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news image Date: 18.07.2016

Technische UniversitÀt Berlin, July 18-22, 2016


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Using mathematical approaches in a Matheon project, Prof. Carsten Hartmann and Dr. Max von Kleist hope to find optimal treatment methods.

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Dangerous bacteria are an increasingly serious social problem. They occur most of all in hospitals and livestock farms, and in the latter among pigs and chickens in particular. These bacteria are so dangerous because antibiotics, our panaceas of many decades, are losing their edge as a weapon. Bacteria have grown resistant to these medicines. The term "multiresistant germs" has become a buzzword of our times. In 2013, nearly 31,000 people became infected with such germs in hospitals in the German federal state of North Rhine Westphalia alone. In terms of overall patient numbers, however, North Rhine Westphalia ranks "only" somewhere in the middle; in ThĂŒringen, almost one in ten hospital patients fell ill with multiresistant germs, and in Saxony one in nine. Of the estimated total of 400,000 people annually infected, around 15,000 die from these germs. These are in fact conservative estimates; other estimates put the death rate much higher.

How is it that antibiotics, which have helped us for so long, are now losing their effectiveness increasingly often? Many experts ascribe it to excessive use of antibiotics in livestock because, by eating the animals treated with these medicines, people also regularly ingest the active substances without actually having an acute infection. In 2013, the Bundesinstitut fĂŒr Risikobewertung, BfR (Federal Institute for Risk Assessment) ascertained that 90 percent of fattened poultry in Germany is contaminated with multiresistant germs.

The bacteria are happy! By now they have developed such efficient defence mechanisms against antibiotics that our medicines are simply ineffective. Apart from that, pressure to reduce antibiotic agents in fattening feed has met with hefty resistance amongst farmers since, without these agents, many animals would not survive on modern livestock farms, and so the margins of fattening farms are narrowing.

The extent of the threat becomes clear when one considers that, on the one hand, the number of multiresistant germs is rapidly rising and, on the other hand, the approval of new antibiotics has been steadily dropping in recent years. Is it a vicious circle? Is drug-based livestock farming really the way of the future, and do we have to simply accept there will be more people who will fall ill or even die? Carsten Hartmann and Max von Kleist see this as a real danger. The mathematicians at the Institute of Mathematics of the Freie UniversitÀt Berlin and of the Matheon research centre in Berlin have therefore come up with an approach that just might break this vicious circle. "Using mathematical approaches, we want to arrive at a more effective use of existing antibiotics because, aside from fattening feed, causes for the spread of multiresistant germs are the too-frequent and improper use of antibiotics," Carsten Hartmann says. Improper use means either the medicines are administered without there being a real need to do so, the wrong agents are used, or, as is frequently the case, the intake of antibiotics is stopped too early. If the medicine is not taken over the prescribed period, then some of the dangerous bacteria can survive and subsequently "remember" how to defend themselves against the medicine. In other words, the bacteria not only continue to multiply, they also mutate, after which the antibiotics can no longer be of any threat to them.

As it stands, there are now a great number of bacterial strains that respond only very slightly or, in some cases, not at all to a given antibiotic. It is being considered, therefore, whether a better approach is to administer a combination of different medicines: "In our research project, we are pondering how we can optimally apply and combine different agents so as to keep the number of bacteria as low as possible, and then kill off the targeted bacteria relatively quickly," the mathematicians explain. This requires a mathematical model and efficient algorithms by which to simulate on the computer how individual bacteria will respond to certain medicines. The mathematicians also want to understand how long the treatment must ideally be continued, and when the highest probability exists that all bacteria have actually been eliminated. "Normally, it should be obvious to take a medicine until the end of the course. It’s common sense, and you don't need mathematics to tell you. But, when the risk of resistance exists, then you should have scientifically substantiated methods that help you to minimise the corresponding risks," Prof. Hartmann advises. So the ultimate aim is mathematical optimisation.

The mathematicians accordingly start with a model that describes the number of bacteria present in a patient, which can constantly change during the course of a disease as well as due to random factors. "We are dealing here with a random variable, which normally varies from one patient to another," the researchers explain. Conventionally, the action of an antibiotic is researched by cultivating a bacterial strain in a petri dish and then seeing if the bacteria die off when the agent is added. If one has a relatively small number of bacteria, then one can still theoretically count them. That means one can actually see when a new bacterium has appeared or an older one has died off. If, however, one has a very large petri dish with very many bacteria, then it is no longer possible to count the pathogens. Yet, the process can still be reasonably approximated by averages and concentrations. One has then a stochastic model for the exact number of bacteria in the body on the one hand and a deterministic model for their averages over a population on the other.

Since the rough, deterministic model is easier to cope with mathematically, it would be nice if it could serve on its own to optimise the antibiotic treatment. It certainly works well when many bacteria are present in the body. Unfortunately, it does not work so well after a highly effective antibiotic has killed off most of the bacteria but left a few survivors. Then you have a case with a small number of bacteria. "With a small number of bacteria, it makes a big difference whether I use the stochastic or the deterministic model, since a single bacterium is either there or it is not, and a declaration of 'there is on average half a bacterium in the body' is not a useful description of the situation for an individuum," Hartmann says. The more precise, stochastic model is always needed when the question arises whether to continue administering the same medicine or instead change to a new one. "As scientists, we tend to prefer the higher precision model; but to calculate an optimum antibiotic therapy from it, this model simply has too many state variables. Therefore, in our project, we are studying approximations between the two models, and their effects on the optimal dose of medicine," von Kleist says.

Currently, Carsten Hartmann and Max von Kleist are investigating how close one can come to an optimum medicine dose using a deterministic model: "Unlike in the stochastic model where the bacteria can be killed off entirely, in the deterministic model, the concentrations of bacteria can ideally become arbitrarily small, and that has an impact on the optimal therapy." The calculations done so far, the two researchers report, show that a dose of medicine optimised on the basis of averages and concentrations achieves a greater probability of success compared to a standardised therapy, and that the optimised therapy can even cope with random fluctuations in the patients' state of health. The researchers call this property ‘robustness of the therapy towards random influences’. "It would naturally be excellent if we could measure the extent of the disease in individual patients and then have an individually adapted strategy at hand that is optimal for those patients. But that is very laborious and so also very expensive. So, we are following a path based on an approximated deterministic model and which leads to "rules of thumb", which give the treating physicians the greatest possible certainty of applying the correct treatment," the mathematicians say.

Their model ought to be useful both for treating with known medicines and for helping develop new antibiotics. For the latter to work, they require better data on the behaviour of certain molecules. The mathematicians are accordingly working with colleagues from the workgroup of Marcus Weber at the Konrad Zuse Centre for Information Technology Berlin. They also have cooperation partners from the medical profession, who provide the necessary patient data. "We also use insights gained from research on HIV patients. So there is justifiable hope that, building on many different parameters, we will come to an applicable model. But we still have a long way to go before we reach that goal," the mathematicians declare.

More information:
Prof. Dr. Carsten Hartmann,
Institute of Mathematics,
Free University of Berlin,
Email: chartman-aet-mi.fu-berlin-punkt-de

Dr. Max von Kleist,
Institute of Mathematics,
Free University of Berlin
Email: max.kleist-aet-fu-berlin-punkt-de

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Mathematicians develop a “virtual hospital” to optimise in-vitro fertilisation.

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The birth of some children can be unplanned, and sadly sometimes even unwanted. On the other hand, many couples long dearly for a child but have their wish unfulfilled due to various medical reasons. Statistically, the problem resides about half the time with the man and half the time with the woman. For many years now, however, medical progress has allowed the artificial fertilisation of female egg cells. The egg cells are taken from the woman and fertilisation is performed in-vitro. Before this can be done, the woman’s normal menstrual cycle first has to be gradually “regulated down” by medication over several days in order to be subsequently restarted – again by medication. The aim is to induce as many egg cells as possible to mature so that they can then be harvested. Exactly what medications are used in the procedure, and at what doses, depends on many individual criteria.

Austrian physician Berthold P. Wiesner is regarded as a pioneer in the field of artificial insemination. He performed many of the first tests between 1940 and 1960 in London. However, Wiesner became a controversial figure after it was discovered that he had contributed most of the donor sperms himself. The first “test tube baby” came into the world in 1978. Since then, the number of births after in-vitro fertilisation (IVF) or intracytoplasmic sperm injection (ICSI) has increased drastically. In Germany, about two percent of all children are born through artificial fertilisation by now. The procedure is never truly risk-free; it is difficult and in many cases unsuccessful. The main reasons for this are variability in the female hormonal cycle and the resulting patient-specific differences; especially in cases of fertility disorders. To find solutions, medical practitioners, biologists, computer scientists and mathematicians have teamed up in the EU project PAEON. The mathematical part in this research consortium is being undertaken by Prof. Susanna Röblitz and Dr. Rainald Ehrig of the Zuse Institute Berlin and members of Matheon. Together with the Zuse Institute, the PAEON project involves members from the Lucerne University of Applied Sciences, ETH Zurich and Hannover Medical School. The project is coordinated by scientists of the UniversitĂ  di Roma Sapienza.

An estimated 12–15 percent of couples have their desire for children unfulfilled. On average, these couples spend around 10 percent of their annual income on in-vitro fertilisation. About 40 percent of all female infertility cases can be attributed to hormonal disorders. The success rates of IVF and ICSI have never exceeded 35 percent. “The goal of our mathematical approach is to develop a model-based decision-making aid for the physicians, which firstly contributes towards a better understanding of the highly complex processes in the female cycle and, secondly, provides the opportunity to simulate and optimise the treatment strategy on the computer, thus making it more efficient. Of course, another important aspect is to reduce the costs,” Susanna Röblitz explains.

The female hormonal cycle follows a fixed series of precisely coordinated processes. It begins with the maturation of the follicle containing the egg cells, followed by ovulation, which is the release of the egg cell. Normally only a single egg cell is released from the largest follicle for fertilisation. After ovulation, the so-called corpus luteum starts to develop. If a pregnancy does not occur, then the corpus luteum degenerates and the uterine lining is shed leading to bleeding at the beginning of the next cycle. All of these processes are controlled by hormones. In artificial fertilisation, the task is to stimulate multiple follicles to ovulation so that several fertile egg cells can be harvested from the woman. This stimulation is done by administering hormones externally, meaning by medication.

Unfortunately, the problems already start here: “The treatment standards of clinics vary very much. Even discounting clinical negligence, which surely exists, use of the right medications and their individual dosing is not easy,” says Rainald Ehrig. This is one of the most important starting points for PAEON. Based on medical data, the mathematicians from Berlin have initially modelled an idealised cycle of a healthy woman. “There is still ambiguity regarding the highly variable nature of this cycle in women – for a woman individually as well as across women,” Susanna Röblitz continues. The next step was therefore to model hormone profiles and follicle development accounting for the differences between individuals, the reasons for dysregulation in hormonally caused fertility disorders, and the influence of external factors (e.g. age and smoking). The aim is to integrate treatment options into the model and simulate them on the computer. Rainald Ehrig describes the mathematical task as follows: “We first develop a model that takes into account the cycle, the maturation of the follicle, the changes under medication, and external influencing factors. This model is continuously validated with real patient data in order to finally allow a meaningful prediction that is ultimately introduced into the actual treatment strategy.” For this to be possible, the model must not be restricted to any of the individual components, instead, it must allow a description of the entire complex system.

The starting point is a conceptual model that predicts how the biological system behaves under given external conditions. This conceptual model is then complemented by data from an experimental model. From these two models, the mathematical model is finally developed. “Here we keep in mind Albert Einstein‘s saying, that mathematical models should be made as simple as possible, but not simpler,” Susanna Röblitz says. A major problem, however, is that many model parameters are unmeasurable and therefore unknown. This is compounded by the fact that most biological processes are largely nonlinear and therefore exhibit no easily predictable patterns of behaviour. “Such complex systems can only be studied using new mathematical methods,” the mathematician continues.

Of course the mathematical models include the known effects of various drugs that are administered during an IVF. This is done by modelling the pharmacodynamics of the ingredients of these drugs, for example their concentrations in the blood over time.

The research will ultimately result in a “virtual hospital” based on virtual patients and complemented with real data collected by physicians. Its output will give physicians a data-substantiated suggestion for a targeted and efficient treatment for the individual. When a real patient comes into the clinic with her data profile, her profile can be compared with the existing data. This yields an initial suggestion for the physician as to how the treatment could be performed. From then on, the system accompanies the treatment and can repeatedly propose suggestions over the course of time. Additionally, one can simulate on the computer new combinations of drugs whose actions would otherwise have to be tested on real patients. Yet, the two mathematicians see this still as a long way to go. “For sure our system ought not only to increase the chance of success, but also reduce the risk and the costs. That is a huge mathematical challenge,” Susanna Röblitz asserts.

The project PAEON has been running since 2013. The project has been recently reviewed for the second time and considered as successful. The EU funding, however, is set to expire next year. The two mathematicians expect to have a demonstration of the “virtual hospital” by then. “Unfortunately, EU projects are usually not prolonged. The project group, however, would very much like to continue its research. So far, all partners have been primarily scientifically oriented. Perhaps there will be a commercial partner in future. The EU has great interest in commercialising its projects,” says Susanna Röblitz.

More information: Prof. Susanna Röblitz,
tel.: +49-30-84185156,
email: susanna.roeblitz-aet-zib-punkt-de or

Dr. Rainald Ehrig,
tel.: +49-30-84185282,
email: ehrig-aet-zib-punkt-de

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