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The   mathematical concepts   that   underlie the   cryosurgical  simulation.

(edited by dr. Giovanni Giorgi).


Introduction.

  In this part, dedicated to mathematical simulation and concepts related to it, will be dealt with specific aspects that are significant and currently studying in top universities of the planet.
  Given the particular thematic topics will be discussed in light of the characteristics of our readers without having to deepen the formal mathematical but describing the choices that are the basis of formalisms treated. We therefore invite readers who want more information biomathematics to request it and we will try to oblige

The cryosurgical planning as a mathematical problem - Part 1.

  Cryosurgery is a medical practice based on the induction of very low temperatures in portions of biological tissue, so as to cause freezing. The control of the temperature of the tissue occurs through the use of special probes such cryoprobes that relate to specific generators are continuously cooled by a continuous recirculation of fluids: the contact of the probe with the surrounding cells and tissue extracts heat determines the freezing. When performing cycles of freezing and thawing of the duration of a few minutes and minimum temperatures of the order of -100 degrees centigrade, the process determines the death of the cells that constitute the portion of the tissue involved.    On this principle is based cryosurgery, which is used in the fight tumors or diseases superficial (skin) or internal organs (prostate, liver, etc.) as compared to more classic techniques, may be less invasive and more simple application according to the cases clinical considered.
  However, if several cycles of freezing and thawing are lethal to portions of the tumor tissue, in the same way that it can be affected   healthy tissue: for this reason, each experiment cryosurgical must be thoroughly planned in order to minimize the consequences of the action of cryoprobes on healthy regions. Because healthy tissue and tumor cells have very similar thermal properties, the lowering of temperature should be targeted and the definition of the free parameters which vary from operation to operation and from patient to patient (i.e., positioning and depth of insertion of cryoprobes, temperature they have reached, etc.) must be carefully designed. When, through the use of some imaging modalities, will be reconstructed precisely the boundaries of the tumor region to be treated and have already been considered possible physiological indications coming from the medical staff, the problem of the definition of the free parameters of an cryosurgical operation becomes tied exclusively to the physical dynamics of the propagation of heat in the tissue. The definition of the most appropriate configuration parameters before each operation is called cryosurgical planning and is generally carried out by automated systems based on complex mathematical and computational tools.
  From the mathematical point of view, the propagation of heat is, like many other physical phenomena, governed by differential equations. In general, the resolution of the differential equation of heat gives rise to a family of solutions: from these, can be extracted (unique) solution to a particular problem by requiring the satisfaction of those which are called initial conditions and boundary conditions, relating to the specific problem. To give an example, in the particular case of cryosurgical application, the temperature distribution at a given instant will be provided by the resolution of a differential equation having, as an initial condition, the temperature at the initial instant of the whole tissue ( 37 degrees C) and, as a boundary condition, the temperature reached on the respective edges from each cryoprobe; each configuration cryoprobes give rise to certain boundary conditions, which will identify the temperature distribution on the particular configuration.
  The problem of planning cryosurgical can be seen as the inverse of the problem described before: the purpose of the schedule is, in fact, to derive the boundary conditions ideal for providing the temperature distribution more suited  to the purpose of the operation that is being cryosurgery to put in place. In other words, assuming that, acquired the edge of the tumor area, the objective of the operation is to bring all the area contained in it to a temperature below a given threshold, however, at the same time, maintaining everything that is place outside it at a temperature above that threshold , the planning cryosurgical must provide those boundary conditions that are functional to getting the temperature distribution desired.

  In general, the direct problem of the determination of the solution of a differential equation given the initial conditions and boundary and the inverse problem of determining from the solution of a differential equation of the corresponding conditions, are radically different. It can be shown that, in the case of the heat equation, the direct problem is always associated with a single solution that depends from continuity of the data, which means that, in the case where there is a lack of precision in the boundary conditions, it will manifest content on the accuracy of the solution. The same statement is not, however, more true when it comes to the inverse problem for the heat equation: here, in fact, small deviations in the temperature data can lead to huge errors in the definition of the parameters boundary and, moreover, cannot always be true that a solution to the inverse problem exists or is unique. These intrinsic characteristics of the inverse problem (i.e., absolutely independent of the mathematical tools) are what make it difficult to realize fast and effective tools that solve the problem of cryosurgical planning. In addition, when (as in cryosurgery) it has to do with biological tissues, the difficulty of the problem is heightened by the fact that the propagation of heat is regulated by non-standard equations that must take account of external contributions of blood and tissues surrounding the passage of state due to freezing.
  The literature on mathematical techniques able to define the most suitable configuration for  cryosurgical operation is not thick and is mostly composed of iterative techniques. These are techniques that work by updating the set of possible solutions moving, iteration by iteration, of optimal directions, i.e. along pathways that lead to the desired solution.


The cryosurgical planning as a mathematical problem - Part 2.


  The problem of the cryosurgical planning is to find, for each patient, the configuration of the parameters most suitable for the purpose of the operation, once it has been defined as the tumor area to be treated through the use of imaging tools. In other words, those parameters must be defined as the number, position and insertion depth of cryoprobes, the optimum temperature of each cryoprobe, etc. necessary to enable the freezing of the entire tumor tissue chipping away at the smallest part of healthy tissue. A technique which solves the problem of planning must be able to operate in a rather elastic, because the parameters to optimize or assemblies in which they must be searched may change according to the patient, the tumor with which it has to do or type of cryosurgical treatment that you want to implement.
  The basic requirements in the operation of a technique of cryosurgical planning are essentially four: first, you need a physical-mathematical model that describes the phenomenon of propagation of heat in the tissue considered, then, serve a mathematical apparatus and computational tools in order to solve, for generic boundary conditions, the problem of heat transfer formulated according to the model referred to in the preceding paragraph, for the third, it is necessary to define a tool able to provide an assessment of each configuration so you can order in terms of effectiveness all configurations considered in order to determine unequivocally the best one, and finally, an essential requirement is to define a technique that, using the three referred to above, is capable of bringing the actual identification of the most appropriate configuration parameters for the specific problem considered.

  For more than sixty years we identify in the equation of heat to biological tissues of Pennes the tool that describes the propagation of heat within a tissue, under certain initial and boundary conditions. This equation is derived from that of classical heat which, in the right member, are added two terms, one describing the heat exchange made by the circulation of blood and another, (least significant), which takes into account the metabolic heat. In the moment in which, at the edge, will reach temperatures below -5 degrees centigrade, the model must also take account of the change of state due to freezing: in this case, the coefficients are part of the equation of Pennes must be updated according to the state ( or temperature range ) of each portion of tissue. The change of state does not occur in biological tissues at a fixed temperature value but in a temperature interval of about 7 degrees to the horse's temperature of -5 Celsius: this is due to the fact that, in general, the biological tissue is not a pure substance.
  In the case of the equation of Pennes (as for most of the partial differential equations) the exact solution can be calculated only in special situations such as, for example, in case you want to calculate the propagation of heat in regions that have particular symmetry properties or tissues whose thermal parameters are not influenced by the temperature value. For all other situations involving the use of software for the calculation of the approximate solutions of the differential problem that, in this case, must be rewritten in a form that is readable to the computer.
  By their nature, the elements that come into play in a differential problem are of infinite dimension that is impossible to be stored in a memory, no matter how large, has finite dimension. For this reason, once you have chosen the most suitable mathematical approach to the need, the problem must be made discrete so it can be stored and sorted by the machine with a minimum margin of error of approximation.




 

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