A Comparison and Catalog of Intrinsic Tumor Growth Models
AA Comparison and Catalog of Intrinsic Tumor GrowthModels
E.A. Sarapata a,1, ∗∗ , L.G. de Pillis a, ∗ a Department of Mathematics, Harvey Mudd College, Claremont, CA 91711
Abstract
Determining the mathematical dynamics and associated parameter valuesthat should be used to accurately reflect tumor growth continues to be of in-terest to mathematical modelers, experimentalists and practitioners. However,while there are several competing canonical tumor growth models that are oftenimplemented, how to determine which of the models should be used for whichtumor types remains an open question. In this work, we determine the bestfit growth dynamics and associated parameter ranges for ten different tumortypes by fitting growth functions to at least five sets of published experimentalgrowth data per type of tumor. These time-series tumor growth data are usedto determine which of the five most common tumor growth models (exponential,power law, logistic, Gompertz, or von Bertalanffy) provides the best fit for eachtype of tumor.
Keywords:
Population dynamics, Parameter fitting, Dynamical systems
1. Introduction
Intrinsic tumor growth functions are a component of nearly all continuous,deterministic, cell-population based cancer models, yet there is no universal con-sensus as to which intrinsic growth function should be used when a new mathe- ∗ Corresponding author ∗∗ Principal corresponding author
Email addresses: [email protected] (E.A. Sarapata), [email protected] (L.G. dePillis) Telephone: 917-216-0573. This author has no affiliations or conflicts of interest.
Preprint submitted to Elsevier October 2, 2018 a r X i v : . [ q - b i o . CB ] D ec atical model is being built. Of the published works which focus exclusively onintrinsic tumor growth, five models are widely used: exponential growth func-tions, power Law functions, logistic growth functions, von Bertalanffy growthfunctions, and Gompertz growth functions [1, 2]. In a study by Hart et al. [2], theauthors compare Gompertz, logistic, exponential and power law growth againstmammography data on human breast cancer. The authors ultimately concludethat the power law should be used to represent breast cancer growth, thoughfuture investigations by the same authors found logistic growth to yield a betterfit to the data of interest [2, 3]. Another study by de Pillis and Radunskaya, inwhich intrinsic tumor growth functions for murine melanoma were compared,concluded that von Bertalanffy and logistic growth models provided the mostaccurate fit to data [1]. A study by Zheng et al. [4] compares exponential andbiexponential models of lung cancer growth, in which the biexponential modelis meant to approximate a tumor with two different speeds of growth. Thatstudy concludes that a biexponential model produces a better fit in all testedcases.The choice of intrinsic growth function is strongly driven by the type of can-cer being modeled, in addition to the environment in which it is growing ( e.g. , invitro, in mice, or in humans). In this work, we carry out a thorough explorationof a large collection of published cancer growth data in both mice and humans,and determine the best fit intrinsic growth functions along with the associatedparameter ranges. For each of the ten tumor types we analyze, we have collectedbetween five and ten separate sets of experimental data. After normalizing thedata sets so they can be compared, we fit the data to exponential, power law,logistic, Gompertz and von Bertalanffy growth models.The process of comparing different growth functions to data naturally yieldsbiologically relevant parameter values and ranges associated with each function.We provide all those parameter ranges in this work. Mathematical modelersmust often be creative in choosing appropriate model parameters: they mayborrow parameter values directly from published sources, or they may fit func-tions to data, if relevant data are available, or they may just have to use an ad oc value, choosing a value that yields biologically reasonable dynamics. A largenumber of studies use experimental data from radically different experimentsto estimate parameters; it is nearly unavoidable to combine data from murineand human sources, or obtained from in vitro and in vivo trials [5, 6, 7, 8]. Thechallenge of function choice and parameter determination will always be presentfor the modeler, and techniques for case-by-case parameter choice will still haveto be pursued [9]. However, the catalog of intrinsic growth laws and associatedparameter ranges we provide for a variety of commonly modeled cancer typesshould provide a helpful starting point as researchers develop new models.
2. Assumptions and Methods
We curated time series tumor growth data sets for ten types of tumor: blad-der cancer, breast cancer, colon cancer, head and neck squamous cell carcinoma,hepatocellular carcinoma, lung cancer, melanoma, ovarian cancer, pancreaticcancer, and renal cell carcinoma. Each group of data sets was collected from atleast five peer-reviewed publications, with the smallest-sized group containingseven data sets and the largest containing seventeen data sets. In addition, atleast one data set collected for each type of cancer was obtained from in vitro trials and at least one data set was collected from in vivo trials. Along with in vitro trials, the range of target organisms included SCID mice, nude mice,normal mice, hamsters and humans. Table Appendix B.1 shows all sources foreach time series data set included in our study, as well as the cell lines for eachtrial.
Among the publications that reported time series tumor growth data, theunits and methods of tumor size measurement varied greatly. At least onepaper per type of tumor was an in vitro trial that reported tumor size as acell number, the preferred unit for our purposes, but all data from in vivo in situ trials were presented in units of mm , mm , mm, cm , or relativevolume. A study by Dempsey et al. demonstrated that unidimensional andbidimensional measures of tumor growth are less accurate as a predictor ofsurvival than volumetric measures, lending to a possible source of pre-analyticerror [10]. In addition, instead of assuming a spherical tumor, volume wasreported in a majority of papers as the product of the height, length and widthof the tumor, overestimating the volume. However, we will also assume that noindividual tumor cells are compressed, which will underestimate the number oftumor cells. The combination of these two assumptions is presumed to bringthe estimated cell number within reasonable error of the real cell number.In many cases, we were able to obtain an estimate of the number of tumorcells in a given volume from murine data sets that reported an initial cell countalong with an initial volume measurement. We then divided the volume by thecell number, allowing for an estimate of the volume of a single tumor cell. Weused this same estimate for data sets on tumor growth for tumors originatingfrom the same organ. The most accurate conversion estimate, requiring thefewest conversions from the original data, was an estimate of 2 . × cells /µ m for pancreatic cancer [11]. For types of tumor that did not have a conversiondata set available, we estimated the conversion ratio at approximately 1 . × cells /µ m [6]. Although these two estimates were obtained from differentsources and for different cells, it should be noted that they are the same orderof magnitude despite the high variability of cell size.This volume estimate of a tumor cell provides a method with which to con-vert volume, area or length measurements to cell number. For those data setswhich reported growth in volume, we normalized each datum by µ T where µ T is the tumor cell volume calculated as above. All of the publications that re-ported an area measurement obtained values by multiplying the minor axis ofthe tumor by the major axis [12, 13, 14, 15]. In this case, we assumed a cubictumor with a volume of a √ a where a is the reported area measurement. Thisallows us to calculate cell number from volume as before. Another set of papersreported only the major axis of the tumor [16, 17, 18]. Here, we assumed a4pherical tumor with the radius being one-half the major axis, using the volumeof the sphere to estimate the cell number. For those papers that reported rela-tive volume, we converted the data to cell number using the information in thesupplemental material sections of each paper [19, 20, 21]. We compare fittings of tumor data for five different ODE growth models; ex-ponential, power Law, logistic, Gompertz, and von Bertalanffy. Let P representan arbitrary population and let t represent time. Exponential growth modelsare the simplest ODE growth model, described by dPdt = rP (1)for some intrinsic growth rate constant r . Exponential growth is actually aspecial case of power law growth, represented by dPdt = rP a , (2)where both r and a are parameters that must be fit to the data. Logistic growth,which incorporates a population carrying capacity, is given by dPdt = rP (cid:18) − PK (cid:19) (3)where r represents the intrinsic growth rate and K represents the carrying ca-pacity. Logistic growth looks very much like exponential exponential growthat low populations, but accounts for the resource-limited slowing of growth forlarger populations. Von Bertalanffy growth, also incorporating a carrying ca-pacity, is given by dPdt = r ( K − P ) . (4)The final commonly used tumor growth model we will include is Gompertzgrowth, one form of which is given by dPdt = r log (cid:18) KP (cid:19) P. (5)5nfortunately, very few data sets track tumor growth long enough to sufficientlyestimate carrying capacities. In order to get a good estimate, therefore, wesought out data sets that recorded large tumor cell populations, and comparedthe former two models against the latter three [20, 22, 16, 23, 11, 24, 25, 26, 12]. The parameters for each tumor growth model were estimated using at leasttwo least-squares distance minimization algorithms. For each ODE model, theODE with parameters was solved numerically using MATLAB’s ode45 function,which adaptively implements a 4th or 5th order Runge-Kutta solver. We thenminimized a least squares distance function between the numerical ODE solutionand a target set of data using either MATLAB’s built-in fminsearch functionor a Markov chain fitting with simulated annealing. MATLAB’s fminsearch is a Nelder-Mead simplex direct search function. Nelder-Mead is one of a classof local-search algorithms. Local algorithms require that the user provide aninitial value sufficiently close to the sought after solution, or the method mayconverge to nearby local minimum, but not to a global minimum. The localminimum found may not produce the best fit [27]. This necessitates the use ofan alternate global data fitting method.The global data fitting method we implement is a Markov chain MonteCarlo (MCMC) fitting with simulated annealing, a non-deterministic search al-gortihm. MCMC evaluates a wider range of values in parameter space than does fminsearch , and also includes a method for escaping from a local minimum tocontinue to search for a global minimum[28, 29, 30]. The stochasticity in thealgorithm yields different outcomes even among trials with the same initial con-ditions [28, 29, 30]. The process is repeated n times, where n is an arbitrarynumber chosen by the user. The algorithm has no standard stopping condi-tion. In our case, we chose n = 200 . This number of runs allowed us to achieverelatively good fits while keeping computational running times reasonable. Im-plementing the algorithm with a larger number of iterations n may increase thechance that a global minimum is located.6imulated annealing is the process of fitting, not to the distance function, butto the distance function raised to successive powers from 0 to 1, where the resultof each fitting is used as the initial condition for the next fitting. The simulatedannealing step reduces the chance that a minimization function will converge to alocal minimum instead of a global minimum, since the act of raising the distancefunction to a power less than 1 reduces the prominence of local minima [28].In our fittings, we ran 10 trials with simulated annealing, corresponding toten iterations. Each iteration of the simulated annealing process involved n repetitions of MCMC, where, as above, n = 200 . While the local search algorithm fminsearch will return parameter valuesthat produce the lowest least-squares fitting within a bounded neighborhood ofthe initial parameters, Markov chain methods such as MCMC return the param-eter values that produce the lowest least-squares fitting over a finite number ofarbitrary parameters from anywhere in the parameter space[31]. This differencein the domain of each algorithm leads to defining behaviors that either help orhinder the goodness of fit. The strength of fminsearch is that will convergerapidly to a local minimum, as long as it is near one. However, it is is known tomiss global minima that may produce a better fit. On the other hand, Markovchain methods like MCMC can locate minima that may be far from the initialstate, but they are less likely to hone in on the exact minimum in a local sink.In order to address the respective shortcomings of these global and localparameter fitting algorithms, we used a hybrid approach that incorporates bothNelder-Mead simplex direct search and Markov chain fitting with simulatedannealing. We start with one round of fminsearch fitting. The resulting pa-rameters are then passed as initial conditions to the MCMC algorithm. MCMCis iterated a sufficient number of times to yield parameters giving a good fit; inthis case, 200 times. Since MCMC is effective at breaking out of local minimaand finding the neighborhood of a global minimum, but less effective at actuallyconverging to the minimum, a second round of fminsearch is then performed,using the results of the Markov chain fitting as initial conditions. This ensuresconvergence to the deepest local minimum. All parameters reported in Section7 were determined using this sequence of fitting algorithms.
To determine the recommended parameter values for each growth functionand each tumor type, we recorded the parameters of the function that bestrepresented all trials with the same model organism at once. However, in orderto determine appropriate parameter ranges, we performed fittings to each dataset individually and recorded the extrema of each set of parameters. It is alsoassumed that in vitro trials are better indicators of intrinsic tumor growth rates,due to the lack of an immune system in the growth environment; and that invivo trials are better indicators of animal carrying capacity, since the growthmedia are closer to conditions the tumor would encounter in a living organism.Thus, when relevant, intrinsic growth rates are determined from in vitro trialsonly and carrying capacities are determined from in vivo trials only. In caseswhere no carrying capacity is given, i.e. , the exponential and power law growthmodels, only in vitro trials are used to determine the growth rate, and the invitro trials are also used to determine the exponent for the power law model.
We can compare the goodness-of-fit between the output of fminsearch and MCMC by comparing the least-squares distances between each parameter-dependent function solution the data set to which it was fit. Using this metric,lower residuals (smaller least squares distances) suggest a better fit. We alsouse the least-squares residuals to calculate the Bayesian Information Criterion(BIC) for each fitting [32]. The BIC guards against over-fitting by accounting forgoodness-of-fit while penalizing models that have larger numbers of parametersto be fit.We carried out two types of parameter sensitivity analysis algorithms onthe individual tumor growth models. A “local” or “one-at-a-time” parametersensitivity analysis was performed to measure what the effect on the modeloutcome is when a single parameter is increased or decreased by some percentage8f its value, while keeping other parameters constant. We also carried out aPartial Rank Correlation Coefficient test (PRCC), which is intended to measurethe statistical influence on the model output of parameters that have monotonicbut nonlinear behavior [33, 34, 35]. Since it is impossible to determine PRCCvalues from a model that has only one parameter, the exponential model isexcluded from PRCC analysis.A PRCC value close to zero implies that parameters are independent of oneanother. If the parameter space is large, Latin Hypercube Sampling can be usedto provide input to the PRCC test by random sampling from an n -dimensionalspace for a model with n parameter values [36, 37, 38]. These techniques are onlyapplicable to models with more than one parameter; thus they are performedfor the power law, logistic, Gompertz and von Bertalanffy models, but excludedfor the exponential model.
3. Results
In order to determine a set of recommended parameters and appropriateranges for each type of cancer and growth model, we fit the parameters of eachgrowth equation to a minimum of five data sets per type of cancer. Theseparameters fall into three different classes: intrinsic growth rates (denoted r ),exponents (denoted a ) and carrying capacities (denoted K .) Two different typesof fittings were performed on each set of related data sets. The in vitro trialsfor each type of cancer were fitted separately for the best fit parameters todetermine an acceptable parameter range, then together with different initialconditions to determine the recommended parameter values.We provide a catalog of suggested parameter values and ranges for ten typesof cancer and five models in Table Appendix A.1. The least squares residualsand BIC values for the combined fittings can be found in Table Appendix A.2.In order to highlight the best fits and the relationship between least squaresresiduals and BIC values, in each row, the lowest least squares residuals values9 ancer Model Ranking The “one-at-a-time” parameter sensitivity analysis was carried out by alter-ing each parameter by 10%, with an initial condition of 1 × tumor cells,running the model for 10 days, and starting with the parameters from the indi-vidually determined in vitro colon trials. The results are presented in Figure 1and Figure 2 (where Figure 2 has the power law exponent removed to increasereadability of the percent changes associated with the other parameters.) We10igure 1: Local Parameter Sensitivity Analysis for Five Models, Altering Pa-rameters by 10%also provide a PRCC analysis over 1000 randomized parameter values usingLatin hypercube sampling, which is presented in Table 2.
4. Discussion
Table 1 provides a summary of the best fit models for each type of tumor.These results suggest that of the models tested, there is no one model that bestapproximates all forms of tumor growth. The power law provides especiallyclose fits to data that do not appear to approach a carrying capacity, mostlikely because it is more flexible in approximating exponential growth dynamics.The Gompertz and logistic models outperform either the von Bertalanffy orexponential models in each case.In some cases, the results of the fitting algorithm may be misleading. Lo-gistic growth fittings sometimes resulted in a carrying capacity with an orderof magnitude much higher than comparable trials but with the same intrinsic11igure 2: Local Parameter Sensitivity Analysis for Five Models, Altering Pa-rameters by 10% (with power law a results removed)Parameter PRCCPower law r a r K K K K - 0.0104Von Bertalanffy K in vivo trial 3 for breast cancer; in vivo trial 4 for head and neck squamous cell carci-noma; in vitro trials 1, 2, and 10 and in vivo trial 1 and the combined in vivo fitfor lung cancer; and in vitro trials 1 and 3 for ovarian cancer. When this hap-pens, it may be that the exponential fit is a better match to the data than thelogistic fit. In such a case, the logistic growth function may be approximatingexponential growth by raising the carrying capacity to a number high enough sothat it does not affect the fitting. This theory is supported by the least squaresresiduals; the least squares residuals from the exponential fit and the residualsfrom the logistic fit are the same when this situation occurs.One concern that must be addressed is whether the best-fit parameters arebiologically accurate [39]. We note that the best-fit von Bertalanffy parameters,which are expected to have intrinsic growth rates similar to all other models,consistently have intrinsic growth rates that are two or three orders of magnitudesmaller. This is enough of an indication to doubt the biological accuracy of thevon Bertalanffy parameters obtained by least-squares fitting. In addition, wehave reason to question the biological relevance of the power law fittings forsimilar reasons. We conclude from the parameter fitting process that it may not be justifiableto alter power law growth parameters, even within the range given by repeatedfits. This is because the best fit power law parameters occasionally have un-characteristically high intrinsic growth rates (e.g. in vivo breast cancer trials 1and 2, the combined head and neck squamous cell carcinoma in vivo trial, invitro lung trial 5) and exponents that are lower than the exponents in trials inthe same cancer. These results suggest that power law fitting is highly sensi-tive, where the intrinsic growth rates rise unpredictably to accommodate lowerexponents and vice versa. Therefore, although power law fits occasionally havelower residuals than the other growth laws, their unstable nature would preventmodelers from changing parameters even slightly within a specified range.13n addition, the sensitivity analyses can be used to provide a basis for ourclaim that the power law is not a viable model. Figure 1 suggests that a , theexponential component of the power law model, affects the model output ata much higher percentage than any other parameter in any other model. Infact, increasing a by only 10% caused the tumor to grow almost 35000% largerin only 10 days. This suggests that altering a individually would change thetumor growth behavior at a massive rate that has no biological justification. Analternative would be to alter a and r in conjunction, such that the relativelylow least squares residuals for the fitting are preserved. However, as the PRCCresults suggest, the relationship between a and r is highly nonlinear. This is notsuggestive in and of itself—none of the other parameters had significant PRCCresults—rather, we draw the conclusion in light of the results of the “one-at-a-time” parameter sensitivity analysis. In practice, a researcher seeking to lowerthe growth rate or raise the exponent of some of the less biologically soundpower law fittings would have difficulty determining a relationship between a and r that allows the parameters to be altered while preserving the behaviorof the original curve. This rigidity and extreme sensitivity is what makes thepower law a less than ideal choice for a tumor growth model.For these reasons, despite the low residual fits we found, we discourage theuse of the power law model. To see that the hybrid fitting algorithm is more effective than either theNelder-Mead simplex direct search or Markov Chain method with simulatedannealing, we note that a set of parameters is only accepted if the least squaresresiduals are lower than they were in the previous fitting. Since fminsearch isused to provide initial values for the Markov Chain method, the residuals of aNelder-Mead simplex direct search on a given data set bound the residuals of thehybrid search from above. Due to the nondeterministic nature of the MarkovChain method, the residuals are not necessarily always greater than those of thehybrid method, but it is true that the residuals returned by a specific iteration of14he Markov Chain method will always be greater than the results of the hybridalgorithm using that specific iteration of the Markov Chain method. We havenoted the inability of the Markov chain method to converge on global minima.One may wonder whether using a hybrid fitting algorithm than is necessarywhen fminsearch may have been sufficient. One issue with fminsearch is theinability to converge to a better minimum once a local minimum is detected bythe algorithm, and to improve the fitting would necessitate changing parame-ters by hand. Since this project required 20 separate parameter fitting trialseach to 70 data sets, not including the 90 combined fittings, manually alteringparameters was not a viable option. Thus, even one instance of fminsearch converging to a non-global minimum would necessitate the use of a strongerparameter fitting algorithm. This hybrid method was adopted after repeateddifficulties with fminsearch which would have remained unfixable otherwise.As a side note, it is possible to fit the equations with carrying capacitiesin two different ways: the first, defining the parameters to be estimated as r and K , and the second, defining the parameters to be estimated as r and 1 /K .Although theoretically equivalent, these two approaches can produce differentoutcomes depending on which fitting metric is used. For the logistic equationdefined as dPdt = rP (cid:18) − PK (cid:19) , (6)it is possible that the fitting algorithm may be slower in converging to the best-fit K , because it is possible for the best-fit K to be several orders of magnitudehigher than the initial condition. However, for the logistic equation defined as dPdt = rP (1 − bP ) (7)where b = K , both the Nelder-Mead simplex direct search and the MCMCmethod occasionally produced results where the carrying capacity was negative.This is a result of the relative distance in parameter space from the negativereal axis; 1 × and − × are much further apart than are 1 × − and − × − , for example. Therefore, fitting to equation (6) makes it more difficult15or either algorithm to reach negative values. We therefore recommend fittinglogistic growth using the parameter forms of equation (6) in order to avoid thefittings from producing a biologically inaccurate carrying capacity. While it may seem odd to perform a sensitivity analysis on a series of modelsthat each have only one or two parameters, these techniques can be interpretedto compare the justifiability of modifying parameters in each growth model.The PRCC values provide a measure of the strength of the relationship betweentwo parameters, while the “one-at-a-time” parameter sensitivity analysis mea-sures the effect of individual parameters on the model output. Therefore, whilethe “one-at-a-time” parameter sensitivity analysis can be used to estimate theeffects of changing the value of a single parameter, the PRCC measure can tellus whether altering a single parameter while leaving the other constant is justi-fiable. If the “one-at-a-time” parameter sensitivity analysis reveals that alteringa parameter by a small amount changes the output of the model by a significantamount, then researchers should be careful when modifying these parameters.
Acknowledgements
We would like to thank Harvey Mudd College for providing us with the re-sources to complete this publication, and Ami Radunskaya, the second reader.
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Supplemental Materials: Tumor Growth Parameters We present a catalog of suggested parameter values and ranges for ten typesof cancer and the five canonical growth functions that were considered. Theparameters found using a hybrid fitting algorithm to a minimum of five datasets per type of tumor are given in Table Appendix A.1. For comparisonpurposes, the least squares residuals and BIC values are presented in TableAppendix A.2. We highlight the lowest least squares residuals in each row witha solid border and the lowest BIC values in each row with a dashed border.27 a n ce r E x p o n e n t i a l P o w e r L a w L og i s t i c G o m p e r t z V o n B e r t a l a n ff y B l a dd e rr . : ( . ) : . . : ( . ) : . . : ( . ) : . . : ( . ) : . . E − : ( . ) : . , K . : ( . ) : . . E : ( . E ) : . E . E : ( . E ) : . E . E : ( . E ) : . E B r e a s t r . : ( . ) : . . : ( . ) : . . : ( . ) : . . : ( . ) : . . E − : ( . E − ) : . , K . : ( . ) : . . E : ( . E ) : . E . E : ( . E ) : . E . E : ( . E ) : . E C o l o n r . : ( . ) : . . : ( . ) : . . : ( . ) : . . : ( . ) : . . E − : ( . E − ) : . E , K . : ( . ) : . . E : ( . E ) : . E . E . E : ( . E ) : . E . E : ( . E ) : . E HN S CC r . : ( . ) : . . : ( . ) : . . : ( . ) : . . : ( . ) : . . E − : ( . E − ) : . E − , K . : ( . ) : . . E : ( . E ) : . E . E : ( . E ) : . E . E : ( . E ) : . E L i v e rr . : ( . ) : . : . ( . ) : . . : ( . ) : . . : ( . ) : . . E − : ( . E − ) : . E − , K . : ( . ) : . . E : ( . E ) : . E . E : ( . R ) : . E . E : ( . E ) : . E L un g r . : ( . ) : . . : ( . ) : . E . : ( . ) : . . : ( . ) : . . E − : ( . E − ) : . , K - . : ( . ) : . . E : ( . E ) : . E . E : ( . E ) : . E . E : ( . E ) : . E M e l a n o m a r . : ( . ) : . . : ( . ) : . . : ( . ) : . . : ( . ) : . . : ( . ) : . , K . : ( . ) : . . E : ( . E ) : . E . E : ( . E ) : . E . E : ( . E ) : . E O v a r i a n r . : ( . ) : . . E − : ( . ) : . E . : ( . ) : . . : ( . ) : . . E − : ( . E − ) : . , K - . : ( . ) : . . E : ( . E ) : . E . E : ( . E ) : . E . E : ( . E ) : . E P a n c r e a t i c r . : ( . ) : . . E − : ( . ) : . . : ( . ) : . . : ( . ) : . . E − : ( . ) : . , K . : ( . ) : . . E : ( . E ) : . E . E : ( . E ) : . E . E : ( . E ) : . E R CC r . : ( . ) : . . : ( . ) : . . : ( . ) : . . : ( . ) : . . E − : ( . E − ) : . , K . : ( . ) : . . E : ( . E ) : . E . E : ( . E ) : . E . E : ( . E ) : . E T a b l e A pp e nd i x A . : R ec o mm e nd e d P a r a m e t e r V a l u e s a nd R a n g e s f o r T e n D i ff e r e n t T y p e s o f C a n ce r a nd F i v e O D E G r o w t h L a w s r i a l E x p o n e n t i a l P o w e r L a w L og i s t i c G o m p e r t z V o n B e r t a l a n ff y R e s i du a l s B I C R e s i du a l s B I C R e s i du a l s B I C R e s i du a l s B I C R e s i du a l s B I C B l a dd e r i n v i t r o . E . . E . . E . . E . . E . B l a dd e r i n v i v o . E . . E . . E . . E . . E . B r e a s t i n v i t r o . E . . E . . E . . E . . E . B r e a s t i n v i v o . E . . E . . E . . E . . E . C o l o n i n v i t r o . E . . E . . E . . E . . E . C o l o n i n v i v o . E . . E . . E . . E . . E . HN S CC i n v i t r o . E . . E . . E . . E . . E . HN S CC i n v i v o . E . . E . . E . . E . . E . L i v e r i n v i t r o . E . . E . . E . . E . . E . L i v e r i n v i v o . E . . E . . E . . E . . E . L un g i n v i t r o . E . . E . . E . . E . . E . L un g i n v i v o . E . . E . . E . . E . . E . M e l a n o m a i n v i t r o . E . . E . . E . . E . . E . M e l a n o m a i n v i v o . E . . E . . E . . E . . E . O v a r i a n i n v i t r o . E . . E . . E . . E . . E . O v a r i a n i n v i v o . E . . E . . E . . E . . E . P a n c r e a t i c i n v i t r o . E . . E . . E . . E . . E . P a n c r e a t i c i n v i v o . E . . E . . E . . E . . E . R CC i n v i t r o . E . . E . . E . . E . . E . R CC i n v i v o . E . . E . . E . . E . . E . T a b l e A pp e nd i x A . : M o d e l E v a l u a t i o n M e t r i c s f o r C o m b i n e d E x p e r i m e n t a l D a t a F i tt i n g s ppendix B. Supplemental Materials: Sources of Data for Parame-ter Values A large number of individual studies were gathered in determining appropri-ate timescale tumor growth data sets to be used in the fitting process. Not onlyare the sources for each type of cancer listed, the individual cell lines used ineach paper are included for posterity. Some papers, which used tissue samplesfrom human subjects as the source of cancerous cells, did not specify a cell line.Cancer and Cell Line Sources
Bladder Cancer
HT1376 [40]UMUC-3 [41]KoTCC-1 [42]EJ-1 [43, 44]
Breast Cancer
MDA-MB-435BAG [45]MCF-7 [46]KPL-1 [47]4T1-GFP-FL [48]
Colon Cancer
KM12L4 [24]Moser [49]HCT116 [49, 50]CX-1 [49]HCA7 [50]LS LiM6 [51]Unspecified [52]
Head and Neck Squamous Cell Carcinoma
UM-SCC-9 [53]Tu-138 [54]Table Appendix B.1: Sources of Timescale Data by Type of Cancer and CellLine 30ancer and Cell Line SourcesTu-167 [54]686LN [54]CAL27 [55]UM-SCC-X [12]PAM-LY2 [13]
Hepatocellular Carcinoma
HCC-26-1004 [56]HCC-2-1318 [56]SH-J1 [57]PLC [58]Hep3B [58]SMMC-7721 [59]Unspecified [60]
Lung Cancer
SW-900 [61]H226 [61]A549 [61][62][63]H460 [62]H1299 [62]U2020 [62]H322a [20]WT226b [20]NCI-H727 [64]3LL [65]NCI-H358 [22]H841 [63]Table Appendix B.1: Sources of Timescale Data by Type of Cancer and CellLine 31ancer and Cell Line Sourcespc14 [63]
Melanoma
M3Dau [14]MIRW5 [66]B16-BL6 [26, 16]A-375 [67]M21 [68]Hs0294 [23]Unspecified [69]
Ovarian Cancer
SKOV-3 [15, 70]HRA [25]A2780 [70]IGROV-1 [70]HCT-116 [70]MA148 [71]
Pancreatic Cancer
PC-1 [17]MIAPaCa-2 [72, 11]PANC-1 [11]PancTu1 [73]HPAC [74]
Renal Cell Carcinoma
Appendix C. Supplemental Materials: Results of Parameter Fit-tings
Individual data sets are labeled with the year and author, and given a uniqueidentifier: either the label they were presented with in the figure from which thedata originated, or the cell line that is used in the paper.In some cases, the line representing the result of the parameter fitting is notvisible. This happens for one of two reasons: a large difference between ordersof magnitude in separate data sets, limiting the available space for data setswith smaller orders of magnitide; or because two or more data sets started withthe same initial condition, causing the combined fitting result to produce thesame curve. A complete list of all of the parameter fittings that are not visible,and the reason for why they cannot be seen, is given below: • In the combined in vitro bladder cancer trials, the AS clusterin and MMcontrol trials of Miyake 2001 share an initial condition, hence only theMM control fitting is visible. • In the combined in vitro breast cancer trials, the three Smith 2004 trialsshare the same initial condition, so the purple curve indicates the fittingto all three of these trials. • In the combined in vivo breast cancer trials, the two Coopman 2000 trialsshare an initial condition, thus the green curve represents the fitting toboth trials. 33
In the combined in vitro colon cancer trials, the Moser and HCT116 trialshave the same initial condition, so the green curve represents the combinedfitting to both. • In the combined in vivo colon cancer trials, two sets of trials have thesame initial condition—the two Reinmuth 2002 trials and the two Warren1995 trials. As a result, the orange curves account for both Reinmuth2002 trials and the pink curves to both Warren 1995 trials. • In the combined in vivo head and neck squamous cell carcinoma trials,the three Liu 1999 trials start with the same initial conditions, hence theteal curve represents the combined fitting to all three data sets. • In the combined in vitro hepatocellular carcinoma trials, the Huynh 2008trials share an initial condition, so the green curve represents the fittingto both data sets. • In the combined in vivo hepatocellular carcinoma trials, the Liu 2005 datasets have the same initial condition, so the teal curve indicates the fittingto both data sets. • In the individual in vitro lung cancer trials, Fig. 4D from Fabbri 2005 wascropped from the graph because it was two orders of magnitude higherthan the next largest tumor, making the other 12 trials impossible todistinguish. Despite its exclusion here, it was used in the fitting analysis. • In the combined in vitro lung cancer trials, not only is Fig. 4D from Fabbri2005 excluded, but several trials from the same study have the same initialconditions ( i.e. , the four visible Fabbri 2005 trials, SW-900 and A549from Esquela-Kerscher, and all three Fujiwara trials.) For this figure, theseafoam green curve is the fit for all 4 visible Fabbri 2005 trials, the SW-900 and A549 trials from Esquela-Kerscher are both represented by theyellow curve, and all three Fujiwara trials are represented by the purplecurve. Additionally, for the von Bertalanffy fitting, the data sets from34ujiwara 1993 and Takahashi 1992 are hidden by Fig. 3 from Tsubouchi2000, presumably because their initial conditions are sufficiently close toeach other. • In the in vivo melanoma trials, the Boucherke 1989 trial is difficult to seebecause of its relatively low order of magnitude, but is visible along thebottom of the graphs. • In the combined in vitro ovarian cancer trials, the A2780 and SKOV-3 trials have the same initial condition, and the IGROV-1 and HCT-116 trials have the same initial condition. As a result, the green curverepresents the fitting to the first two trials, and the purple curve is thefitting to the last two trials. 35 i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g s t o I n V i t r o B l a dd e r C a n ce r T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g s t o I nd i v i du a l I n V i v o B l a dd e r C a n ce r T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g t o C o m b i n e d I n V i v o B l a dd e r C a n ce r T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g s t o I nd i v i du a l I n V i t r o B r e a s t C a n ce r T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g t o C o m b i n e d I n V i t r o B r e a s t C a n ce r T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g s t o I nd i v i du a l I n V i v o B r e a s t C a n ce r T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g t o C o m b i n e d I n V i v o B r e a s t C a n ce r T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g s t o I nd i v i du a l I n V i t r o C o l o n C a n ce r T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g t o C o m b i n e d I n V i t r o C o l o n C a n ce r T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g s t o I nd i v i du a l I n V i v o C o l o n C a n ce r T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g t o C o m b i n e d I n V i v o C o l o n C a n ce r T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g s t o I nd i v i du a l I n V i t r o H e a d a nd N ec k S q u a m o u s C e ll C a r c i n o m a T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g t o C o m b i n e d I n V i t r o H e a d a nd N ec k S q u a m o u s C e ll C a r c i n o m a T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g s t o I nd i v i du a l I n V i v o H e a d a nd N ec k S q u a m o u s C e ll C a r c i n o m a T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g t o C o m b i n e d I n V i v o H e a d a nd N ec k S q u a m o u s C e ll C a r c i n o m a T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g s t o I nd i v i du a l I n V i t r o H e p a t o ce ll u l a r C a r c i n o m a T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g t o C o m b i n e d I n V i t r o H e p a t o ce ll u l a r C a r c i n o m a T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g s t o I nd i v i du a l I n V i v o H e p a t o ce ll u l a r C a r c i n o m a T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g t o C o m b i n e d I n V i v o H e p a t o ce ll u l a r C a r c i n o m a T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g s t o I nd i v i du a l I n V i t r o L un g C a n ce r T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g t o C o m b i n e d I n V i t r o L un g C a n ce r T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g s t o I nd i v i du a l I n V i v o L un g C a n ce r T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g t o C o m b i n e d I n V i v o L un g C a n ce r T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g s t o I nd i v i du a l I n V i t r o M e l a n o m a T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g t o C o m b i n e d I n V i t r o M e l a n o m a T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g s t o I nd i v i du a l I n V i v o M e l a n o m a T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g t o C o m b i n e d I n V i v o M e l a n o m a T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g s t o I nd i v i du a l I n V i t r o O v a r i a n C a n ce r T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g t o C o m b i n e d I n V i t r o O v a r i a n C a n ce r T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g s t o I nd i v i du a l I n V i v o O v a r i a n C a n ce r T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g t o C o m b i n e d I n V i v o O v a r i a n C a n ce r T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g s t o I n V i t r o P a n c r e a t i c C a n ce r T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g s t o I nd i v i du a l I n V i v o P a n c r e a t i c C a n ce r T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g t o C o m b i n e d I n V i v o P a n c r e a t i c C a n ce r T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g s t o I nd i v i du a l I n V i t r o R e n a l C e ll C a r c i n o m a T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g t o C o m b i n e d I n V i t r o R e n a l C e ll C a r c i n o m a T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g s t o I nd i v i du a l I n V i v o R e n a l C e ll C a r c i n o m a T r i a l s i g u r e A pp e nd i x C . : P a r a m e t e r F i tt i n g t o C o m b i n e d I n V i v o R e n a l C e ll C a r c i n o m a T r i a l ss