Category Archives: cancer models

Misapplication of statistical tests to simulated data: Mathematical Oncologists join Cardiac Modellers

In a previous blog post we highlighted the pitfalls of applying null hypothesis testing to simulated data, see here.  We showed that modellers applying null hypothesis testing to simulated data can control the p-value because they can control the sample size. Thus it’s not a great idea to analyse simulations using null hypothesis tests, instead modellers should focus on the size of the effect.  This problem has been highlighted before by White et al.  which is well worth a read, see here.

Why are we blogging about this subject again? Since that last post, co-authors of the original article we discussed there have repeated the same misdemeanour (Liberos et al., 2016), and a group of mathematical oncologists based at Moffitt Cancer Center has joined them (Kim et al., 2016).

The article by Kim et al., preprint available here, describes a combined experimental and modelling approach that “predicts” new dosing schedules for combination therapies that can delay onset of resistance and thus increase patient survival.  They also show how their approach can be used to identify key stratification factors that can determine which patients are likely to do better than others. All of the results in the paper are based on applying statistical tests to simulated data.

The first part of the approach taken by Kim et al. involves calibrating a mathematical model to certain in-vitro experiments.  These experiments basically measure the number of cells over a fixed observation time under 4 different conditions: control (no drug), AKT inhibitor, Chemotherapy and Combination (AKT/Chemotherapy).  This was done for two different cell lines. The authors found a range of parameter values when trying to fit their model to the data. From this range they took forward a particular set, no real justification as to why that certain set, to test the model’s ability to predict different in-vitro dosing schedules. Unsurprisingly the model predictions came true.

After “validating” their model against a set of in-vitro experiments the authors proceed to using the model to analyse retrospective clinical data; a study involving 24 patients.  The authors acknowledge that the in-vitro system is clearly not the same as a human system.  So to account for this difference they perform an optimisation method to generate a humanised model.  The optimisation is based on a genetic algorithm which searched the parameter space to find parameter sets that replicate the clinical results observed.  Again, similar to the in-vitro situation, they found that there were multiple parameter sets that were able to replicate the observed clinical results. In fact they found a total of 3391 parameter sets.

Having now generated a distribution of parameters that describe patients within the clinical study they are interested in, the authors next set about generating stratification factors. For each parameter set the virtual patient exhibits one of four possible response categories. Therefore for each category a distribution of parameter values exists for the entire population. To assess the difference in the distribution of parameter values across the categories they perform a students t-test to ascertain whether the differences are statistically significant. Since they can control the sample size the authors can control the standard error and p-value, this is exactly the issue raised by White et al. An alternative approach would be to state the difference in the size of the effect, so the difference in means of the distributions. If the claim is that a given parameter can discriminate between two types of responses then a ROC AUC (Receiver Operating Characteristic Area Under Curve) value could be reported. Indeed a ROC AUC value would allow readers to ascertain the strength of a given parameter in discriminating between two response types.

The application of hypothesis testing to simulated data continues throughout the rest of the paper, culminating in applying a log-rank test to simulated survival data, where again they control the sample size. Furthermore, the authors choose an arbitrary cancer cell number which dictates when a patient dies. Therefore they have two ways of controlling the p-value.  In this final act the authors again abuse the use of null hypothesis testing to show that the schedule found by their modelling approach is better than that used in the actual clinical study.  Since the major results in the paper have all involved this type of manipulation, we believe they should be treated with extreme caution until better verified.

References

Liberos, A., Bueno-Orovio, A., Rodrigo, M., Ravens, U., Hernandez-Romero, I., Fernandez-Aviles, F., Guillem, M.S., Rodriguez, B., Climent, A.M., 2016. Balance between sodium and calcium currents underlying chronic atrial fibrillation termination: An in silico intersubject variability study. Heart Rhythm 0. doi:10.1016/j.hrthm.2016.08.028

White, J.W., Rassweiler, A., Samhouri, J.F., Stier, A.C., White, C., 2014. Ecologists should not use statistical significance tests to interpret simulation model results. Oikos 123, 385–388. doi:10.1111/j.1600-0706.2013.01073.x

Kim, E., Rebecca, V.W., Smalley, K.S.M., Anderson, A.R.A., 2016. Phase i trials in melanoma: A framework to translate preclinical findings to the clinic. Eur. J. Cancer 67, 213–222. doi:10.1016/j.ejca.2016.07.024

 

Time-dependent bias of tumour growth rate and time to tumour re-growth

The title of this blog entry refers to a letter published in the journal entitled, CPT: Pharmacometrics & Systems Pharmacology. The letter is open-access so those of you interested can read it online here.  In this blog entry we will go through it.

The letter discusses a rather strange modelling practice which is becoming the norm within certain modelling and simulation groups in the pharmaceutical industry. There has been a spate of publications citing that tumour re-growth rate (GR) and time to tumour re-growth (TTG), derived using models to describe imaging time-series data, correlates to survival [1-6]. In those publications the authors show survival curves (Kaplan-Meiers) highlighting a very strong relationship between GR/ TTG and survival.  They either split on the median value of GR/TTG or into quartiles and show very impressive differences in survival times between the groups created; see Figure 2 in [4] for an example (open access).

Do these relationships seem too good to be true? In fact they may well be. In order to derive GR/TTG you need time-series data. The value of these covariates are not known at the beginning of the study, and only become available after a certain amount of time has passed.  Therefore this type of covariate is typically referred to as a time-dependent covariate. None of the authors in [1-6] describe GR/TTG as a time-dependent covariate nor treat it as such.

When the correlations to survival were performed in those articles the authors assumed that they knew GR/TTG before any time-series data was collected, which is clearly not true. Therefore survival curves, such as Figure 2 in [4], are biased as they are based on survival times calculated from study start time to time of death, rather than time from when GR/TTG becomes available to time of death.  Therefore, the results in [1-6] should be questioned and GR/TTG should not be used for decision making, as the question around whether tumour growth rate correlates to survival is still rather open.

Could it be the case that the GR/TTG correlation to survival is just an illusion of a flawed modelling practice?  This is what we shall answer in a future blog-post.

[1] W.D. Stein et al., Other Paradigms: Growth Rate Constants and Tumor Burden Determined Using Computed Tomography Data Correlate Strongly With the Overall Survival of Patients With Renal Cell Carcinoma, Cancer J (2009)

[2] W.D. Stein, J.L. Gulley, J. Schlom, R.A. Madan, W. Dahut, W.D. Figg, Y. Ning, P.M. Arlen, D. Price, S.E. Bates, T. Fojo, Tumor Regression and Growth Rates Determined in Five Intramural NCI Prostate Cancer Trials: The Growth Rate Constant as an Indicator of Therapeutic Efficacy, Clin. Cancer Res. (2011)

[3] W.D. Stein et al., Tumor Growth Rates Derived from Data for Patients in a Clinical Trial Correlate Strongly with Patient Survival: A Novel Strategy for Evaluation of Clinical Trial Data, The Oncologist.  (2008)

[4] K. Han, L. Claret, Y. Piao, P. Hegde, A. Joshi, J. Powell, J. Jin, R. Bruno, Simulations to Predict Clinical Trial Outcome of Bevacizumab Plus Chemotherapy vs. Chemotherapy Alone in Patients With First-Line Gastric Cancer and Elevated Plasma VEGF-A, CPT Pharmacomet. Syst. Pharmacol. (2016)

[5] J. van Hasselt et al., Disease Progression/Clinical Outcome Model for Castration-Resistant Prostate Cancer in Patients Treated With Eribulin, CPT Pharmacomet. Syst. Pharmacol. (2015)

[6] L. Claret et al., Evaluation of Tumor-Size Response Metrics to Predict Overall Survival in Western and Chinese Patients With First-Line Metastatic Colorectal Cancer, J. Clin. Oncol. (2013)

The exponential growth effect

A common critique of biologists, and scientists in general, concerns their occasionally overenthusiastic tendency to find patterns in nature – especially when the pattern is a straight line. It is certainly notable how, confronted with a cloud of noisy data, scientists often manage to draw a straight line through it and announce that the result is “statistically significant”.

Straight lines have many pleasing properties, both in architecture and in science. If a time series follows a straight line, for example, it is pretty easy to forecast how it should evolve in the near future – just assume that the line continues (note: doesn’t always work).

However this fondness for straightness doesn’t always hold; indeed there are cases where scientists prefer to opt for a more complicated solution. An example is the modelling of tumour growth in cancer biology.

Tumour growth is caused by the proliferation of dividing cells. For example if cells have a cell cycle length td, then the total number of cells will double every td hours, which according to theory should result in exponential growth. In the 1950s (see Collins et al., 1956) it was therefore decided that the growth rate could be measured using the cell doubling time.

In practice, however, it is found that tumours grow more slowly as time goes on, so this exponential curve needed to be modified. One variant is the Gompertz curve, which was originally derived as a model for human lifespans by the British actuary Benjamin Gompertz in 1825, but was adapted for modelling tumour growth in the 1960s (Laird, 1964). This curve gives a tapered growth rate, at the expense of extra parameters, and has remained highly popular as a means of modelling a variety of tumour types.

However, it has often been observed empirically that tumour diameters, as opposed to volumes, appear to grow in a roughly linear fashion. Indeed, this has been known since at least the 1930s. As Mayneord wrote in 1932: “The rather surprising fact emerges that the increase in long diameter of the implanted tumour follows a linear law.” Furthermore, he noted, there was “a simple explanation of the approximate linearity in terms of the structure of the sarcoma. On cutting open the tumour it is often apparent that not the whole of the mass is in a state of active growth, but only a thin capsule (sometimes not more than 1 cm thick) enclosing the necrotic centre of the tumour.”

Because only this outer layer contains dividing cells, the rate of increase for the volume depends on the doubling time multiplied by the volume of the outer layer. If the thickness of the growing layer is small compared to the total tumour radius, then it is easily seen that the radius grows at a constant rate which is equal to the doubling time multiplied by the thickness of the growing layer. The result is a linear growth in radius. This  translates to cubic growth in volume, which of course grows more slowly than an exponential curve at longer times – just as the data suggests.

In other words, rather than use a modified exponential curve to fit volume growth, it may be better to use a linear equation to model diameter. This idea that tumour growth is driven by an outer layer of proliferating cells, surrounding a quiescent or necrotic core, has been featured in a number of mathematical models (see e.g. Checkley et al., 2015, and our own CellCycler model).  The linear growth law can also be used to analyse tumour data, as in the draft paper: “Analysing within and between patient patient tumour heterogenity via imaging: Vemurafenib, Dabrafenib and Trametinib.” The linear growth equation will of course not be a perfect fit for the growth of all tumours (no simple model is), but it is based on a consistent and empirically verified model of tumour growth, and can be easily parameterised and fit to data.

So why hasn’t this linear growth law caught on more widely? The reason is that what scientists see in data often depends on their mental model of what is going on.

I first encountered this phenomenon in the late 1990s when doing my D.Phil. in the prediction of nonlinear systems, with applications to weather forecasting. The dominant theory at the time said that forecast error was due to sensitivity to initial condition, aka the butterfly effect. As I described in The Future of Everything, researchers insisted that forecast errors showed the exponential growth characteristic of chaos, even though plots showed they clearly grew with slightly negative curvature, which was characteristic of model error.

A similar effect in cancer biology has again changed the way scientists interpret data. Sometimes, a straight line really is the best solution.

References

Collins, V. P., Loeffler, R. K. & Tivey, H. Observations on growth rates of human tumors. The American journal of roentgenology, radium therapy, and nuclear medicine 76, 988-1000 (1956).

Laird A. K. Dynamics of tumor growth. Br J of Cancer 18 (3): 490–502 (1964).

W. V. Mayneord. On a Law of Growth of Jensen’s Rat Sarcoma. Am J Cancer 16, 841-846 (1932).

Stephen Checkley, Linda MacCallum, James Yates, Paul Jasper, Haobin Luo, John Tolsma, Claus Bendtsen. Bridging the gap between in vitro and in vivo: Dose and schedule predictions for the ATR inhibitor AZD6738. Scientific Reports, 5(3)13545 (2015).

Yorke, E. D., Fuks, Z., Norton, L., Whitmore, W. & Ling, C. C. Modeling the Development of Metastases from Primary and Locally Recurrent Tumors: Comparison with a Clinical Data Base for Prostatic Cancer. Cancer Research 53, 2987-2993 (1993).

Hitesh Mistry, David Orrell, and Raluca Eftimie. Analysing within and between patient patient tumour heterogenity via imaging: Vemurafenib, Dabrafenib and Trametinib. (Working paper)

The CellCycler

Tumour modelling has been an active field of research for some decades, and a number of approaches have been taken, ranging from simple models of an idealised spherical tumour, to highly complex models which attempt to account for everything from cellular chemistry to mechanical stresses. Some models use ordinary differential equations, while others use an agent-based approach to track individual cells.

A disadvantage of the more complex models is that they involve a large number of parameters, which can only be roughly estimated from available data. If the aim is to predict, rather than to describe, then this leads to the problem of overfitting: the model is very flexible and can be tuned to fit available data, but is less useful for predicting for example the effect of a new drug.

Indeed, there is a rarely acknowledged tension in mathematical modelling between realism, in the sense of including lots of apparently relevant features, and predictive accuracy. When it comes to the latter, simple models often out-perform complex models. Yet in most areas there is a strong tendency for researchers to develop increasingly intricate models. The reason appears to have less to do with science, than with institutional effects. As one survey of business models notes (and these points would apply equally to cancer modelling) complex models are preferred in large part because: “(1) researchers are rewarded for publishing in highly ranked journals, which favor complexity; (2) forecasters can use complex methods to provide forecasts that support decision-makers’ plans; and (3) forecasters’ clients may be reassured by incomprehensibility.”

Being immune to all such pressures (this is just a blog post after all!) we decided to develop the CellCycler – a parsimonius “toy” model of a cancer tumour that attempts to capture the basic growth and drug-response dynamics using only a minimal number of parameters and assumptions.  The model uses circa 100 ordinary differential equations (ODEs) to simulate cells as they pass through the phases of the cell cycle; however the equations are simple and the model only uses parameters that can be observed or reasonably well approximated. It is available online as a Shiny app.

CellCyclerScreenshot1
Screenshot of the Cells page of the CellCycler. The plot shows how a cell population is affected by two different drugs.

The CellCycler model divides the cell cycle into a number of discrete compartments, and is therefore similar in spirit to other models that for example treat each phase G1, S, G2, and mitosis as a separate compartment, with damaged cells being shunted to their own compartment (see for example the model by Checkley et al. here). Each compartment has its own set of ordinary differential equations which govern how its volume changes with time due to growth, apoptosis, or damage from drugs. There are additional compartments for damaged cells, which may be repaired or lost to apoptosis. Drugs are simulated using standard PK models, along with a simple description of phase-dependent drug action on cells. For the tumour growth, we use a linear model, based like the Checkley et al. paper on the assumption of a thin growing layer (see also our post on The exponential growth effect).

The advantages of compartmentalising

Dividing the cell cycle into separate compartments has an interesting and useful side effect, which is that it introduces a degree of uncertainty into the calculation. For example, if a drug causes damage and delays progress in a particular phase, then that drug will tend to synchronize the cell population in that state. However there is an obvious difference between cells that are affected when they are at the start of the phase, and those that are already near the end of the phase. If the compartments are too large, that precise information about the state of cells is lost.

The only way to restore precision would be to use a very large number of compartments. But in reality, individual cells will not all have exactly the same doubling time. We therefore want to have a degree of uncertainty. And this can be controlled by adjusting the number of compartments.

This effect is illustrated by the figure below, which shows how a perturbation at time zero in one compartment tends to blur out over time, for models with 25, 50, and 100 compartments, and a doubling time of 24 hours. In each case a perturbation is made to compartment 1 at the beginning of the cell cycle (the magnitude is scaled to the number of compartments so the total size of the perturbation is the same in terms of total volume). For the case with 50 compartments, the curve after one 24 hours is closely approximated by a normal distribution with standard deviation of 3.4 hours or about 14 percent. In general, the standard deviation can be shown to be approximately equal to the doubling time divided by the square root of N.

PertSimulationPlot
The solid lines show volume in compartment 1 following a perturbation to that compartment alone, after one cell doubling period of 24 hours. The cases shown are with N=25, 50, and 100 compartments. Dashed lines are the corresponding normal distributions.

A unique feature of the CellCycler is that it exploits this property as a way of adjusting the variability of doubling time in the cell population. The model can therefore provide a first-order approximation to the more complex heterogeneity that can be simulated using agent-based models. While we don’t usually have exact data on the spread of doubling times in the growing layer, the default level of 50 compartments gives what appears to be a reasonable degree of spread (about 14 percent). Using 25 compartments gives 20 percent, while using 100 compartments decreases this to 10 percent.

Using the CellCycler

The starting point for the Shiny web application is the Cells page, which is used to model the dynamics of a growing cell population. The key parameters are the average cell doubling time, and the fraction spent in each phase. The number of model compartments can be adjusted in the Advanced page: note that, along with doubling time spread, the choice also affects both the simulation time (more compartments is slower), and the discretisation of the cell cycle. For example with 50 compartments the proportional phase times will be rounded off to the nearest 1/50=0.02.

The next pages, PK1 and PK2, are used to parameterise the PK models and drug effects. The program has a choice of standard PK models, with adjustable parameters such as Dose/Volume.  In addition the phase of action (choices are G1, S, G2, M, or all), and rates for death, damage, and repair can be adjusted. Finally, the Tumor page (shown below) uses the model simulation to generate a plot of tumor radius, given an initial radius and growing layer. Plots can be overlaid with experimental data.

CellCyclerScreenshot2
Screenshot of the Tumor page, showing tumor volume (black line) compared to control (grey). Cell death due to apoptosis by either drug (red and blue) and damage (green) are also shown.

We hope the CellCycler can be a useful tool for research or for exploring the dynamics of tumour growth. As mentioned above it is only a “toy” model of a tumour. However, all our models of complex organic systems – be they of a tumor, the economy, or the global climate system – are toys compared to the real things. And of course there is nothing to stop users from extending the model to incorporate additional effects. Though whether this will lead to improved predictive accuracy is another question.

Try the CellCycler web app here.

References:

Stephen Checkley, Linda MacCallum, James Yates, Paul Jasper, Haobin Luo, John Tolsma, Claus Bendtsen. “Bridging the gap between in vitro and in vivo: Dose and schedule predictions for the ATR inhibitor AZD6738,” Scientific Reports.2015;5(3)13545.

Green, Kesten C. & Armstrong, J. Scott, 2015. “Simple versus complex forecasting: The evidence,” Journal of Business Research, Elsevier, vol. 68(8), pages 1678-1685.

Complexity versus simplicity in relating tumour size change to survival in oncology drug development

Every pharmaceutical company would like to be able to predict the survival benefit of a new cancer treatment compared to an existing treatment as early as possible in drug development.  This quest for the “holy grail” has led to tremendous efforts from the statistical modelling community to develop models that link variables related to change in disease state to survival times.  The main variable of interest, for obvious reasons, is tumour size measured via imaging.  The marker derived from imaging is called the Sum of Longest Diameters (SLD).  It represents the sum of longest diameters of target lesions, which end up being large lesions that are easy to measure.  Therefore the marker is not representative of the entire tumour burden within the patient.  However, a change within the first X weeks of treatment in SLD is used within drug development to make decisions regarding whether to continue the development of a drug or not.  Therefore, changes in SLD have been the interest of most, if not all, statistical models of survival.

There are two articles that currently analyse the relationship between changes in SLD and survival in quite different ways across multiple studies in non-small cell lung cancer.

The first approach (http://www.ncbi.nlm.nih.gov/pubmed/19440187) by the Pharmacometrics (pharmaco-statistical modelling) group within the FDA involved quite a complex approach.  They used a combination of semi-parametric and parametric survival modelling techniques together with a mixed modelling approach to develop their final survival model.  The final model was able to fit to all past data but the authors had to generate different parameter sets for different sub-groups.  The amount of technical ability required to generate these results is clearly out of the realms of most scientists and requires specialist knowledge.  This approach can quite easily be defined as being complex.

The second approach (http://www.ncbi.nlm.nih.gov/pubmed/25667291) by the Biostatistics group within the FDA involved a simple plotting approach!  In the article the authors categorise on-treatment changes in SLD using a popular clinical approach to create drug response groups.  They then assess whether the ratio of drug response between the arms of clinical studies related to the final outcome of the study.  The outcomes of interest were time to disease progression and survival.  The approach actually worked quite well!  A strong relationship was found between ratio of drug response and the differences in disease progression.  Although not as strong, the relationship to survival was also quite promising.  This approach simply involved plotting data and can be clearly done by most if not all scientists once the definitions of variables are understood.

The two approaches are clearly very different when it comes to complexity: one involved plotting while the other required degree-level statistical knowledge!  It could also be argued that the results of the plotting approach are far more useful for drug development than the statistical modelling approach as it clearly answers the question of interest.  These studies show how sometimes thinking about how to answer the question through visualisation and also taking simple approaches can be incredibly powerful.