a lot of this. It's required to replace the LDR implant of 60-gray at 0.6 grade per hour by a 10-fraction HDR implant. This, again, typical dose that we used to give to service cancer patients was 60-gray at 0.6 grade per hour. Very typical low-dose rate treatment. And I want to now come in with a 10-fraction HDR treatment. What dose per fraction do I need to use for HDR? So solution, the time is 100 hours. We can use the simplified form of the BED equation quite accurately. And I put the numbers in, the various parameters in. And I came up with the BED. The BED that we used to use with low-dose rate was 65.1. What about the BED? For the high-dose rate now, I put 65.1 equals. And then it just solved the linear quadratic equation. You get a quadratic equation in dose per fraction. And 4.49-gray per fraction. I did this about 25 years ago when we first started doing high-dose rate brachytherapy for service cancer. And they're still using the same dose per fraction now. And actually, they do get better results. Because I haven't thrown in things like geometrical sparing here, which would have been nice. But I didn't, because it makes it a bit more complicated. But this is the regime that they use. 10 fractions of 4.5-gray. They've used it for probably 25 years now and still getting good results. And publishing actually better results than they previously had with low-dose rate brachytherapy. Yes, yes. His experience is that most people use five or six fractions, not 10 fractions. 10 fractions is very inconvenient. And why did we use 10 fractions and not five or six or seven fractions? We used 10 because the radiation oncologist and the gynecologist got together and said, the last thing we want to do is complicate our patients. So give us a fractionation scheme where you're more confident that you won't complicate your patients anymore. Because if we complicate our patients, then gynecologists are not going to refer their patients to us. They're going to refer them to the hospital down the road. They're still doing low-dose rate. That was the argument, OK? And it seems to work. Well, how do you put 10 fractions of high-dose rate into a patient? In those days, it was very difficult for several reasons. And I talked about these a little bit yesterday. One is the complication of putting this applicator into the patient 10 times. And the other problem was anesthesia that you needed in order to put the applicator into the uterine canal, which is very painful for the patient. And that's when we came up with this stent that you put in under anesthesia before you start the treatment. And then from then on, it slides in very easily into the patient without pain. So that all came at the same time. And they just keep doing the same thing, keep using 10 fractions, because it's not difficult to treat with 10 fractions when it doesn't hurt the patient. You don't need to do anesthesia. Probably they could have got as good a result with six fractions as most people in the world have been doing, but they hate to change when they're getting such good results. Is this better or worse for normal tissues? And here we go. We look at it. The BED for low-dose rate was 100 for normal tissues at 112. The BED for the tumor was 112. And that's how I did the calculation. So, and I say, well, amazing by pure luck, it wasn't pure luck, it was because I knew the answer before I started, because I'd done the calculation many years ago, OK? But it's interesting. Can you actually do better if you have geometrical sparing? And so I did this calculation. And this is what I got when I threw in geometrical sparing. So I've got the, let's have a look, OK. So I've got geometrical sparing thrown in here. Oh, this is without geometrical sparing. So there's the answer that we just got 4.5 grade. Here's the answer that we just got in that calculation. 4.5 grade per fraction in 10 fractions. And if you look at 4.5 grade per fraction in 10 fractions, you find it's equivalent to 0.6 grade per hour. That's that point there. Now what happens when we throw in geometrical sparing of the normal tissues? Which we get, because this is highly conformal therapy. And that's the curve that you get with geometrical sparing of normal tissues. And it makes a big difference. So the curve we just looked at was geometrical sparing of one, which is this red curve. So let's now throw in an HDR of 6 grade per fraction and see what it's equivalent to with some sparing. And what you find is with a fairly modest geometrical sparing of normal tissues that, for instance, 0.6, that's this yellow curve here. So your sparing, your rectum and bladder are getting equivalent uniform doses of 0.6 of what the tumor is getting. And what you find is, if you look that up, you can give about 6 grade per fraction instead of 4.5 grade per fraction than I showed before. So it's great. And that's why people are using 6 grade per fraction for high dose rate rectum therapy for a lot of things. But cervix cancer would be a good example. It works. That's why you do it, because of geometrical sparing of the normal tissues. So these models are really useful for new concepts that are coming into a new field like hypofractionation or high dose rate rectum therapy. Let's now look at permanent implants. What total dose for a palladium permanent prostate implant producing the same tumor control as 145 grade of iodine? So I'm changing from iodine to palladium. I did this calculation when palladium first started, and the company came to me and said, what dose can we tell our users to give? And I did the calculation. And you put the values in with the different decay rate, lambda, the different decay rate of palladium. So you do this for, as the equation, you do this for iodine. And you come up with 0.07 grade per hour, which is what we all use for a permanent implants for iodine. We've been doing that for many years. What do you have to use now for palladium? You put all that in to the equation for palladium, and you come up with the initial dose rate, palladium is 0.21, 0.209 grade per hour. So the total dose of palladium is that divided by the dose rate, and it's 122.9 grade, about 123 grade. And actually, when I first did this, I came up with 124, I remember, and I told the company, and then they told their users, use 124 grade, if they're changing to palladium, and that's still the number that they use today. That's the number recommended by the American Brachytherapy Society. So the model worked amazingly well, considering it's an approximation. So let me summarize. The BED model's useful for solution of radiotherapy problems changes in dose per fraction or dose rate. But remember, biology is much more complex than we've assumed when we derived the linear quadratic model. It was a very simple derivation. If you remember, a little bit of hand waving and creating breaks, double strand breaks with a single passage of a particle or two separate particles, way too simple. But it seems to work pretty well. It's an approximate model. The parameters are even approximate because we haven't yet discovered what parameters to put into that model. But the model is useful, and it seems to work really well. And I always remember the statement, I think it was a statistician box who said that all models are wrong. They're not accurate. All models are wrong, but some are useful. And this is a useful one that's wrong. Okay, thank you very much.