A model for lung cancer risk prediction that combines deep learning features from the Sybil model with clinical and epidemiological factors.
The Sybil-Epi model was calibrated using isotonic regression with 5-fold cross validation (default setting), from the scikit-learn package.
Given a subject with the factors below1:
| Factor | Value |
|---|---|
| 6-year Risk Sybil2 | 0.123949024 |
| Age (years) | 63 |
| BMI (kg/m2) | 28.88 |
| COPD (0-yes, 1-no) | 0 |
| Education level3 | 6 |
| Ethnicity - White (0-yes, 1-no) | 1 |
| Ethnicity - Black (0-yes, 1-no) | 0 |
| Ethnicity - Asian (0-yes, 1-no) | 0 |
| Ethnicity - Others (0-yes, 1-no) | 0 |
| Family lung cancer history (0-yes, 1-no) | 0 |
| Personal cancer history (0-yes, 1-no) | 0 |
| Smoking duration (years) | 40 |
| Smoking intensity (cigarrettes per day) | 1.0 |
| Smoking quit time (years) | 40 |
| Smoking status (0-former, 1-current) | 0 |
| Age x Smoking duration (years2) | 2520 |
1All factors were measured using the units indicated in the PLCOm2012 model.
2The 6-year Risk Sybil value can be calculated from a single low-dose CT image, analyzed using the Sybil model.
3Education was measured in six ordinal levels: less than high-school graduate (level 1), high-school graduate (level 2), some training after high school (level 3), some college (level 4), college graduate (level 5), and postgraduate or professional degree (level 6).
First, calculate the sum after multiplying each factor by the coefficients indicated in section 1. Coefficients for each factor included in the Sybil-Epi model
Note that the intercept must be added to the calculation. Also, note that the sum must be calculated for each of the indicated 5 models.
Calculation for Model 1: -20.79038191000 + 0.123949024 x 7.02617157000 + 63 x 0.21847996200 + 28.88 x -0.01732394310 + 0 x 0.18149021100 + 6 x -0.08116632920 + 0 x 1 + 0 x 0.23084095400 + 0 x 0.17212853500 + 0 x 0.36442735200 + 0 x 0.09914501700 + 0 x 0.20821163600 + 40 x 0.36915862700 + 1.0 x 0.01823174660 + 40 x 0.00262488699 + 0 x 0.03382896010 + 2520.11 x -0.00458450762 = -3.805957544
Calculation for Model 2: -0.24621993000 + 0.123949024 x 6.96079105000 + 63 x -0.09406773960 + 28.88 x -0.01994318480 + 0 x 0.23223599200 + 6 x -0.08115994020 + 0 x 1 + 0 x 0.15766851600 + 0 x 0.43564514700 + 0 x 0.30574606700 + 0 x 0.08925881800 + 0 x 0.24459386000 + 40 x -0.04855800000 + 1.0 x 0.01969007140 + 40 x 0.01330713690 + 0 x 0.14971355100 + 2520.11 x 0.00171930035 = -3.430330657
Calculation for Model 3: -7.40740562000 + 0.123949024 x 6.92597482000 + 63 x 0.01853234780 + 28.88 x -0.02247213930 + 0 x 0.18960467600 + 6 x -0.05201051400 + 0 x 1 + 0 x 0.23894223900 + 0 x 0.31138165200 + 0 x 0.59204369500 + 0 x -0.00728941902 + 0 x 0.17568551900 + 40 x 0.13150375700 + 1.0 x 0.01734555020 + 40 x 0.00645552283 + 0 x 0.11961590300 + 2520.11 x -0.00103727472 = -3.420673907
Calculation for Model 4: -14.98083140000 + 0.123949024 x 7.24776249000 + 63 x 0.13195954400 + 28.88 x -0.01906357390 + 0 x 0.37200519100 + 6 x -0.06836542110 + 0 x 1 + 0 x 0.22021814400 + 0 x -0.05802796240 + 0 x 0.54687080400 + 0 x 0.05297350630 + 0 x 0.17857532900 + 40 x 0.30506137200 + 1.0 x 0.01915918910 + 40 x 0.00959059658 + 0 x 0.17588838900 + 2520.11 x -0.00367140897 = -3.376488254
Calculation for Model 5: 0.01625732000 + 0.123949024 x 7.47072426000 + 63 x -0.10165534300 + 28.88 x -0.02823116440 + 0 x 0.47694046700 + 6 x -0.05069236120 + 0 x 1 + 0 x 0.45120666500 + 0 x 0.02304009340 + 0 x -0.03478278100 + 0 x 0.02851307510 + 0 x 0.07705976050 + 40 x -0.16755627900 + 1.0 x 0.01929939040 + 40 x 0.00837984637 + 0 x 0.14322364400 + 2520.11 x 0.00381010544 = -3.327802709
Second, find the corresponding score for each sum using the isotonic function tables in section 2. Isotonic function for each model
| Model | Sum | Range in corresponding isotonic function | Calibrated score |
|---|---|---|---|
| Model 1 | -3.805957544 | [-3.80976013309103, -3.76522223698408] | 0.0519480519480519 |
| Model 2 | -3.430330657 | [-3.50322676511143, -3.38260458926028) | 0.0252427184466019 |
| Model 3 | -3.420673907 | [-3.4288999459988, -3.41768951227193) | 0.0212765957446808 |
| Model 4 | -3.376488254 | [-3.58668557313358, -3.36313485155923) | 0.0199155099577549 |
| Model 5 | -3.327802709 | [-3.49548891109088, -3.19647540660684) | 0.00674915635545556 |
*If the sum does not fall within any of the listed ranges, then linear interpolation between the closest ranges must be applied.
Finally, the lung cancer risk score assigned to the subject used in this example is calculated as the average of the calibrated scores from all models
Calibrated score = (0.0519480519480519 + 0.0252427184466019 + 0.0212765957446808 + 0.0199155099577549 + 0.00674915635545556) / 5 = 0.025026
| Factor | Model 1 | Model 2 | Model 3 | Model 4 | Model 5 |
|---|---|---|---|---|---|
| Intercept | -20.79038191000 | -0.24621993000 | -7.40740562000 | -14.98083140000 | 0.01625732000 |
| 6-year Risk Sybil | 7.02617157000 | 6.96079105000 | 6.92597482000 | 7.24776249000 | 7.47072426000 |
| Age | 0.21847996200 | -0.09406773960 | 0.01853234780 | 0.13195954400 | -0.10165534300 |
| BMI | -0.01732394310 | -0.01994318480 | -0.02247213930 | -0.01906357390 | -0.02823116440 |
| COPD | 0.18149021100 | 0.23223599200 | 0.18960467600 | 0.37200519100 | 0.47694046700 |
| Education level | -0.08116632920 | -0.08115994020 | -0.05201051400 | -0.06836542110 | -0.05069236120 |
| Ethnicity - White | 0.0 (Reference) | 0.0 (Reference) | 0.0 (Reference) | 0.0 (Reference) | 0.0 (Reference) |
| Ethnicity - Black | 0.23084095400 | 0.15766851600 | 0.23894223900 | 0.22021814400 | 0.45120666500 |
| Ethnicity - Asian | 0.17212853500 | 0.43564514700 | 0.31138165200 | -0.05802796240 | 0.02304009340 |
| Ethnicity - Others | 0.36442735200 | 0.30574606700 | 0.59204369500 | 0.54687080400 | -0.03478278100 |
| Family lung cancer history | 0.09914501700 | 0.08925881800 | -0.00728941902 | 0.05297350630 | 0.02851307510 |
| Personal cancer history | 0.20821163600 | 0.24459386000 | 0.17568551900 | 0.17857532900 | 0.07705976050 |
| Smoking duration | 0.36915862700 | -0.04855800000 | 0.13150375700 | 0.30506137200 | -0.16755627900 |
| Smoking intensity | 0.01823174660 | 0.01969007140 | 0.01734555020 | 0.01915918910 | 0.01929939040 |
| Smoking quit time | 0.00262488699 | 0.01330713690 | 0.00645552283 | 0.00959059658 | 0.00837984637 |
| Smoking status | 0.03382896010 | 0.14971355100 | 0.11961590300 | 0.17588838900 | 0.14322364400 |
| Age x Smoking duration | -0.00458450762 | 0.00171930035 | -0.00103727472 | -0.00367140897 | 0.00381010544 |
| Range | Score |
|---|---|
| [-6.83084097765182, -5.54535145507885] | 0.0 |
| [-5.54443742079984, -5.31175599485575] | 0.00157480314960629 |
| [-5.31143200626821, -5.08199266314459] | 0.00488997555012224 |
| [-5.0816819409446, -5.03867510600285] | 0.0104166666666666 |
| [-5.03822641268466, -4.9376550049408] | 0.0143540669856459 |
| [-4.93695653610866, -4.76914044854747] | 0.0175849941383352 |
| [-4.76895524803483, -4.52447036706167] | 0.0215404699738903 |
| [-4.52429109634284, -4.1851070855342] | 0.0225340136054421 |
| [-4.18507857823729, -4.13856683549135] | 0.0234604105571847 |
| [-4.13834600108628, -4.12227312343759] | 0.0241935483870967 |
| [-4.1222142981056, -4.05303979153328] | 0.0417536534446764 |
| [-4.05303564896005, -3.86110437983613] | 0.0432653061224489 |
| [-3.86110032761937, -3.80980785285466] | 0.050314465408805 |
| [-3.80976013309103, -3.76522223698408] | 0.0519480519480519 |
| [-3.76469486998651, -3.63506267680873] | 0.0521885521885521 |
| [-3.63504437844557, -3.58218684183874] | 0.0588235294117647 |
| [-3.58212663522492, -3.41422959054526] | 0.062700964630225 |
| [-3.41407934535584, -3.39395767263892] | 0.0862068965517241 |
| [-3.3936194461336, -3.35075959553484] | 0.0983606557377049 |
| [-3.35049833366132, -3.32526202667457] | 0.126984126984126 |
| [-3.32425346311377, -3.16743242146143] | 0.132832080200501 |
| [-3.16648078040121, -3.11974481801039] | 0.158730158730158 |
| [-3.11949547782089, -2.88716205883827] | 0.187943262411347 |
| [-2.88679264910134, -2.28351814304314] | 0.192737430167597 |
| [-2.28341771334039, -2.25075681964111] | 0.222222222222222 |
| [-2.24722264484806, -1.81503763638207] | 0.3125 |
| [-1.80981471342024, -1.56220188001589] | 0.510204081632653 |
| [-1.56203367410988, -0.883730771581255] | 0.528089887640449 |
| [-0.881204139655352, -0.747571012833969] | 0.588235294117647 |
| [-0.747516347831869, 0.26902159779528] | 0.60204081632653 |
| [0.347617076263695, 0.350744122092173] | 0.666666666666666 |
| [0.364813256341062, 1.32971502224201] | 0.790697674418604 |
| [1.33205349683115, 2.25257322093028] | 0.8 |
| [2.28793489335508, 3.22150837109364] | 1.0 |
| Range | Score |
|---|---|
| [-5.13976202645897, -4.45855247691666) | 0.0 |
| [-4.4582698193259, -4.2518450533451) | 0.00130378096479791 |
| [-4.25157297578535, -4.12103947494856) | 0.00404312668463611 |
| [-4.12091177902061, -4.02640561549037) | 0.0045662100456621 |
| [-4.02639931241959, -3.87850590132921) | 0.00699300699300699 |
| [-3.87836333800783, -3.78265108085928) | 0.0122767857142857 |
| [-3.78263902318789, -3.66874059942404) | 0.0186005314437555 |
| [-3.66865649809458, -3.50333686239594) | 0.0245398773006134 |
| [-3.50322676511143, -3.38260458926028) | 0.0252427184466019 |
| [-3.38239050905076, -3.36134026536294) | 0.0261780104712041 |
| [-3.3611225958752, -3.30651259082491) | 0.0295983086680761 |
| [-3.30631233790642, -3.20914725751849) | 0.0464135021097046 |
| [-3.20902487646987, -3.11694062505718) | 0.0606617647058823 |
| [-3.11679462879763, -2.96532175570692) | 0.0787096774193548 |
| [-2.96511797871301, -2.69559229246829) | 0.0954598370197904 |
| [-2.69501884194289, -2.62274546546793) | 0.124260355029585 |
| [-2.6223546843081, -2.48658932345086) | 0.129921259842519 |
| [-2.48610328068047, -2.30092176611133) | 0.151658767772511 |
| [-2.30071351946817, -2.24700122492978) | 0.166666666666666 |
| [-2.24684490923774, -2.01801620372871) | 0.2 |
| [-2.01495607993973, -1.66317335999902) | 0.266272189349112 |
| [-1.6569219881803, -1.08676074827303) | 0.284360189573459 |
| [-1.08249227836323, -1.07366923763311) | 0.333333333333333 |
| [-1.07010276090708, 0.0705479290336056) | 0.503816793893129 |
| [0.0738831826353788, 0.401184842679566) | 0.558823529411764 |
| [0.402345918309606, 0.645414076527883) | 0.676470588235294 |
| [0.653619752845769, 1.38234013948577) | 0.709090909090909 |
| [1.38787187448386, 1.48660866194891) | 0.75 |
| [1.50162544808152, 1.69133780551668) | 0.833333333333333 |
| [1.70614331310756, 2.0612330181618) | 0.923076923076923 |
| [2.14202917083523, 3.91208317660163) | 1.0 |
| Range | Score |
|---|---|
| [-5.40896695555443, -4.44557905591795) | 0.0 |
| [-4.44556769824729, -4.20796960678975) | 0.0050251256281407 |
| [-4.20739595368818, -3.91552915782963) | 0.00583657587548638 |
| [-3.91548039595661, -3.5345198348362) | 0.0096698899633211 |
| [-3.53431434149028, -3.43490951844732) | 0.0177725118483412 |
| [-3.43488087385878, -3.42904077520323) | 0.0204081632653061 |
| [-3.4288999459988, -3.41768951227193) | 0.0212765957446808 |
| [-3.41754960133783, -3.39211171288659) | 0.0241545893719806 |
| [-3.39205300926651, -3.32995536982051) | 0.0264227642276422 |
| [-3.32990172496423, -3.24685531071719) | 0.0287277701778385 |
| [-3.24671206819021, -3.11901522252616) | 0.0375275938189845 |
| [-3.11886226305581, -2.99974728760598) | 0.0395061728395061 |
| [-2.99956474429094, -2.91765833557168) | 0.0506607929515418 |
| [-2.91750760894448, -2.90054878707901) | 0.0666666666666666 |
| [-2.90028915863532, -2.6475189963683) | 0.0817120622568093 |
| [-2.64741434406712, -2.62150132519831) | 0.0952380952380952 |
| [-2.6209657404627, -2.59858231770701) | 0.123076923076923 |
| [-2.59812553384104, -2.4554499391278) | 0.126262626262626 |
| [-2.45529021102402, -2.39397293199902) | 0.130081300813008 |
| [-2.39363220142586, -1.87769220955232) | 0.134304207119741 |
| [-1.87523366083048, -1.84412706174444) | 0.17391304347826 |
| [-1.84244882776309, -1.59368802685856) | 0.211678832116788 |
| [-1.59106147350401, -1.56382787161001) | 0.222222222222222 |
| [-1.55982641375981, -1.4549350688779) | 0.24074074074074 |
| [-1.44902791926516, -1.40284906648665) | 0.25 |
| [-1.40284160717211, -1.31017627785731) | 0.263157894736842 |
| [-1.30232455720342, -1.19292612431877) | 0.333333333333333 |
| [-1.18992323507933, -1.17861833654162) | 0.428571428571428 |
| [-1.17606128036306, -0.413588502397382) | 0.439759036144578 |
| [-0.402302355080826, -0.278037063043519) | 0.5 |
| [-0.252188292713815, 0.28318091010164) | 0.52 |
| [0.284280188256867, 0.828005605867875) | 0.625 |
| [0.860334523461037, 0.996858651407914) | 0.692307692307692 |
| [1.00118109607741, 1.05344457993507) | 0.777777777777777 |
| [1.0754991844332, 1.66608973697844) | 0.878048780487804 |
| [1.76795244660686, 2.91795097929127) | 0.923076923076923 |
| [2.94294776493842, 3.53725043203523) | 1.0 |
| Range | Score |
|---|---|
| [-5.40297596576861, -4.145046360106) | 0.0 |
| [-4.14399863519635, -3.96469127014186) | 0.00229621125143513 |
| [-3.96433456464248, -3.79821097842455) | 0.00884955752212389 |
| [-3.79779347620638, -3.78276999952974) | 0.00952380952380952 |
| [-3.7827334089108, -3.71202032064926) | 0.0116731517509727 |
| [-3.71195047309248, -3.58669159781696) | 0.0128055878928987 |
| [-3.58668557313358, -3.36313485155923) | 0.0199155099577549 |
| [-3.36309455610468, -3.34856928840831) | 0.0253164556962025 |
| [-3.34843950750268, -3.25175025722636) | 0.0260416666666666 |
| [-3.25173129770055, -3.24480312347089) | 0.0476190476190476 |
| [-3.244576314838, -2.89445996228474) | 0.0486707566462167 |
| [-2.89442920543924, -2.43294566623595) | 0.0659340659340659 |
| [-2.43273732882454, -2.35438840883121) | 0.072072072072072 |
| [-2.35399267156437, -2.32665326581679) | 0.0806451612903225 |
| [-2.32644944266306, -2.00073280666356) | 0.0842293906810035 |
| [-1.99758423146622, -1.87517611131506) | 0.102739726027397 |
| [-1.87334395146985, -1.47428118372159) | 0.131832797427652 |
| [-1.47404665717457, -1.2238632031688) | 0.158730158730158 |
| [-1.22335885613264, -0.94569571043684) | 0.288659793814433 |
| [-0.943034578788692, -0.170516290240955) | 0.343915343915343 |
| [-0.159465347170856, 0.00728976386126056) | 0.4 |
| [0.0145462944545293, 0.0674686168349953) | 0.428571428571428 |
| [0.0689045131453163, 0.794428399251971) | 0.541176470588235 |
| [0.794991274019949, 0.860753576674929) | 0.545454545454545 |
| [0.861091367961949, 1.1731265029034) | 0.567567567567567 |
| [1.17327917862875, 1.20119869829976) | 0.6 |
| [1.20950964955107, 1.65124958904896) | 0.612244897959183 |
| [1.6735448032525, 1.86887592828489) | 0.695652173913043 |
| [1.88419092062132, 2.65929028108868) | 0.741935483870967 |
| [2.67248359202892, 2.76664158066157) | 0.8 |
| [2.78591249999188, 3.84403802653575) | 1.0 |
| Range | Score |
|---|---|
| [-5.34372288616349, -4.00900324181277) | 0.0 |
| [-4.008262731788, -3.84370310240735) | 0.00617283950617283 |
| [-3.84285955438415, -3.49562213214767) | 0.00625 |
| [-3.49548891109088, -3.19647540660684) | 0.00674915635545556 |
| [-3.19574010845561, -3.14630729811392) | 0.0114285714285714 |
| [-3.14629122540069, -2.95688490723843) | 0.0181347150259067 |
| [-2.95664182722659, -2.31745946097859) | 0.0225442834138486 |
| [-2.31720653772112, -2.30899511048998) | 0.0227272727272727 |
| [-2.3089006058332, -1.72097525226115) | 0.0361819434872501 |
| [-1.72092762782885, -1.68686596037886) | 0.0410958904109589 |
| [-1.68654701114382, -1.236526434842) | 0.0479812755997659 |
| [-1.23644413938852, -1.11165732156823) | 0.0509915014164305 |
| [-1.11144435180836, -1.06707253269346) | 0.0787401574803149 |
| [-1.0656848188659, -0.899850540104916) | 0.0828729281767955 |
| [-0.899122118493487, -0.235687989432131) | 0.102428722280887 |
| [-0.231686727236445, -0.0682802062167959) | 0.152 |
| [-0.0655954118333348, -0.0517199399968599) | 0.153846153846153 |
| [-0.0515587696700816, 0.496278828219814) | 0.175 |
| [0.496381921331663, 0.670826030444628) | 0.212765957446808 |
| [0.672084076515664, 0.773830880197755) | 0.258064516129032 |
| [0.786770287499513, 2.20130937840007) | 0.391705069124424 |
| [2.25738212058063, 2.39931731317832) | 0.5 |
| [2.42177386174513, 2.68056055833383) | 0.527777777777777 |
| [2.68100312409866, 2.88201473762726) | 0.607142857142857 |
| [2.91232541743944, 3.95594449481789) | 0.648351648351648 |
| [3.97198777284055, 4.33025189595118) | 0.75 |
| [4.33776582233737, 5.81635295182968) | 1.0 |