Modeling the Survival of Retrospective Clinical Data from Prostate Cancer Patients in Komfo Anokye Teaching Hospital, Ghana

Transcription

1 Internatonal Journal of Appled Scence and Technology Vol. 5, No. 6; December 2015 Modelng the Survval of Retrospectve Clncal Data from Prostate Cancer Patents n Komfo Anokye Teachng Hosptal, Ghana Asedu-Addo, Samuel KKwes Nw-Mozu, Isaac Al, Mohammed Unversty of Educaton, Wnneba Ghana Abstract In ths study, we model and estmate how long a patent at Komfo Anokye Hosptal can lve wth prostate cancer dseases knowng the status of the cancer. A total of 199 patents ( ) were retreved. Varables consdered were age, treatment and stage of the cancer. AIC was used to select the best parametrc model. By comparng dfferent parametrc models usng AIC, the result ndcated that Webull model ft best. A straght lne graph of K-M survval aganst tme {log (-log S (t)) vs. log t)} produced confrmed Webull assumpton. A plot of Kaplan Meer hazard wth the Webull hazard further confrmed the Webull assumpton. Keywords: Model, Survval, Prostate Cancer, Kaplan Meer 1. Introducton As scence and mathematcs progress nto technology, abstract thnkng becomes real to the physcal world. The applcaton of mathematcs and statstcs n the area of bologcal and medcal felds has ganed root n recent studes. See for example Van Erruggen et. al. (2005) and Holbrook & Longest (2013). One area n statstcs that makes use of statstcal language to explan the characterstcs behavor of a system s statstcal modelng. As bologcal and medcal problems are vewed by scentsts as quanttatve, statstcal models are formulated to solve those problems. One of the major challenges n medcne s the ablty to determne the best treatment for dagnoss. Thus, the best treatment s one that has passed through a statstcal test to prove sgnfcant effect on a patent. Another challenge s the ablty to estmate how long a patent can recover from a treatment or can lve wth a partcular dsease. One strategy that has contrbuted to resolve such problems s survval modelng. Cox (1972) formulated a model that could help medcal statstcans to estmate probablstc future behavor of a patent based on ether retrospectve cohort or prospectve cohort studes. Accordng to Cleves et al. (2010), formulatng a model helps to obtan some estmates for use n desgnng a future survval study. In a study to determne the predctor varables contrbutng to the survval of prostate cancer n a hosptal n Ghana, Asedu-Addo et. al. (2014) dentfed the stage of cancer at whch a patent s dagnosed as the man contrbutor to the survval of a patent and recommended that future work must be done to model the survval of a prostate cancer patent. Thus, the am of ths paper was to formulate a model that would help to estmate how long a prostate cancer patent can lve wth the dsease. The model was guded by Murray (1993) recommendaton that the art of good modelng must consst of the followng: () a good understandng and apprecaton of the bologcal problem, () the mathematcal model representng dealzaton of the physcal laws takng nto account assumpton n order to make the model tractable, () approprate soluton must rely on usng numercal technques and (v) the soluton obtaned should be consstent wth physcal ntuton and evdence. 2. Data Descrpton Data for ths study consst of all prostate cancer patents dagnosed at Oncology Department at Komfo Anokye Teachng Hosptal (KATH) n Kumas, Ghana wthn a 9 year perod (2005 to 2013). Patent s records were evaluated for the endponts of ether survval or death. In 11 cases out of 210, no nformaton of patent s records was retrevable and hence excluded from the analyss. Ths approach s smlar to the study conducted by Sddqu et al (2001) n Pakstan breast cancer patents. 79

2 ISSN (Prnt), (Onlne) Center for Promotng Ideas, USA Thus, n all 199 prostate cancer patents were used n ths study. Generally, n research, data are taken over a fnte perod of tme. Sometmes, tme to event may not be observed for most varables n the study populaton. Ths result s called censored data. Such observaton s predomnant n the area of medcne. In medcne, durng dagnoss of a partcular dsease, all patents n the study do not have the same entry tme. Thus, the amount of follow up vares from patent to patent. Also the tme to follow up untl patents de wth the dseases under treatment may not be observed due to a patent stll alve before the study ends, a patent s lost to follow-up durng the study perod or patents wthdraws from treatment. Hence, censorng and dfferental followup cause dffcultes n the analyss of such data that cannot be handled by the standard statstcal methods such as logstc regresson or ordnary regresson model. Survval analyss s dfferent from the other statstcal methods. In survval analyss, dependent varable s always tme untl occurrence of an event of nterest and therefore survval data are generally skewed and thus, the assumpton of normalty does not hold. In ths study, the followng category of patents were censored: patents who were alve and dsease-free; patents who ded from other causes other than prostate cancer; patents who were stll alve wth the dsease as at ther last follow up date; and, patents who were lost to follow-up. 3. Mathematcal Formulaton The survval and hazard functons are key concepts n survval analyss for descrbng the dstrbuton of event tmes. The survval functon gves, for every tme, the probablty of survvng (or not experencng the event) up to that tme. The survval functon denoted by S s defned by S( t) Pr( T t) Where, t s some tme, T s a random varable denotng the tme of death, and "Pr" stands for probablty. That s, the survval functon s the probablty that the tme of death s later than some specfed tme t. The hazard functon ( t) gves the potental that the event wll occur, per tme unt, gven that an ndvdual has survved up to the specfed tme. For contnuous random varables, the hazard functon s gven by 1 Pr( t T t t) S( t) ( t) lm. t0 t Pr( T t) S( t) A number of models are avalable to analyze the relatonshp of a set of predctor varables wth the survval tme. Methods nclude parametrc, nonparametrc and sem parametrc approaches. In ths work parametrc method was employed Parametrc Survval Analyss Parametrc analyss s a branch of statstcs that assumes that data has come from a type of probablty dstrbuton. In order to perform a parametrc survval analyss, we need to dentfy an approprate dstrbuton functon that characterzes the behavor of the survval tmes. An accelerated falure tme model s a parametrc model that consders the lnear relatonshp between the logarthm of survval tme and covarates of nterests. Furthermore, we assume that the survval tme follows a gven theoretcal probablty dstrbuton. Let T denote a contnuous non-negatve random varable representng the survval tme and X be the ndependent varables (covarates) n the model. The analytcal structure of the falure model s gven by log( T ) X (1) 80 Where s an error term that has a sutable probablty dstrbuton such as normal, Webull or gamma and s a constant to be determned. By exponentaton of (1), we obtan a model for survval tme T exp( ) T0 X (2) Where T 0 s the exponentated error term? The three most wdely used parametrc models; exponental, Webull and Log-logstc were consdered n ths study. The defnton of the dstrbutons consdered n ths study was taken from Table man and Km (2003) as follows: 3.2 The Exponental Dstrbuton Exponental dstrbuton has one parameter wth hazard functon assumed to be constant over tme. Consder the densty functon f ( t ) gven by:

3 Internatonal Journal of Appled Scence and Technology Vol. 5, No. 6; December 2015 f ( t) exp( t) (3) The survval functon S( t ) s gven by S( t) f ( t) dt (4) 0 The correspondng hazard functon s f ( t) ( t) S( t) (5) 3.3 The Webull Dstrbuton The Webull model has two parameters and t s obtaned f the error term ϵ has an extreme value dstrbuton wth the followng densty: x f ( x) exp( x exp ), x Equvalently, T has the Webull dstrbuton wth the followng densty: k 1 k f ( t) k( t) exp( ( t) ), t 0 (6) The survval functon s: S( t) f ( t) dt 0 exp( ( t) k ) (7) The hazard functon s gven as: f ( t) ( t) S( t) 1 k( t) k (8) Where s the scale parameter and k s the shape parameter. The Webull dstrbuton s convenent because of ts smple form. It ncludes several hazard shapes and reduces to exponental dstrbuton when k Logs-Logstc Model The lfetme T s log-logstcally dstrbuted f Y log( T ) s logstcally dstrbuted wth locaton parameter and scale parameter. Hence, Y s also of the form Y Z Where Z s a standard logstc random varable wth densty exp( z) (1 exp( z)) 2, z Equvalently, T has the log-logstc dstrbuton wth the followng densty: 1 ( ) ( ) k f t k t, t 0, k 0, 0 (9) We have the survval functon as S( t) f ( t) dt 0 1 (10) 1 ( t) k 81

4 ISSN (Prnt), (Onlne) Center for Promotng Ideas, USA The hazard functon s gven as: f ( t) ( t) S( t) k 1 k( t) ( t) (11) k 1 ( t) 4. Model Selecton In ths study, the Akake Informaton Crteron (AIC), whch s one of the most wdely used n selectng the best competng fttng parametrc models, was employed. The AIC s defned n Dez (2012) as: AIC 2 Log(maxmum lkelhood) k p Where p s number of parameters n each model under consderaton and k a predetermned constant. The lower the AIC, the better the model. 4.1 Dstrbuton Assumptons If the Kaplan-Meer survval functon can be estmated, then a straght lne graph of survval aganst tme ndcate a partcular parametrc dstrbuton. A straght lne graphcal assumpton for each dstrbuton s summarzed n Table 1. Tale 1: Parametrc Survval Models Assumpton Exponental Weull Log-Logstcs Survval S(t) S ( t ) exp( t ) k S( t) exp( ( t) ) 1 S( t) 1 ( t) k Assumpton log S( t) vs. tme t log( log S( t)) vs. log t S( t) log vs. log t 1 S( t) both Korosteleva (2008) and Machn et al. (2006) ndcated that Kaplan Meer survval graph can be used to check model approprateness. Thus, f ether exponental or Webull or log-logstc dstrbuton survval graph behaves smlarly to Kaplan Meer survval curve, then that partcular model ft the data best. Thus, from ths suggeston, the reverse s also true snce the complment of survval s hazard. That s, f any of the dstrbuton behaves smlarly to Kaplan Meer hazard graph then that partcular dstrbuton ft the data best. 5 Results and Analyss Table 2 provdes the AICs for each parametrc model under study. The AIC result gave approxmate equal values for both the Webull and Log-logstc models for patent survval whch does not depend on the predctor under consderaton. Table 2: Model selecton usng AIC for no predctor Exponental Weull Log-Logstcs AIC: no predctor AIC: full predctor The AIC for full predctors n Table 2 ndcates that the Webull dstrbuton s reasonable to model the survval and hazard functon. 82

5 Internatonal Journal of Appled Scence and Technology Vol. 5, No. 6; December 2015 Fgure 1: Webull Assumpton graph for no predctor A graph of log t aganst log( log( S( t))) should produce an approxmate straght lne f the Webull assumpton s true where t s the survval tme and S( t ) s the Kaplan Meer survval of tme. Fgure 1 produces an approxmate straght lne suggestng that the Webull dstrbuton s approprate. Fgure 2: Webull and K-M survval curve for no predctor Further assumpton s checked f the Webull model agrees wth the Kaplan Merer survval functon. From Fgure 2, the parametrc curve goes ncely through the Kaplan-Meer estmates. The output n Tale 3 gves the estmaton of the Webull model. Thus, the output of the Webull dstrbuton n table 3 lead to the survval and the hazard models wth no predctor for equaton (10 and 11) respectvely. The output n Tale 3 gves the estmaton of the Webull model. Tale 3: Weull output for no covarate Value Std. Error Z P Intercept e log (scale) e Thus, the output of the Webull dstrbuton n table 3 lead to the survval and the hazard models wth no predctor for equaton (10 and 11) respectvely S( t) exp( t ) (10) h( t) t (11) It can be seen that the hazard rate of a prostate cancer patent ncreases slowly wth tme. Ths means that the longer a patent stay n the study wthout any nfluence by covarate, the hgher hs chance of experencng the event. The Webull model gves a close approxmaton to the Kaplan Meer estmator by comparson. For nstance, the probablty that a patent wll survve n the 250 days s estmated by the Webull as (approxmately 1, hgher survval) and that of the Kaplan Meer estmate s about

6 ISSN (Prnt), (Onlne) Center for Promotng Ideas, USA The result also shows that a prostate cancer patent wll eventually experence death at the long run. The parametrc model become most powerful by most researchers than the non parametrc technque n the sense that, the parametrc models gve an exact estmate than the non-parametrc estmator whch can only gve up to two decmal places. Also, the parametrc technque can gve a future estmate whch the Kaplan Meer non parametrc cannot do. For nstance, we can estmate the survval and hazard tme for a patent n the next 5000 days for parametrc model but cannot be estmated by the Kaplan Meer estmator. The model only explan the survval and hazard rate of a patent but does not account for the factors that yeld ths results. It s therefore, mportant to nvestgate f covarate (age, treatment, cancer stage) have sgnfcant nfluence on a prostate cancer survval or death. The nvestgaton can be done n two ways. One s to add possble covarates one at a tme and select those whch are sgnfcant. The second s to start wth the full model whch ncludes all possble covarates, and then knock those whch show no sgnfcant effect on the survval functon. For the purpose of ths study, the second strategy s adopted. Earler study by Asedu-Addo et. al. (2014) gave the output of sgnfcant predctor as shown n Tale 4. Predctor Intercept Age Treatment Stage 2 Stage 3 Table 4: P-Values for full predctors P-Value e e e e e 198 Stage e It can be seen from Table 4 that the stage 4 of the prostate cancer appear to be sgnfcant but age at whch patents were detected prostate cancer was not sgnfcant. Patents treatments shows no sgnfcant as reported by the Wald test. If the stage of the cancer of a patent s sgnfcant t must satsfy the Webull assumpton. The log-log for both stage one and stage two cannot be plotted and so have been omtted. Ths s because no patent experences the event for stage one cancer and so ts log-log graph cannot be estmated. Only one patent experences the event and so ts log-log graph s also omtted. Fgure 3: Webull assumpton for prostate cancer stages (stages 3 and 4) Agan a graph of log( t) aganst log( log( S( t))) should produce an approxmate straght lne f the Webull assumpton s true. From fgure 3 t can be observed that both graphs gve an approxmate straght lne, suggestng that Webull assumpton s met. The valdaton of the Webull assumpton s further cheeked wth the K-M. graph as show n Fgure 4. We observe that both graphs gve an approxmate straght lne, suggestng that Webull assumpton s met. The valdaton of the Webull assumpton s further checked wth the K-M 84

7 Internatonal Journal of Appled Scence and Technology Vol. 5, No. 6; December 2015 Fgure 4: Webull and K-M hazard graph for prostate cancer stages It can be observed that the Webull dstrbuton goes farly smoothly wth the Kaplan-Meer hazard functon gvng ndcaton to use the Webull model. Accordng to Korosteleva (2008), t s wse to rerun the model wthout the nsgnfcant predctors. Based on ths suggeston, only the predctor (stage of the cancer) s ncluded n the model and the result of the Webull output s as shown n Table 5. Table 5: Webull estmaton Value Std. Error Z P Intercept e+00 Stage e 59 Stage e+00 Stage Inf 0.00e+00 Log (scale) e 05 The survval and hazard functons for ths model are summarzed n the Table 6: Survval Stage model S t x (, stage 1) exp( t ) Table 6: Webull survval and hazard model Hazard stage model h( t, x ) stage 1 t S t x (, stage 2) exp( t ) S t x (, stage 3) exp( t ) S t x (, stage 4) exp( t ) h( t, x ) stage 2 h( t, x ) stage 3 h( t, x ) stage 4 t t t The estmated acceleraton factor γ comparng the cancer stage 2, stage 3 and stage 4 to stage 1: [(stage 1 vs. stage 2),(stage 1 vs. stage 3),(stage 1, stage 4) ] s gven by: exp( ), where s the coeffcent of each cancer stage. The estmated acceleraton factor s gven by Table 7. Table 7: Estmated acceleraton factor for patents cancer stages Cancer stage exp( ) Stage e Stage e Stage e From Table 7, t can e seen that the estmated coeffcent for stage 2, stage 3 and stage 4 are negatves mplyng that the survval for patent wth prostate cancer stage 2, stage 3 and stage 4 are decreased by a factor of e 16, e 16 and e respectvely. 85

8 ISSN (Prnt), (Onlne) Center for Promotng Ideas, USA In all, the survval rate decreases wth ncrease n prostate cancer stages. Alternatvely, the correspondng hazard can have ts nterpretaton as havng a prostate cancer stage 2 accelerated death by a factor of e relatve to a stage 1 prostate cancer whle havng a prostate cancer stage 3 accelerated death by a factor of e relatve to a stage 1 prostate cancer and a prostate cancer stage 4 accelerated death by a factor of e relatve to a stage 1 prostate cancer. Overall, the hazard of the stages of the cancer s seen to ncrease n cancer stages relatve to stage 1. For nstance, the hazard or rsk of a patent dyng wth a prostate cancer n the next 15 years (5475 days) wth stage 1, stage 2, stage 3 and stage 4 are , , and respectvely. 6 Dscusson and Concluson Usng the AIC to compare the three most common used dstrbutons, t s found that the Webull dstrbuton s consdered the best ft for the prostate cancer data snce the AIC value for the Webull model s smaller than the Exponental dstrbuton. Agan by the law of parsmony, the Webull dstrbuton was consdered over the Loglogstc snce s smply used. The Webull dstrbuton s seen to be very close to the non-parametrc Kaplan-Meer dstrbuton, suggestng model accuracy. Table 4 shows patents survval does not depend on age or treatment gven. Ths means a patent of age 89 years wth stage 3 prostate cancer wll lkely have the same survval wth patent wth age 40 years. The nterpretaton of the exact value of the co-effcent cancer stage s ntutve. However, the negatve sgn ndcate that, the survvor rate of a patent decreases wth ncrease n prostate cancer stage. The value of stage 1 cancer s gven by the Webull ntercept whch gave a postve co-effcent sgn ndcatng a perfect survval or low rsk. References Asedu-Addo, S. K., Nw-Mozu, I., Assuah, C. K. & Asumadu, O. R. (2014). Determnaton of predctor varables contrbutng to the survval of prostate cancer patents n Komfo Anokye teachng Hosptal, Ghana. Internatonal Journal of Appled Scence and Technology. Vol 4 No. 6 Cleves, M., Gould, W., Guterrez, R. G. & Marchenko, Y. V. (2010). An Introducton to Survval Analyss Usng Stata. Stata Press, Thrd Edton Cox, D. R. (1972). Regresson models and lfe tables. Journal of the Royal Statstcal Socety, 34(2): Seres B (Methodologcal). Dez, D. (2012). Survval Analyss n R. Holbrook, L.T. & Longest, P.W. (2013). Valdatng CFD predctons of hghly localzed aerosol deposton n arway models: n vtro data and effects of surface propertes. Journal of Aerosol Scence, Vol. 59 pp Korosteleva, O. (2008). Clncal Statstcs: Introducng Clncal Trals, Survval Analyss, and Longtudnal Data Analyss. Second edton. Machn, D., Cheung, Y. B. & Mahesh K. B. (2006). Survval Analyss: A Practcal Approach. Murray, J. D. (1993). Mathematcal Bology: Berln: Sprnger-Verlag Sddque, T., Sabh, M., Khan, S. & Salam, A. (2001). A survval analyss of metastatc breast cancer n Pakstan patents. Journal of Pakstan Medcal Assocaton, 54(67). Tableman, M. & Km, J. S. (2004). Survval Analyss Usng S: Analyss of Tme-to-Event Data. Chapman and Hall/CRC Van Ertbruggen, C., Hrsch, C. & Pava, M. (2005). Anatomcally based three-dmensonal model of arways to smulate flow and partcle transport usng computatonal flud dynamcs. Journal of Appled Physology 98(3),