Ablation, scars and insights from parameter lumping

This is quite a long post elaborating on a bit of work I’ve been doing with Greg Morley and his group, which was published in Scientific Reports. I hope it provides an accessible introduction to a fairly complicated set of experiments and simulations.

Background

Ablation is a treatment for some abnormal heart rhythms, which works by killing areas of the heart that we think are misbehaving and giving rise to abnormal rhythms. The idea behind ablation is that if you identify and kill a region of tissue that is allowing electrical signals to propagate ‘the wrong way’, then afterwards you will have created a non-conducting wall of scar tissue that prevents any abnormal electrical activity. As a treatment strategy, there’s clearly room for improvement; I hope we’ll soon look back on it as Dr McCoy does for dialysis in the scene in Star Trek IV. But at the moment, it’s the best we’ve got for certain conditions.

Usually, and ideally, you stop ablating when the arrhythmia terminates, so it is very effective in the short term. Unfortunately the success rate in the long term isn’t great – after a single round of atrial ablation people stay (for at least 3 years) arrhythmia free in only about 50% of cases¹. So why do people re-lapse after ablation? Well it can be because electrical waves somehow get through the scar regions (see ¹ for further references to this).

You also get similar scars forming after a heart attack – in this case the cardiac muscle itself doesn’t get a blood supply, and dies because of that, leaving a scar region. This damage leads to scars with larger border zones than in ablation, as in these border zones have low but not-zero blood supply, so you get semi-functioning tissue where some of the muscle cells (myocytes) survive, and others don’t. The problem now is the opposite to ablation – you’d like to restore conduction in these regions to get the rest of the heart beating as well as it can, but electrical activity is disturbed by the presence of these scar regions and their border zones, and arrhythmias become more likely (or even occur instantly in the case of big heart attacks).

Greg works on optical mapping for whole hearts in various species to study what happens around these scars. This is a clever technique where you put a dye into a tissue, it sticks to the membrane of cells, and when excited by an external light source, it emits light at an intensity dependent on the voltage across the cell membrane. So you can excite the dye with light at a certain wavelength, and record the light emitted at another wavelength in a camera to create a map of electrical activity in your sample, and you can do this in real time to see how electrical waves move across the heart. Here’s a simulation of electrical activity that Pras did with our Chaste software, and it includes a comparison with optical mapping at 39 seconds in the video below (this is for a completely different situation – just to show you how optical mapping works!):

Experimental Results

Greg found that in some intact hearts from mice recovering from ablation he could observe the optical mapping signal appearing to go straight through the middle of the scars. This is unexpected, to say the least, and required a lot of further investigation. I’ll let you read the full story in the paper, but suffice to say we confirmed that:

• there aren’t any myocytes (normal heart muscle cells) left in the scars (shown by doing histology and using an electron microscope) and there are lots of what are largely fibroblasts left there, the cells in the scars are definitely ‘non myocytes’ anyway;
• the observations aren’t just optical mapping artefacts (shown by doing direct micro-electrode recordings, showing that local electrical pulses from a suction electrode diffuse in, and even putting black foil in the middle of the hearts!);
• the observations depend on electrical coupling between myocytes and fibroblasts, as it goes away if you don’t have this (shown by specifically breeding a strain of mice where you could knock out Connexin43 in just the non myocytes – clever stuff!). In fibroblasts, Connexin 43 is thought to form gap junctions electrically coupling them to only myocytes, not to other fibroblasts. If you knock out Connexin43 proteins in the non myocytes with this special mouse strain, then the optical mapping signal you record is much smaller in the scar region; the electrical signal no longer diffuses in from a suction electrode, indicating that fibroblast-myocyte junctions are required to see conduction into the scar.*

To quote from the article, this is “the first direct evidence in the intact heart that the cells in myocardial scar tissue are electrically coupled to the surrounding myocardium and can support the passive conduction of action potentials.”

Where mathematical modelling comes in

When Greg presented the experimental results at conferences he had a hard time convincing people that the experimental results were real, and not some odd artefacts of the experimental set up! It seemed that nobody expected electrical waves to travel through these regions, even if they were populated with cells, as these cells weren’t cardiac myocytes. It was counter-intuitive to most cardiac electrophysiologists that any signal would get across this gap, because that’s the whole point of ablation! This is where we thought it would be quite interesting to see what you would expect, from the standard model of electrical conduction, if there are neutral non-myocyte cells in a lesion that are coupled together. By neutral we mean not providing any ‘active’ transmembrane currents, and just providing passive electrical resistance.

Our standard simplest model for the reaction-diffusion of voltage in cardiac tissue is the monodomain equation (so-called because there’s an two-domain extension called the bidomain equation):

There are a few terms here that need defining: V is voltage across the cell membrane – which is the quantity that we consider to be diffusing around, rather than the individual types of ions themselves. I_ion is the current across a unit surface area of the cell membrane at any point in space (varies across space), I_stim is any external stimulus current which is applied to a unit volume of the tissue, C_m is the capacitance of a unit surface area of the membrane (the bigger this is, the more current is required to charge up a bit of membrane the same amount), and σ is the conductivity of the tissue – how easy it is for voltage to diffuse (expressed as a vector as there can be preferential conduction in different directions). χ is a scaling factor of surface-area-of-membrane-per-unit-volume-of-tissue, required to change the current across a unit-of-surface-area membrane into current across all-the-surface-area-of-membrane-in-a-unit-volume of tissue. So you can think of χ as the amount of membrane packed into a little cube of the tissue.

So, what will be different in the lesion?:

• 𝜎 , the cell-cell conductivity, could be varying. This would represent how much Connexin was present to link the lesion cells together with gap junctions.
• 𝜒 , the surface area of membrane in a unit volume of tissue, could also be varying. This would represent non-myocyte membrane density in the lesion.
• 𝐶_𝑚, the capacitance of a unit of membrane area, probably won’t change (membrane made of same stuff). Beware though, a maths/biology language barrier means experimentalists might call 𝜒 ‘capacitance’ to confuse us all!
• I_ion will be small compared to myocytes, these cells aren’t actively trying to propagate electrical signals. So we did the study twice, once with this set to zero in the scar, and once with it set to use a previously published model of fibroblast electrophysiology (didn’t make any visual difference to the results).

Now in the scar region we aren’t applying an external stimulus so I_stim = 0, and we can divide through by χ to get:

A nice property springs out – that the entire behaviour of the scar region (in terms of difference to the rest of the heart) is determined by the ratio of 𝜎/χ. So we introduce a factor ρ that will scale this quantity.

Even before we run any simulations, we’ve learnt something here by writing down the model and doing this “parameter lumping” (see nondimensionalisation for a framework in which to do this kind of thing rigorously!). Just by looking at ρ = 1 we see that there are an infinite number of ways we could get exactly the same behaviour. The scar cells could be just as well coupled (𝜎) and just as densely packed (χ) as the rest of the heart (incredibly unlikely to Greg), or they could be 1% as well coupled with 1% as much membrane present (more plausible to Greg); before we even do a simulation we can state definitively that this would give exactly the same behaviour: as ρ = 1 in both cases, and we’re solving the same equations! So this scaling is interesting, as we didn’t know whether the value of ρ would be increased or decreased in the scar, despite strong suspicion that the cells in the scar will have both reduced gap junctions and reduced membrane surface area available.

So we did a simulation with a mouse ventricular myocyte model, on a realistic sized bit of tissue 5mm x 5mm, with a 2mm diameter lesion in the middle. There are no preferential fibre directions here, something that could easily make the conduction more or less likely.

So what behaviours did we predict with different values of ρ? First for ρ = 1; the 𝜎/χ takes the same value as usual case (even though both could be reduced/increased by any factor):

In the video we see that conduction does indeed ‘get across’ the scar, even though it is not ‘driven’ as such as it is in the rest of the tissue, but instead is simply diffusing across that region. We predict the wave will get across a 2mm scar easily with this value of ρ.

What about with more membrane (or less conductivity): ρ = 0.1 (Remembering this would happen whenever 𝜎/χ is ten times less that the normal muscle: so both when 𝜎 is 1% of normal and χ is 10% of normal; or when 𝜎 is 0.1% of normal and χ is 1% of normal, for instance).

This time, you can see that the wave struggles to get across the lesion, but the membrane voltage still gets high, and that would probably still be high enough to record in optical mapping.

With less membrane density relative to the well-coupled-ness of the lesion cells (ρ = 10)? (Remembering this would happen whenever 𝜎/χ is ten times more that the normal muscle: so both when 𝜎 is 10% of normal and χ is 1% of normal; or when 𝜎 is 1% of normal and χ is 0.1% of normal, for instance). We see that the wave even appears to accelerate across the lesion region and conduction sets off earlier than before at the far side, as there is less membrane to charge so it is easier to do it:

So our conclusion was that it’s perfectly possible that a voltage signal could be recorded in the lesion, and a voltage signal could effectively travel straight through the scar, and conduction carry on out the other side. We’re not entirely sure about the value that ρ should take – but this behaviour was fairly robust, and matched what we saw in the experiments.

This simple model predicted many of the features of the recordings that were made from the scar region – see Figure 7 in the paper, and compare with experiments in Figure 4. So, it helped Greg answer accusations of “This is impossible!” that he got when he presented stuff at conferences, as he could reply that “If the cells have gap junctions, even without any voltage-gated ion channels of their own, this is exactly what we’d expect – see!”.

As usual, all the code is up on the Chaste website if anyone wants to have more of a play with this.

References

1 “Long-term Outcomes of Catheter Ablation of Atrial Fibrillation: A Systematic Review and Meta-analysis” Ganesan et al. J Am Heart Assoc.(2013)2:e004549 doi:10.1161/JAHA.112.004549

*Incidentally, I’d like to do a sociology experiment where I give biologists and mathematicians  logic problems to do mentally. My hypothesis is that biologists would beat mathematicians, as they are always carrying around at least ten “if this then that” assumptions in their heads. Then I’d give them a pen and paper and see if the situation reversed…

Should I work with IC50s or pIC50s?

When you are expressing how much a drug inhibits something, it’s common to fit a Hill curve through a graph of concentration against % inhibition as shown here:

Figure 1: A concentration-effect curve showing % inhibition as a function of drug concentration. The Inhibitory Concentration 50% value is simply defined as the concentration that causes 50% block, here I’ve expressed it as the ratio of the drug concentration over IC50, so IC50 is defined to be 1 here. Notice we’ve plotted this on a log scale, which perhaps answers the question in the title already…

In our case this is often ‘% inhibition’ for a given ionic current, a consequence of a drug molecule binding to, and blocking, ion channels of a particular type on a cell’s membrane.

A while back we were interested what the distribution of IC50s would be if you repeated an experiment lots and lots of times. We asked AstraZeneca and GlaxoSmithKline if they had ever done this, and it turned out both companies had hundreds, if not thousands, of repeats as they did positive controls as part of their ion channel screening programmes. We plotted histograms of the IC50 values and Hill coefficients from the concentration – effect curves they had fitted, and found these distributions:

Figure 2: Histograms of pIC50s and Hill coefficients fitted to over 12,000 repeats of a screen examining whether Cisapride inhibits the hERG potassium channel current. The red lines are distributions fitted to these distributions: logistic for pIC50 and log-logistic for Hill coefficients.

After some investigation, we found the standard probability distributions that both the IC50s and Hill coefficients from these experiments seemed to follow very well were Log-logistic.

Now, where do pIC50s come in? A pIC50 value is simply a transformation of an IC50, defined with a logarithm as:

pIC50 [log M] = -log10(IC50 [M]) = -log10(IC50 [μM]) + 6

At this point it is good to note that logarithms to the base 10 (also known as common logarithms) were invented by Henry Briggs who, like me, was born in Halifax!

The ‘pIC50’ is analogous to ‘pH‘ in terms of its negative log relationship with Hydrogen ion concentration, which is more familiar to most of us.

The logarithm means that pIC50s end up being distributed Logistically, as:

ln(X) ~ Loglogistic(α, β).                       ...(1)

implies that

X ~ Logistic(μ, σ),                              ...(2)

as shown below:

Figure 3: Logistic pIC50 and Log-logistic IC50s. Histogram of samples from the same simulated distribution pIC50 ~ Logistic(6, 0.2), with red lines indicating fitted Logistic and Log-logistic distributions (see note on implementation below!).

You can see how nicely these distributions work across a lot of different ion currents and drug compounds in the Elkins et al. paper supplement.

More recently, we ran into an interesting question: “Given just a handful of IC50 values  should we take the mean value as ‘representative’, or the mean value once they are converted to pIC50s”?

Well, ideally, you’d try and do rigorous inference on the parameters of the underlying distribution – as we did in the original paper – and as our ApPredict project will try to do that for you if you install that. To do that across multiple experiments you probably want to use mixed-effect or hierarchical inference models. But it’s fair enough to want a rough and ready answer in some cases, especially if you don’t know much about the spread of your data. A tool to try and infer the spread parameter σ from a decent number of repeated runs of the same experiment is something I’ll try to provide soon.

But, let’s just say you want to have this ‘representative’ effect of a drug, given a handful of dose-response curves. You’ve got a few options: you could take a load of IC50 values, and take the mean of those; or you could take a load of pIC50 values, and take the mean of those. Or perhaps the median values? But which distribution should you use? Which would be more representative, and what is the behaviour you’re looking for? Answering this was a bit more interesting and involved than I expected…

First off, let’s look at some of the properties of the distributions shown in Figure 3. Here are the theoretical properties of both distributions (N.B. there are analytic formulae for these entries, which you can get off the wikipedia pages for each distribution (Loglogistic, Logistic). I converted the answers back to pIC50, for easy comparison, but the IC50s were really taken from the right hand distribution in Figure 3 in IC50 units.).

Theoretical Results	Mean	Median	Mode
From IC50 distbn	5.836	6 (μ)	6.199
From pIC50 distbn	6 (μ)	6 (μ)	6 (μ)

So as you might expect (from looking at it), there’s certainly a skew introduced into the Loglogistic distribution on the right of Figure 3 for the IC50 values.

So what to measure depends what kind of ‘representative’ behaviour we are after. Or in other words, what is a drug really doing when we get these distributions for observations of its action? Well, you really want to be inferring the ‘centering’ distribution parameter (μ), which in this case would be the mean/median/mode pIC50 or the median IC50. You will already get the impression that it’s most useful to think in pIC50s – as the distribution is symmetric the mean, median and mode are all the same.

But what about the more realistic case of the properties of a handful of samples of those distributions? I just simulated that and show the results in Figure 4 (I think you could probably do this analytically on paper, given time, but I haven’t yet!).

Figure 4:distributions of (Top row) Mean, and (Bottom row) median from N=3 samples of (Left) pIC50 and (Right) IC50. Simulated by taking 3 samples from Figure 3, and repeating the process a million times to build up these histograms. Note that the mean can be taken from IC50 or pIC50 transformed variables, and these give different answers.

It seems the only estimates for which you are likely to get back an unbiased and consistent estimate is the mean/median of pIC50 values, since the distribution is symmetric. The IC50 distribution isn’t symmetric, and so taking the mean of IC50 samples leads to a bias, it does however seem to give you a good estimate for the median of the IC50 distribution (better than the median of a sample of IC50s does!) for a low N – see top right plot of Figure 4. As N increases you do eventually get a distribution whose peak is at the mean, but N needs to be quite a lot larger than your average (no pun intended) experimental N. As you might expect, the median pIC50 is not quite as good a measure as the mean for the centre of the pIC50 distribution (but that’s hard to see visually in Fig 4, it does almost as well here).

You could point out that none of these “make that much difference” to the plots above, but you will introduce a bias if you use the wrong statistic, and for the semi-realistic distributions that we’ve got as an example here, your estimate of pIC50 = 5.836 versus the true pIC50 = 6 does give a block error of almost 10% when you substitute it back into a Hill curve of slope 1 at 1μM.

Importantly, pIC50s are much nicer numbers to deal with: they are always given in the same units of log M, and you can recognise at a glance for your average pharmaceutical compound whether you’re likely to have no block (<1 ish), very weak activity (2-3ish) which is perhaps just noise, low block (4-5ish) or strong block (>6ish) – give or take the concentration that the compound is going to be present at – see Figure 1! These easy-to-grasp numbers make it much easier to spot typos than it is when you’re looking at IC50s in different units. We’ve also seen parameter fitting methods that struggle with IC50s, but are happier working in log-space with pIC50s, as searching (in some sense ‘linearly’) for a pIC50 within say 9 to 2 is easier than searching for an IC50 in 1nM to 10,000μM.

So my conclusion is that I’ll try and work with pIC50s, and if you need a quick summary statistic, use the mean of pIC50s. They seem to occur in symmetric distributions with samples that behave nicely, and therefore are generally much easier to have sensible intuition about!

Is your model really ‘comprehensive’?

I’ve noticed more and more papers using the phrase “comprehensive model”. This phrase grates every time, and this post is about why.

I thought this might just be me getting old and grumpy, so I plotted the mentions of the phrases “comprehensive mathematical model” or “comprehensive model” in ‘Topic’ in Web of Science over the years in Figure 1. Sure enough, more papers in recent years feature models that are comprehensive*.

Figure 1: the rise of Comprehensive Models! Mentions of “Comprehensive Models” or “Comprehensive Mathematical Models” in Web of Science 1945 – 2015.

So why don’t I think a mathematical model is ever comprehensive?

Usually by comprehensive the authors mean something like this describes most of what we’ve seen well enough to say/predict something (or even to within the observational error bounds in some physics experiments). Perhaps their model integrates information/theories that weren’t part of a single conceptual framework/mathematical model before. That’s great – but it isn’t comprehensive!

This post is really an excuse to plug and discuss the following quotation from James Black (physiologist and Nobel Prize winner for developing the first beta-blockers). He summarises what mathematical models are and aren’t, and what they are for, beautifully:

[Mathematical] models in analytical pharmacology are not meant to be descriptions, pathetic descriptions, of nature; they are designed to be accurate descriptions of our pathetic thinking about nature. They are meant to expose assumptions, define expectations and help us to devise new tests.

Sir James Black (1924 – 2010)
Nobel Prize Lecture, 1988**

A mathematical model isn’t supposed to be a comprehensive representation of a system – it’s always going to be a pathetic representation in many ways! Models make simplifying assumptions (by definition***), generally ignoring things that we think will make a smaller difference to the model’s predictions than the things that we have included (usually things that happen really fast/slow or that are really small/big).

What models do allow us to do is see exactly what we would expect to happen if the system works in the simple(ish!) way we think it does. Then that can teach us loads about whether the system does work as we thought it did, or whether something fundamental is missing from our understanding.

We’ll always be able to come back and add more detail as time goes on, we learn more, and can measure more things. So that means that a model is never finished, and never comprehensive. So I’d say let’s avoid using the word comprehensive to describe any kind of model!

 

A Large Confession: I thought the word comprehensive implied ‘includes everything’. Purely to back up my point I looked it up, and sure enough one of the OED’s definitions is “grasps or understands (a thing) fully” which a model never does; but another is “having the attribute of comprising or including much; of large content or scope” which might be OK! Hmmmm… given the ambiguity I still conclude we’d better avoid the word anyway! But I’ll let you make your own minds up:

 

 

* Yeah, I know I really need to divide by the total number of papers that mention models etc. but that’s harder to search sensibly – and WoS doesn’t summarise data for over 10,000 papers!

** You can watch his Nobel Prize lecture online. He’s quite a remarkable man and invented/discovered lots of classes of completely new drugs, interestingly – as you can see in the video – he used modelling a lot. His modelling would now be part of the trendy new field of Quantitative Systems Pharmacology (although I’d say we’ve been doing it for years😉 ), and if QSP people ever have a prize for anything I think they should name it after him!

*** ‘Model’ means a simplification of reality, so it’s confusing to say they are “comprehensive” whilst there are lots of things we know about that aren’t included. I double-checked this one too, and indeed our kind of model is defined as “A simplified or idealized description or conception of a particular system, situation, or process, often in mathematical terms, that is put forward as a basis for theoretical or empirical understanding, or for calculations, predictions, etc.“.

 

Are 30 compounds enough to test our simulations?

I’ve just contributed to a new review ‘Recent developments in using mechanistic cardiac modelling for drug safety evaluation’, which discusses where we are with using mathematical models of the electrical activity of cardiac muscle cells to predict whether new drugs will cause any dangerous changes to heart rhythm. Quite a bit of the review brings up challenges that we’re going to come across as we assess whether or not the simulations are ready to help regulators decide on the safety of new pharmaceutical compounds.

How do we quantify ‘predictive power’?

As part of this review I got thinking about how reliable our assessments of simulation predictions are. We often think about evaluating predictions in terms of binary (yes/no) outcomes: e.g. do we keep developing a drug or not; do we allow a drug onto the market or not? To look at predictions of this kind of thing we often use a confusion matrix, also known as a contingency matrix/table, as shown in Figure 1.

Figure 1: a ‘confusion matrix’, adapted from Wikipedia. For us the ‘Condition’ / Gold Standard would be the experimental measurement, and the ‘Test Outcome’ would be the simulation prediction.

In Kylie‘s study evaluating simulation predictions of whether or not compounds caused 10% or more prolongation of the rabbit wedge QT interval, we looked at a decent number of compounds, 77 for simulations based on PatchXpress automated patch clamp technology (and even more for other sources of screening data). Here’s a slightly more elaborate confusion matrix from that study:

Table 1: Simulation performance when evaluating predictions of whether or not a drug compound will cause over 10% prolongation of the ECG waveform in the rabbit wedge, the numbers are the numbers of validation compounds which fall into each box. Note this isn’t a standard confusion matrix, as we’ve got three categories instead of two, and I’ve got the rows and columns swapped relative to the one above(!) sorry. But you can see the calculations that give us a sensitivity, specificity and accuracy.

In Table 1 I separated out the predictions into three categories (>= 10% shortening, >= 10% prolongation, or ‘no effect’ as between the two), to allow us to evaluate predictions of shortening as well. But you just need to add up the relevant boxes as shown to evaluate predictions of prolongation. So here we do quite well, and it feels as if we’ve got enough compounds in the validation set to say something concrete.

But when we’re evaluating a new simulation’s performance, how many validation compounds do we need? Clearly more than 1 or 2, since you could easily get these right by accident, or design. But are 10 validation compounds enough? Or do we need 20?, 30?, 50?, 77 as we used here?, 100?, or even more?

How does our evaluation depend on the number of validation compounds?

In Figure 2 I’ve plotted how our estimates of the prediction sensitivity and accuracy change (and converge to a roughly steady value) as we increase the number of validation compounds that we consider. Here the estimates happen to ‘level out’ at about N=60 compounds. But by simply permuting the order in which we consider these 77 compounds, and repeating the process, we get another possible graph (two permutations shown in Figure 2).

Figure 2: sensitivity and accuracy with increasing numbers of compounds. Two random permutations of the 77 available compounds are considered.

So if we didn’t have the big picture here, and only had 30 compounds to work with (solid vertical lines in Fig. 2), we could equally well conclude that the sensitivity of the simulation prediction is 35% (which you’d consider rubbish) or 65% (pretty good)! So we have a problem with N=30 validation compounds.

Can we estimate our uncertainty in the performance of our model?

How do we know how far away the ‘real’ sensitivity and accuracy is likely to be? And by ‘real’ I mean the sensitivity and accuracy if we had an infinitely large set of validation compounds to test.

Luckily, the stats for this is well established (and Dave remembered it when we were doing Kylie’s study!). These measures are a sum of binary yes/no measurements, i.e. do we get the prediction right or not? This means each validation compound can be thought of as a sample from a binomial distribution for the sensitivity or accuracy, and the confidence interval for the underlying parameter ‘p’ (in this case the sensitivity or accuracy themselves) has a nice simple analytic expression first worked out by Wilson in 1927. So this post is certainly nothing new for a statistician! But you very rarely see these confidence intervals presented alongside assessments of sensitivity, specificity or accuracy. Perhaps this is because the confidence intervals often look quite big?!?

So I’ve plotted the Wilson’s Score Interval on top of the sensitivity and accuracy graphs for the two permutations of the compounds in Figure 3, which also appears in the new review paper.

Figure 3: same as Figure 2, but this time with 95% confidence intervals overlaid, calculated using Wilson’s score interval.

Fig. 3 nicely shows how the intervals narrow with increasing numbers of compounds, and also how the 95% confidence intervals do indeed include the N=77 performance evaluations (our best guess at the ‘real’ numbers) about 95% of the time. So now when we look at just 30 compounds we still get the 35% or 65% sensitivity estimates, but we know that the ‘real’ sensitivity measure could easily be somewhere in 12-63% for the blue compounds, or 35-85% for the red ones, and indeed the N=77 estimate is at 60% – within both of these. In fact, the confidence interval applies at N=77 too, so the ‘real’ value (if we kept evaluating more and more compounds) could still be anywhere in approximately 41-77%; the accuracy estimate has narrowed more, to around 63-82%.

Note that because the sensitivity calculation, shown in Fig. 1, effectively operates with a smaller sample size than the accuracy (just the experimental prolonging compounds instead of all of them) you get wider confidence intervals. Note this means you can’t tell a-priori what number of compounds you’ll need to evaluate, unless you have a fair idea of the positive/negative rate for the Gold Standard measurements.

If you are getting appropriately sceptical, you’ll now wonder why I only showed you two permutations of the compound ordering, isn’t this as bad as only looking at two compounds? Quite right! So here are 50 random permutations overlaid for Sensitivity, Specificity, Cohen’s Kappa value, and Accuracy:

Figure 4: 50 repeats of this permutation of compound ordering exercise, with black dots being the assessed predictivity index, and overlaid confidence intervals for each (click for a bigger version, sorry I couldn’t make MatLab play nicely with the sizes of the axes labels).

So, is testing 30 compounds enough?

Well, you can use as many or few compounds as you like, as long as you evaluate the confidence in your prediction scores (sensitivity, specificity, accuracy etc.)! This little exercise has persuaded me it’s vital to present confidence alongside the prediction scores themselves (something I’ve not always done myself). In terms of getting confidence intervals that are the kind of width we want for deciding whether or not predictions of QT prolongation are good enough (probably down to around 10%), this little exercise suggests we probably need to be doing validation studies on around 100 compounds.

Model Complexity

We’ve been thinking about how to parameterise/train/calibrate models, and also how to select an appropriate model to begin with. This raises all sorts of interesting questions on the details, but this post is just about setting out a big overview of how this sort of thing might work, in an ideal scenario.

We’ll go through the concepts with the help of the diagram I’ve drawn in Figure 1. We initially sketched this out for Kylie‘s thesis, she’s the one doing all the work on this for the hERG channel models at the moment, but it didn’t make the final cut there, so here it is instead.

Figure 1: model complexity. The relationship between the complexity of your model (* relative to the available training data’s information content); the error you will get in calibration/training; the uncertainty in parameters (the inverse of ‘identifiability’ if you like, how ill-posed the parameter fitting problem is); and the fit that you will get when predicting validation data.

First off, if you have an overly simple model, sitting at the top of Figure 1, you probably won’t have much ambiguity in the parameter values it should take for the best fit to training data, but the fit to training data isn’t likely to be very good, and the prediction of new cases in validation is likely to be worse.

But, it is always possible to get a better fit to training data by increasing the complexity of your model (adding free parameters, extra equations etc.):

“With four parameters I can fit an elephant, and with five I can make him wiggle his trunk”
John von Neumann

In statistical models/machine learning approaches this is called “overfitting”, but the same applies to adding lots of complexity (postulated reactions or ion channel states etc.) into biophysical models. For some examples of overfitting see a google image search on ‘overfitting’ to give you a good idea how it happens, and why it might lead to really bad predictions later on.

Figure 2: Overfitting. From Twitter, possibly attributable to @ResearchMark, but I couldn’t find the original source, sorry!

So you can make your model overly-complicated, but this is subtle: there’s the possibility that if you try and fit too many parameters to a dataset that isn’t “information rich” enough, you’ll also run into problems. I’ve shown an example of this in Figure 3, now we’re at the bottom of Figure 1, the model is too complex for the training data to constrain in panel A, and so gives rubbish predictions in panel B (and a whole range of predictions depending at random on our initial guess at the parameters! This is a bad thing). But if we change the training data to a more ‘information-rich’ protocol that allows us to identify the parameters better, then we can make good predictions and we are back in the ‘optimal’ setting. So “Model Complexity” really means ‘model complexity relative to a given training protocol’.

Figure 3: Predicting with too much uncertainty in parameters (click to see a bigger version). This is an example using synthetic data generated from the ten Tusscher IKr model, but that doesn’t really matter for now. Blue dots – data, Red lines – fits (in A and C) and predictions (in B and D). A & B are from one fitting procedure – fitting to data in A, and predicting data in B, repeated from 20 different initial guesses at parameters. You get 20 different optimal parameter sets, all of which give a great fit to training data (A), but then give 20 different predictions in validation data (B), with inset showing how bad it can be. Conversely, if we train to the data in C, 20 different guesses again, then all parameters converge to the same guess (there is low uncertainty in parameters), and all predict the same thing, the right behaviour, in D. For more info on this see Fink & Noble (2009), and Fink et al. (2011) where a version of this figure first appeared.

So that was an example of a model with too many free parameters to constrain in a certain situation. It is quite possible, and easy, to write down a model that has too many free parameters to constrain with any of your available data. Or you could even write down a whole range of models of high complexity that fit the data perfectly, but will lead to different predictions. What should you do in this case? I am still a fan of simplifying the model down to one with identifiable parameters*, and testing out the simplest model that’s a decent fit with identifiable parameters. I make that sound easy, but in general it isn’t.

So which model should you use? Well, things like Information Criteria will try and let you guess from just the training data fit and model complexity. But you’ve no guarantee that they are going to accurately predict the profile of your validation error on the right of Figure 4, although that is what they’re effectively trying to do. So it is absolutely crucial to do the last step of making some validation predictions.

Making validation predictions sounds simple enough, but opens up more questions: how do I choose which validation experiment to perform? If I choose one that is too simple/similar to the training/calibration experiment, then I could trick myself into thinking I’m doing really well (and kid myself that the green shape on the right is really the same shape as the red training triangle). Most of the literature will tell you to do something based on the ‘context of use’, that is, situations as similar as possible to where you want to use the model ‘for real’. e.g. if I want to use a model to predict cardiac effects at 1Hz, then test it out at just 1Hz. This makes perfect sense for statistical/machine learning models.

With biophysical models, I don’t think it is that simple, and as usual it probably depends on what you are making the model for. If you want to predict some behaviour in future, then yes, do the ‘context of use’ simulations to see how well you are likely to do. But you might want to know whether your biophysical model is a good description of the biophysics – perhaps the biophysics itself is what you are trying to learn about, or perhaps you can’t be sure what the ‘context of use’ is (voltage waveforms in arrhythmias?). The biophysics should hold – not only outside the training data regime – in any regime where the underlying assumptions you built in about physical processes hold. In this case, perhaps it makes sense to throw the wackiest validation protocol you can think of at the model, to see if it really does capture the underlying biophysics? So we’re trying that, and I’ll let you know how we get on!

You also have to avoid falling into the trap of using validation predictions to perform the model selection for you. Sounds like an obvious thing to do, but then you have turned your ‘validation data’ into ‘model selection training data’! This is easy to do by accident, and really if you do this you should have a third set of data – sometimes called ‘test data’ to see how well the calibrated and selected model performs. Often machine learning people do work with three datasets like this – see Chapter 7 of Hastie et al.’s Elements of Statistical Learning book for a good intro.

*As an aside, people often talk about a model being “identifiable” (i.e. “the parameters of the model are identifiable”). But we have to remember that only makes any sense as shorthand for the statement “the parameters of the model are identifiable, given this training/calibration data“. Which immediately raises the question of “What if we calibrate the parameters to the results of a different experiment?”, as the example in Figure 3 shows, which is getting me into the whole new field of optimal experimental design…

If computational biologists did astronomy…

Here is a cheeky post along the lines of Can a biologist fix a radio? (well worth a read). The discovery that Pluto wasn’t the size we expected got me thinking about how we would have tackled it. So in our case: Can a computational biologist do astronomy?

Our protagonists are a Mathematical / Computational Biologist MCB, and a Noteworthy Astronomy Scholar with Antiquated ideas, NASA. (If anyone’s initials are MCB, I forgot and genuinely didn’t mean you!)

MCB: I am going to make a model of how Pluto moves under gravity.

NASA: we know that Pluto is subject to gravitational forces, so its orbit depends on its total mass according to Newton’s law of universal gravitation.

MCB: Great, so if I could measure the density and radius of Pluto then I could work out its mass with M = density x volume, and then I could work out the gravitational forces on it.

NASA: Well, if we say it is a uniform sphere, then I suppose so yes.

MCB: Good. I will do some calculations…

MCB: I have varied the radius and density of Pluto and discovered that Pluto can have almost any orbit you like.

NASA: By the way, we have watched the orbit, and it is very consistent.

MCB: Interesting! I will do some more calculations to see which densities and radii it uses.

NASA: Er, we don’t really have any evidence that Pluto is varying its radius and density. It might make sense just to work with its total mass for now.

MCB: That’s all very well, but we don’t have any evidence that it isn’t changing its radius and density all the time do we?

NASA: Er…

MCB: In fact this would make sense, because Pluto’s orbit must be robust to fluctuations in radius and density that occur when space dust hits it stochastically. It probably varies them continuously to adapt to getting hotter when it is closer to the sun too, and all sorts of other things. I will do some more calculations…

MCB: The radius and density of Pluto could take almost any values and still give that orbit! Pluto is robust to changes in its radius and density [as long as they fall near a line that looks a bit like 1/density is proportional to radius^3].

NASA: [You’ll probably find that constant of proportionality is 4/3 pi], I think we might need new equipment to measure the radius and density, you should probably stick with a simple model with just total mass for now.

MCB: All the evidence we have is completely consistent with Pluto taking a wide range of radii and densities, I can make some interesting predictions for what happens to Pluto in new situations at different radii and densities and publish lots of papers about them.

NASA: Well it might take various values, but we’d better go and check.

[Some time later…]

NASA: We’ve just measured Pluto, its radius and density weren’t what we guessed, but they take constant values!

[The End]

In case you didn’t get my not-so-subtle point, I have read quite a few papers linking un-identifiability of parameters in inverse problems as plausibility that the parameters really do take a wide range of values, backed up with an argument that they must to confer biological robustness, or explain variability. Biological systems obviously do have to be robust to stochastic transcription, noise and a wide range of conditions. But lack of information about a quantity is not evidence that it varies in reality: certain things are very carefully controlled by numerous feedback mechanisms e.g. blood temperature. Don’t get me wrong, I think understanding variability and uncertainty is absolutely crucial. But if you are relying on something varying a lot for your results, I would say the onus is on you to design an experiment where you can measure it as well as possible, and do it lots of times to show how it varies.

Job Advert

A quick shameless plug for a job we are advertising.

This is a 3 year postdoctoral position to work with me in the Computational Biology Group in the Department of Computer Science, University of Oxford. It would best suit someone with a PhD in a subject such as: Bayesian inference; model selection; model calibration; uncertainty quantification; or optimal experimental design.

The idea will be to apply these kind of techniques to various problems in electrophysiology modelling, to allow us to begin to assign probabilities to predictions. The post would suit someone who is looking to apply their statistical skills to real problems, and to learn about biophysically-based modelling and its applications for biological problems. No prior experience of modelling these kind of things is necessary, or even desirable!

More details and a link to the official Oxford job portal can be found on the CS news feed. Please contact me at gary.mirams@cs.ox.ac.uk for any informal enquiries about the post.