157:
126:
95:
3808:
try to fit an inverse
Gompertz function to data directly, especially if one only has relatively few data points available from which to calculate the fit. Instead one can fit the transposed relationship of the data to a forward Gompertz function, and then convert it to the equivalent inverse function using the relationship between the two given above.
70:
in 1819. The function was first presented in his June 16, 1825 paper at the bottom of page 518. The
Gompertz function reduced a significant collection of data in life tables into a single function. It is based on the assumption that the mortality rate increases exponentially as a person ages. The
3807:
between its two asymptotes, which are now vertical instead of horizontal like in the forward
Gompertz function. Outside of the range defined by the vertical asymptotes, the inverse function requires computing the logarithm of negative numbers. For this and other reasons it is often impractical to
2984:
Based on the above considerations, Wheldon proposed a mathematical model of tumor growth, called the Gomp-Ex model, that slightly modifies the
Gompertz law. In the Gomp-Ex model it is assumed that initially there is no competition for resources, so that the cellular population expands following the
2441:
The theoretical study by
Fornalski et al. showed the biophysical basis of the Gompertz curve for cancer growth except very early phase where parabolic function is more appropriate. They found also that the Gompertz curve describes the most typical case among the broad family of the cancer dynamics’
1688:
The differentiation between energy used at rest and metabolic rate work allows for the model to more precisely determine the rate of growth. The energy at rest is lower than the energy used to maintain a tissue, and together represent the energy required to maintain the existing tissue. The use of
1411:
The metabolic function is particularly concerned with accounting for the rate of metabolism within an organism. This function can be applied to monitor tumor cells; metabolic rate is dynamic and is greatly flexible, making it more precise in detailing cancer growth. The metabolic curve takes in to
1701:
In the 1960s A.K. Laird for the first time successfully used the
Gompertz curve to fit data of growth of tumors. In fact, tumors are cellular populations growing in a confined space where the availability of nutrients is limited. Denoting the tumor size as X(t) it is useful to write the Gompertz
2437:
As noticed by Steel and by
Wheldon, the proliferation rate of the cellular population is ultimately bounded by the cell division time. Thus, this might be an evidence that the Gompertz equation is not good to model the growth of small tumors. Moreover, more recently it has been noticed that,
2208:
is the instantaneous proliferation rate of the cellular population, whose decreasing nature is due to the competition for the nutrients due to the increase of the cellular population, similarly to the logistic growth rate. However, there is a fundamental difference: in the logistic case the
1394:
and grow rapidly in numbers. Despite its popularity, the function initial rate of tumor growth is difficult to predetermine given the varying microcosms present with a patient, or varying environmental factors in the case of population biology. In cancer patients, factors such as age, diet,
3823:
by a
Gompertz function. Once the standards are thus fit to a Gompertz function, calculating the unknown concentration of samples in the assay from their measured optical density is achieved using the inverse of the Gompertz function that was produced when fitting the standard curve.
2433:
2974:
2315:
1531:
1139:
Population biology is especially concerned with the
Gompertz function. This function is especially useful in describing the rapid growth of a certain population of organisms while also being able to account for the eventual horizontal asymptote, once the
2660:
1814:
3528:
2201:
2786:
3644:
1390:, which is the most widely accepted convention of generally detailing a population's growth. Moreover, the function makes use of initial growth rate, which is commonly seen in populations of bacterial and cancer cells, which undergo the
357:
3130:
2539:
2081:
504:
944:
2867:
2324:
3199:
1271:
2876:
3410:
257:
2214:
1425:
699:
2551:
1925:
3869:"Deciphering death: a commentary of Gomperz (1825)'On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies'"
1008:
3799:
2090:
786:
2695:
1073:
78:(1667–1754) in the 1750s. However, de Moivre assumed that the mortality rate was constant. An extension to Gompertz's work was proposed by the English actuary and mathematician
1412:
consideration the energy the body provides in maintaining and creating tissue. This energy can be considered as metabolism and follows a specific pattern in cellular division.
3313:
3271:
845:
559:
1708:
3422:
1689:
these two factors, alongside the energy required to create new tissue, comprehensively map the rate of growth, and moreover, lead in to an accurate representation of the
1403:
play a role in determining the tumor growth rate. The carrying capacity is also expected to change based on these factors, and so describing such phenomena is difficult.
3043:
2688:
3057:
1650:
3538:
3228:
3010:
609:
2455:
1997:
1951:
878:
1982:
722:
1682:
1611:
1582:
1354:
1325:
1849:
178:
147:
116:
55:. The function was originally designed to describe human mortality, but since has been modified to be applied in biology, with regard to detailing populations.
269:
2438:
including the interaction with immune system, Gompertz and other laws characterized by unbounded F(0) would preclude the possibility of immune surveillance.
1871:
1553:
1376:
1296:
1097:
uptake, where costs were initially high (so uptake was slow), followed by a period of rapid growth, followed by a slowing of uptake as saturation was reached
4463:
d'Onofrio A (2005). "A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences".
1930:
independently on X(0)>0. Note that, in absence of therapies etc.. usually it is X(0) < K, whereas, in presence of therapies, it may be X(0) > K;
2544:
883:
2804:
3139:
1152:
3341:
397:
194:
614:
3918:"On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies"
1876:
47:
of the function is approached much more gradually by the curve than the left-side or lower valued asymptote. This is in contrast to the
3336:
can be explicitly expressed in traditional functional notation as a single continuous function. Given a
Gompertz function of the form:
1420:
share a similar growth pattern and this model, as a result, considers cellular division, the foundation of the development of a tumor.
962:
3688:
4204:
Caravelli F, Sindoni L, Caccioli F, Ududec C (August 2016). "Optimal growth trajectories with finite carrying capacity".
2428:{\displaystyle \lim _{X\rightarrow 0^{+}}F(X)=\lim _{X\rightarrow 0^{+}}\alpha \log \left({\frac {K}{X}}\right)=+\infty }
3045:. The assumption that there is no competition for resources holds true in most scenarios. It can however be affected by
4335:
791:
4447:
4417:
2969:{\displaystyle X(t)={\frac {K}{e}}=K\cdot \lim _{\nu \rightarrow +\infty }\left({\frac {\nu }{\nu +1}}\right)^{\nu }}
954:
43:
which describes growth as being slowest at the start and end of a given time period. The right-side or future value
2310:{\displaystyle F(X)=\alpha \left(1-\left({\frac {X}{K}}\right)^{\nu }\right)\Rightarrow F(0)=\alpha <+\infty }
1873:
is the carrying capacity, i.e. the maximum size that can be reached with the available nutrients. In fact it is:
1526:{\displaystyle B=\sum _{C}(N_{C}B_{C})+\left(E_{C}{\operatorname {d} \!N_{C} \over \operatorname {d} \!t}\right)}
82:(1826–1891) in 1860, who added a constant background mortality rate to Gompertz's exponentially increasing one.
1125:
the size of the English Knowledge can be modelled with Gompertz function and a modified function to some extent
727:
3965:
A second edition was issued in 1743; a third edition was issued in 1750; a fourth edition was issued in 1752.
2655:{\displaystyle X^{\prime }(t)=\alpha \nu \left(1-\left({\frac {X(t)}{K}}\right)^{\frac {1}{\nu }}\right)X(t)}
957:, which states the rate of absolute mortality (decay) falls exponentially with current size. Mathematically,
52:
4055:
Islam T, Fiebig DG, Meade N (2002). "Modelling multinational telecommunications demand with limited data".
1100:
Population in a confined space, as birth rates first increase and then slow as resource limits are reached
4603:
4518:"Possible Source of the Gompertz Law of Proliferating Cancer Cells: Mechanistic Modeling of Tumor Growth"
1382:
This function consideration of the plateau cell number makes it useful in accurately mimicking real-life
1018:
71:
resulting Gompertz function is for the number of individuals living at a given age as a function of age.
74:
Earlier work on the construction of functional models of mortality was done by the French mathematician
87:
Graphs of Gompertz curves, showing the effect of varying one of a,b,c while keeping the others constant
4183:
Montero-Camacho, Paulo; Li, Yin; Cranmer, Miles (2024). "Five parameters are all you need (in ΛCDM)".
1809:{\displaystyle X(t)=K\exp \left(\log \left({\frac {X(0)}{K}}\right)\exp \left(-\alpha t\right)\right)}
509:
3838:
3672:
3523:{\textstyle \lim _{t\to -\infty }a\mathrm {e} ^{-\mathrm {e} ^{b-ct}}+d=a\mathrm {e} ^{-\infty }+d=d}
3278:
3236:
382:
4142:
Sottoriva A, Verhoeff JJ, Borovski T, McWeeney SK, Naumov L, Medema JP, et al. (January 2010).
2196:{\displaystyle X^{\prime }(t)=F\left(X(t)\right)X(t),\quad {\mbox{with}}\quad F^{\prime }(X)\leq 0,}
4144:"Cancer stem cell tumor model reveals invasive morphology and increased phenotypical heterogeneity"
3843:
2781:{\displaystyle \lim _{\nu \rightarrow +\infty }\nu \left(1-x^{1/\nu }\right)=-\log \left(x\right)}
66:(1779–1865) was an actuary in London who was privately educated. He was elected a fellow of the
3015:
51:
in which both asymptotes are approached by the curve symmetrically. It is a special case of the
4608:
3639:{\textstyle \lim _{t\to \infty }a\mathrm {e} ^{-\mathrm {e} ^{b-ct}}+d=a\mathrm {e} ^{0}+d=a+d}
2667:
4025:
1992:
It can be shown that the dynamics of X(t) are governed by the Gompertz differential equation:
1416:
can be used to model such growth, irrespective of differing masses and development times. All
3833:
3958:
1618:
4529:
4482:
4289:
4268:"Is the public sector of your country a diffusion borrower? Empirical evidence from Brazil"
4223:
4095:
3206:
2988:
1116:
Detailing population growth in animals of prey, with regard to predator-prey relationships
571:
8:
4598:
1936:
1413:
1383:
850:
352:{\textstyle \lim _{t\to \infty }a\mathrm {e} ^{-b\mathrm {e} ^{-ct}}=a\mathrm {e} ^{0}=a}
4533:
4486:
4293:
4227:
4099:
1958:
704:
4498:
4472:
4385:
4360:
4312:
4279:
4267:
4247:
4213:
4184:
4001:
3976:
3939:
3893:
3868:
1660:
1589:
1560:
1332:
1303:
163:
132:
101:
28:
4118:
4083:
4068:
1825:
4561:
4443:
4413:
4390:
4317:
4239:
4165:
4123:
4006:
3943:
3898:
2796:
1141:
75:
63:
48:
36:
4502:
4251:
4108:
4564:
4537:
4490:
4380:
4372:
4307:
4297:
4231:
4155:
4113:
4103:
4064:
4037:
3996:
3988:
3929:
3888:
3880:
3848:
3333:
2792:
1856:
1653:
1538:
1387:
1361:
1281:
562:
40:
4160:
4143:
4302:
3820:
3046:
79:
4578:
4494:
3125:{\displaystyle F(X)=\max \left(a,\alpha \log \left({\frac {K}{X}}\right)\right)}
4542:
4517:
4235:
3816:
4041:
3992:
4592:
4081:
2534:{\displaystyle X^{\prime }(t)=\alpha \log \left({\frac {K}{X(t)}}\right)X(t)}
2076:{\displaystyle X^{\prime }(t)=\alpha \log \left({\frac {K}{X(t)}}\right)X(t)}
67:
1113:
Modelling market impact in finance and aggregated subnational loans dynamic.
4394:
4321:
4243:
4169:
4127:
4010:
3934:
3902:
3884:
1107:
1094:
499:{\textstyle f(t)=a\mathrm {e} ^{-b\mathrm {e} ^{-ct}}=ae^{-e^{-ct+\ln b}}}
3804:
1985:
32:
4516:
Fornalski KW, Reszczyńska J, Dobrzyński L, Wysocki P, Janiak MK (2020).
4376:
4082:
Zwietering MH, Jongenburger I, Rombouts FM, van 't Riet K (June 1990).
3811:
In this way the inverse function has many uses. For instance, certain
3328:
The Gompertz function is a one-to-one correspondence (also known as an
1400:
1396:
4569:
4515:
3329:
1690:
1391:
44:
3917:
4284:
4218:
4189:
4477:
3803:
The inverse function only produces numerical values in the set of
2319:
whereas in the Gompertz case the proliferation rate is unbounded:
939:{\displaystyle \max \left({\frac {df}{dt}}\right)={\frac {ac}{e}}}
4583:
4141:
4030:
The Assurance Magazine, and Journal of the Institute of Actuaries
4026:"On the law of mortality and the construction of annuity tables"
2862:{\displaystyle X(t)=\left({\frac {\nu }{\nu +1}}\right)^{\nu }K}
1953:
is a constant related to the proliferative ability of the cells.
3963:. London, England: Francis Fayram, Benj. Motte, and W. Pearson.
3316:
3194:{\displaystyle X_{C}=K\exp \left(-{\frac {a}{\alpha }}\right).}
4203:
1684:= energy required to create new tissue from an individual cell
3812:
3535:
is the distance from the base to the second asymptote, since
2985:
exponential law. However, there is a critical size threshold
1417:
1266:{\displaystyle N(t)=N_{0}\exp(\ln(N_{I}/N_{0})(1-\exp(-bt)))}
156:
125:
94:
2209:
proliferation rate for small cellular population is finite:
3873:
Philosophical Transactions of the Royal Society of London B
3405:{\displaystyle f(t)=a\mathrm {e} ^{-\mathrm {e} ^{b-ct}}+d}
3922:
Philosophical Transactions of the Royal Society of London
252:{\displaystyle f(t)=a\mathrm {e} ^{-b\mathrm {e} ^{-ct}}}
4182:
3819:
whose concentrations can be fit extremely well to their
3683:
the corresponding inverse function can be expressed as:
4559:
3049:, that requires the creation of sub-factors variables.
694:{\displaystyle t_{hwp}={\frac {\ln(b)-\ln(\ln(2))}{c}}}
3541:
3425:
3281:
3239:
2445:
2158:
1961:
1939:
1859:
1828:
1663:
1621:
1592:
1563:
1541:
1364:
1335:
1306:
1284:
1021:
853:
730:
707:
574:
400:
272:
3977:"Laws of Mortality from the Biological Point of View"
3691:
3344:
3209:
3142:
3060:
3018:
2991:
2879:
2807:
2698:
2670:
2554:
2458:
2327:
2217:
2093:
2000:
1879:
1711:
1428:
1155:
965:
886:
794:
617:
512:
197:
166:
135:
104:
2871:and one in the graph of the Gompertz function when
1920:{\displaystyle \lim _{t\rightarrow +\infty }X(t)=K}
1851:
is the tumor size at the starting observation time;
3793:
3655:-axis (translates the graph to the left or right).
3638:
3522:
3404:
3307:
3265:
3222:
3193:
3124:
3037:
3004:
2968:
2861:
2780:
2682:
2654:
2533:
2427:
2309:
2195:
2075:
1976:
1945:
1919:
1865:
1843:
1808:
1676:
1644:
1605:
1576:
1547:
1525:
1370:
1348:
1319:
1290:
1265:
1067:
1002:
938:
872:
839:
780:
716:
693:
603:
553:
498:
368:-axis (translates the graph to the left or right).
351:
251:
172:
141:
110:
1511:
1495:
4590:
4054:
3543:
3427:
3076:
2915:
2700:
2367:
2329:
1881:
1144:is determined (plateau cell/population number).
887:
274:
4265:
1090:Examples of uses for Gompertz curves include:
1003:{\displaystyle k^{r}\propto {\frac {1}{y(t)}}}
4579:https://archive.org/details/philtrans04942340
4462:
3794:{\displaystyle f^{-1}(t)={\frac {1}{c}}\left}
1119:Modelling bacterial cells within a population
3323:
3203:Here there are some numerical estimates for
4266:Rocha LS, Rocha FS, Souza TT (2017-10-05).
1652:= energy required to maintain the existing
4433:
4431:
4429:
2545:generalized logistic differential equation
781:{\textstyle {\frac {d^{2}}{dt^{2}}}f(t)=0}
4541:
4509:
4476:
4384:
4311:
4301:
4283:
4217:
4188:
4159:
4117:
4107:
4000:
3974:
3956:
3933:
3892:
953:The function curve can be derived from a
84:
4456:
4336:"Knowledge:Modelling Knowledge's growth"
4084:"Modeling of the bacterial growth curve"
3915:
3866:
3419:is the base horizontal asymptote, since
2979:
155:
124:
93:
4437:
4426:
4352:
4023:
1584:= number of cells in the given organism
701:The point of maximum rate of increase (
4591:
4440:Mathematical Models in Cancer Research
4088:Applied and Environmental Microbiology
2085:i.e. is of the form when broken down:
1613:= metabolic rate of an individual cell
1356:is the plateau cell/population density
568:The halfway point is found by solving
4560:
4407:
4401:
4358:
3052:the growth follows the Gompertz Law:
1395:ethnicity, genetic pre-dispositions,
4057:International Journal of Forecasting
1068:{\textstyle r={\frac {y'(t)}{y(t)}}}
2450:The Gompertz differential equation
2446:Gompertz growth and logistic growth
1696:
1386:. The function also adheres to the
1378:is the initial rate of tumor growth
13:
3608:
3573:
3563:
3553:
3503:
3495:
3460:
3450:
3440:
3375:
3365:
2928:
2713:
2560:
2464:
2422:
2304:
2170:
2099:
2006:
1894:
1508:
1492:
1406:
434:
421:
333:
307:
294:
284:
231:
218:
14:
4620:
4553:
4024:Makeham, William Matthew (1860).
2690:is a positive real number) since
1134:
3651:sets the displacement along the
3308:{\textstyle X_{C}\approx 10^{6}}
3266:{\textstyle X_{C}\approx 10^{9}}
2795:in the graph of the generalized
840:{\displaystyle t_{max}=\ln(b)/c}
554:{\displaystyle f(t)=e^{-e^{-t}}}
364:sets the displacement along the
4328:
4259:
4197:
4109:10.1128/AEM.56.6.1875-1881.1990
2164:
2156:
1327:is the initial density of cells
1129:
1085:
4584:http://chemoth.com/tumorgrowth
4176:
4135:
4075:
4048:
4017:
3968:
3950:
3909:
3860:
3711:
3705:
3550:
3434:
3354:
3348:
3070:
3064:
2922:
2889:
2883:
2817:
2811:
2707:
2649:
2643:
2611:
2605:
2571:
2565:
2528:
2522:
2509:
2503:
2475:
2469:
2374:
2360:
2354:
2336:
2289:
2283:
2277:
2227:
2221:
2181:
2175:
2150:
2144:
2133:
2127:
2110:
2104:
2070:
2064:
2051:
2045:
2017:
2011:
1971:
1968:
1908:
1902:
1888:
1838:
1832:
1763:
1757:
1721:
1715:
1555:= energy organism uses at rest
1468:
1445:
1260:
1257:
1254:
1242:
1227:
1224:
1196:
1187:
1165:
1159:
1059:
1053:
1045:
1039:
994:
988:
826:
820:
769:
763:
682:
679:
673:
664:
652:
646:
584:
578:
522:
516:
410:
404:
281:
207:
201:
1:
4161:10.1158/0008-5472.CAN-09-3663
4069:10.1016/S0169-2070(02)00073-0
3854:
1103:Modelling of growth of tumors
948:
389:
53:generalised logistic function
4303:10.1371/journal.pone.0185257
2543:is the limiting case of the
7:
4495:10.1016/j.physd.2005.06.032
4412:. Oxford: Clarendon Press.
4359:Laird AK (September 1964).
3957:de Moivre, Abraham (1725).
3916:Gompertz, Benjamin (1825).
3827:
16:Asymmetric sigmoid function
10:
4625:
4543:10.12693/APhysPolA.138.854
4361:"Dynamics of Tumor Growth"
4236:10.1103/PhysRevE.94.022315
3038:{\displaystyle X>X_{C}}
1399:, lifestyle and origin of
1147:It is modeled as follows:
187:
58:
4410:Growth Kinetics of Tumors
4365:British Journal of Cancer
4042:10.1017/S204616580000126X
3993:10.1017/S002217240000961X
3324:Inverse Gompertz function
2791:In addition, there is an
2683:{\displaystyle \nu >0}
1081:is an arbitrary constant.
955:Gompertz law of mortality
4442:. Bristol: Adam Hilger.
3844:Von Bertalanffy function
1122:Examining disease spread
49:simple logistic function
4522:Acta Physica Polonica A
1645:{\textstyle N_{C}B_{C}}
266:is an asymptote, since
80:William Matthew Makeham
3975:Greenwood, M. (1928).
3960:Annuities upon Lives …
3935:10.1098/rstl.1825.0026
3885:10.1098/rstb.2014.0379
3867:Kirkwood, TBL (2015).
3795:
3661:sets the growth rate (
3640:
3524:
3406:
3309:
3267:
3224:
3195:
3126:
3039:
3006:
2970:
2863:
2782:
2684:
2656:
2535:
2429:
2311:
2197:
2077:
1978:
1947:
1921:
1867:
1845:
1810:
1678:
1646:
1607:
1578:
1549:
1527:
1372:
1350:
1321:
1292:
1267:
1069:
1004:
940:
874:
841:
782:
724:) is found by solving
718:
695:
605:
555:
506:has the same shape as
500:
374:sets the growth rate (
353:
253:
180:
174:
149:
143:
118:
112:
3834:Gompertz distribution
3796:
3641:
3525:
3407:
3310:
3268:
3225:
3223:{\displaystyle X_{C}}
3196:
3127:
3040:
3007:
3005:{\displaystyle X_{C}}
2980:Gomp-ex law of growth
2971:
2864:
2783:
2685:
2657:
2536:
2430:
2312:
2198:
2078:
1979:
1948:
1922:
1868:
1846:
1811:
1679:
1647:
1608:
1579:
1550:
1528:
1373:
1351:
1322:
1293:
1268:
1075:is the rate of growth
1070:
1005:
941:
875:
842:
783:
719:
696:
606:
604:{\textstyle f(t)=a/2}
556:
501:
354:
254:
175:
159:
144:
128:
113:
97:
39:(1779–1865). It is a
3689:
3539:
3423:
3342:
3279:
3237:
3207:
3140:
3058:
3016:
2989:
2877:
2805:
2696:
2668:
2552:
2456:
2325:
2215:
2091:
1998:
1959:
1946:{\textstyle \alpha }
1937:
1877:
1857:
1826:
1709:
1661:
1619:
1590:
1561:
1539:
1426:
1362:
1333:
1304:
1282:
1153:
1019:
963:
884:
873:{\textstyle t_{max}}
851:
792:
728:
705:
615:
572:
510:
398:
270:
195:
164:
133:
102:
4534:2020AcPPA.138..854F
4487:2005PhyD..208..220D
4438:Wheldon TE (1988).
4377:10.1038/bjc.1964.55
4294:2017PLoSO..1285257R
4228:2016PhRvE..94b2315C
4100:1990ApEnM..56.1875Z
1977:{\textstyle \log()}
1414:Energy conservation
1384:population dynamics
1108:cosmic reionization
717:{\textstyle 0.368a}
89:
4604:Time series models
4562:Weisstein, Eric W.
3981:Journal of Hygiene
3791:
3636:
3557:
3520:
3444:
3402:
3330:Bijective function
3305:
3263:
3220:
3191:
3122:
3035:
3002:
2966:
2932:
2859:
2778:
2717:
2680:
2652:
2531:
2425:
2388:
2350:
2307:
2193:
2162:
2073:
1974:
1943:
1917:
1898:
1863:
1841:
1806:
1702:Curve as follows:
1677:{\textstyle E_{C}}
1674:
1642:
1606:{\textstyle B_{C}}
1603:
1577:{\textstyle N_{C}}
1574:
1545:
1523:
1444:
1368:
1349:{\textstyle N_{I}}
1346:
1320:{\textstyle N_{0}}
1317:
1288:
1263:
1065:
1000:
936:
870:
837:
778:
714:
691:
601:
551:
496:
349:
288:
249:
181:
170:
150:
139:
119:
108:
85:
29:mathematical model
4408:Steel GG (1977).
4206:Physical Review E
3775:
3725:
3542:
3426:
3181:
3111:
2954:
2914:
2903:
2844:
2797:logistic function
2699:
2632:
2618:
2513:
2410:
2366:
2328:
2260:
2161:
2055:
1880:
1844:{\textstyle X(0)}
1770:
1516:
1435:
1142:carrying capacity
1063:
998:
934:
912:
758:
689:
273:
185:
184:
173:{\displaystyle c}
142:{\displaystyle b}
111:{\displaystyle a}
76:Abraham de Moivre
64:Benjamin Gompertz
37:Benjamin Gompertz
25:Gompertz function
4616:
4575:
4574:
4565:"Gompertz Curve"
4548:
4547:
4545:
4513:
4507:
4506:
4480:
4471:(3–4): 220–235.
4460:
4454:
4453:
4435:
4424:
4423:
4405:
4399:
4398:
4388:
4356:
4350:
4349:
4348:
4347:
4332:
4326:
4325:
4315:
4305:
4287:
4278:(10): e0185257.
4263:
4257:
4255:
4221:
4201:
4195:
4194:
4192:
4180:
4174:
4173:
4163:
4139:
4133:
4131:
4121:
4111:
4079:
4073:
4072:
4052:
4046:
4045:
4021:
4015:
4014:
4004:
3972:
3966:
3964:
3954:
3948:
3947:
3937:
3913:
3907:
3906:
3896:
3864:
3849:Sigmoid function
3800:
3798:
3797:
3792:
3790:
3786:
3785:
3781:
3780:
3776:
3774:
3760:
3726:
3718:
3704:
3703:
3645:
3643:
3642:
3637:
3617:
3616:
3611:
3593:
3592:
3591:
3590:
3576:
3566:
3556:
3529:
3527:
3526:
3521:
3507:
3506:
3498:
3480:
3479:
3478:
3477:
3463:
3453:
3443:
3411:
3409:
3408:
3403:
3395:
3394:
3393:
3392:
3378:
3368:
3334:inverse function
3314:
3312:
3311:
3306:
3304:
3303:
3291:
3290:
3273:for human tumors
3272:
3270:
3269:
3264:
3262:
3261:
3249:
3248:
3229:
3227:
3226:
3221:
3219:
3218:
3200:
3198:
3197:
3192:
3187:
3183:
3182:
3174:
3152:
3151:
3131:
3129:
3128:
3123:
3121:
3117:
3116:
3112:
3104:
3047:limiting factors
3044:
3042:
3041:
3036:
3034:
3033:
3011:
3009:
3008:
3003:
3001:
3000:
2975:
2973:
2972:
2967:
2965:
2964:
2959:
2955:
2953:
2939:
2931:
2904:
2896:
2868:
2866:
2865:
2860:
2855:
2854:
2849:
2845:
2843:
2829:
2793:inflection point
2787:
2785:
2784:
2779:
2777:
2754:
2750:
2749:
2748:
2744:
2716:
2689:
2687:
2686:
2681:
2661:
2659:
2658:
2653:
2639:
2635:
2634:
2633:
2625:
2623:
2619:
2614:
2600:
2564:
2563:
2540:
2538:
2537:
2532:
2518:
2514:
2512:
2495:
2468:
2467:
2434:
2432:
2431:
2426:
2415:
2411:
2403:
2387:
2386:
2385:
2349:
2348:
2347:
2316:
2314:
2313:
2308:
2276:
2272:
2271:
2270:
2265:
2261:
2253:
2202:
2200:
2199:
2194:
2174:
2173:
2163:
2159:
2140:
2136:
2103:
2102:
2082:
2080:
2079:
2074:
2060:
2056:
2054:
2037:
2010:
2009:
1983:
1981:
1980:
1975:
1952:
1950:
1949:
1944:
1926:
1924:
1923:
1918:
1897:
1872:
1870:
1869:
1864:
1850:
1848:
1847:
1842:
1815:
1813:
1812:
1807:
1805:
1801:
1800:
1796:
1775:
1771:
1766:
1752:
1697:Growth of tumors
1683:
1681:
1680:
1675:
1673:
1672:
1651:
1649:
1648:
1643:
1641:
1640:
1631:
1630:
1612:
1610:
1609:
1604:
1602:
1601:
1583:
1581:
1580:
1575:
1573:
1572:
1554:
1552:
1551:
1546:
1532:
1530:
1529:
1524:
1522:
1518:
1517:
1515:
1506:
1505:
1504:
1490:
1488:
1487:
1467:
1466:
1457:
1456:
1443:
1388:sigmoid function
1377:
1375:
1374:
1369:
1355:
1353:
1352:
1347:
1345:
1344:
1326:
1324:
1323:
1318:
1316:
1315:
1297:
1295:
1294:
1289:
1272:
1270:
1269:
1264:
1223:
1222:
1213:
1208:
1207:
1180:
1179:
1074:
1072:
1071:
1066:
1064:
1062:
1048:
1038:
1029:
1009:
1007:
1006:
1001:
999:
997:
980:
975:
974:
945:
943:
942:
937:
935:
930:
922:
917:
913:
911:
903:
895:
879:
877:
876:
871:
869:
868:
847:The increase at
846:
844:
843:
838:
833:
810:
809:
787:
785:
784:
779:
759:
757:
756:
755:
742:
741:
732:
723:
721:
720:
715:
700:
698:
697:
692:
690:
685:
638:
633:
632:
610:
608:
607:
602:
597:
563:affine transform
560:
558:
557:
552:
550:
549:
548:
547:
505:
503:
502:
497:
495:
494:
493:
492:
451:
450:
449:
448:
437:
424:
385:(e = 2.71828...)
358:
356:
355:
350:
342:
341:
336:
324:
323:
322:
321:
310:
297:
287:
258:
256:
255:
250:
248:
247:
246:
245:
234:
221:
179:
177:
176:
171:
148:
146:
145:
140:
117:
115:
114:
109:
90:
41:sigmoid function
4624:
4623:
4619:
4618:
4617:
4615:
4614:
4613:
4589:
4588:
4556:
4551:
4514:
4510:
4461:
4457:
4450:
4436:
4427:
4420:
4406:
4402:
4357:
4353:
4345:
4343:
4334:
4333:
4329:
4264:
4260:
4212:(2–1): 022315.
4202:
4198:
4181:
4177:
4148:Cancer Research
4140:
4136:
4080:
4076:
4053:
4049:
4022:
4018:
3973:
3969:
3955:
3951:
3914:
3910:
3865:
3861:
3857:
3830:
3821:optical density
3764:
3759:
3755:
3748:
3744:
3731:
3727:
3717:
3696:
3692:
3690:
3687:
3686:
3612:
3607:
3606:
3577:
3572:
3571:
3567:
3562:
3561:
3546:
3540:
3537:
3536:
3499:
3494:
3493:
3464:
3459:
3458:
3454:
3449:
3448:
3430:
3424:
3421:
3420:
3379:
3374:
3373:
3369:
3364:
3363:
3343:
3340:
3339:
3326:
3299:
3295:
3286:
3282:
3280:
3277:
3276:
3257:
3253:
3244:
3240:
3238:
3235:
3234:
3214:
3210:
3208:
3205:
3204:
3173:
3169:
3165:
3147:
3143:
3141:
3138:
3137:
3103:
3099:
3083:
3079:
3059:
3056:
3055:
3029:
3025:
3017:
3014:
3013:
2996:
2992:
2990:
2987:
2986:
2982:
2960:
2943:
2938:
2934:
2933:
2918:
2895:
2878:
2875:
2874:
2850:
2833:
2828:
2824:
2823:
2806:
2803:
2802:
2767:
2740:
2736:
2732:
2725:
2721:
2703:
2697:
2694:
2693:
2669:
2666:
2665:
2624:
2601:
2599:
2595:
2594:
2587:
2583:
2559:
2555:
2553:
2550:
2549:
2499:
2494:
2490:
2463:
2459:
2457:
2454:
2453:
2448:
2402:
2398:
2381:
2377:
2370:
2343:
2339:
2332:
2326:
2323:
2322:
2266:
2252:
2248:
2247:
2240:
2236:
2216:
2213:
2212:
2169:
2165:
2157:
2123:
2119:
2098:
2094:
2092:
2089:
2088:
2041:
2036:
2032:
2005:
2001:
1999:
1996:
1995:
1960:
1957:
1956:
1938:
1935:
1934:
1884:
1878:
1875:
1874:
1858:
1855:
1854:
1827:
1824:
1823:
1786:
1782:
1753:
1751:
1747:
1740:
1736:
1710:
1707:
1706:
1699:
1668:
1664:
1662:
1659:
1658:
1636:
1632:
1626:
1622:
1620:
1617:
1616:
1597:
1593:
1591:
1588:
1587:
1568:
1564:
1562:
1559:
1558:
1540:
1537:
1536:
1507:
1500:
1496:
1491:
1489:
1483:
1479:
1478:
1474:
1462:
1458:
1452:
1448:
1439:
1427:
1424:
1423:
1409:
1407:Metabolic curve
1363:
1360:
1359:
1340:
1336:
1334:
1331:
1330:
1311:
1307:
1305:
1302:
1301:
1283:
1280:
1279:
1218:
1214:
1209:
1203:
1199:
1175:
1171:
1154:
1151:
1150:
1137:
1132:
1088:
1049:
1031:
1030:
1028:
1020:
1017:
1016:
984:
979:
970:
966:
964:
961:
960:
951:
923:
921:
904:
896:
894:
890:
885:
882:
881:
858:
854:
852:
849:
848:
829:
799:
795:
793:
790:
789:
751:
747:
743:
737:
733:
731:
729:
726:
725:
706:
703:
702:
639:
637:
622:
618:
616:
613:
612:
593:
573:
570:
569:
540:
536:
532:
528:
511:
508:
507:
470:
466:
462:
458:
438:
433:
432:
425:
420:
419:
399:
396:
395:
392:
337:
332:
331:
311:
306:
305:
298:
293:
292:
277:
271:
268:
267:
235:
230:
229:
222:
217:
216:
196:
193:
192:
190:
165:
162:
161:
134:
131:
130:
103:
100:
99:
61:
17:
12:
11:
5:
4622:
4612:
4611:
4606:
4601:
4587:
4586:
4581:
4576:
4555:
4554:External links
4552:
4550:
4549:
4528:(6): 854–862.
4508:
4455:
4448:
4425:
4418:
4400:
4371:(3): 490–502.
4351:
4327:
4258:
4196:
4175:
4134:
4094:(6): 1875–81.
4074:
4063:(4): 605–624.
4047:
4036:(6): 301–310.
4016:
3987:(3): 267–294.
3967:
3949:
3908:
3858:
3856:
3853:
3852:
3851:
3846:
3841:
3836:
3829:
3826:
3817:standard curve
3789:
3784:
3779:
3773:
3770:
3767:
3763:
3758:
3754:
3751:
3747:
3743:
3740:
3737:
3734:
3730:
3724:
3721:
3716:
3713:
3710:
3707:
3702:
3699:
3695:
3681:
3680:
3673:Euler's Number
3666:
3656:
3646:
3635:
3632:
3629:
3626:
3623:
3620:
3615:
3610:
3605:
3602:
3599:
3596:
3589:
3586:
3583:
3580:
3575:
3570:
3565:
3560:
3555:
3552:
3549:
3545:
3530:
3519:
3516:
3513:
3510:
3505:
3502:
3497:
3492:
3489:
3486:
3483:
3476:
3473:
3470:
3467:
3462:
3457:
3452:
3447:
3442:
3439:
3436:
3433:
3429:
3401:
3398:
3391:
3388:
3385:
3382:
3377:
3372:
3367:
3362:
3359:
3356:
3353:
3350:
3347:
3325:
3322:
3321:
3320:
3319:(mouse) tumors
3302:
3298:
3294:
3289:
3285:
3274:
3260:
3256:
3252:
3247:
3243:
3217:
3213:
3190:
3186:
3180:
3177:
3172:
3168:
3164:
3161:
3158:
3155:
3150:
3146:
3120:
3115:
3110:
3107:
3102:
3098:
3095:
3092:
3089:
3086:
3082:
3078:
3075:
3072:
3069:
3066:
3063:
3032:
3028:
3024:
3021:
3012:such that for
2999:
2995:
2981:
2978:
2963:
2958:
2952:
2949:
2946:
2942:
2937:
2930:
2927:
2924:
2921:
2917:
2913:
2910:
2907:
2902:
2899:
2894:
2891:
2888:
2885:
2882:
2858:
2853:
2848:
2842:
2839:
2836:
2832:
2827:
2822:
2819:
2816:
2813:
2810:
2776:
2773:
2770:
2766:
2763:
2760:
2757:
2753:
2747:
2743:
2739:
2735:
2731:
2728:
2724:
2720:
2715:
2712:
2709:
2706:
2702:
2679:
2676:
2673:
2651:
2648:
2645:
2642:
2638:
2631:
2628:
2622:
2617:
2613:
2610:
2607:
2604:
2598:
2593:
2590:
2586:
2582:
2579:
2576:
2573:
2570:
2567:
2562:
2558:
2530:
2527:
2524:
2521:
2517:
2511:
2508:
2505:
2502:
2498:
2493:
2489:
2486:
2483:
2480:
2477:
2474:
2471:
2466:
2462:
2447:
2444:
2424:
2421:
2418:
2414:
2409:
2406:
2401:
2397:
2394:
2391:
2384:
2380:
2376:
2373:
2369:
2365:
2362:
2359:
2356:
2353:
2346:
2342:
2338:
2335:
2331:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2285:
2282:
2279:
2275:
2269:
2264:
2259:
2256:
2251:
2246:
2243:
2239:
2235:
2232:
2229:
2226:
2223:
2220:
2192:
2189:
2186:
2183:
2180:
2177:
2172:
2168:
2155:
2152:
2149:
2146:
2143:
2139:
2135:
2132:
2129:
2126:
2122:
2118:
2115:
2112:
2109:
2106:
2101:
2097:
2072:
2069:
2066:
2063:
2059:
2053:
2050:
2047:
2044:
2040:
2035:
2031:
2028:
2025:
2022:
2019:
2016:
2013:
2008:
2004:
1990:
1989:
1984:refers to the
1973:
1970:
1967:
1964:
1954:
1942:
1928:
1927:
1916:
1913:
1910:
1907:
1904:
1901:
1896:
1893:
1890:
1887:
1883:
1866:{\textstyle K}
1862:
1852:
1840:
1837:
1834:
1831:
1817:
1816:
1804:
1799:
1795:
1792:
1789:
1785:
1781:
1778:
1774:
1769:
1765:
1762:
1759:
1756:
1750:
1746:
1743:
1739:
1735:
1732:
1729:
1726:
1723:
1720:
1717:
1714:
1698:
1695:
1686:
1685:
1671:
1667:
1656:
1639:
1635:
1629:
1625:
1614:
1600:
1596:
1585:
1571:
1567:
1556:
1548:{\textstyle B}
1544:
1521:
1514:
1510:
1503:
1499:
1494:
1486:
1482:
1477:
1473:
1470:
1465:
1461:
1455:
1451:
1447:
1442:
1438:
1434:
1431:
1408:
1405:
1380:
1379:
1371:{\textstyle b}
1367:
1357:
1343:
1339:
1328:
1314:
1310:
1299:
1291:{\textstyle t}
1287:
1262:
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1221:
1217:
1212:
1206:
1202:
1198:
1195:
1192:
1189:
1186:
1183:
1178:
1174:
1170:
1167:
1164:
1161:
1158:
1136:
1135:Gompertz curve
1133:
1131:
1128:
1127:
1126:
1123:
1120:
1117:
1114:
1111:
1104:
1101:
1098:
1087:
1084:
1083:
1082:
1076:
1061:
1058:
1055:
1052:
1047:
1044:
1041:
1037:
1034:
1027:
1024:
996:
993:
990:
987:
983:
978:
973:
969:
950:
947:
933:
929:
926:
920:
916:
910:
907:
902:
899:
893:
889:
867:
864:
861:
857:
836:
832:
828:
825:
822:
819:
816:
813:
808:
805:
802:
798:
777:
774:
771:
768:
765:
762:
754:
750:
746:
740:
736:
713:
710:
688:
684:
681:
678:
675:
672:
669:
666:
663:
660:
657:
654:
651:
648:
645:
642:
636:
631:
628:
625:
621:
600:
596:
592:
589:
586:
583:
580:
577:
546:
543:
539:
535:
531:
527:
524:
521:
518:
515:
491:
488:
485:
482:
479:
476:
473:
469:
465:
461:
457:
454:
447:
444:
441:
436:
431:
428:
423:
418:
415:
412:
409:
406:
403:
391:
388:
387:
386:
383:Euler's Number
379:
369:
359:
348:
345:
340:
335:
330:
327:
320:
317:
314:
309:
304:
301:
296:
291:
286:
283:
280:
276:
244:
241:
238:
233:
228:
225:
220:
215:
212:
209:
206:
203:
200:
189:
186:
183:
182:
169:
152:
151:
138:
121:
120:
107:
60:
57:
35:, named after
21:Gompertz curve
15:
9:
6:
4:
3:
2:
4621:
4610:
4609:Growth curves
4607:
4605:
4602:
4600:
4597:
4596:
4594:
4585:
4582:
4580:
4577:
4572:
4571:
4566:
4563:
4558:
4557:
4544:
4539:
4535:
4531:
4527:
4523:
4519:
4512:
4504:
4500:
4496:
4492:
4488:
4484:
4479:
4474:
4470:
4466:
4459:
4451:
4449:0-85274-291-6
4445:
4441:
4434:
4432:
4430:
4421:
4419:0-19-857388-X
4415:
4411:
4404:
4396:
4392:
4387:
4382:
4378:
4374:
4370:
4366:
4362:
4355:
4341:
4337:
4331:
4323:
4319:
4314:
4309:
4304:
4299:
4295:
4291:
4286:
4281:
4277:
4273:
4269:
4262:
4253:
4249:
4245:
4241:
4237:
4233:
4229:
4225:
4220:
4215:
4211:
4207:
4200:
4191:
4186:
4179:
4171:
4167:
4162:
4157:
4153:
4149:
4145:
4138:
4129:
4125:
4120:
4115:
4110:
4105:
4101:
4097:
4093:
4089:
4085:
4078:
4070:
4066:
4062:
4058:
4051:
4043:
4039:
4035:
4031:
4027:
4020:
4012:
4008:
4003:
3998:
3994:
3990:
3986:
3982:
3978:
3971:
3962:
3961:
3953:
3945:
3941:
3936:
3931:
3927:
3923:
3919:
3912:
3904:
3900:
3895:
3890:
3886:
3882:
3878:
3874:
3870:
3863:
3859:
3850:
3847:
3845:
3842:
3840:
3837:
3835:
3832:
3831:
3825:
3822:
3818:
3814:
3809:
3806:
3801:
3787:
3782:
3777:
3771:
3768:
3765:
3761:
3756:
3752:
3749:
3745:
3741:
3738:
3735:
3732:
3728:
3722:
3719:
3714:
3708:
3700:
3697:
3693:
3684:
3679:= 2.71828...)
3678:
3674:
3670:
3667:
3664:
3660:
3657:
3654:
3650:
3647:
3633:
3630:
3627:
3624:
3621:
3618:
3613:
3603:
3600:
3597:
3594:
3587:
3584:
3581:
3578:
3568:
3558:
3547:
3534:
3531:
3517:
3514:
3511:
3508:
3500:
3490:
3487:
3484:
3481:
3474:
3471:
3468:
3465:
3455:
3445:
3437:
3431:
3418:
3415:
3414:
3413:
3399:
3396:
3389:
3386:
3383:
3380:
3370:
3360:
3357:
3351:
3345:
3337:
3335:
3332:) and so its
3331:
3318:
3300:
3296:
3292:
3287:
3283:
3275:
3258:
3254:
3250:
3245:
3241:
3233:
3232:
3231:
3215:
3211:
3201:
3188:
3184:
3178:
3175:
3170:
3166:
3162:
3159:
3156:
3153:
3148:
3144:
3135:
3132:
3118:
3113:
3108:
3105:
3100:
3096:
3093:
3090:
3087:
3084:
3080:
3073:
3067:
3061:
3053:
3050:
3048:
3030:
3026:
3022:
3019:
2997:
2993:
2977:
2961:
2956:
2950:
2947:
2944:
2940:
2935:
2925:
2919:
2911:
2908:
2905:
2900:
2897:
2892:
2886:
2880:
2872:
2869:
2856:
2851:
2846:
2840:
2837:
2834:
2830:
2825:
2820:
2814:
2808:
2800:
2798:
2794:
2789:
2774:
2771:
2768:
2764:
2761:
2758:
2755:
2751:
2745:
2741:
2737:
2733:
2729:
2726:
2722:
2718:
2710:
2704:
2691:
2677:
2674:
2671:
2662:
2646:
2640:
2636:
2629:
2626:
2620:
2615:
2608:
2602:
2596:
2591:
2588:
2584:
2580:
2577:
2574:
2568:
2556:
2547:
2546:
2541:
2525:
2519:
2515:
2506:
2500:
2496:
2491:
2487:
2484:
2481:
2478:
2472:
2460:
2451:
2443:
2439:
2435:
2419:
2416:
2412:
2407:
2404:
2399:
2395:
2392:
2389:
2382:
2378:
2371:
2363:
2357:
2351:
2344:
2340:
2333:
2320:
2317:
2301:
2298:
2295:
2292:
2286:
2280:
2273:
2267:
2262:
2257:
2254:
2249:
2244:
2241:
2237:
2233:
2230:
2224:
2218:
2210:
2207:
2203:
2190:
2187:
2184:
2178:
2166:
2153:
2147:
2141:
2137:
2130:
2124:
2120:
2116:
2113:
2107:
2095:
2086:
2083:
2067:
2061:
2057:
2048:
2042:
2038:
2033:
2029:
2026:
2023:
2020:
2014:
2002:
1993:
1987:
1965:
1962:
1955:
1940:
1933:
1932:
1931:
1914:
1911:
1905:
1899:
1891:
1885:
1860:
1853:
1835:
1829:
1822:
1821:
1820:
1802:
1797:
1793:
1790:
1787:
1783:
1779:
1776:
1772:
1767:
1760:
1754:
1748:
1744:
1741:
1737:
1733:
1730:
1727:
1724:
1718:
1712:
1705:
1704:
1703:
1694:
1692:
1669:
1665:
1657:
1655:
1637:
1633:
1627:
1623:
1615:
1598:
1594:
1586:
1569:
1565:
1557:
1542:
1535:
1534:
1533:
1519:
1512:
1501:
1497:
1484:
1480:
1475:
1471:
1463:
1459:
1453:
1449:
1440:
1436:
1432:
1429:
1421:
1419:
1415:
1404:
1402:
1398:
1393:
1389:
1385:
1365:
1358:
1341:
1337:
1329:
1312:
1308:
1300:
1285:
1278:
1277:
1276:
1273:
1251:
1248:
1245:
1239:
1236:
1233:
1230:
1219:
1215:
1210:
1204:
1200:
1193:
1190:
1184:
1181:
1176:
1172:
1168:
1162:
1156:
1148:
1145:
1143:
1124:
1121:
1118:
1115:
1112:
1109:
1106:Modelling of
1105:
1102:
1099:
1096:
1093:
1092:
1091:
1080:
1077:
1056:
1050:
1042:
1035:
1032:
1025:
1022:
1015:
1014:
1013:
1010:
991:
985:
981:
976:
971:
967:
958:
956:
946:
931:
927:
924:
918:
914:
908:
905:
900:
897:
891:
865:
862:
859:
855:
834:
830:
823:
817:
814:
811:
806:
803:
800:
796:
775:
772:
766:
760:
752:
748:
744:
738:
734:
711:
708:
686:
676:
670:
667:
661:
658:
655:
649:
643:
640:
634:
629:
626:
623:
619:
598:
594:
590:
587:
581:
575:
566:
564:
544:
541:
537:
533:
529:
525:
519:
513:
489:
486:
483:
480:
477:
474:
471:
467:
463:
459:
455:
452:
445:
442:
439:
429:
426:
416:
413:
407:
401:
384:
380:
377:
373:
370:
367:
363:
360:
346:
343:
338:
328:
325:
318:
315:
312:
302:
299:
289:
278:
265:
262:
261:
260:
242:
239:
236:
226:
223:
213:
210:
204:
198:
167:
158:
154:
153:
136:
127:
123:
122:
105:
96:
92:
91:
88:
83:
81:
77:
72:
69:
68:Royal Society
65:
56:
54:
50:
46:
42:
38:
34:
30:
27:is a type of
26:
22:
4568:
4525:
4521:
4511:
4468:
4464:
4458:
4439:
4409:
4403:
4368:
4364:
4354:
4344:, retrieved
4342:, 2023-03-18
4339:
4330:
4275:
4271:
4261:
4209:
4205:
4199:
4178:
4154:(1): 46–56.
4151:
4147:
4137:
4091:
4087:
4077:
4060:
4056:
4050:
4033:
4029:
4019:
3984:
3980:
3970:
3959:
3952:
3925:
3921:
3911:
3876:
3872:
3862:
3839:Growth curve
3813:ELISA assays
3810:
3805:real numbers
3802:
3685:
3682:
3676:
3668:
3662:
3658:
3652:
3648:
3532:
3416:
3338:
3327:
3202:
3136:
3133:
3054:
3051:
2983:
2873:
2870:
2801:
2790:
2692:
2663:
2548:
2542:
2452:
2449:
2440:
2436:
2321:
2318:
2211:
2205:
2204:
2087:
2084:
1994:
1991:
1929:
1818:
1700:
1687:
1422:
1410:
1381:
1274:
1149:
1146:
1138:
1130:Applications
1095:Mobile phone
1089:
1086:Example uses
1078:
1011:
959:
952:
567:
393:
375:
371:
365:
361:
263:
191:
86:
73:
62:
24:
20:
18:
3928:: 513–585.
2442:functions.
1986:natural log
1110:of hydrogen
33:time series
4599:Demography
4593:Categories
4346:2023-03-23
4285:1604.07782
4219:1510.05123
4190:2405.13680
3855:References
1401:metastasis
1397:metabolism
949:Derivation
394:The curve
390:Properties
4570:MathWorld
4478:1309.3337
4465:Physica D
4340:Knowledge
3944:145157003
3769:−
3753:
3742:
3736:−
3698:−
3582:−
3569:−
3554:∞
3551:→
3504:∞
3501:−
3469:−
3456:−
3441:∞
3438:−
3435:→
3384:−
3371:−
3293:≈
3251:≈
3179:α
3171:−
3163:
3134:so that:
3097:
3091:α
2962:ν
2945:ν
2941:ν
2929:∞
2923:→
2920:ν
2912:⋅
2852:ν
2835:ν
2831:ν
2765:
2759:−
2746:ν
2730:−
2719:ν
2714:∞
2708:→
2705:ν
2672:ν
2630:ν
2592:−
2581:ν
2578:α
2561:′
2488:
2482:α
2465:′
2423:∞
2396:
2390:α
2375:→
2337:→
2305:∞
2296:α
2278:⇒
2268:ν
2245:−
2234:α
2185:≤
2171:′
2100:′
2030:
2024:α
2007:′
1966:
1941:α
1895:∞
1889:→
1791:α
1788:−
1780:
1745:
1734:
1691:lag phase
1437:∑
1392:log phase
1246:−
1240:
1234:−
1194:
1185:
977:∝
818:
671:
662:
656:−
644:
561:after an
542:−
534:−
487:
472:−
464:−
440:−
427:−
313:−
300:−
285:∞
282:→
237:−
224:−
45:asymptote
4503:15031322
4395:14219541
4322:28981532
4272:PLOS ONE
4252:35578084
4244:27627325
4170:20048071
4128:16348228
4011:20475000
3903:25750242
3879:(1666).
3828:See also
3665:scaling)
1036:′
378:scaling)
160:Varying
129:Varying
98:Varying
4530:Bibcode
4483:Bibcode
4386:2071101
4313:5628819
4290:Bibcode
4224:Bibcode
4096:Bibcode
4002:2167778
3894:4360127
3815:have a
2664:(where
1819:where:
1298:is time
1275:where:
788:for t.
611:for t.
188:Formula
59:History
4501:
4446:
4416:
4393:
4383:
4320:
4310:
4250:
4242:
4168:
4126:
4119:184525
4116:
4009:
3999:
3942:
3901:
3891:
3412:where
3317:murine
1654:tissue
1012:where
259:where
31:for a
4499:S2CID
4473:arXiv
4280:arXiv
4248:S2CID
4214:arXiv
4185:arXiv
3940:S2CID
2799:when
709:0.368
381:e is
4444:ISBN
4414:ISBN
4391:PMID
4318:PMID
4240:PMID
4166:PMID
4124:PMID
4007:PMID
3899:PMID
3315:for
3023:>
2675:>
2299:<
2206:F(X)
2160:with
1418:taxa
19:The
4538:doi
4526:138
4491:doi
4469:208
4381:PMC
4373:doi
4308:PMC
4298:doi
4232:doi
4156:doi
4114:PMC
4104:doi
4065:doi
4038:doi
3997:PMC
3989:doi
3930:doi
3926:115
3889:PMC
3881:doi
3877:370
3671:is
3544:lim
3428:lim
3160:exp
3094:log
3077:max
2916:lim
2762:log
2701:lim
2485:log
2393:log
2368:lim
2330:lim
2027:log
1963:log
1882:lim
1777:exp
1742:log
1731:exp
1237:exp
1182:exp
888:max
880:is
275:lim
23:or
4595::
4567:.
4536:.
4524:.
4520:.
4497:.
4489:.
4481:.
4467:.
4428:^
4389:.
4379:.
4369:13
4367:.
4363:.
4338:,
4316:.
4306:.
4296:.
4288:.
4276:12
4274:.
4270:.
4246:.
4238:.
4230:.
4222:.
4210:94
4208:.
4164:.
4152:70
4150:.
4146:.
4122:.
4112:.
4102:.
4092:56
4090:.
4086:.
4061:18
4059:.
4032:.
4028:.
4005:.
3995:.
3985:28
3983:.
3979:.
3938:.
3924:.
3920:.
3897:.
3887:.
3875:.
3871:.
3750:ln
3739:ln
3297:10
3255:10
3230::
2976:.
2788:.
1693:.
1191:ln
815:ln
668:ln
659:ln
641:ln
565:.
484:ln
4573:.
4546:.
4540::
4532::
4505:.
4493::
4485::
4475::
4452:.
4422:.
4397:.
4375::
4324:.
4300::
4292::
4282::
4256:.
4254:.
4234::
4226::
4216::
4193:.
4187::
4172:.
4158::
4132:.
4130:.
4106::
4098::
4071:.
4067::
4044:.
4040::
4034:8
4013:.
3991::
3946:.
3932::
3905:.
3883::
3788:]
3783:)
3778:)
3772:d
3766:t
3762:a
3757:(
3746:(
3733:b
3729:[
3723:c
3720:1
3715:=
3712:)
3709:t
3706:(
3701:1
3694:f
3677:e
3675:(
3669:e
3663:y
3659:c
3653:x
3649:b
3634:d
3631:+
3628:a
3625:=
3622:d
3619:+
3614:0
3609:e
3604:a
3601:=
3598:d
3595:+
3588:t
3585:c
3579:b
3574:e
3564:e
3559:a
3548:t
3533:a
3518:d
3515:=
3512:d
3509:+
3496:e
3491:a
3488:=
3485:d
3482:+
3475:t
3472:c
3466:b
3461:e
3451:e
3446:a
3432:t
3417:d
3400:d
3397:+
3390:t
3387:c
3381:b
3376:e
3366:e
3361:a
3358:=
3355:)
3352:t
3349:(
3346:f
3301:6
3288:C
3284:X
3259:9
3246:C
3242:X
3216:C
3212:X
3189:.
3185:)
3176:a
3167:(
3157:K
3154:=
3149:C
3145:X
3119:)
3114:)
3109:X
3106:K
3101:(
3088:,
3085:a
3081:(
3074:=
3071:)
3068:X
3065:(
3062:F
3031:C
3027:X
3020:X
2998:C
2994:X
2957:)
2951:1
2948:+
2936:(
2926:+
2909:K
2906:=
2901:e
2898:K
2893:=
2890:)
2887:t
2884:(
2881:X
2857:K
2847:)
2841:1
2838:+
2826:(
2821:=
2818:)
2815:t
2812:(
2809:X
2775:)
2772:x
2769:(
2756:=
2752:)
2742:/
2738:1
2734:x
2727:1
2723:(
2711:+
2678:0
2650:)
2647:t
2644:(
2641:X
2637:)
2627:1
2621:)
2616:K
2612:)
2609:t
2606:(
2603:X
2597:(
2589:1
2585:(
2575:=
2572:)
2569:t
2566:(
2557:X
2529:)
2526:t
2523:(
2520:X
2516:)
2510:)
2507:t
2504:(
2501:X
2497:K
2492:(
2479:=
2476:)
2473:t
2470:(
2461:X
2420:+
2417:=
2413:)
2408:X
2405:K
2400:(
2383:+
2379:0
2372:X
2364:=
2361:)
2358:X
2355:(
2352:F
2345:+
2341:0
2334:X
2302:+
2293:=
2290:)
2287:0
2284:(
2281:F
2274:)
2263:)
2258:K
2255:X
2250:(
2242:1
2238:(
2231:=
2228:)
2225:X
2222:(
2219:F
2191:,
2188:0
2182:)
2179:X
2176:(
2167:F
2154:,
2151:)
2148:t
2145:(
2142:X
2138:)
2134:)
2131:t
2128:(
2125:X
2121:(
2117:F
2114:=
2111:)
2108:t
2105:(
2096:X
2071:)
2068:t
2065:(
2062:X
2058:)
2052:)
2049:t
2046:(
2043:X
2039:K
2034:(
2021:=
2018:)
2015:t
2012:(
2003:X
1988:.
1972:)
1969:(
1915:K
1912:=
1909:)
1906:t
1903:(
1900:X
1892:+
1886:t
1861:K
1839:)
1836:0
1833:(
1830:X
1803:)
1798:)
1794:t
1784:(
1773:)
1768:K
1764:)
1761:0
1758:(
1755:X
1749:(
1738:(
1728:K
1725:=
1722:)
1719:t
1716:(
1713:X
1670:C
1666:E
1638:C
1634:B
1628:C
1624:N
1599:C
1595:B
1570:C
1566:N
1543:B
1520:)
1513:t
1509:d
1502:C
1498:N
1493:d
1485:C
1481:E
1476:(
1472:+
1469:)
1464:C
1460:B
1454:C
1450:N
1446:(
1441:C
1433:=
1430:B
1366:b
1342:I
1338:N
1313:0
1309:N
1286:t
1261:)
1258:)
1255:)
1252:t
1249:b
1243:(
1231:1
1228:(
1225:)
1220:0
1216:N
1211:/
1205:I
1201:N
1197:(
1188:(
1177:0
1173:N
1169:=
1166:)
1163:t
1160:(
1157:N
1079:k
1060:)
1057:t
1054:(
1051:y
1046:)
1043:t
1040:(
1033:y
1026:=
1023:r
995:)
992:t
989:(
986:y
982:1
972:r
968:k
932:e
928:c
925:a
919:=
915:)
909:t
906:d
901:f
898:d
892:(
866:x
863:a
860:m
856:t
835:c
831:/
827:)
824:b
821:(
812:=
807:x
804:a
801:m
797:t
776:0
773:=
770:)
767:t
764:(
761:f
753:2
749:t
745:d
739:2
735:d
712:a
687:c
683:)
680:)
677:2
674:(
665:(
653:)
650:b
647:(
635:=
630:p
627:w
624:h
620:t
599:2
595:/
591:a
588:=
585:)
582:t
579:(
576:f
545:t
538:e
530:e
526:=
523:)
520:t
517:(
514:f
490:b
481:+
478:t
475:c
468:e
460:e
456:a
453:=
446:t
443:c
435:e
430:b
422:e
417:a
414:=
411:)
408:t
405:(
402:f
376:y
372:c
366:x
362:b
347:a
344:=
339:0
334:e
329:a
326:=
319:t
316:c
308:e
303:b
295:e
290:a
279:t
264:a
243:t
240:c
232:e
227:b
219:e
214:a
211:=
208:)
205:t
202:(
199:f
168:c
137:b
106:a
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