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# Moment Generating Function Gamma Distrib

• Apr 8th 2011, 08:22 PM
dwsmith
Moment Generating Function Gamma Distrib
$\displaystyle\frac{1}{\sqrt{7}\Gamma\left(\frac{1} {2}\right)}\lim_{b\to\infty}\int_0^b\left(x^{-1/2}\exp\left[-x\left(\frac{1}{7}-t\right)\right]\right) \ dx$

$\displaystyle\frac{1}{\sqrt{7}\Gamma\left(\frac{1} {2}\right)}\lim_{b\to\infty}\int_0^b\left(x^{-1/2}\exp\left[\frac{-x}{\frac{1}{\left(\frac{1}{7}-t\right)}}\right]\right) \ dx$

$\displaystyle\alpha=\frac{1}{2} \ \beta=\frac{1}{\frac{1}{7}-t}$

$\displaystyle =\frac{1}{\sqrt{7}\Gamma\left(\frac{1}{2}\right)}\ left(\frac{1}{\frac{1}{7}-t}\right)^{1/2}\Gamma\left(\frac{1}{2}\right)$

Correct?
• Apr 8th 2011, 08:38 PM
mr fantastic
Quote:

Originally Posted by dwsmith
$\displaystyle\frac{1}{\sqrt{7}\Gamma\left(\frac{1} {2}\right)}\lim_{b\to\infty}\int_0^b\left(x^{-1/2}\exp\left[-x\left(\frac{1}{7}-t\right)\right]\right) \ dx$

$\displaystyle\frac{1}{\sqrt{7}\Gamma\left(\frac{1} {2}\right)}\lim_{b\to\infty}\int_0^b\left(x^{-1/2}\exp\left[\frac{-x}{\frac{1}{\left(\frac{1}{7}-t\right)}}\right]\right) \ dx$

$\displaystyle\alpha=\frac{1}{2} \ \beta=\frac{1}{\frac{1}{7}-t}$

$\displaystyle =\frac{1}{\sqrt{7}\Gamma\left(\frac{1}{2}\right)}\ left(\frac{1}{\frac{1}{7}-t}\right)^{1/2}\Gamma\left(\frac{1}{2}\right)$

Correct?

After simplification, does it agree with the general result given here: Gamma distribution - Wikipedia, the free encyclopedia ?
• Apr 9th 2011, 10:14 AM
dwsmith
It does.

$\displaystyle\frac{1}{(1-7t)^{1/2}}$

But when I take the second derivative as t = 0 and subtract that from the mean squared, I don't obtain the correct variance.
• Apr 9th 2011, 11:21 AM
theodds
Quote:

Originally Posted by dwsmith
It does.

$\displaystyle\frac{1}{(1-7t)^{1/2}}$

But when I take the second derivative as t = 0 and subtract that from the mean squared, I don't obtain the correct variance.

I do :) You should try again.


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i Linux basic -part 2 – All things in moderation

# Linux basic -part 2

##### 4.Linux Scheduler ( Cron Job)
• Cron is a utility that helps us create schedule to perform a certain task/command. As we know that /etc having configuration files for most of services same as for cron.
• We will jush go through a quick review of how dose it work and how do we set it up. The follwing is the hierarchy for it.
First * represent min 0-59
Second * represent hour 0-23
Third * represent day of month 1-31
Forth * represent month 1-12
Fifth * represent day of week 0-6

##### * How to use crontab

Let’s say you have a script which backs up important files, or creates a report about system statistics, for example. Let’s say the script is called /home/myname/scripts/do-every-day.sh, and you want to run it every morning at 7 A.M.
To edit the crontab, use this command:

crontab -e

This will open the crontab in a text editor (Usually this is **vi** or **vim**, but it may be something else depending on your **Linux distribution**).

The default crontab file looks like this:

These lines all start with a # because they are comments; they are ignored by cron, and are just there for you to read.
So, now let’s add our job to the crontab. Each job you add should take up a single line.
We want our job to run at 7 A.M., which would be minute 0, hour 5, every day of the month, every month, every day of the week. We need to add a line to the bottom of the file which looks like this:

0 7 * * * /home/myname/scripts/do-every-day.sh

The asterisks (“*”) in our entry tell cron that for that unit of time, the job should be run “every”. You can now save the file and exit the text editor. In vi, this is done by pressing ESCAPE and then typing :wq (for “write and quit”) and pressing ENTER. crontab will give you the following message:

crontab: installing new crontab

…and return you to the command line. Your script will now run automatically at 7 A.M., every day.
To view your crontab, you can use this command:

crontab -v

…or, to remove your crontab so that there no jobs are ever executed by cron, use this command:

crobtab -r

###### More examples
15 6 2 1 * /home/melissa/backup.sh

Run the shell script /home/melissa/backup.sh on January 2 at 6:15 A.M.

15 06 02 Jan * /home/melissa/backup.sh

Same as the above entry. Zeroes can be added at the beginning of a number for legibility, without changing their value.

0 9-18 * * * /home/carl/hourly-archive.sh

Run /home/carl/hourly-archive.sh every hour, on the hour, from 9 A.M. through 6 P.M., every day.

0 9,18 * * Mon /home/wendy/script.sh

Run /home/wendy/script.sh every Monday, at 9 A.M. and 6 P.M.

30 22 * * Mon,Tue,Wed,Thu,Fri /usr/local/bin/backup

Run /usr/local/bin/backup at 10:30 P.M., every weekday.

###### * Cron Permission

Two files play important role in cron.
Cron Permission
Two files play important role in cron.

cron.allow
cron.deny

If these files exist, then they impose some restriction accordingly on users. That is, if a user is in deny list, so he/she won’t be able to schedule any job/task and if user is in allowed list then she/he will be able to add schedule job/task. All we have to do is just add user name in either of these two files
Cron Files

/etc/crontab: system-wide crontab
[email protected]:~#cat /etc/crontab
# /etc/crontab: system-wide crontab
# Unlike any other crontab you don’t have to run the 'crontab'
# command to install the new version when you edit this file
# and files in /etc/cron.d. These files also have username fields,
# that none of the other crontabs do.
SHELL=/bin/sh
PATH=/usr/local/sbin:/usr/local/bin:/sbin:/bin:/usr/sbin:/usr/bin
# m h dom mon dow user command
17 * * * * root cd / && run-parts --report /etc/cron.hourly
25 6 * * * root test -x /usr/sbin/anacron || ( cd / && run-parts
--report /etc/cron.daily )
47 6 * * 7 root test -x /usr/sbin/anacron || ( cd / && run-parts
--report /etc/cron.weekly )
52 6 1 * * root test -x /usr/sbin/anacron || ( cd / && run-parts
--report /etc/cron.monthly )

##### REFERENCES

http://www.computerhope.com/unix/ucrontab.htm

format: plain


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Rutherford Scattering Formula // outdoor-experience.info

Rutherford scattering formula derivation in Hindi.

Alpha Scattering by Charge Cloud. The Rutherford scattering experiment put to rest the Thomson model of the atom, because it could be shown that a positive charge distributed throughout the classical volume of the atom could not deflect the alpha particles by more than a small fraction of a degree. 05.12.2017 · Rutherford Scattering Tonya Coffey. Loading. Unsubscribe from Tonya Coffey?. Impact parameter and scattering angle - Relation by Dilip Sir - Duration: 26:59.

Rutherford correctly deduced that in the large angle scattering, which corresponded to closer approach to the nucleus, the alpha was actually hitting the nucleus. This meant that the size of the nucleus could be worked out by finding the maximum angle for which the inverse square scattering formula worked, and finding how close to the center of the nucleus such an alpha came. RUTHERFORD FORMULA FOR POINT-LIKE COULOMBIC SCATTERING Orientation A point-like charged projectile approaches a charged point-like target along a straight line that would pass by the nucleus at a distance b, the impact parameter, if there were no interaction between them. However, the incoming particle is de. Rutherford’s Formula a formula for the effective scattering cross section of nonrelativistic charged point particles interacting according to Coulomb’s law. It was derived in 1911 by E. Rutherford. In a center-of-mass system, that is, a system in which the total momentum of the colliding particles is equal to zero, Rutherford’s formula has the.

Rutherford scattering Introduction to Nuclear Science Simon Fraser University Spring 2011 NUCS 342 March 16, 2011 NUCS 342 Lecture 22 March 16, 2011 1 / 29. But Rutherford derived his formula based on Classical Mechanics, which truly is not applicable to the scattering. Title Author €Ö/ Subject Created Date: üò. Rutherford Scattering Discussion 3 2015/04/15Daniel Ben-Zion 1 Derivations The setup for the Rutherford scattering calculation is shown in Figure1. Figure 1: A diagram of the parame-ters in the scattering experiment We have an incoming particle, for example an, which is going to de ect o the nucleus of an atom in the material. How did Rutherford figure out the structure of the atom without being able to see it? Simulate the famous experiment in which he disproved the Plum Pudding model of the atom by observing alpha particles bouncing off atoms and determining that they must have a small core. Rutherford Scattering_. Rutherford scattering formula suggests that the number of particles detected by the detector depend on the deflection/scattering angle. However, the angle of deflection further depends upon several factors like Impact parameter. Okay! First of all let's try to understand what is Impact parameter.

1. I have been trying to look for a derivation of the Rutherford scattering formula from Hyperphysics, but cannot find one. It doesn't show up in the original paper How is this. In the Wikipedia article about rutherford scattering the derivation of the scattering cross section  \fracd\sigmad\Omega =\left\frac Z_1 Z_2 e^28\pi.
2. 02.12.2018 · Explanation Rutherford scattering formula derivation in Hindi Rutherford scattering cross section Rutherford Rqphysics nuclearchemistry.
3. 06.10.2014 · Derivation of Rutherford Scattering Formula A particle arrive at point A, which is still distant from the nucleus, with velocity. If it were not deflected, it will pass the nucleus at distance b. Between them there us a repulsive Coulumb force.
4. Rutherford Scattering - PhET Interactive Simulations.

Rutherford Scattering - Georgia State University.

Rutherford Scattering of α-Particles. as expressed by the Balmer formula and the Ritz combi-nation principle for atomic spectra. Rutherford came upon the idea of probing the struc-ture of atom by ﬁring α-particles, or helium ions, at thin metal foils. B. Scattering from a Rutherford atom Evidently a head-on collision with the heavy nucleus of a Rutherford atom could drastically alter the path of the much lighter alpha particle. Conversely, much of the atomic volume was now assumed to contain only the. Rutherford Scattering Cross-Section Consider a particle of type, incident with relative velocity onto an ensemble of particles of type with number density. If is the probability per unit time of the particle being scattered into the range of solid angle to, then the differential scattering cross-section,. The complete agreement between the results of the measurements and the point- Rutherford scattering cross-section formula shows this in accordance with Gauss's law of electrostatics: a finite charge distribution in the external space beyond the charges cannot be distinguished from a point charge with an potential. 07.02.2020 · In 1911 Ernest Rutherford published a formula which indicated that the number of particles that would be deflected by an angle θ due to scattering from fixed nuclei is inversely proportional to the fourth power of the sine function of one half the angle of deflection; i.e.,.

Rutherford Backscattering Spectrometry RBS is a widely used method for the surface layer analysis of solids. This lecture gives a briefintroduction into the method, and describes scatteringkinematics, scattering cross-section data, stopping power data, detector resolution issues, and electronic energy loss straggling in some detail. Computer. Modul Atomic/Nuclear Physics Rutherford Scattering In this experiment, it comes to understanding the scattering experiment of Rutherford. It should also help with understanding the charge dis-tribution in an atom from the of scattering of α-particles. This experi-ment has signiﬁcantly contributed to the understanding of the structure. Rutherford and colleagues were able to calculate the number of alpha particles which would be scattered into any angle based upon the number of nuclei and their spacing in the gold foil. The observations agreed with these calculations up to a cert. Rutherford Scattering Formula The scattering of alpha particles from nuclei can be modeled from the Coulomb force and treated as an orbit. The scattering process can be treated statistically in terms of the cross-section for interaction with a nucleus which is considered to be a point charge Ze. For a detector at a specific angle with respect to the incident beam, the number of particles per.

The Rutherford formula then becomes dσ dΩ = 4Zα cm 2 q4 In QM scattering theory we normally express the differential cross section in terms of q 2 or the Lorentz invariant quantity q 2. These are the same up to a sign for elastic scattering i E−E f. 2 General Formula for Scattering in Nonrelativistic QM. Rutherford leitete die rutherfordsche Streuformel aus der klassischen Physik her. Eine vollständige quantenmechanische Behandlung des Problems mit Hilfe der bornschen Näherung ergibt, dass die rutherfordsche Streuformel in erster Ordnung korrekt ist und quantenmechanische Effekte nur kleine Korrekturen darstellen. Ein weiteres Problem der rutherfordschen Formel ist der Grenzfall =, für die. With high enough alpha energies, however, the projectile punches in close enough to the nuclear center to come into range of the nuclear strong force and the distribution of scattered alphas departs from the Rutherford formula. Index Rutherford concepts Scattering concepts Blatt ref, Sec 14.2. Rutherford scattering, first discovered by Lord Rutherford, refers to the elastic scattering of a heavy charged particle such as an α-particle from a nucleus. In his famous scattering experiment, Rutherford bombarded a thin gold foil with α-particles and studied how many of them deflected from their original direction of motion.

Rutherford Scattering¶. Today’s understanding of the atom, as a structure whose positive charge and majority of mass are concentrated in a minute nucleus, is due to the -particle scattering experiments conducted by Ernest Rutherford and his colleagues 1909-1914.The essential features of Rutherford’s apparatus are shown in Fig. 1: -particle emitted from a radioactive source strike a thin. Rutherford scattering definition, the scattering of an alpha particle through a large angle with respect to the original direction of motion of the particle, caused by an atom Rutherford atom with most of the mass and all of the positive electric charge concentrated at a center or nucleus. See more. source for Rutherford Scattering. The bottom line is that you will need to nd a relationship between what you measure - number of counts in a given time interval as a function of scattering angle - to the charge on the nucleus. The Rutherford formula after some algebra gives Z= K p Nsin2 2. Rutherford Scattering Apparatus, Different Diameter Targets, Touch Sensitive Tape, Large Protractor. Procedure: Place the scattering unit on a table and tape a new length of.


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# Align the number of an enumerated list with an equation with cases

How do I change the second item (the equation with cases) in the list so - the label (b) is vertically aligned with the lhs of the equation, - the equation in (b) is left aligned with the start of equations in the list, and - there isn't a blank line before the (b) equation begins?

While similar to the question Alignment for Equations with Numbering, Align Number from Enumerate with Question, and Displaying an Equation in a List, I am unable to figure out how to use either minipage or parbox to get alignments right.

I include my output, my code, and a (fuzzy) picture of the exercise I am trying to mimic.

\documentclass[11pt]{amsart}
\usepackage{mathtools}
\usepackage{enumitem}
\DeclarePairedDelimiter\floor{\lfloor}{\rfloor} % floor function fences
\begin{document}
Prove that the greatest-integer function, $\floor{x}$, has the properties indicated.
\begin{enumerate}[label=(\alph*)]
\item $\floor{x + n} = \floor{x} + n,$ for every integer $n$.
\item $\floor{-x} = \begin{cases} - \floor{x} & \text{if } x \text{ is an integer}, \\ - \floor{x} -1 & \text{otherwise.} \end{cases}$
\item $\floor{x+y} = \floor{x} + \floor{y} \qquad \text{or} \qquad \floor{x} + \floor{y} +1$.
\item $\floor{2x} = \floor{x} + \floor{x + \frac{1}{2}}$
\item $\floor{3x} = \floor{x} + \floor{x + \frac{1}{3}} + \floor{x + \frac{1}{3}}$
\end{enumerate}
\end{document}

## 2 Answers

In this case, i.e. in the case of an equation without label, it is very easy. EDIT Comments by @SandyG and @ HenkMetselaar built in.

\documentclass[11pt]{amsart}
\usepackage{mathtools}
\newcommand{\SandyFloor}[1]{\ensuremath{\left\lfloor#1\right\rfloor}}
\usepackage{enumitem}
\DeclarePairedDelimiter\floor{\lfloor}{\rfloor} % floor function fences
\begin{document}
Prove that the greatest-integer function, $\floor{x}$, has the properties indicated.
\begin{enumerate}[label=(\alph*)]
\item $\floor{x + n} = \floor{x} + n,$ for every integer $n$.
\item $\displaystyle \floor{-x} = \begin{cases} - \floor{x} & \text{if } x \text{ is an integer}, \\ - \floor{x} -1 & \text{otherwise.} \end{cases}$
\item $\floor{x+y} = \floor{x} + \floor{y} \qquad \text{or} \qquad \floor{x} + \floor{y} +1$.
\item $\displaystyle\floor{2x} = \floor{x} + \SandyFloor{x + \frac{1}{2}}$.
\item $\displaystyle\floor{3x} = \floor{x} + \SandyFloor{x + \frac{1}{3}} + \SandyFloor{x + \frac{2}{3}}$.
\end{enumerate}
\end{document}

• You may wish to use \big\lfloor and \big\rfloor to get the floor to the bottom of the fractions. – Sandy G Dec 12 '17 at 3:14
• \floor{x + \frac{2}{3}} as last term in the last line. – Henk Metselaar Dec 12 '17 at 3:14
• Yes, @HenkMetselaar. I have changed \floor to \floor* to get the better versions. – Bryan M-H Dec 12 '17 at 3:28
• Thank you, @marmot, Such a simple change! Not only did I change \floor to \floor* to get the better fence-size, I've also added a few points to the spacing between items in the list with \itemsep=3pt to keep things from running into each other. – Bryan M-H Dec 12 '17 at 3:55
• For now I have taken out the \displaystyle command. If it weren't an exercise, I would probably use it, but the floor fences with the fractions seem too big with it. – Bryan M-H Dec 12 '17 at 4:08

I have currently changed my code to the following. It doesn't enlarge the floor fences as much as @Sandy G suggested and as @marmot used in his example.

Prove that the greatest-integer function, $\floor{x}$, has the properties indicated.

\begin{enumerate}[label=(\alph*), itemsep=4pt]
\item $\floor{x + n} = \floor{x} + n,$ for every integer $n$.
\item $\floor{-x} = \begin{cases} - \floor{x} & \text{if } x \text{ is an integer}, \\ - \floor{x} -1 & \text{otherwise.} \end{cases}$
\item $\floor{x+y} = \floor{x} + \floor{y} \qquad \text{or} \qquad \floor{x} + \floor{y} +1$.
\item\label{exFormula1} $\floor{2x} = \floor{x} + \floor*{x + \frac{1}{2}}$.
\item\label{exFormula2} $\floor{3x} = \floor{x} + \floor*{x + \frac{1}{3}} + \floor*{x + \frac{1}{3}}$.
\end{enumerate}

Great advice, thanks!


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Article | Open | Published:

# On the potential of in vitro organ-chip models to define temporal pharmacokinetic-pharmacodynamic relationships

## Abstract

Functional human-on-a-chip systems hold great promise to enable quantitative translation to in vivo outcomes. Here, we explored this concept using a pumpless heart only and heart:liver system to evaluate the temporal pharmacokinetic/pharmacodynamic (PKPD) relationship for terfenadine. There was a time dependent drug-induced increase in field potential duration in the cardiac compartment in response to terfenadine and that response was modulated using a metabolically competent liver module that converted terfenadine to fexofenadine. Using this data, a mathematical model was developed to predict the effect of terfenadine in preclinical species. Developing confidence that microphysiological models could have a transformative effect on drug discovery, we also tested a previously discovered proprietary AstraZeneca small molecule and correctly determined the cardiotoxic response to its metabolite in the heart:liver system. Overall our findings serve as a guiding principle to future investigations of temporal concentration response relationships in these innovative in vitro models, especially, if validated across multiple time frames, with additional pharmacological mechanisms and molecules representing a broad chemical diversity.

## Introduction

Integrative pharmacology is a discipline that builds an understanding of the inter-relationship between pharmacokinetics (PK), the drug’s time course for absorption, distribution, metabolism and excretion and pharmacodynamics (PD), the biological effect of a drug. In drug discovery, this multi-variate approach guides medicinal chemists to modify structural properties of a drug molecule to improve its chance of becoming a medicine in a process known as “lead optimization”. Pharmacological effect is driven by drug concentration however the seminal work of Segre1 and Sheiner et al.2, identified that the maximum drug effect is not always driven by the peak drug concentration but that in some cases time is a critical factor influencing drug effect, especially in in vivo studies. This is known as hysteresis and is driven by limited access to the site of drug action or slow receptor kinetics. In the main, current drug discovery programs are only able to truly investigate concentration-effect-time relationships during the advanced stages of the preclinical program. At this stage there are typically between 1 and 3 potential drug candidates progressed to animal studies to confirm efficacy and safety. It is therefore costly and time consuming to discover that such potential candidates may have poor therapeutic qualities preventing their onward progression, not to mention the significant limitations of extrapolating animal data to humans because drug behavior in animals can be dramatically different to that in humans.

Over the last two decades there has been an explosion in the development of microphysiological systems (MPS) or “body-on-a-chip” models3,4. These microengineered in vitro models aim to recreate the in vivo tissue microenvironment enabling cells, typically human in origin, to maintain viability and function. The hype that these systems could recapitulate organ function5 is gradually being replaced by hope that such systems might enable more eloquent models for drug discovery reducing the reliance on animals which has ethical, monetary, time and translational advantages.

Using a ‘heart-on-a-chip’ that was fluidically connected to a ‘liver-on-a-chip’ we set out to discover whether it was possible to derive in vitro temporal PKPD relationships with the well characterized small molecule drug terfenadine. This multi-organ platform utilized a pumpless system and serum-free medium with interconnected compartments6,7. The absence of a pump and the accompanying tubing enabled a bubble free, low volume system providing the ability to detect metabolite formation that might otherwise be diluted in higher-volume systems8. The sinusoidal oscillatory rocking profile for driving fluid flow was designed via a transient flow model driven by gravitational acceleration to produce shear stresses on the cellular layers of no more than 0.05 dynes/cm2 (within acceptable physiological ranges)8,9 throughout the rocking profile, as described previously6. This PK model additionally includes parameters for convective transport of compounds throughout the system (mixing) and compound-specific adsorption to the housing materials and has been used for both hydrophobic and hydrophilic compounds7. Currently, the adsorption is determined experimentally for each compound, though this framework is expected to be expanded to create a prediction of the adsorption based on the physical characteristics of the compounds. This PK model is used to inform our predictive PKPD model and the effect of metabolites on toxicity in multi-organ systems has also been explored at single timepoints in this model10,11. The functional cardiac system utilized patterned human cardiomyocytes that had previously been shown to allow non-invasive measurements of field potential duration (FPD), conduction velocity and beat rate12.

We reproduced the terfenadine-induced increase in QT-interval in the heart only model and demonstrated that in the presence of a metabolically competent liver compartment, this effect was diminished. We also discovered that this pharmacodynamic effect was driven by the intracellular drug concentration in cardiomyocytes, consistent with the intracellular binding site of terfenadine to the hERG channel13,14. Furthermore, we have been able to build a mathematical model that predicts drug response in this MPS model and additionally that these approaches can also be used to qualitatively predict the effect of terfenadine in preclinical species. Finally, we demonstrated the value of MPS models in drug discovery by testing an AstraZeneca proprietary small molecule which was identified as having a hERG liability, due to the formation of a hERG active metabolite, during the final stages of lead optimization. Taken together, these results provide the first description of integrative pharmacology in MPS models and builds confidence towards the transformative effect these models can have on drug discovery.

## Results

### Characteristics of heart:liver and heart only MPS models

Figure 1 shows the composition of the heart:liver and heart only models. Each model encompassed a microfluidic system (total volume 2 mL), 2 media chambers and 2 (heart model) or 3 (heart:liver model) cellular chambers (1 liver and 2 cardiac). These cellular chambers contain electrodes (multi electrode array; MEA) and flexible cantilevers to assess cardiac electrical and mechanical function, respectively (Fig. 1a). Only the MEA cardiac chamber was utilized in this study. Following cell attachment with human primary hepatocytes in the liver chamber and patterned human induced pluripotent stem cell derived cardiomyocytes (hiPS-CMs) in the heart chamber, each model was placed on an interval rocker that induced flow by reciprocal levelling between reservoirs resulting in a closed loop recirculating flow12. This recirculating serum-free system supports the maintenance of liver (cytochrome (CYP) P450 activity and albumin production) and cardiac function for up to 28-days15. The presence and cellular structure of key hepatocyte (albumin) and cardiomyocyte (myosin heavy chain) markers were confirmed following 14 days by immunofluorescence (Fig. 1b–e). Cardiomyocyte resting membrane potential was shown to be physiologically relevant at approximately −85 to −90 mV, representative of the adult human heart16,17,18. Taken together this provides evidence that the heart:liver model is a suitable system to investigate both parent compound and metabolite driven cardiovascular changes. Importantly the integrated technology also allows in vitro temporal PKPD relationships to be derived.

### Value of temporal PK and FPD measurements of parent compound driven changes

To build insight into whether an MPS model could be used to derive temporal PKPD relationships, the well-characterized small molecule, terfenadine was utilized. Terfenadine is a hERG inhibitor resulting in QT prolongation in both preclinical species and in humans, but it is rapidly metabolized, mainly by CYP3A4, to fexofenadine which does not cause QT prolongation19,20. The integrated MEA electrodes in both the heart only and heart:liver models were utilized to measure the external field potential. This allows the quantification of FPD, a surrogate of the action potential duration at 90% repolarization (APD90), hence enabling an assessment of the impact of terfenadine on the QT interval21.

Heart:liver and heart only models were exposed to 5 µM terfenadine or vehicle (0.1% DMSO). Simultaneous heart FPD, beat rate, terfenadine and fexofenadine concentrations in the media and cellular lysates were measured at 0, 0.5, 1, 4, 12, 16 and 24 hours. Figure 2a depicts typical field potential traces obtained from the heart only model. Using these traces FPD and beat rate were quantified, as represented in Fig. 2b–e in the heart only (Fig. 2b,c) and heart:liver model (Fig. 2d,e). In the heart only model, time dependent increases in FPD were noted, with peak measurable changes (8-fold increase) observed at 4 hours. A 5 µM nominal concentration of terfenadine in heart only systems had a profound effect on cardiac repolarization at 0.5 to 1 hour that rhythmic cardiac beating ceased and therefore, FPD (Fig. 2b) and beat rate (Fig. 2c) were unquantifiable. These changes in FPD were reduced in the heart:liver model, over the time course of the experiment (Fig. 2d). This data provides evidence that the FPD responses observed were driven by terfenadine (the parent drug). No changes in cell viability in either the heart or liver chambers at 24 hours were present in either model (Fig. 2f), confirming these changes in FPD were not the result of gross cell death.

### Qualitative and quantitative translational modelling

Qualitatively the terfenadine response was comparable across standard preclinical and clinical cardiovascular safety studies (Table 1). It is important to note that quantitative differences in the magnitude of QT prolongation across pre-clinical species and human is known, the occurrence of QT prolongation can be qualitatively compared, but predicting the magnitude of change from either Guinea pig or dog pre-clinical studies is variable. For example, dogs have been found to be approximately 2.5 times less sensitive to QT prolongation in terms of the magnitude of response22.

Due to the measurement of both the PK and PD response simultaneously, a PKPD relationship was developed to predict the FPD changes obtained in these models. To develop a quantitative translational model, temporal data across the MPS platform for terfenadine, fexofenadine, and FPD measurements were used to develop a mathematical PKPD model using Phoenix NLME 7.0 software. The mathematical model could accurately capture terfenadine and fexofenadine exposure levels in the MPS model over time (Fig. 4a and Supplementary Fig. 2). It was further able to capture changes in FPD signals produced by the heart chamber (Fig. 4a). Corresponding peak shifts in cardiac terfenadine PK profiles and FPD profiles verify the assumption that changes in FPD are associated with cardiac bioaccumulation of terfenadine. The goodness of fit captured by the model parameter estimates was further reinforced through percent coefficient of variation (% CV) rates below 26% (Supplement Table 1).

FPD simulations were performed based on published in vivo PK profiles to demonstrate the potential of the MPS model to predict cardiotoxicity. Representative PK profiles for Guinea pig23, dogs24 and human25 were used as an input for the simulation to predict corresponding cardiac terfenadine bioaccumulation and therefore predicted FPD profiles (Fig. 4b). The predicted FPD values were qualitatively comparable to the reported dog and human QTc prolongation seen in Table 1; however, the FPD predictions failed to reach published Guinea pig QTc prolongation levels (Table 1). The quantitative translation is likely to have been limited by several potential sources of variability; (1) differences between in vitro and in vivo: the chip is made from human cells, and although stem-cell models are commonly used to explore FPD, there remains a possibility that the exact response magnitude differs between the chip and in vivo; (2) donor to donor variability has not been accounted for and could potentially be a factor affecting the quantitative responses measured; (3) variability between species: there are clearly different responses observed in vivo between different species and to address the impact of this would require development and validation of species-relevant chip models which is beyond the scope of this manuscript and (4) variability between in vivo responses: the in vivo models do not all agree from study to study, and different observed exposure profiles and FPD could be another confounding layer in the interpretation of the results. Nonetheless, these in vivo simulations show the potential for the MPS model to qualitatively predict cardiotoxicity in humans and in animal models, especially the dog and could be improved to reach quantitative predictions by reducing sources of variability.

### Application in drug discovery

Application of MPS model data to build further confidence in the human risk of industrial scientists involved in drug safety disciplines of pharmaceutical companies. Following the characterization of the MPS model with terfenadine and a proprietary AstraZeneca compound (AZ12818677) was evaluated in both the heart only and heart:liver model. Prolongation of the QT interval on the electrocardiogram can lead to a fatal arrhythmia known as Torsade de Pointes26. Consequently, it is a mandatory requirement prior to first in human administration of potential new drug candidates that their ability to inhibit the hERG channel, a biomarker for QT prolongation, is investigated. This is typically conducted in vitro at a very early stage in drug discovery because of high-throughput cellular technologies27. During the lead optimization phase of drug discovery this in vitro data is supplemented by an in vivo assessment in small animals, typically either the Guinea pig or rat, providing an integrated assessment of cardiovascular safety. These studies allow compounds to be profiled while chemical diversity is available, facilitating the selection of optimal molecules prior to mandated regulatory evaluation of cardiovascular function in non-rodents. AZ12818677 (parent compound) was assessed for hERG inhibition as is standard practice and no quantifiable IC50 value could be derived at concentrations up to 100 µM. It was concluded that this compound had a low potential to cause in vivo QT prolongation. AZ12818677 was therefore progressed to a small animal cardiovascular in vivo study to assess QT prolongation, in this case the anaesthetized Guinea pig. However, unexpectedly the corrected QT interval increased by 22% (Fig. 5a) at a concentration of 3 µM. Due to the lack of activity of AZ12818677 at the hERG channel, it was concluded that the QT prolongation in the Guinea pig was not being driven by AZ12818677-induced hERG inhibition. Subsequent investigations identified a major metabolite, AZ12864610, which was produced in humans and Guinea pig (among other preclinical species). Retrospective analysis of AZ12864610 on the in vitro hERG assay, revealed it was a potent inhibitor with an IC50 of 1.14 µM (95% confidence interval (CI) 0.86 µM – 2.2 µM). Due to this metabolite driven effect on hERG activity and the in vivo QT prolongation we investigated whether the heart:liver model could prospectively detect this metabolite driven effect (Fig. 5b).

Figure 5c highlights representative FP traces following the parent (AZ12818677) compound in the heart:liver model at 24 hours. These traces were quantified, resulting in an EC50 of 2.7 µM (95% CI 0.9–7.6 µM) and 13.2 µM (95% CI 9.8–17.9 µM) on FPD in the heart:liver and heart only models, respectively (Fig. 5d). To confirm the changes in the FPD in the heart:liver model were a result of the formation of the metabolite AZ12864610, it was chemically synthesized and evaluated in the heart only model. As expected, the FPD was prolonged resulting in an EC50 of 1.2 µM (95% CI 0.7–1.9 µM) (Fig. 5g). At concentrations above 3 µM, there were incidents of extreme FPD prolongation (Fig. 5e–f) to the point of arrhythmogenesis suggesting a profound effect on the electrical activity of the cells. This arrhythmogenic activity resulted in large error associated with FPD measurements which were removed from the graphical representation for the sake of clarity (Fig. 5f), but clearly a pharmacodynamic effect was detectable. In addition, cell viability in either the heart or liver chambers at 24 hours was not reduced (Fig. 5h), confirming these changes are not the result of gross cell death. The FPD data obtained for these AstraZeneca compounds and terfenadine demonstrate that these MPS model systems can detect cardiovascular changes driven by either the parent molecule or the metabolite.

The ability of the liver cellular component to metabolize AZ12818677 to AZ12864610 was assessed. The measured concentration of AZ12818677 remained relatively stable over 24 hours. For example, following a nominal concentration of 300 µM, 251 µM (SD 0 µM) was measured in the media of the liver component at 0 hours, and 251 µM (SD 5.7 µM) was measured at 24 hours (Fig. 6a). Associated with this, only 0.5 µM and 1.4 µM of AZ12864610 (the metabolite) was detected at 6 and 24 hours respectively (Fig. 6b). Taken together this suggests a complex PKPD relationship, with additional factors influencing the PK and metabolism of the parent compound. Due to the presence of a PD effect in the heart:liver and heart only model despite minimal concentrations of the metabolite, we investigated the potential of cardiomyocytes to produce the metabolite. Since this metabolite is a product of amide hydrolysis (Fig. 6c) it is possible that cardiomyocytes can produce it through cholinesterases28. Formation of this metabolite in cardiomyocytes (Fig. 6d) is a significant observation which confirms the PD response in the cardiac only system is derived from a metabolite formed by cardiomyocytes and the response observed in heart:liver model could be a cumulative effect. Although we did not investigate if AZ12864610 undergoes an N-acetylation reaction to form AZ12818677, it could be a possible explanation for the limited detection of the metabolite in incubations with hepatocytes (Fig. 6b), which can potentially convert the metabolite back to parent via N-acetyl transferase enzymes.

## Discussion

Human-on-a-chip models are gaining increasing popularity as tools that will improve preclinical translation thus ensuring potential new drug candidates have a higher probability of success during the clinical trial process. Indeed, the pharmaceutical industry have pointed to defining their context of use to realize their full impact29. But to date, these models have been largely characterized only with well-studied reference compounds and so the performance of these models in a ‘real-world’ setting remains to be determined30,31. Moreover, little attention has been paid to exploiting the dynamic nature of these models to derive an understanding of the temporal relationship between measured pharmacodynamic response and pharmacokinetics30. The ability to build PKPD relationships in vitro and associated quantitative translational understanding would enable scientists to understand compound behavior prior to in vivo testing, offering a cost and time saving as well as a credible option to reduce and eventually replace animal models. Furthermore, although several groups have developed computational models to analyze data from MPS models7,8,32,33, no-one has yet published data on quantitative translation of MPS data to in vivo outcome whether that be in animals or humans34. Here we report first on the use of a heart-liver MPS model to develop a temporal PKPD relationship for the anti-histamine, terfenadine. Using the quantitative data, we then show that we can build a mathematical model that correctly predicts the PK and PD in the MPS, accounting for parent and metabolite concentration across compartments of the device (e.g. conservation of mass) as well as the relationship between target tissue exposure and pharmacodynamic outcomes. Because the model allows the PK and PD responses to be evaluated separately, we were then able to compare the in vitro PD response with in vivo outcome in commonly used preclinical species. Finally, we demonstrated the application of this MPS model to drug discovery by recreating the parent and metabolite effects of a proprietary AstraZeneca small molecule compound, which in contrast to terfenadine the cardiovascular effects were driven by the metabolite.

Our simultaneous measurement of cardiomyocyte response to the cardiotoxic drug terfenadine, together with matched quantification of drug concentration in the circulating media and in the lysates of both cardiomyocytes and hepatocytes, show that the pharmacodynamic response was not driven by the concentration of drug in the media but rather by the concentration of drug within the cardiomyocytes. This was also supported by the lack of response of cardiomyocytes to terfenadine in the presence of metabolically competent hepatocytes and the time-dependent increase in the concentration of the hERG-inactive metabolite fexofenadine in the circulating media. Although it is known that terfenadine binds to an intracellular site of hERG13,14, we have for the first time in a dynamic in vitro system convincingly proven this35. With this knowledge, we can determine the peak effect of terfenadine in the model, rather than at a fixed time point36. This principle is fundamental to developing an understanding of drug behavior and being able to define this during early drug discovery will be a valuable contribution to the optimization of potential new drug candidates.

The commitment to measure concentration and effect over time enables an accurate description of drug behavior37 and is a critical component of quantitative translation. Most of the current in vitro literature assumes that the concentration of drug added to the media drives the effects measured38 and that the administered concentration is equivalent to the effective concentration. It has been established that polydimethylsiloxane (PDMS), commonly used in MPS models, absorbs drug compounds, especially those that are highly lipophilic39 and measuring actual drug concentration also addresses these limitations. As demonstrated with terfenadine, the PKPD modelling approach was critical for understanding both the flux of compound between compartments as well as the resulting PD response in the context of dynamic exposure profiles of parent and metabolite. Further evaluation of a proprietary AstraZeneca small molecule compound highlights the complexity of changes observed with novel molecules during drug discovery, providing an example of a ‘real-world’ application. This serves as an illustration of the complexity and diversity of PKPD relationships different compounds can drive. This diversity and ability of both PK and PD effects to occur through both direct and indirect interactions, either instantaneously or delayed further highlights the importance of assessing both the PD and PK effects in conjunction with each other to allow multiple influences on hysteresis to be evaluated40.

The application of mathematical modelling of cardiovascular safety data is well established in terms of PKPD modelling of in vivo data to predict and translate human effects at expected therapeutic doses41. However, the use of these modelling approaches to MPS data is still in its infancy but has been shown to enhance the value of the MPS models7,42,43,44. Here we demonstrate the potential to use mathematical modelling for translational assessment in the same well-established manner as is commonly applied to in vivo study data. Just as with translating in vivo data across species, the model-derived PK profile and PKPD relationships are built and extrapolated to in vivo scenarios for a single compound at a time by substituting exposure profiles. Further, as with translational validation of in vivo models, proof of concept is initially established with a small number of compounds and confidence in the translational utility of the experimental platform will grow as additional data is generated across differential chemistry, pharmacology and mechanisms. This joint experimental-computation approach provides the prospect of creating quantitative translational understanding for cardiovascular changes based on a human model that can be applied within drug discovery, predicting the preclinical and clinical changes.

Within this study we have assessed one element of cardiovascular safety: changes in ECG waveforms and intervals. However, drug-induced cardiovascular toxicity can also result in modulation of hemodynamics (heart rate, blood pressure and cardiac contractility) and cardiac pathology45. The integrated measurement technologies within this model system (other MEA parameters and flexible cantilevers in combination with a liver module)6,12, combined with the ability to collect media samples over time provides the prospect of assessing both changes in cardiac contractility via assessment of force utilizing the flexible cantilevers and pathology via assessment of released soluble cardiac biomarkers, for example, cardiac troponin46. Such an approach if validated with a diverse panel of pharmacological agents, and coupled with liver cells of varying metabolic competencies, representing interindividual variability, could enable a holistic assessment of cardiac safety or efficacy in one combined human-based model not only for acute responses, as measured here, but potentially for repeated dose toxicity as cell function can be maintained within these systems for up to 28-days15. This would not only allow detection of changes but potentially provide phenotypic fingerprints of individual compounds to be generated.

In conclusion, our findings serve as a guiding principle to future MPS investigations. Whilst these data are robust, we strongly advocate that further compounds, ideally working across different time frames, through other pharmacological mechanisms and representing a broader chemical diversity, should be evaluated to increase the validation of this heart-liver model but also to build mathematical models that enable quantitative translation to animal and human outcome.

## Methods

### Microfluidic housing design and assembly

Microfluidic housings were designed and fabricated as previously described6,7 using poly (methyl methacrylate) and PDMS elastomer sheets. The medium recirculates between 2 reservoirs through a microfluidic network formed by the PDMS gaskets, with flow driven by a rocking platform producing a sinusoidal rocking profile with amplitude of ±1° and period of 60 seconds. The medium flows over the cell-containing areas of the liver coverslips and cardiac cantilever and MEA chips within the microfluidic channel. The system contained a total of 2 mL of medium throughout the flow channel and reservoirs. Compound dosing was applied at the reservoir closest to the liver chamber, such that the compound would first pass over and interact with the liver before being recirculated throughout the system.

### Preparation of chips

Cantilever array chips for force measurements were fabricated following protocols previously described47,48,49,50,51,52. Each cantilever chip contained an array of two rows of microscale cantilevers (4 µm thick, 100 µm wide, and 750 µm long), with 16 cantilevers per row. Custom patterned MEA chips, for electrical measurements, were fabricated using standard microfabrication procedures following a previously described protocol7. Each MEA chip included 10 platinum recording electrodes separated by 1 mm. The surface of each chip were coated with poly (ethylene glycol) (PEG)-containing silane, and this PEG layer was patterned by selective ablation using a 193 nm ArF excimer laser (Lambda Physik, Santa Clara, CA). The ablated PEG surfaces were coated with human plasma fibronectin (Millipore) diluted in 1X phosphate buffered saline (PBS) (Thermo Scientific) to concentrations of 10 ug/mL for MEAs and 50 ug/mL for cantilevers, then incubated at 37 °C for 30 minutes followed by a PBS rinse. Printed circuit boards and flexible elastomeric connectors (zebra connectors, FUJIPOLY, 1 mm wide × 18.2 mm long × 9 mm tall) were incorporated into each system to create an interface between the MEA and a commercially available 60 electrode amplifier (MEA1060, Multichannel-systems).

### Cell culture

hiPS-CMs (purchased from Cellular Dynamics) were thawed and seeded directly onto the surfaces at 50,000 cells per MEA and 500,000 cells per cantilever6. Primary human hepatocytes were purchased from the Cell Resources Core (Massachusetts General Hospital, Boston, MA, USA). Once thawed, 150,000 hepatocytes were cultured in the hepatic chamber which was coated with rat tail collagen type I (60 ug/mL) as previously described6. The same batch of primary human hepatocytes and hiPS-CMs were used throughout the course of these experiments to block for any potential batch to batch variation within these cells.

### Drug preparation

Tefenadine (Sigma Aldrich, T9652, St. Louis MO, USA) and the proprietary AstraZeneca compounds AZ12818677 and AZ10565970 were dissolved in DMSO to a concentration 1000x the desired final nominal dosing concentration. Drug was added through the reservoir closest to the liver chamber to allow for first contact of the drug with the liver. 2 μL of the drug stock solution (or vehicle) was added to each system to achieve the desired working concentration in 2 mL of total volume within the platform.

### Cardiac activity as measured by MEA

Custom printed circuit boards were produced to interface the MEAs inside the system with a commercially available 60 electrode amplifier (MEA1060, Multichannel-systems) using a Zebra® elastomeric connector (Fujipoly)7. A stimulus generator (STG 1002, Multichannel Systems) was used to stimulate the cells (800 mV rectangular pulse, 0.5 Hz to 3.0 Hz in 0.25 Hz increments). The multichannel systems software suite was used to control both the amplifier and stimulator and was used to record action potentials.

Baseline recordings of spontaneous and stimulated cell activity were taken immediately prior to drug application and at 0.5, 1, 4, 12, 16 and 24 hours after exposure to terfenadine and at 0.5, 1, 2, 4 and 24 hours after exposure to AZ12818677. FPD, spontaneous beat frequency and CV were extracted using Clampfit (Axon Instruments). The FPD was calculated by measuring the time from the initiation of the Q-wave until the apex of t-wave. Spontaneous beat frequency was obtained from spontaneous recordings of cardiac activity 30 seconds in length.

### Immunofluorescence

Hepatocytes and cardiomyocytes were fixed with 4% paraformaldehyde for 15 minutes at room temperature, washed three times with PBS, permeabilized with 0.1% Triton-X for 15 minutes, then blocked with a PBS solution containing 5% donkey serum and 1% BSA for one hour at room temperature. Afterwards, hepatocytes were incubated with sheep anti-human serum albumin primary antibody (Abcam) at a 1:100 dilution overnight at 4 °C, washed 3 times with PBS, then incubated with Alexa Fluor 568 conjugated donkey anti-sheep secondary antibody (Thermo Fisher) at a 1:300 dilution for 2 hours at room temperature. Cardiomyocytes were treated similarly with mouse anti-myosin heavy chain primary antibody (Developmental Studies Hybridoma Bank) at a 1:10 dilution, Alexa Fluor 568 conjugated donkey anti-mouse secondary (Thermo Fisher) at a 1:300 dilution. Both cell types were subsequently stained with DAPI (Thermo Fisher) at a 300 nM concentration for 5 minutes. Cells were imaged on a TCS SP8 confocal microscope (Leica Microsystems) and processed using LAS X software.

### Cell viability

Hepatocyte viability was assessed via a MTT assay. Briefly, cells were incubated in 500 µL of growth medium containing 5 mg/mL MTT powder for 90 minutes at 37 °C, 5% CO2. The resultant crystals were dissolved in 100 µL of lysis buffer (10% SDS with 0.5% acetic acid in dimethyl sulfoxide (DMSO)) and was placed into a 96 well plate. Absorbance was read at 570 nm using a BioTek Synergy HT plate reader. Cardiomyocyte viability was assessed using 10% solution (v:v) of alamar blue (Thermofisher) in media at 37 °C, 5% CO2 for 4 hours. 100 µL of this solution was then placed into a 96 well plate and read at fluorescence excitation wavelength 570 nm and emission at 590 nm using the BioTeK Synergy HT plate reader.

### LC-MS/MS analysis of AZ12818677 and metabolite in cardiomyocyte incubations

hiPS-CMs (iCell™ Cardiomyocytes), cell culture thawing, and maintenance media were purchased from Cellular Dynamics International (cat# CMM-100-120-005) (Madison, WI). hiPS-CMs were cultured on gelatin coated 96 well plates according to the manufacturers protocol http://www.cellulardynamics.com/products/lit/CDI_iCellCMUsersGuide110927.pdf) at a density of 20000 cells/well. Cell culture media was replaced every 48 hours for 10 days. Following 10 days in culture, hiPS-CMs were treated with 0.1% DMSO, AZ12818677 (10, 100 or 200 µM), media and hiPS-CM samples were collated at 0, 2, 6 and 24 hours. Concentration of all analytes in the incubations was determined by LC-MS/MS. Acquity ultra performance liquid chromatography [UPLC] system, (Waters, UK) coupled to a triple-quadrupole mass spectrometer (Xevo TQ-S; Waters, Milford, MA) was used to carry out the sample analysis. The analytes were separated by reverse-phase liquid chromatography using Waters Atlantis® T3, 3 µm, 2.1 × 50 mm column (Waters, UK). Mobile Phases A and B consisted of water (containing 0.1% ammonium formate w/v) and ACN (containing 0.1% FA v/v), respectively. The flow rate was held constant at 0.9 mL/min throughout the gradient run. The initial mobile phase composition of 99% A and 1% B was held for 1 min. Mobile phase B was then increased linearly to 70% until 1.5 min, followed by further increase to 40% B until 3.5 min. At 1.51 min the composition of A and B was made to 1% A and 99% B and was held until 4 minutes. At 4.1 min A and B were reversed to the initial 99% A and 1% B and held for 1 minute. Analytes quantitation was achieved by MS–MS detection in positive electrospray ionization mode. The MS operating conditions were as following: the capillary voltage was 1.14 kV and source offset was 50 V. The desolvation temperature was set to 600 °C. Nitrogen was used as the desolvation gas (800 L/h) and cone gas (150 L/h). Argon was used as the collision gas at a flow rate of 0.15 mL/min. Detection of the ions was performed in the MRM mode using the transitions of m/z 424.21 → 187.98 for AZ12864610 and m/z 455.1984 → 165.2068 for verapamil, the internal standard, with a Dwell time of 0.03, 0.05 and 0.02 s, respectively. Peak integration and calibrations were performed using TargetLynx software (Version 4.1, Waters, Milford, MA).

### hERG inhibition electrophysiology

A functional electrophysiology assay in CHO cells stably transfected with Kv11.1 (hERG) was conducted on the IonWorks™ platform as described previously53. AZ12818677 and AZ12864610 were tested at 0.01, 0.03, 0.1, 0.33, 1, 3, 10, 33 and 100 µM in 0.33% DMSO. Concentration-effect curves were plotted for each compound and expressed relative to vehicle (0.33% DMSO) ± SD. The molar concentration of test compound producing 50% inhibition (IC50) were calculated using Graphpad Prism™ (La Jolla, CA) to fit data to a 4-parameter non-linear regression curve. The maximum (DMSO control) values were fixed. IC50s/EC50s were generated in quadruplet for each compound and geometric means and 95% CI were subsequently calculated.

### Anesthetized Guinea pig

This study was ethically reviewed by the establishments’ Animal Welfare and Ethical Review Body (AWERB), following the principles of the 3Rs. All work was then conducted under an approved project license, governed by the UK home office under The Animals (Scientific Procedures) Act 1986. All animal identification, conditions of housing, acclimatization, environment, diet and water were in accordance with the standard AstraZeneca operating procedures. The effects on cardiac electrophysiology and bioanalysis were assessed in the anaesthetized Guinea pig (Dunkin Hartley) via recording of the ventricular monophasic action potential at 90% repolarization (MAPD90) as described previously54. AZ12818677 or vehicle was tested (I.V.) at 0.744 and 2.015 mg/kg (4 animals per group), formulated as a solution in 30% v/v dimethylacetamide in water (vehicle). Recordings were obtained for 20 minutes following the start of each dose level followed by a 30-minute washout.

### Heart only and heart-liver chip PKPD modelling and in vivo simulation

Time course terfenadine and fexofenadine concentration profiles were modelled using a 4-compartment model in Phoenix NLME Software version 7.0. Heart, liver, media, and PDMS compartments and their corresponding parameters can be seen outlined in Fig. 4a. Flow of terfenadine for medium-PDMS and medium-heart pathways was modelled using a first order process as defined by parameters KP1/KP2 and KL1/KL2, respectively. Flow of terfenadine from medium and heart compartments was assumed to be saturable as defined by Eqs 1 and 2. Metabolism of terfenadine to fexofenadine within the heart and liver compartments was modelled using a first order process as defined by parameters Kfc and Kfl respectively. It was assumed that there was a delay of fexofenadine release into the system as defined by parameter Kint.

A fifth compartment, FPD, was added to the Phoenix model to represent a PD profile linked to the defined PK model. This PD profile was modelled as a first order process in which the change in effect on FPD was a function of terfenadine heart bioaccumulation as defined by parameters KFPDin and KFPDout. An indirect modelling approach was used for FPD because a hysteresis relationship was found between cardiac terfenadine concentration and FPD values (Supplementary Fig. 1). This approach was further confirmed when %CV values increased when modelled directly.

Using the experimental data collected (Fig. 3), parameters (except for Kint) were simultaneously fit across heart only and heart-liver systems. Parameters were then fixed and the fexofenadine parameter, Kint, was fit using the experimental fexofenadine concentration data collected (Fig. 3). In the heart only case, the liver compartment was eliminated resulting in relevant parameters being equal to 0. Volumes of each compartment was defined by system parameters described above. The liver volume was initialized based on the system estimates but allowed to vary to account for any differences between assumed and actual cell numbers or sizes. Once parameters were estimated, the model was used to simulate FPD profiles given published in vivo terfenadine PK profiles. Published Guinea pig23, dog24 and human25, PK profiles were used as inputs into the established model in order to predict cardiac bioaccumulation over time, and therefore predict change in predicted FPD. While Guinea pig and dog temporal PKPD data is available in the literature, matching human data is not. Instead, the model made use of a hypothesized dosing schedule that would simulate a PK profile matching published Cmax and AUC data in humans orally dosed with terfenadine, as this information is available in the literature25. The reported standard deviation values in PK profile were used to capture the associated error in predicted FPD values. Percent free protein in plasma was assumed to be 2% for dog and 3% for Guinea pig and human for the FPD simulation23,24.

$${{\boldsymbol{K}}}_{{\boldsymbol{medium}}{\boldsymbol{to}}{\boldsymbol{heart}}}={\boldsymbol{KC}}{\bf{1}}$$
(1)
$${{\boldsymbol{K}}}_{{\boldsymbol{heart}}{\boldsymbol{to}}{\boldsymbol{medium}}}={\boldsymbol{KC}}{\bf{2}}\ast (\frac{{\bf{1}}}{{\bf{1}}+\frac{{\boldsymbol{Cardia}}{{\boldsymbol{c}}}_{{\boldsymbol{Terf}}}}{{\boldsymbol{I}}{{\boldsymbol{C}}}_{{\bf{50}}}}})$$
(2)

## Data Availability

All data supporting the findings of this study are available within the article and its Supplementary Information Files.

## Ethics declarations

### Competing Interests

The authors declare no competing interests.

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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50. 50.

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51. 51.

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52. 52.

Wilson, K. et al. Measurement of contractile stress generated by cultured rat muscle on silicon cantilevers for toxin detection and muscle performance enhancement. PLoS One 5(6), e11042 (2010).

53. 53.

Bridgland-Taylor, M. H. et al. Optimisation and validation of a medium-throughput electrophysiology-based hERG assay using IonWorks HT. J Pharmacol Toxicol Methods 54(2), 189–99 (2006).

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55. 55.

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## Acknowledgements

The authors would like to thank Karin Skansen, Matthew Bridgland-Taylor and Karen Philips for conduct of modeling, hERG inhibition experiments and Guinea pig studies on the AstraZeneca compound, respectively. The work at Hesperos was supported by AstraZeneca as well as the National Institutes of Health, SBIR grant number R44TR001326-02, from the National Center for Advancing Translational Sciences (NCATS).

## Author information

A.P., D.P.W., J.J.H. and L.E. conceived the study and were involved in the design of experiments. C.W.M., C.J.L., R.B., B.W., L.R.B., N.N.S., M.T.S., M.S., J.R. and A.S. conducted MPS model experiments or LCMS and analysed the data. R.M., V.P.M., C.J.L. and J.T.M. were involved in the development of the mathematical model and conducted the associated modelling. A.P., L.E., C.W.M., J.J.H., M.L.S. wrote the manuscript which was critically reviewed by D.P.W. and K.F.

### Competing Interests

The authors declare no competing interests.

Correspondence to James J. Hickman or Lorna Ewart.


text:
College Algebra (10th Edition)

$(x-6)(x+6)$
The given expression can be written as: $x^2-6^2$ RECALL: $a^2-b^2=(a-b)(a+b)$ Use the formula above with $a=x$ and $b=6$ to obtain: $=(x-6)(x+6)$


text:
# Equation of motion coursework

#### KateyLou

1. The problem statement, all variables and given/known data

I have coursework on rhe convolution integral, however i am struggling to find the equation of motion to start the whole thing off with.

I will attach a picture of the problem, is is simply a cantilevered bar with a concentrated mass on the end and a damper.

2. Relevant equations

I am assuming you find the deflection at the end of the beam as a result of the force P(t), which i think is
P(t)L^3/3EI

However i am not sure where to go from here

3. The attempt at a solution

Thinking it may possibly be
m$$\ddot{x}$$+c$$\dot{x}$$=P(t)L^3/3EI
however i think i may need to add in stiffness somewhere else..

#### Attachments

• 5.8 KB Views: 150

#### Dr.D

Your simple model for the cantilever, looks OK, and it gives you the stiffness if you look at it properly. You have a deflection relation that is usually written as
defl = P*L^3/(3*E*I)
P = (3*E*I)/(L^3) * defl
and from there you can see that the stiffness of this system is
K = 3EI/L^3
Now back to your equation of motion, which will look like
m*xddot + c*xdot + K*x = F(t)
where F(t) is whatever applied forcing function acts on the mass.

See if that will get you going!

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text:
UVA {EGAnet} R Documentation

## Unique Variable Analysis

### Description

Identifies redundant variables in a multivariate dataset using a number of different association methods and types of significance values (see Christensen, Garrido, & Golino, 2020 for more details)

### Usage

UVA(
data,
n = NULL,
model = c("glasso", "TMFG"),
corr = c("cor_auto", "pearson", "spearman"),
method = c("cor", "pcor", "wTO"),
sig,
key = NULL,
reduce = TRUE,
auto = TRUE,
label_latent = TRUE,
reduce.method = c("latent", "remove", "sum"),
lavaan.args = list(),
plot.redundancy = FALSE,
plot.args = list()
)

### Arguments

 data Matrix or data frame. Input can either be data or a correlation matrix n Numeric. If input in data is a correlation matrix, then sample size is required. Defaults to NULL model Character. A string indicating the method to use. Current options are: glasso Estimates the Gaussian graphical model using graphical LASSO with extended Bayesian information criterion to select optimal regularization parameter. This is the default method TMFG Estimates a Triangulated Maximally Filtered Graph corr Type of correlation matrix to compute. The default uses cor_auto. Current options are: cor_auto Computes the correlation matrix using the cor_auto function from qgraph. pearson Computes Pearson's correlation coefficient using the pairwise complete observations via the cor function. spearman Computes Spearman's correlation coefficient using the pairwise complete observations via the cor function. method Character. Computes weighted topological overlap ("wTO" using EBICglasso), partial correlations ("pcor"), or correlations ("cor") Defaults to "wTO" type Character. Type of significance. Computes significance using the standard p-value ("alpha"), adaptive alpha p-value (adapt.a), or some threshold "threshold". Defaults to "threshold" sig Numeric. p-value for significance of overlap (defaults to .05). Defaults for "threshold" for each method: "wTO" .25 "pcor" .35 "cor" .50 key Character vector. A vector with variable descriptions that correspond to the order of variables input into data. Defaults to NULL or the column names of data reduce Boolean. Should redundancy reduction be performed? Defaults to TRUE. Set to FALSE for redundancy analysis only auto Boolean. Should redundancy reduction be automated? Defaults to TRUE. Set to FALSE for manual selection label_latent Boolean. Should latent variables be labelled? Defaults to TRUE. Set to FALSE for arbitrary labelling (i.e., "LV_") reduce.method Character. How should data be reduced? Defaults to "latent" "latent" Redundant variables will be combined into a latent variable "remove" All but one redundant variable will be removed "sum" Redundant variables are combined by summing across cases (rows) lavaan.args List. If reduce.method = "latent", then lavaan's cfa function will be used to create latent variables to reduce variables. Arguments should be input as a list. Some example arguments (see lavOptions for full details): estimator Estimator to use for latent variables (see Estimators) for more details. Defaults to "MLR" for continuous data and "WLSMV" for mixed and categorical data. Data are considered continuous data if they have 6 or more categories (see Rhemtulla, Brosseau-Liard, & Savalei, 2012) missing How missing data should be handled. Defaults to "fiml" std.lv If TRUE, the metric of each latent variable is determined by fixing their (residual) variances to 1.0. If FALSE, the metric of each latent variable is determined by fixing the factor loading of the first indicator to 1.0. If there are multiple groups, std.lv = TRUE and "loadings" is included in the group.label argument, then only the latent variances i of the first group will be fixed to 1.0, while the latent variances of other groups are set free. Defaults to TRUE adhoc Boolean. Should adhoc check of redundancies be performed? Defaults to TRUE. If TRUE, adhoc check will run the redundancy analysis on the reduced variable set to determine if there are any remaining redundancies. This check is performed with the arguments: method = "wTO", type = "threshold", and sig = .20. This check is based on Christensen, Garrido, and Golino's (2020) simulation where these parameters were found to be the most conservative, demonstrating few false positives and false negatives plot.redundancy Boolean. Should redundancies be plotted in a network plot? Defaults to FALSE plot.args List. Arguments to be passed onto ggnet2. Defaults: vsize = 6Changes node size alpha = 0.4Changes transparency label.size = 5Changes label size edge.alpha = 0.7Changes edge transparency

### Value

Returns a list:

 redundancy A list containing several objects: redudant Vectors nested within the list corresponding to redundant nodes with the name of object in the list data Original data correlation Correlation matrix of original data weights Weights determine by weighted topological overlap, partial correlation, or zero-order correlation network If method = "wTO", then the network computed following EGA with EBICglasso network estimation plot If redundancy.plot = TRUE, then a plot of all redundancies found descriptives basic A vector containing the mean, standard deviation, median, median absolute deviation (MAD), 3 times the MAD, 6 times the MAD, minimum, maximum, and critical value for the overlap measure (i.e., weighted topological overlap, partial correlation, or threshold) centralTendency A matrix for all (absolute) non-zero values and their respective standard deviation from the mean and median absolute deviation from the median method Returns method argument type Returns type argument distribution If type != "threshold", then distribution that was used to determine significance reduced If reduce = TRUE, then a list containing: data New data with redundant variables merged or removed mergedA matrix containing the variables that were decided to be redundant with one another methodMethod used to perform redundancy reduction adhoc If adhoc = TRUE, then the adhoc check containing the same objects as in the redundancy list object in the output

### Author(s)

Alexander Christensen <alexpaulchristensen@gmail.com>

### References

# Simulation using UVA
Christensen, A. P., Garrido, L. E., & Golino, H. (under review). Unique Variable Analysis: A novel approach for detecting redundant variables in multivariate data. PsyArXiv.

# Implementation of UVA (formally node.redundant)
Christensen, A. P., Golino, H., & Silvia, P. J. (2020). A psychometric network perspective on the validity and validation of personality trait questionnaires. European Journal of Personality, 34, 1095-1108.

# wTO measure
Nowick, K., Gernat, T., Almaas, E., & Stubbs, L. (2009). Differences in human and chimpanzee gene expression patterns define an evolving network of transcription factors in brain. Proceedings of the National Academy of Sciences, 106, 22358-22363.

# Selection of CFA Estimator
Rhemtulla, M., Brosseau-Liard, P. E., & Savalei, V. (2012). When can categorical variables be treated as continuous? A comparison of robust continuous and categorical SEM estimation methods under suboptimal conditions. Psychological Methods, 17, 354-373.

### Examples

# Select Five Factor Model personality items only
idx <- na.omit(match(gsub("-", "", unlist(psychTools::spi.keys[1:5])), colnames(psychTools::spi)))
items <- psychTools::spi[,idx]

# Change names in redundancy output to each item's description
key.ind <- match(colnames(items), as.character(psychTools::spi.dictionary$item_id)) key <- as.character(psychTools::spi.dictionary$item[key.ind])

# Automated selection of redundant variables (default)

uva.results <- UVA(data = items, key = key)

# Manual selection of redundant variables
if(interactive()){
uva.results <- UVA(data = items, key = key, type = "adapt")
}

[Package EGAnet version 1.1.0 Index]


text:
# ch.1 proposition 14 in Lang's Algebraic Number Theory

The proposition is as follows :

Let $A$ be integrally closed in its quotient field $K$, and $B$ be integral closure of $A$ in finite Galois extension $L$ of $B$, with group $G$. Let $\mathfrak p$ be maximal ideal of $A$, $\mathfrak q$ be maximal ideal of B lying above $\mathfrak p$. Then $B/\mathfrak{q}$ is normal over $A/\mathfrak p$, and mapping $\sigma \mapsto \bar{\sigma}$ is homomorphism of $G_{\mathfrak q}$ (decomposition group of $\mathfrak q$) onto the Galois group of $B/\mathfrak q$ over $A/\mathfrak p$.

I have no problem with the part proving $\bar{B}/\bar{A}$ is normal extension. As the automorphism group is referred to as 'Galois group of $\bar{B}/\bar{A}$', I assume the proposition means that $\bar{B}/\bar{A}$ is normal and separable extension, but it seems that the separability part of the proof doesn't appear in the text.

I find it hard to prove the separability myself and actually it seems that it is not generally separable.

The separability does not always hold. It is true, however, if the residue field $A/\mathfrak{p}$ is finite.


text:
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## AP®/College Calculus AB

### Course: AP®/College Calculus AB>Unit 3

Lesson 1: The chain rule: introduction

# Identifying composite functions

AP.CALC:
FUN‑3 (EU)
,
FUN‑3.C (LO)
,
FUN‑3.C.1 (EK)
Review of composite functions and how to recognize them. This is a valuable skill when approaching the Chain Rule in calculus.

## Want to join the conversation?

• Can't anything be considered a composite function?
• Well, yes, you can have u(x)=x and then you would have a composite function. In calculus, we should only use the chain rule when the function MUST be a composition. This is the only time where the chain rule is necessary, but you can use it whenever you want, technically.

Example - d/dx(3x+2). Clearly, the answer is 3, but we could use the chain rule:
u(x) = 3x
v(x) = x+2
d/dx(3x+2)=d/dx[v(u(x))])=1*3=3.
So the chain rule is usable but completely unnecessary.
• So at Sal said:
"So this would be cosine of u of x plus one." //where u of x is equal to sin(x)
But couldn't we say that u of x is equal to (sin(x) +1)?
Why did we leave the "1" out?
Or do we get to choose?
• At Sal states "I also want to state that there's oftentimes more than one way to compose ..."

So, yes you could choose to set u(x) = sin(x) + 1.

Try determining the derivative both the way Sal shows and this other way — you will find that you get the same answer.
• Hi... anything can be a composite function and chain rule can be applied but when is a function NECESSARILY a composite function?

• Well, a function is necessarily a composite function when you can't simplify it with known simplification techniques to arrive at a function comprised of a single function (a lot of functions have alternate definitions that you can use for simplification/complication later in your math education, like the link between exp(x) and trig functions). For example, (x^2)^2 isn't necessarily a composite function because with simplification, you can end up with x^4, which is a single function. Something that is necessarily a composite function is something like ln(sin(x)), which you can't simplify with elementary techniques to end up with a single function.
• say we have w(u(x)) we can take d/dx of this using the composite rule formula which it becomes
w'(u(x)) * u'(x)

But since this example starts talking about situations where you can use three functions at instead of two, how would you find the derivate then?

if g(x) = h(w(u(x))
does d/dx of g(x) become
= h'(w(u(x))) * w'(u(x)) * u'(x) ?

I did some working out and have seen that it can not and it seems that it becomes:
=h'(w(u(x))) * u'(x)

Without the middle term I had above w'(u(x))
I don't understand why I thought it should be in there. Is this correct? What is the proper way to think about the chain rule with more than 2 functions involved in the composition?
• You were right the first time, why did you decide it was: h'(w(u(x))) • u'(x)?

One way to remember this is to think of taking the derivative in steps:

Let f(x) = w(u(x)), so g(x) = h(f(x))

Apply the chain rule once:
g'(x) = h'(f(x)) • f'(x)

f'(x) = w'(u(x)) • u'(x)

Which gives us:
g'(x) = h'(w(u(x))) • w'(u(x)) • u'(x)
• Hey can't I make h(x)=Cos x and v(x)=sin x +1 then g(x)=h(v(x))=h(sin x +1)=cos(sin x +1) ??
• Yep, that looks about right. As a challenge, could you figure out what v(h(x)) would be?
• Sal says at that it would be hard to express sin(x)cos(x) as a composite function. What about:

Let f(x) = x*sin(cos^-1(x))

Let g(x) = cos(x)

Then f(g(x)) = cos(x)*sin(cos^-1(cos(x)) = cos(x)sin(x)
• You for sure can do that, but the only problem is, xsin(cos^-1 x) will not be defined for values greater than 1 or less than -1, which makes it "hard" to define it in some other way
• Easy tip.
In a composite function, one is INSIDE another.

Example: ln(sin(x)) because sin(x) is INSIDE ln(x),
except that we take the natural log of sin(x) instead of x in this composite function.
• can I apply the chain rule when I have a composition of three functions ? if it's possible, please write the formula for that.
• You can apply the chain rule to f(g(h(x))) by treating g(h(x)) as one function, so you get
f'(g(h(x))·[g(h(x))]'
Then evaluate the derivative of g(h(x)) by using chain rule again.


text:
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History
 Subject: M::B on Win32 produces 'EXE_FILES' option in Makefile.PL that fails under *nix
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Using Module::Build 0.30 on WinXP SP3, AS Perl 5.10: If I put a perl script (e.g. "check_perldiag") in the 'bin' folder of my distribution, Module::Build creates a Makefile.PL with the following option: 'EXE_FILES' => [ 'bin\\check_perldiag' ] This, however, apparently fails under *unix, where make says: "No rule to make target `bin\check_perldiag', needed by `blib/script/bin\check_perldiag'. Stop." If I change the above option to: 'EXE_FILES' => [ 'bin/check_perldiag' ] both make on *nix and nmake on Win32 run without any problems, but Module::Build keeps overriding this line every time I do "./Build dist". I cannot change it back from within the resulting archive, since this messes up the permissions inside the archive and CPAN refuses to index the distribution (due to the "no world-writable dirs" rule). Petar Shangov
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On Mon Jan 05 16:04:31 2009, pshangov wrote: Show quoted text
> If I change the above option to: > > 'EXE_FILES' => [ > 'bin/check_perldiag' > ] > > both make on *nix and nmake on Win32 run without any problems, but > Module::Build keeps overriding this line every time I do "./Build > dist". I cannot change it back from within the resulting archive, > since this messes up the permissions inside the archive and CPAN > refuses to index the distribution (due to the "no world-writable > dirs" rule).
If you disable the Makefile generation, the Makefile.PL becomes just like any other file (but you have to manually maintain this.) It sounds like you're using the 'traditional' form of Makefile.PL. I think an alternative would be to use the passthrough form. Another alternative is to eliminate the Makefile.PL, or just send to fix its generation. Thanks, Eric

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