A New Fuzzy Control Model for Kidney Patients

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1 Appled Mathematcs, 0, 3, Pblshed Onlne Septembe 0 ( A ew Fzzy Contol Model fo Kdney Patents Mna Lagzan, Mohammad Shazan, Al Vahdan Kamyad, Malhe Layeghan, Reza ekmat Depatment of Appled Mathematcs, Fedows Unvesty of Mashhad, Mashhad, Ian Depatment of ephology, Ghaem osptal, Mashhad, Ian Emal: math.lagzan64@gmal.com, shazan65@yahoo.com Receved Jne 3, 0; evsed Agst 6, 0; accepted Agst 3, 0 ABSTRACT The fnal dagnoss of some dseases depends on many factos and nfomaton. Consdeng all these factos and evewng all of them s a dffclt pocess, ths povdng a mathematcal model that can smltaneosly consde all these factos s a geat help fo physcans to dagnose and teat these dseases. In ths pape, we popose a new fzzy contol model fo kdney patents. Fo the nfeence and conclson of ths model, we se Mamdan appoach. Also we employ a new method to smooth the well-known non-smooth pecewse membeshp fnctons,.e. tapezodal and half tapezodal membeshp fnctons. Keywods: Mathematcal Modelng; Fzzy Contol Model; Kdney Tansplantaton; Membeshp Fnctons. Intodcton Thee ae many papes abot mathematcal applcatons n medcal feld. These papes sally ted to popose mathematcal models to contol dseases, adstng the optmal dosage of medcne to the bette teatment of dsease and etc. As we know, one of the most mpotant and vtal ogans nsde the hman body ae kdneys, whch ae esponsble fo blood fltaton. If ths ogan doesn t wok popely, t can case many poblems fo the body and even the hman de. One way to teat kdney fales s kdney. Befoe the kdney, t s mpotant to know whethe the hman body wll accept the gaft o wll eect t, o whethe the gaft wll wok well o not. If one can pedct them by data obtaned fom the pevos patents nde kdney, t can mpove the confdence of physcans to do the tansplantng o not. In ths pape we popose a mathematcal model to detemne whethe patent s tansplanted kdney wll wok well o not, n the fst week afte, o t has delay n the fncton sng povded medcal nfomaton, befoe. Snce medcal concepts ae not clea and thee s no tanspaent and complete fnalty n medcal sses, the fzzy logc has been sed to popose ths model. To desgn ths fzzy contol system, the most mpotant factos that egadng medcal teats and atcles, physcan s decsons, they can affect the eslts of kdney ae gende, nsln esstance, age and so on. Categozng each of these factos, we offe the fzzy contol les as IF-TE. Afte detemnng these les, whch ae the bass of the fzzy contol model, we exet the npts (ndces) that ae elated to a patent who has been selected fo kdney, to the system of fzzy les and then we se one of the defzzfcaton methods to the fnal conclsons accodng to the Mamdan appoach. That s, we specfy that the kdney tansplanted to the patent, wll woks well o patent wll have delayed gaft fncton.. The Man Factos o Vaables n Kdney Tansplantaton In ths secton, we consde some ndcatos o factos that ae consdeed to have effect on patent s kdney. Then we dvde each of these factos fo se n the fzzy contol model to the bette classfcaton of patents. Some of the most mpotant factos n the kdney ae gende, nsln esstance, knd of dono, (alve o cadave), mpaed panceatc beta cell o beta cell fncton,, length of dalyss befoe, systolc and dastolc blood pesse befoe and afte the. Consdeng these factos, fst we classfy them accodng to the gende. Then each of the above-mentoned gops s dvded nto two gops accodng to tansplanted kdney s taken fom alve o cadave. Othe factos ae classfed as follows:.. Resstance Facto Wth egads to the expet s sght, f nsln esstance s Copyght 0 ScRes.

2 098 M. LAGZIA ET AL. mease s less than one, a body hasn t nsln esstance. Othewse the body has nsln esstance. If patent s nsln esstance s and fo othe patent s.00, can we conclde that the fst patent has nsln esstance and the othe one hasn t? It seems that t sn t a te conclson. Fo solvng ths mstake and havng moe clealy and closely conclson, wth egads to expet s pont of vew, we popose a fzzy fncton fo nsln esstance facto. ow, when we want to esponse a qeston abot has patent's body have nsln esstance o not, the answe s what membeshp degee of nsln esstance has the patent s body. Fge shows the membeshp fncton of ths facto... Impaed Panceatc Beta Cells In the case of does patent s body have an mpaed panceatc beta cell o not, we do the same as nsln esstance facto. It means that we explan the vale of mpaed panceatc beta cell by ts membeshp degee. Fge shows the membeshp fncton of ths facto: We explan othe factos as the same as and, wth fzzy membeshp fnctons except that these factos have moe than one membeshp fncton (Table ). Fges 3-5, show the membeshp fnctons. Table. Some of the man factos fo wth the classfcatons. Gops Factos Low (L) omal () gh () Age Less than 40 Between 35 and 55 Moe than 50 Less than 8 Between 7 and 6 Moe than 5 Dal month Less than 40 - Moe than 36 L Fge 3. Age membeshp fnctons. L t(yea).3. Systolc and Dastolc Pesse omal (): Less than 0 and less than 80. Slghtly hgh (S): 0-39 o gh (): o Fge 4. membeshp fnctons. L.75 esstance vale Fge. esstance membeshp fncton. 75 Beta cell vale Fge. Membeshp fncton of beta cell fncton Fge 5. Dal month membeshp fnctons. Dal Month Vey hgh (V): Geate than o eqal to 60 o 00. Accodng to the physcans sght abot the patent s blood pesse, snce the nomal case has been consdeed as and and the othe cases as o, we se the podct and maxmm of membeshp degees, espectvely. The coespondng membeshp fnctons ae depcted n Fges 6 and Appoxmaton of Classcal Membeshp Fnctons by Smooth Membeshp Fnctons As we know, membeshp fnctons play a key ole n fzzy contol method. On the othe hand, wokng wth Copyght 0 ScRes.

3 M. LAGZIA ET AL. 099 S V Systolc blood pesse Fge 6. Systolc blood pesse membeshp fncton. Fge 8. Vey hgh age membeshp fncton. S V Dastolc blood pesse Fge 7. Dastolc blood pesse membeshp fncton. smooth fnctons, fo the eason of applyng t-nom and s-nom opeatos, especally n Mamdan contol s mch ease than wokng wth seveal cteon fnctons and classcal pecewse lnea ones. Theefoe, n ths secton we smooth all of the membeshp fnctons shown n Fges 3-7. Fo ths ppose, we se the method pesented n [], that mplemented the stochastc and goal fzzy pogammng. In ths method, each fzzy membeshp fncton n the fom of tangla, tapezodal and half tapezodal can be appoxmated by the smooth fncton n geneal fom. We show 3 of these dagams n Fges 8-0 whch ae appoxmated smooth fnctons of age membeshp fncton shown n Fge 3: x tan p x whee px s a polynomal ee, fo example, the age (low, nomal and hgh) membeshp fnctons obtaned as follows: Low age: x tan x omal age: 4 x tan x 45 gh age: x tan x x Fge 9. Low age membeshp fncton. Fge 0. omal age membeshp fncton. 4. Poposed Fzzy Contol System and Basc Rles Fzzy easonng eqes some basc les (nfeence les) whch can be desgned accodng to expet s decson. Desgnng these basc les that ae n the fom of f-then, the emaned steps wll be as follows: ) Detemne membeshp degee n antecedent of each le; ) Calclate ence of les; 3) Aggegate ence of les. Befoe explanng the appopate IF-TE les, t s mpotant to menton that the tlzed nfeence pocede s Mamdan method. The man dea of Mamdan Copyght 0 ScRes.

4 00 M. LAGZIA ET AL. contol system s explanaton of pocess cases by lngstc vaables and demonstatng them wth tangla o tapezodal fzzy sets. Fo example, we can take nto consdeaton the age lngstc vaable as low, medm (nomal) and hgh tems. Then by the method mentoned n Secton, we cold change these lngstc vaables nto fzzy sets (membeshp fnctons) and then change them nto smooth membeshp fnctons by the method pesented n Secton 3. In geneal, the les that connect the npt vaables to otpt vaables, ae n the followng fom: Rle : f x s A and x s A and x s A then s A n n n fo otpt o TE secton to only takes vales 0 and. We consdeed ths membeshp fncton n the followng fom: 0 x, x x x whee Bt the fst step n fzzy contol models s calclaton of antecedent membeshp vale. Mn opeato s sed fo a sample of and. Becase of podct opeato havng compensaton popety, we se ths opeato nstead of mn opeato. m le s vale s calclated as follows: npt mn,,, n x mn, whee, A s -th tem of -th lngstc vaable, coespondng to membeshp fncton x and A s the otpt whch detemnes whethe the patent s kdney has been delayed o not. In the seqel, a table of ffteen fzzy les have been sed n ths atcle s gven The eslt of ths pocess, s obtaned by aggegaton of all eslts sng max opeato: Fo example, the fst le of Table s explaned as max follows: Snce the conclson mst gve a cetan vale, 0 o, If age s nomal, s low, dal month s low, blood.e. the gaft has been delayed o not, we eqe one of pesse s vey hgh befoe and hgh afte the adeqate defzzfcaton pocedes. ee the cente, then the gaft has been delayed. of aea (COA) defzzfcaton method has been em- Fo sng the Mamdan contol method, t s eqed ployed fo conclson, that s: Table. Some of the le bases employed n the poposed method. o. Gende Dono (Alve =, Cadava = ) esstance (Yes =, o = 0) Age Beta cell fncton (Yes =, o = 0) Dal month Blood pesse befoe Blood pesse afte Gaft s delayed o = ) 0 0 L L V L V S 0 L 0 4 L L S S 0 L S S 7 0 L L L 0 9 V V S V 3 0 L 0 S 0 4 L S S S 0 Copyght 0 ScRes.

5 M. LAGZIA ET AL. 0 Table 3. Smlaton eslts of the poposed method fo fve patents. o. Gende Dono (Alve =, Cadava = ) esstance (Yes =, o = 0) Age Beta cell fncton (Yes =, o = 0) Dal month Blood pesse befoe Blood pesse afte Gaft s delayed o = ) Smlaton eslt o = ) U d d f U s close to zeo, t s consdeed as zeo, and f t s close to, t s consdeed as. The smlaton eslts whch wee done by MATLAB softwae appoved the accacy of the poposed model. In fact, sng medcal data obtaned fom some new patents who wee ndegong kdney, the smlatons show the consstent of 90 pecent of cases wth the medcal test eslts. The smlaton eslts ae gven n Table 3, fo fve patents. 5. Conclson In ths pape, accodng to medcal data obtaned fom patents ndegong kdney, fzzy les have been wtten and Mamdan fzzy contol model s pesented to detemne enal fncton s delayed o not. Ths eslt whch s gven by zeo o one s sed to ase confdence of medcal doctos fo. Accacy of the poposed model s appoved sng medcal data obtaned fom some new patents who wee ndegong kdney. Fo fte woks, one can pedct the fastng blood glcose sng the ndcatos mentoned n ths pape. 6. Acknowledgements Ths eseach was sppoted by a gant fom Fedows Unvesty of Mashhad, o. MA8749K. Also, we appecate the cente of excellence modelng of lnea and non-lnea systems of depatment of appled mathematcs of Fedows Unvesty of Mashhad fo the sppot to acheve ths eseach. REFERECES [] A. V. Kamyad,. assan Zadeh and J. Cha, An Effcent Appoach fo Solvng a Wde Class of Fzzy Lnea Pogammng Poblems, Intenatonal Confeence on Comptatonal Intellgence and Softwae Engneeng, Whan, -3 Decembe 009. Copyght 0 ScRes.