# Numerical simulation of self-oscillating catalytic reaction in a plug-flow
reactor

N.V. Peskov<EMAIL_ADDRESS>M.M. Slinko  Faculty of Computational
Mathematics and Cybernetics,
Lomonosov Moscow State University, Moscow, Russian Federation.  Semenov
Institute of Chemical Physics, Moscow, Russian Federation.

###### Abstract

Self-oscillations in some oxidation reactions on metal catalysts are
associated with periodic oxidation/reduction of the catalyst surface. If the
reaction proceeds in a flow reactor at atmospheric pressure, then the reaction
dynamics is complicated by the mass transfer in the gas phase. In this paper,
the reaction in a flow reactor is described by an 1D system of advection-
diffusion-reaction equations with an abstract STM-type kinetic model of the
surface catalytic reaction. The dependencies of the solution of the system on
the reaction parameters, the intensity of diffusion in the adsorption layer,
and on the gas flow rate are studied numerically. Solutions such as localized
oscillations (”breathing” structures) and mobile oxidation/reduction fronts on
the catalyst surface are constructed.

###### keywords:

nonlinear dynamics , advection-diffusion-reaction equations , spatiotemporal
structures

##  1 Introduction

Heterogeneous catalytic reactions are highly non-linear systems that can
operate far from thermodynamic equilibrium. They are known to exhibit complex
temporal and spatiotemporal behavior including self-sustained oscillations,
chemical waves and deterministic chaos [1, 2, 3]. Spatiotemporal pattern
formation has been observed both under UHV conditions over single crystal
surfaces [4] and under atmospheric pressure conditions on pure and supported
metallic surfaces [5]. To image pattern formation on surfaces under UHV
conditions the photoelectron emission microscopy (PEEM) and later low energy
electron microscopy (LEEM) were developed [6, 7]. Under atmospheric conditions
infrared video thermography (IRT) has been widely used to detect and measure
temperature patterns in the oxidation of CO and ethylene on a Rh/SiO2 catalyst
[8], oxidation of NH3 on a Pt ribbon [9], CO oxidation over Pd/Al2O3 [10], H2
oxidation on a Pt/SiO2 catalyst [11] and a Ni foil [12].

Under normal pressure conditions external transport limitations can
significantly influence the reaction rate and the dynamic behavior of the
system. In most cases IRT clearly showed that certain areas of the catalyst
were hotter than others. The temperature difference between the cold and hot
regions was up to 54 ℃ during H2 oxidation on a Pt/SiO2 [10], 150 ℃ for CO
oxidation over supported Pd [10] and 190 ℃ during CO oxidation over Rh/SiO2
catalyst [8]. That is why first simulations of spatiotemporal temperature
patterns used generic kinetics, like first order exothermic reaction coupled
with a slow change in surface activity [13], or polynomial kinetics [14, 15],
that were claimed to have similar properties to a detailed model.

Later spatiotemporal patterns during catalytic oxidation of CO over Pd
supported on a glass-fiber disk-shaped cloth in a continuous reactor with feed
flowing perpendicular to and through the disk were reported [16]. Breathing
pattern were detected representing a hot spot that expanded and contracted
continuously causing the oscillations in exit CO2 concentration. The
temperature gradient between the hot and cold domains could be relatively
small reaching only few degrees at low gas feed and low CO concentration (0.5
vol. %). Therefore, simulations were carried out assuming that the oscillatory
behavior was driven by kinetic effects which were reflected by temperature
oscillations [17]. The model was based on the oscillatory kinetics model of
the Sales-Turner- Maple (STM) type.

The originally suggested STM model assumes that the reaction of CO oxidation
proceeds according to the Langmuir-Hinshelwood mechanism and includes the
following elementary steps [18]: dissociative adsorption of oxygen, O2;
adsorption and desorption of carbon monoxide, CO; surface reaction of CO
oxidation, CO + O → CO2, and dissolution and segregation of adsorbed atoms of
oxygen, O. It is also suggested that dissolved oxygen blocks the O2 adsorption
and thus creates a feedback mechanism necessary for the origin of reaction
rate self-sustained oscillations. The dissolution and segregation of oxygen
can be interpreted as the oxidation and reduction of the catalyst surface. In
Ref. 17 the STM type model was coupled with an enthalpy and gas-phase balances
in a CSTR reactor. Although the authors did not aim to carry out a
comprehensive parameter adjustment of the model, the distributed model
simulated experimentally detected breathing patterns. It was demonstrated that
the counteracting interaction of the reaction kinetics which tried to enlarge
the hot domain and the effect of the cold boundary could lead to the formation
of breathing patterns [1].

Recently it was found out that under atmospheric pressure conditions
spatiotemporal structures on the catalyst surface during reaction could be
monitored by naked eye and recorded by a video camera [19, 20]. It was shown
that such method was successful for methane, ethane, and CO oxidation over Ni
and Co metallic catalysts, due the different colors of metallic and oxidic
states [21, 22, 23]. It was shown that in all reactions the propagation of
oxidation (dark color) and reduction (light color) fronts was relevant in the
oscillatory phenomena. The most active state was the metallic surface, while
the oxidic state was less reactive. The synchronization of video observations
with the in situ x-ray photoelectron spectroscopy also proved the connection
of the spatiotemporal patterns during propane oxidation over Ni foil with the
reversible bulk oxidation of Ni to NiO [24]. Mathematical modeling of kinetic
isothermal waves has practically not been carried out so far. There is only
one work describing the occurrence of isothermal traveling waves of oxidation
and reduction during the oxidation of CO on a nickel catalyst which arise when
CO and oxygen gradients in the reactor were taken into account due to the
influence of mass transfer processes on the reaction rate [25].

The main goal of this work is to study the effect of mass transfer processes
on the oscillating reaction, following STM type mechanism in a plug flow
reactor. Both gradients in the gas phase and catalyst surface will be
considered. In order to get the most general results that are not related to
any particular reaction, an abstract kinetic model will be used.

##  2 Kinetic model

The abstract model describes the reaction between an oxidizing agent X and a
reducing agent Y on the surface of a metal catalyst M. The elementary stages
of this reaction are shown in Table 1, where $C_{x}$ and $C_{y}$ denote the
concentrations of X and Y in the gas phase, and, respectively, x and y denote
adsorbed particles of the X and Y species. Designations $x$ and $y$, ($x\geq
0$, $y\geq 0$, $x+y\leq 1$) are the fraction of the catalyst surface covered
with particles x and y, respectively; $z$ ($0\leq z\leq 1$) is the fraction of
the oxidized surface Mx; $k_{i}$ is the rate constant of corresponding stage.
It should be noted that since the most common oxidizing agent is oxygen, two
free active sites are necessary for X adsorption. Moreover as the oscillating
mechanism of STM type will be used it is suggested, that the adsorption of X
occurs only on the reduced catalyst surface.

Table 1: Elementary stages of catalytic reaction X + Y $\to$ XY stage | description | stage rate
---|---|---
X adsorption | X + 2M $\to$ 2x | $R_{1}=C_{x}k_{1}(1-x-y)^{2}(1-z)^{2}$
Y adsorption | Y + M $\to$ y | $R_{2}=C_{y}k_{2}(1-x-y)$
y desorption | y $\to$ Y + M | $R_{3}=k_{3}y$
x + y reaction | x + y $\to$ XY + 2M | $R_{4}=k_{4}xy$
catalyst oxidation | M + x $\to$ Mx | $R_{5}=k_{5}x(1-z)$
catalyst reduction | Mx + y $\to$ M + XY | $R_{6}=k_{6}yz$

The corresponding system of kinetic equations is written as:

$\displaystyle\dot{x}=C_{x}k_{1}(1-x-y)^{2}(1-z)^{2}-k_{4}xy-k_{5}x(1-z),$
$\displaystyle\dot{y}=C_{y}k_{2}(1-x-y)-k_{3}y-k_{4}xy-k_{6}yz,$ (1)
$\displaystyle\dot{z}=k_{5}x(1-z)-k_{6}yz.$

Following [18], the dimensionless time $\tau=C_{x}k_{1}t$ is introduced and
the dimensionless system is obtained:

$\displaystyle\dot{x}=(1-x-y)^{2}(1-z)^{2}-\kappa_{4}xy-\kappa_{5}x(1-z),$
$\displaystyle\dot{y}=p(1-x-y)-\kappa_{3}y-\kappa_{4}xy-\kappa_{6}yz,$ (2)
$\displaystyle\dot{z}=\kappa_{5}x(1-z)-\kappa_{6}yz,$

where $p=(C_{y}k_{2})/(C_{x}k_{1})$ is the new parameter and
$\kappa_{i}=k_{i}/(C_{x}k_{1})$.

Using the values of the parameters for the STM model determined in [18] as the
initial approximation, the following values of the dimensionless rate
constants of the stages were chosen for model (2):

$\kappa_{3}=10^{-1},\;\kappa_{4}=10^{3},\;\kappa_{5}=\kappa_{6}=10^{-2}.$ (3)

In what follows, the parameter $p$ plays the role of an external control
parameter. Figure 1 shows the solution of the system (2) for $p=0.5$ starting
from zero initial conditions $x(0)=y(0)=z(0)=0$.

Figure 1: Top: solution of system (2); bottom: corresponding reaction rate
$R_{4}$ at $p=0.5$ and other parameters from (3).

The steady solution has the form of regular oscillations, the period of which
is clearly divided into two successive parts or stages: oxidation
($\dot{z}>0$) and reduction ($\dot{z}<0$) of the catalyst. During the
oxidation stage, the reaction rate is much higher than during the reduction
stage. Of the rate constants (3), the solution is most sensitive to a change
in the value of $\kappa_{3}$. For fixed values of the remaining parameters,
self-oscillations exist in the interval $0.085<\kappa_{3}<0.28$. The
parameters $\kappa_{4},\,\kappa_{5},\,\kappa_{6}$ can be changed by several
orders of magnitude without a qualitative variation in the solution. The
oscillation period is determined by constants $\kappa_{5},\>\kappa_{6}$ and
can vary by several orders of magnitude. The ratio of parameters
$\kappa_{5}/\kappa_{6}$ determines the ratio of the duration of the of
oxidation and reduction stages in the oscillation period.

Figure 2 gives a general idea of the special solutions (attractors) of the
system (2) depending on the parameter $p$.

Figure 2: Bifurcation diagram of stationary solution of system (2) in
dependence on $p$. Solid lines – stable solutions, dotted lines – unstable
solutions.

Figure 2 shows a bifurcation diagram of a non-trivial stationary solution of
system (2) in dependence on $p$. In order not to obscure the picture, only the
$y$ and $z$ components of the stationary solution are shown. The stationary
value of the component $x$ is close to zero in the considered range of $p$.
There are three bifurcation points of the stationary solution on the interval
$0<p<1$. The first point, marked ‘$\circ$’, is the supercritical Hopf
bifurcation H at $p=p_{1}=0.347$. At the point H, the stability of the
stationary solution changes and a stable limit cycle is born. The range of
oscillations in the oxidation degree $z$ in the limit cycle is shown in Figure
2 by the gray stripe. The limit cycle vanishes in a small neighborhood of the
point, marked ‘$\times$’, of the saddle-node bifurcation SN at
$p=p_{2}=0.532$. In this work, the type of bifurcation, in which the limit
cycle disappears, and its exact position are unimportant – one can assume that
self-oscillations exist in the interval of $p$ values between the points H and
SN. Passing through the point SN, three stationary states appear: one stable
and two unstable. The multiplicity of stationary states exists till the third
bifurcation point T at $p$ = 0.815, after which there is only one stable
stationary state.

In addition to the non-trivial stationary solution discussed above, there is
one trivial solution: $x=1,\,y=0,\,z=1$, which is unstable for any value of
$p$. There is no other stationary solution of system (2).

##  3 The model of the flow reactor

A flow reactor is modeled as a rectangular channel with a width of $L$ m and a
height of $L/2$ m. The flat square catalyst of size $L\times L$ m2 is located
at the bottom of the channel. A gas mixture containing an oxidant X and a
reducing agent Y flows through the channel at a linear velocity $u$ m/s.

Let $\xi$ be the coordinate along the channel in the direction of the gas
flow, while the catalyst is located on the segment $0<\xi<L$. It is assumed
that the distributions of reagent concentrations, $c_{x}(\xi,t)$ and
$c_{y}(\xi,t)$ [mol/m3], in the gas flow along the reactor are described by a
system of advection-diffusion-reaction equations:

$\displaystyle\frac{\partial c_{x}(\xi,t)}{\partial t}+u\frac{\partial
c_{x}(\xi,t)}{\partial\xi}=D_{x}\frac{\partial^{2}c_{x}(\xi,t)}{\partial\xi^{2}}-q_{x}(\xi,t),$
$\displaystyle\frac{\partial c_{y}(\xi,t)}{\partial t}+u\frac{\partial
c_{y}(\xi,t)}{\partial\xi}=D_{y}\frac{\partial^{2}c_{y}(\xi,t)}{\partial\xi^{2}}-q_{y}(\xi,t),$
(4)

where $D_{x}$ and $D_{y}$ [m2/s] are the diffusion coefficient of X and Y,
$q_{x}$ and $q_{y}$, [mol/(m${}^{3}\cdot$s)], describe the mass transfer
between the gas phase and the catalyst.

The exchange terms, $q_{x}(\xi,t)$ and $q_{y}(\xi,t)$, can be estimated as
follows. The strip of the catalyst surface with area $L\Delta\xi$, located
between $\xi$ and $\xi+\Delta\xi$ across the gas flow, contains
$N_{c}L\Delta\xi$ of active surface centers, $N_{c}$ [mol/m2] is the surface
density of active centers. Each center adsorbs $R_{1}\Delta t$ molecules X and
$(R_{2}-R_{3})\Delta t$ molecules Y during the time interval $\Delta t$.
Therefore, the change in the concentration X in the volume of the gas phase
$L\times\Delta\xi\times L/2$ above the strip is equal to
$2(R_{1}N_{c}L\Delta\xi\Delta t)/(L^{2}\Delta\xi)$ = $2(R_{1}N_{c}/L)\Delta t$
and

$q_{x}(\xi,t)=\frac{2N_{c}}{L}R_{1}(\xi,t),\;q_{y}(\xi,t)=\frac{2N_{c}}{L}(R_{2}(\xi,t)-R_{3}(\xi,t)).$
(5)

To close the system (3), a system of equations of the type (2) for distributed
surface coverages $x(\xi,t)$, $y(\xi,t)$, $z(\xi,t)$, should be added to it.
For the generality of the model, the mobility of adsorbed particles must be
taken into account. It is assumed that the adsorbed atoms of the oxidizing
agent x and reducing agent y can migrate over the catalyst surface, while the
oxidized surface atoms of the catalyst Mx are immobile. This paper considers
migration of the hopping type, i.e. an atom x or y can jump from a surface
center below it to another free center. Thus, the intensity of migration is
determined not only by the number of surface atoms, but also by the number of
free centers on the surface [26].

To quantify the surface migration, let us consider three successive intervals
(cells) of width $\Delta\xi$ on the $\xi$ axis with centers at the points
$\xi_{-}=\xi-\Delta\xi$, $\xi$ and $\xi_{+}=\xi+\Delta\xi$. Let’s assume that
the frequency of jumps from one cell to the next is proportional to the
product of the number of atoms in the given cell by the number of free places
in the neighboring cell, divided by the distance between the cell centers.
Then, the change of concentration of x atoms in the cell $\xi$ during time
$\Delta t$ can be expressed as

$\Delta(x\Delta\xi)\approx\frac{H_{x}}{\Delta\xi}(x_{-}s+x_{+}s-xs_{-}-xs_{+})\Delta
t,$

where $H_{x}$ [m2/s] is the coefficient similar to the diffusion coefficient
and $s=1-x-y$ is the surface coverage of empty centers in corresponding cell.
In the limit at $\Delta\xi\to 0$, $\Delta t\to 0$, one obtains

$\frac{\partial x}{\partial
t}=H_{x}\left(s\frac{\partial^{2}x}{\partial\xi^{2}}-x\frac{\partial^{2}s}{\partial\xi^{2}}\right)=H_{x}\left((1-y)\frac{\partial^{2}x}{\partial\xi^{2}}+x\frac{\partial^{2}y}{\partial\xi^{2}}\right).$
(6)

Similar expression can be formulated for y migration.

Finally, the complete model of flow reactor can be written as:

$\displaystyle\frac{\partial c_{x}}{\partial t}+u\frac{\partial
c_{x}}{\partial\xi}=D_{x}\frac{\partial^{2}c_{x}}{\partial\xi^{2}}-ac_{x}k_{1}(1-x-y)^{2}(1-z)^{2},$
$\displaystyle\frac{\partial c_{y}}{\partial t}+u\frac{\partial
c_{y}}{\partial\xi}=D_{y}\frac{\partial^{2}c_{y}}{\partial\xi^{2}}-a[c_{y}k_{2}(1-x-y)-k_{3}y],$
$\displaystyle\frac{\partial x}{\partial
t}=J_{x}(x,y)+c_{x}k_{1}(1-x-y)^{2}(1-z)^{2}-k_{4}xy-k_{5}x(1-z),$ (7)
$\displaystyle\frac{\partial y}{\partial
t}=J_{y}(y,x)+c_{y}k_{2}(1-x-y)-k_{3}y-k_{4}xy-k_{6}yz,$
$\displaystyle\frac{\partial z}{\partial t}=k_{5}x(1-z)-k_{6}yz,$

where $a=2N_{c}/L$ and migratory terms of the form (6) are denoted by $J_{x}$
and $J_{y}$.

Considering the concentrations of reagents in the gas phase, the model (3) is
supplemented by the following Danckwerts boundary conditions at the inflow
boundary ($\xi=0$)

$u(c_{x}-C_{x})|_{\xi=0}=D_{x}\frac{\partial
c_{x}}{\partial\xi}\bigg{|}_{\xi=0},\;u(c_{y}-C_{y})|_{\xi=0}=D_{y}\frac{\partial
c_{y}}{\partial\xi}\bigg{|}_{\xi=0};$ (8)

and the Neumann boundary conditions at the outflow boundary ($\xi=L$)

$\frac{\partial c_{x}}{\partial\xi}\bigg{|}_{\xi=L}=0,\;\frac{\partial
c_{y}}{\partial\xi}\bigg{|}_{\xi=L}=0.$ (9)

For surface coverages $x,y,z$ the Neumann boundary conditions are set both at
the inlet and at the outlet of the reactor.

In dimensionless time $\tau=C_{x}k_{1}t$ and dimensionless coordinate
$\zeta=\xi/L$ ($0<\zeta<1$) the corresponding system for dimensionless
concentrations $v=c_{x}/C_{x}$, $w=c_{y}/C_{y}$ is as follows:

$\displaystyle\frac{\partial v}{\partial\tau}+\frac{1}{\tau_{L}}\frac{\partial
v}{\partial\zeta}=d_{x}\frac{\partial^{2}v}{\partial\zeta^{2}}-a_{x}v(1-x-y)^{2}(1-z)^{2},$
$\displaystyle\frac{\partial w}{\partial\tau}+\frac{1}{\tau_{L}}\frac{\partial
w}{\partial\zeta}=d_{y}\frac{\partial^{2}w}{\partial\zeta^{2}}-a_{y}\big{[}pw(1-x-y)-\kappa_{3}y\big{]},$
$\displaystyle\frac{\partial
x}{\partial\tau}=j_{x}(x,y)+v(1-x-y)^{2}(1-z)^{2}-\kappa_{4}xy-\kappa_{5}x(1-z),$
(10) $\displaystyle\frac{\partial
y}{\partial\tau}=j_{y}(y,x)+pw(1-x-y)-\kappa_{3}y-\kappa_{4}xy-\kappa_{6}yz,$
$\displaystyle\frac{\partial z}{\partial\tau}=\kappa_{5}x(1-z)-\kappa_{6}yz,$

Here the following notations are used:

$\tau_{L}=C_{x}k_{1}t_{L},\;t_{L}=L/u;$
$d_{x}=\frac{D_{x}}{C_{x}k_{1}L^{2}},\;d_{y}=\frac{D_{y}}{C_{x}k_{1}L^{2}};$
$a_{x}=\frac{a}{C_{x}},\;a_{y}=\ \frac{a}{C_{y}};$
$j_{x}(x,y)=\frac{J_{x}(x,y)}{C_{x}k_{1}L^{2}},\;j_{y}(y,x)=\frac{J_{y}(y,x)}{C_{x}k_{1}L^{2}}.$

##  4 Parameter estimates

To carry out numerical calculations, it is necessary to set the values of the
invariable parameters of the model and determine the ranges of change in the
values of the variable parameters.

In this work, the values of the rate constants $\kappa_{i}$ (see Eq. (3)),
adsorption constants $k_{1,2}$, and the oxidant concentration $C_{x}$ in the
inlet gas flow are fixed. The ideal gas theory is usually used to estimate the
adsorption constants. From the Maxwell distribution, an estimate of the
average frequency of collisions of molecules with an adsorption site (with
area $1/N_{c}$) on the catalyst surface follows, which is used to estimate the
adsorption constant

$k_{i}=\frac{s_{i}}{N_{c}}\sqrt{\frac{RT}{2\pi M_{i}}},\;i=1,2,$

where $R$ [J/(K$\cdot$mol)] is the ideal gas constant, $T$ [K] is the gas
temperature, $M_{i}$ [kg/mol] is the mass of 1 mol of gas and $s$ is the
sticking coefficient. The calculations will use the values of the constants
calculated for O2 ($M_{1}=0.032$ kg/mol) and CO ($M_{2}=0.028$ kg/mol) at the
gas temperature $T=850$ K. Also, the value of $N_{c}=2.66\times 10^{-5}$ for
Ni surface will be used. For simplicity, we assume that $s_{1}=s_{2}=0.1$.
Then, one has

$k_{1}=7.048\times 10^{5}.\;k_{2}=7.534\times 10^{5}\;\mathrm{m^{3}/(mol\times
s).}$

Calculations are carried out for a constant concentration of the oxidizing
agent $C_{x}$, which corresponds to a partial pressure of 2 kPa in the inlet
gas mixture. Using the equation of state for the ideal gas, the following
value of $C_{x}$ concentration was calculated:

$C_{x}=2.830\times 10^{-1}\,\mathrm{mol/m^{3}}.$

Therefore, the frequency of X adsorption per site can be obtained as:

$C_{x}k_{1}=1.995\times 10^{5}\,\mathrm{s^{-1}}.$

The concentration of the reducing agent, $C_{y}$, at a given value of the
parameter $p$ is determined from the equation $p=(C_{y}k_{2})/(C_{x}k_{1})$.

Diffusion of reagents in the gas phase reduces the stiffness of the system of
equations and facilitates the numerical solving. A reasonable change in the
diffusion coefficients does not have a noticeable effect on the solutions of
system (3). Therefore, all calculations were carried out at constant and
identical values of the diffusion coefficients $D_{x}=D_{y}=D=5\times
10^{-5}\,\mathrm{m^{2}/s}$.

While the estimates of the diffusion coefficients in the gas phase of a number
of species are well known, the values of the diffusion coefficients of the
reactants over the catalyst surface differ significantly. Therefore, we assume
$H_{x}=H_{y}=H$, and take $H=D$ as the base value. To demonstrate the
dependence of the solution on $H$, its value will be decreased and increased
by an order of magnitude.

The reaction dynamics in a flow reactor is greatly influenced by the residence
time $t_{L}$. At $t\to 0$ the dynamics behavior of the reaction will approach
the description by the kinetic model, while at a sufficiently large $t_{L}$
the reaction will stop due to a lack of reagents. In the calculations,
$t_{L}=0.01$ s was chosen as the main value, which at $L=0.01$ m corresponds
to the gas flow velocity $u=1$ m/s.

The frequency of oscillations in the kinetic model is determined mainly by the
oxidation-reduction rates of the catalyst. For most of the calculations
associated with regular oscillations, sufficiently high redox rates are chosen
($\kappa_{5}=\kappa_{6}=10^{-2}$), which produce a high frequency of
oscillations and, consequently, reduce the computational time. In our computer
simulation, the most interesting dynamics is observed at
$\kappa_{5}=10^{-5},\,10^{-6}$, which give oscillation frequencies comparable
with the experimental data. Therefore, at the end of the paper, several
results for such values of rates will be presented.

##  5 Numerical method

For the numerical solution of system (3), the method of finite differences is
used. Derivatives with respect to $\zeta$ on the right side of the system are
replaced by finite differences on a uniform grid of $N$ nodes,
$\zeta_{k}=h/2+(k-1)h$, $k=1,2,\dots,N$, where $h=1/N$ is the grid step. The
first order derivative is replaced by the backward difference,
$f^{\prime}_{\zeta}\approx(f_{k}-f_{k-1})/h$ (1st order of accuracy), the
second order derivative is approximated by the central difference,
$f^{\prime\prime}_{\zeta\zeta}=(f_{k-1}-2f_{k}+f_{k+1})/h^{2}$ (2nd order of
accuracy), where $f_{k}=f(\zeta_{k})$.

The entries $f_{0}$ and $f_{N+1}$ are determined using the boundary conditions
(8),(9). From the Danckwerts conditions at $\zeta=0$ one obtains

$u(f_{0}-1)=\frac{D}{Lh}(f_{1}-f_{0}).$

From the Neumann conditions at $\zeta=0,1$ it is followed that

$f_{0}=f_{1},\;f_{N+1}=f_{N}.$

Thus, system (3) transforms into a system of $5N$ ODEs for a $5N$-dimensional
vector of unknowns $q^{N}=\\{v_{k},w_{k},x_{k},y_{k},z_{k},\,k=1,2,\dots,N\\}$

$\dot{q}^{N}(\tau)=F(q^{N}(\tau)).$ (11)

The Jacobi matrix of the system (11) is 13-diagonal, so it is reasonable to
use the technique of sparse matrices to solve the system numerically.
Moreover, it follows from our computational experience that the numerical
solution of the system ceases to noticeably depend on $N$ when $N$ is not less
than 1000. Consequently, all numerical solutions of system (11) presented
below were obtained with $N=1000$.

To solve the system (11), the Matlab ‘ode15s’ solver was used.

##  6 Results and discussion

This section presents the results of numerical simulation of system (11) for
various values of the model parameters in order to demonstrate the influence
of parameters on the dynamics of the model. First of all, the dependence their
solution on the parameter $p$, which was a bifurcation parameter in the
kinetic model, is shown. Then, the role of another external parameter,
$t_{L}$, is explained. From the internal parameters of the model, three
parameters were chosen to demonstrate the influence upon the model behavior,
the surface migration coefficient $H$ and the catalyst oxidation rate
constants $\kappa_{5}$, $\kappa_{6}$. In all cases, the solutions start from
the ”zero” initial condition $q^{N}(0)={\bm{0}}$.

###  6.1 High frequency oscillations

As it was noted above, the frequency of oscillations is determined mainly by
the $\kappa_{5},\,\kappa_{6}$. As a first step, we take the same values of
parameters as in the kinetic model, namely $\kappa_{5}=\kappa_{6}=10^{-2}$. In
this case, the model (11) produces oscillations with (dimensionless) period of
the order of several $\tau_{L}$ that corresponds to the real time oscillation
frequency about $10^{2}$ s-1.

####  6.1.1 Dependence of the solution on the parameter $p$

The dependence of the solution of the flow reactor model on the parameter $p$
qualitatively reproduces the dependence on this parameter of the solution of
the kinetic model. Here, too, for small $p$ (more precisely, for $p<0.446$)
the solution from zero initial data tends to a stationary state. Figure 3
shows the graphs of the stationary solution at $p=0.445$, just before the
onset of oscillations. Curve $z$ on the lower panel of Figure shows that most
of the catalyst plate, with the exception of the left part (front relative to
the gas flow), is completely oxidized. The reaction proceeds only in the
relatively narrow frontal strip of the plate, where all the reducing agent is
consumed.

For some $p$ in the interval (0,445, 0.446), the stationary solution (solution
to system $F(q^{N})=0$) undergoes a Hopf bifurcation, as a result of which it
becomes unstable and a stable limit cycle is born. The unstable steady state
is continued numerically up to $p$ = 0.8723. With a further increase in $p$,
the steady state is rapidly restructured, and the numerical solution of the
equation on the uniform grid with $N=1000$ becomes impossible.

Figure 3: Stationary state of the system (11) at $p=0.445$. Top: relative
concentration of reagents in the gas flow. Bottom: concentrations of surface
species.

The top panel of Figure 4 shows the space-time plot of the degree of oxidation
of the catalyst surface, $z(\zeta,\tau)$, at $p$ = 0.446, that is just after
the onset of the oscillations. At the front of the catalyst plate, rapid,
regular oscillations in the oxidation degree are visible, which gradually
decay as one moves to the right along the gas flow. The lower part of the
Figure shows the range of oxidation degree oscillations depending on the
dimensionless coordinate $\zeta$. Note that the oscillations have a relatively
high frequency (especially in real time), which is associated with a large
value of the oxidation constant $\kappa_{5}$ and the proximity of $p$ to the
Hopf bifurcation point.

Figure 4: Top: the space-time plot of $z(\zeta.\tau)$ at $p$=0.446. Bottom:
the range of $z$ oscillations in dependence on $\zeta$. Figure 5: Same as in
Figure 4 for $p$ = 0.82.

As $p$ increases, the oscillations shift deeper into the plate, their
frequency decreases, and the amplitude increases. Fig. 5 shows the
oscillations in the oxidation degree $z(\zeta,\tau)$ at $p$ = 0.82. In this
case, the front part of the plate is constantly in a reduced state, while the
back part is in an oxidized state.

Figure 6: Same as in Figure 4 for $p$ = 0.981. Figure 7: The concentration of
reagents at the reactor outlet in dependence on time.

With a further increase in the parameter $p$, the region of oscillations moves
to the right (rear) edge of the plate, as shown in Figure 6 for $p$ = 0.981.
In this case, almost the entire surface of the plate is reduced. Thus, the
range of values of the parameter $p$, in which oscillations exist, in the
model of a flow reactor is more than 2 times wider than in the kinetic model.

Figure 7 shows plots of reactant concentrations $v$ and $w$ in the gas phase
at the outlet of the reactor (at $\zeta=1$) as a function of time for the
three values of the parameter $p$ presented in Figures 4-6. When oscillations
of the model variables occur far from the right edge of the plate, they are
hardly noticeable in the gas flow at the outlet of the reactor. In particular,
the graphs do not show oscillations in concentrations at $p$ = 0.446. And at
$p$ = 0.82, despite the large amplitude of oscillations in a wide band in the
central part of the plate, there are only weak fluctuations in the
concentration of the oxidizing agent at the outlet flow.

Figure 8: Stationary state of the system (3) at $p=0.982$. Top: relative
concentration of reagents in the gas flow. Bottom: concentrations of surface
species.

At $p$ = 0.982, there are no more oscillations in the model and the solution
of system (11) tends to the stationary state shown in Figure 8. Now the
surface of the catalyst is completely reduced and covered with x. The reaction
proceeds only at the right (rear) edge of the plate.

The stationary solution at $p$ = 0.982 was continued to the left by parameter
$p$ up to $p$ = 0.9815, and the solution remained stable. In the interval
$0.8723<p<0.9815$, where the stationary solution undergoes radical
reconstruction, using the methods of this work and our computational
capabilities, it seems impossible to obtain a stationary solution of system
(11).

####  6.1.2 The effect of the residence time $t_{L}$

The time $t_{L}=L/u$ during which the gas molecules in the flow pass over the
surface of the catalyst determines the rate of replenishment of the mass of
reagents to participate in the reaction. If this time is less than the
characteristic reaction time, then the reaction proceeds at almost constant
concentrations of reactants in the gas phase over the whole catalyst surface
and the reaction can be described by a kinetic model. An example of
oscillations at small $t_{L}$ is shown in Figure 9.

Figure 9: Oscillations of the oxidation degree $z$ at $p$ = 0.5 and $t_{L}$ =
0.0001s.

Figure 9 represents the coherent oscillations of the oxidation degree (and
other model variables) over the entire surface of the catalyst. Moreover, the
amplitude and frequency of oscillations are close to the amplitude and
frequency of oscillations obtained in the kinetic model.

With an increase in the time of flight $t_{L}$, the coherence of the
oscillations is violated. Different parts of the surface along the gas flow
are in significantly different conditions – at different concentration of
reagents in the gas phase. Alternating “fronts” of oxidation and reduction
appear on the surface, moving in opposite directions. The oxidation front
moves from the rear edge of the plate to the front, and the reduction front
from the front to the back. An example of such fronts is shown in Figure 10.

Figure 10: Moving fronts of oxidation and reduction at $p$ = 0.72, $t_{L}$ =
0.005 s.

With a long time of flight, the flow does not have time to replenish the
consumption of reagents for the reaction, and a non-uniform distribution of
reagent concentrations along the catalyst surface occurs in the reactor.
Increasing the time of flight leads to at least two effects. First, the range
of values of the parameter $p$, in which oscillations exist, increases.
Secondly, the width of the band on the catalyst surface decreases, on which
oscillations in the concentrations of reagents are possible.

Oscillation localization is observed. Note that, by essence, Figures 4-6 show
examples of such localization. As time $t_{L}$ increases, the localization
becomes clearer. Figure 11 shows an example of localization at $t_{L}$ = 0.02
s.

Some details of localization are shown in Figure 12. This Figure shows graphs
of the minimum and maximum values of the relative concentrations of reagents
in the gas phase, depending on the coordinate $\zeta$. On the lower panel,
green lines mark the boundaries $p_{1}$ and $p_{2}$ of the range of $p$
values, in which there are oscillations in the kinetic model.

Figure 11: Localization of surface oscillations at $p$ = 1, $t_{L}$ = 0.02 s.
Figure 12: Localization of surface oscillations at $p$ = 1, $t_{L}$ = 0.02 s.

It can be seen that oscillations on the catalyst surface arise when the graph
of $pw$, which stands in place of $p$ in the kinetic model, crosses the upper
boundary of this interval. The continuation of oscillations beyond the lower
boundary of the interval $(p_{1},p_{2})$ is due to the gas flow. Thus, one can
conclude that oscillations arise where condition $p_{1}<pw(\zeta,\tau)<p_{2}$
is satisfied on a fairly wide band of the catalyst surface.

####  6.1.3 Impact of surface migration

Recall that all previous results were obtained under the assumption that the
surface migration coefficient $H$ is equal to the diffusion coefficient in the
gas phase $D$. Here we give two examples of solving a system (11) at $p=0.82$,
$t_{L}=0.01s$ and with a migration coefficient an order of magnitude greater
and an order of magnitude less than $D$.

Figure 13: Space-time plot of the surface oxidation degree at $H=10D$.

Figure 13 shows the space-time plot of $z(\zeta,\tau)$ at $H=10D$. The
comparison with Figure 5, which presents a similar picture for $H=D$, shows
that with increasing $H$, the oscillation frequency increases and the
oscillation band on the plate surface expands. This conclusion is supported by
Figure 14, which shows a similar plot for $H=0.1D$. It can be seen that as $H$
decreases, the oscillation frequency and the oscillatory domain also decrease.

Figure 14: Space-time plot of the surface oxidation degree at $H=0.1D$. Figure
15: Space-time plot of the surface oxidation degree at $H=0.1D$ and classical
Fick diffusion.

In this paper, the surface migration is described by a nonlinear equation (6).
If one removes the nonlinear terms from this equation, then the classical Fick
diffusion remains. For comparison, Figure 15 shows the space-time diagram for
$z$ calculated with the classic diffusion. The pictures are very similar, but
compared to the nonlinear migration, the frequency of oscillations here
somewhat decreases and, on the contrary, the oscillation band on the catalyst
surface expands. From a computational point of view, the scheme with the
nonlinear migration is somewhat softer than the scheme with the classic
diffusion – in the first case, the computer time for solving the problem is
somewhat less than in the second case.

###  6.2 Moving fronts and reaction rate distribution

On the space-time sweeps of the oscillations shown above, the moving fronts of
the oxidation and reduction of the catalyst are clearly visible. The oxidation
front moves against the gas flow, reaches the edge of the plate or some point
inside the plate, and stops. Then, from the place where the oxidation front
stops, the reduction front begins to move, which moves in the opposite
direction – along the flow. Note that the surface reaction x + y $\to$ XY
occurs mainly at the oxidation front, as shown in Figure 16.

Figure 16: Top: space-time plot of the reaction rate $R_{4}=\kappa_{4}xy$.
Bottom: the oxidation degree $z$ and reaction rate $R_{4}$ at the point in
time marked with the horizontal yellow line in the top figure.
$p=0.82,H=0.1D$.

Figure 16 shows the space-time plot of the reaction rate $R_{4}$ for the
solution of the system shown in Figure 14, which of all the shown solutions
represents the reaction rate distribution most expressively. The oxidation
front advances on the reduced surface. The reduced surface is covered with
adsorbed molecules of the reducing agent y, there is very little oxidizing
agent x on it, and the x + y reaction practically does not occur. At the
oxidation front, the concentration y decreases due to the reduction reaction y
+ Mx. The possibility opens up for the adsorption of the oxidizing agent X
and, consequently, the reaction x + y. With advancement into the oxidized
surface behind the oxidation front, the reaction decays due to a decrease in
the concentrations of y on the surface and Y in the gas phase.

###  6.3 Low frequency oscillations

In this section, we shall take $\kappa_{5}=\kappa_{6}=10^{-6}$, so the
oscillation frequency decreases by 4 orders of magnitude compared to the
previous section and becomes about $10^{-2}$ s-1. At the same time, the
bifurcation diagram of the kinetic model with such constant values differs
little from Figure 2. In particular, now the limit cycle is observed on the
interval 0.339 $<p<$ 0.532, while in the previous case it was 0.347 $<p<$
0.532.

Figure 17: Top: space-time plot of $z$ oscillations. Bottom: time series of
$v(1,\tau)$ – blue line, $w(1,\tau)$ – magenta line, at
$\kappa_{5}=\kappa_{6}=10^{-6}$, $p$ = 0.7. Figure 18: The same as in Fig. 17
for $p$ = 0.8. Figure 19: The same as in Fig. 17 for p = 0.827. Figure 20: The
same as in Fig. 17 for p = 0.829. Figure 21: The same as in Fig. 17 for p =
0.84.

The high-frequency oscillations for the considered values of the parameters
have the form of (simple, one-turn) regular oscillations. The form of low-
frequency oscillations changes with a change in the value of the parameters,
in particular, the parameter $p$. Oscillations can become more complex, and
even changes are observed that resemble Feigenbaum scenario of transition to
deterministic chaos. An example of such complication is shown in Figures
17-21. These figures, like the previous ones, consist of two parts. The upper
part shows the space-time plot of the degree of oxidation of the catalyst
surface $z(\zeta,\tau)$, and the lower part shows the time dependence of the
relative concentration of reactants in the gas phase at the outlet of the
reactor.

As in the high frequency case, low frequency oscillations arise at a certain
value of $p$ at the frontal (left) edge of the plate and, with increasing $p$,
quickly propagate to the entire surface of the plate. The space-time plot in
Figure 17 ($p$ = 0.7) shows alternating moving fronts of oxidation and
reduction. Moreover, both fronts propagate from right to left, from the rear
edge of the plate to the front.

As the parameter $p$ approaches the right edge of the interval of existence of
the oscillatory solution in the flow reactor model, the oscillations quickly
become more complex from the ”simple” oscillations shown in Figure 17 to the
irregular oscillations shown in Figure 21. Due to limited computational
capabilities, we cannot present a complete picture of how the solution changes
with increasing parameter $p$, so we present several examples of solutions.
First, a “period-doubling” occurs – each second oxidation front does not reach
the left edge of the plate. An example of a ”doubled period” oscillation is
shown in Figure 18 ($p$ = 0.8). Then the period doubles again, – Figure 19
($p$ = 0.827). The next example, Figure 20 ($p$ = 0.829), is not part of the
period-doubling sequence, but shows a ”7-cycle”. Note, however, that the
resemblance to the Feigenbaum scenario here is purely superficial. The process
of complication of oscillations and the final irregular oscillations require a
detailed analysis, which is beyond the scope of this work.

In this work, we limited ourselves to the assumption that the rate of
oxidation of the catalyst $\kappa_{5}$ is equal to the rate of its reduction
$\kappa_{6}$. However, the relation $\kappa_{5}/\kappa_{6}$ plays a
significant role in the dynamics of the model, which we do not discuss here.
The only example shown in Figure 22 demonstrates the appearance of complex
mixed mode oscillations for different values of oxidation and reduction rates,
namely for $\kappa_{5}=10^{-6}$, $\kappa_{6}=10^{-7}$, at p = 0.82.

Figure 22: The same as in Fig. 17 for p = 0.82, $r_{5}=10^{-6}$,
$r_{6}=10^{-7}$ .

##  7 Concluding remarks

The results obtained in this work show that complex regimes can be realized in
a flow reactor for the reaction mechanism described by the simplest kinetic
model. It was clearly demonstrated that the region of oscillations in the
parameter space of the distributed model of the flow reactor is much larger
than in the point kinetic model. The spatial-temporal behavior of the
distributed system (3) is highly dependent upon the frequency of oscillations
in the point kinetic model.

High-frequency oscillations with a period of not more than several times of
flight $t_{L}$ of the gas flow over the catalyst surface have a regular shape
over the entire range of values of the parameter $p$, on which oscillations
exist. For small values of $p$, the solutions of the system tend over time to
a stationary state in which the catalyst surface is oxidized. The reaction
proceeds only at the leading edge of the catalyst plate. As $p$ increases,
oscillations arise when the inequality $pw(\zeta,\tau)>p_{1}$ is satisfied in
a sufficiently wide band of $\zeta$ near the left edge of the plate
($\zeta=0$). In this case, the Hopf bifurcation occurs in the discrete system
(11). The bifurcation occurs at a larger value of $p$, then in the kinetic
model, because due to the Dankwerts condition, $w(0,\tau)<1$. At further
increase in $p$, the oscillation band on the plate expands in the direction of
the flow and shifts towards the back edge of the plate. The width of the
oscillation band depends on the time of flight $t_{L}$ (or on the flow
velocity $u$), – the larger $t_{L}$, the narrower the band.

The oscillations arising in the band of fulfillment of inequality
$p_{1}<pw(\zeta,\tau)<p_{2}$ are transferred by the gas flow to a wider band
and even to the entire plate. Therefore, the observed region of oscillations
on the plate is much wider than the band on which they originate.

Low-frequency oscillations with a period of several thousand times of flight
$t_{L}$ demonstrate, in contrast to high-frequency ones, diverse dynamics. In
particular, as the parameter $p$ approaches the right end of the oscillation
interval, oscillations become more complicated. In this work, the complication
of oscillations is demonstrated by the example of the concentration of
reagents at the reactor outlet, which can be measured experimentally. As $p$
increases, the oscillations take the form of mixed mode oscillations with a
gradual increase in the number of maxima in the period. When $p$ is close to
the boundary of the oscillatory interval, the oscillations become aperiodic.

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# CliniDigest: A Case Study in Large Language Model Based Large-Scale
Summarization of Clinical Trial Descriptions

Renee D. White<EMAIL_ADDRESS>0009-0009-1225-1580 Stanford
University450 Serra MallStanfordCaliforniaUSA94305 ,  Tristan Peng
<EMAIL_ADDRESS>0000-0001-7907-574X Stanford University450 Serra
MallStanfordCaliforniaUSA94305 ,  Pann Sripitak<EMAIL_ADDRESS>0009-0003-1352-0494 Stanford University450 Serra
MallStanfordCaliforniaUSA94305 ,  Alexander Rosenberg Johansen
<EMAIL_ADDRESS>0000-0002-4993-7916 Stanford University450 Serra
MallStanfordCaliforniaUSA94305 and  Michael Snyder<EMAIL_ADDRESS>0000-0003-0784-7987 Stanford University450 Serra
MallStanfordCaliforniaUSA94305

(2023; 1 June 2023)

###### Abstract.

A clinical trial is a study that evaluates new biomedical interventions. To
design new trials, researchers draw inspiration from those current and
completed. In 2022, there were on average more than 100 clinical trials
submitted to ClinicalTrials.gov every day, with each trial having a mean of
approximately 1500 words (cli, [n. d.]). This makes it nearly impossible to
keep up to date. To mitigate this issue, we have created a batch clinical
trial summarizer called CliniDigest using GPT-3.5. CliniDigest is, to our
knowledge, the first tool able to provide real-time, truthful, and
comprehensive summaries of clinical trials. CliniDigest can reduce up to 85
clinical trial descriptions (approximately 10,500 words) into a concise
200-word summary with references and limited hallucinations. We have tested
CliniDigest on its ability to summarize 457 trials divided across 27 medical
subdomains. For each field, CliniDigest generates summaries of $\mu=153,\
\sigma=69$ words, each of which utilizes $\mu=54\%,\ \sigma=30\%$ of the
sources. A more comprehensive evaluation is planned and outlined in this
paper.

Summarization, Clinical Trials, Prompt Engineering, Wearables

††journalyear: 2023††copyright: rightsretained††conference: ACM International
Conference on Information Technology for Social Good; September 6–8, 2023;
Lisbon, Portugal††booktitle: ACM International Conference on Information
Technology for Social Good (GoodIT ’23), September 6–8, 2023, Lisbon,
Portugal††doi: 10.1145/3582515.3609559††isbn: 979-8-4007-0116-0/23/09††ccs:
Applied computing Health care information systems††ccs: Applied computing
Document searching††ccs: Human-centered computing Accessibility systems and
tools

Figure 1. CliniDigest integrated into Wearipedia. The text in the orange box
on the left of the page is automatically synthesized by CliniDigest using data
from ClinicalTrials.gov. For this particular example, the text summarizes the
Fitbit Sense wearable’s use in clinical research in bariatrics.

Wearipedia home page.

##  1\. Introduction

Clinical trials are the de facto method for developing new treatments and
innovations within most medical fields. On average, clinical trials take 21
months from being posted on MEDLINE and ClinicalTrials.gov to publication
(Ross et al., 2013). Moreover, only about 46% of completed studies publish
articles. Reasons for not publishing vary from publication bias, commercial
interest, to lack of significance (Ross et al., 2009). However,
ClinicalTrials.gov maintains a list of all trials published, unpublished, and
in progress; clinical research coordinators have to register their studies
with ClinicalTrials.gov when they start recruiting participants and keep the
entry updated until the trial’s completion or withdrawal. This makes
ClinicalTrials.gov a valuable resource to get up-to-date information on
current clinical trials. However, the increase in clinical trial submissions
creates a need for search tools to compile the vast amount of information. We
present CliniDigest, a prompt engineering technique for GPT-3.5 that condenses
up to 85 clinical trials into a 200-word summary with references. The 457
clinical trials used as our testing dataset for CliniDigest are scraped from
ClinicalTrials.gov and are annotated into 14 medical fields and two
subdomains: completed and new. Tables 1 and 2 show example prompts and figure
1 and table 3 show summaries of CliniDigest. Since GPT-3.5 is limited to 4,096
tokens, we use multiple levels of summarization and prompting to get truthful
and concise summaries, explained in more detail in section 3. Upon completion,
CliniDigest, the subject of this paper, will be available on Wearipedia
(https://www.wearipedia.com), which is an open-sourced, encyclopedic directory
of biomedical fitness sensors and trials using them. Currently, Wearipedia
only includes a set of links to raw clinical trials scraped for each device
using an advanced text-search algorithm (Marra et al., 2020).

Beyond providing in-depth information about wearables, Wearipedia also
provides a Python package for a uniformed data extraction process across all
supported wearables.111Code for the library can be found here:
https://github.com/Stanford-Health/wearipedia. Accompanying the Python
libraries are premade Jupyter Notebooks that fully demonstrate the
capabilities of Wearipedia’s data extraction pipeline. Additionally, they
provide an explanatory boilerplate for users more unfamiliar with programming
or those who want a quick start to easily extract data from their wearables.
Given the pedigree of Wearipedia, accessibility and comprehensiveness of
wearable data and information are the core tenets of the project. Thus, it is
crucial that the information that Wearipedia presents on each wearable’s
device page is as accurate, complete, and digestible as possible. The tensions
between Wearipedia’s goals as a whole with the enormity of data and amount of
wearables suggest that the conventional solution of manually crafting thorough
abstracts for the purpose of Wearipedia’s goals does not cater well to the
overall scale and progression of the project, motivating the development of
CliniDigest. CliniDigest will be used by the clinical team, which often
includes physicians, researchers, clinical research coordinators worldwide.
Since these team members come from a wide spectrum of academic backgrounds and
need the information in a timely manner, we have the following goals:
accessibility, efficiency, accuracy, comprehensiveness, and timeliness.

First, the summaries should be accessible by any member of the clinical trial
community, which covers a diverse group of individuals. Accessibility is
determined by readability and empirical evaluation by clinical team members.
Second, as a part of Wearipedia, CliniDigest should expedite the process of
choosing the right wearable for a clinical question. Furthermore, a summary
should not misrepresent clinical trials and although a summary may not be able
to include all trials, it should cover as broadly as possible. Moreover, as a
major benefit of ClinicalTrials.gov is access to trials before official
publication, CliniDigest should reflect updates and changes when they happen.

CliniDigest represents an example of integrating GPT-3.5 into a text-centric
database. It is able to solve text summarization in a fashion that GPT-3.5,
information retrieval, or text summarization algorithm wouldn’t be able to do
on their own.

We have done preliminary evaluations on the length, reference usage, and
readability of the summaries. We have also planned an extensive evaluation
that can empirically test the efficacy of CliniDigest’s large-scale
compression. This would test the ability of clinical research coordinators to
understand the breadth and depth of their investigative field. A successful
evaluation would entail the integration of CliniDigest into the pipeline that
transforms raw, web-scraped clinical trial data from various sources into a
refined summary displayed in an intuitive user interface on Wearipedia’s
website.

Figure 2. Batch summarization method: we circumvent the token limit through a
series of cascading inputs through GPT-3.5. First, we split the raw data into
batches of 15 studies each. Then, we prepend the intermediate prompt shown in
table 1 as a prompt to GPT-3.5. Thus, each batch will produce one summary.
After each batch has been processed, we combine the intermediary summaries
through the prompt shown in 2 to further condense the summary into a concise
description that encompasses the extent of our data.

Batch Summarization Diagram

##  2\. Related Work

Text summarization of medical literature is an active field of research. Due
to the exponential rise in registered clinical trials, research has shifted
from single-document summarization to multi-document summarization, with
techniques trending to those incorporating machine learning (See et al., 2017;
Mishra et al., 2014). There are four main types of summarization, grouped on
two axes: extractive (summarization through extracting source material) versus
abstractive (summarization through paraphrasing the main idea), and indicative
(providing a general overview of the topic) versus informative (providing in-
depth information) (Mishra et al., 2014). For CliniDigest, we strive to create
indicative abstractive summaries, differing from the extractive paradigm that
has already shown to be effective in summarization of clinical trials (Gulden
et al., 2019).

Recent advances in summarization have been led by the emergence of general
purpose large language models (Kieuvongngam et al., 2020; Afantenos et al.,
2005; Mishra et al., 2014). Through prompt engineering, the style of
summarization can be adapted to many types of writing and summary styles.
Prompt engineering allows users to efficaciously converse with large language
models: through meticulous design of these instructions, users can ensure the
quality and form of a response. This rose out of recent developments in
natural language processing that found it more cost-effective and efficient to
train generalizable prompt-based models based on evaluating text probability,
one that GPT-3.5 falls under (Liu et al., 2023b; Lester et al., 2021; Brown et
al., 2020). Although the field is still relatively new, much scholarly work
has already been done in the standardization and categorization of effective
prompt engineering paradigms (White et al., 2023). Other scholarly endeavors
have examined the ability of large language models to enhance efficiency and
performance in various information-transfer based applications, such as
content creation via refining social media and crowdsourcing pitches,
assisting with the writing of important sections in radiology reports,
generation of high-quality and empathetic answers to patient questions, and
low variance summarization and zero-shot de-identification of patient records
(Short and Short, 2023; Ma et al., 2023; Ayers et al., 2023; Wang et al.,
2023; Chuang et al., 2023; Liu et al., 2023a).

Despite the promises of using prompt engineering, there are many pitfalls to
watch out for. Coined by research in neural machine translation, but similarly
applied to prompt-based large language models, the GPT-4 technical paper
describes hallucinations as the tendency to “ ‘produce content that is
nonsensical or untruthful in relation to certain sources’ ” (Agarwal et al.,
2018; OpenAI, 2023). This type of output is categorically antithetical to the
goals of the project. Although formally defined in the GPT-4 technical paper,
GPT-3 (and subsequently GPT-3.5) also faces similar limitations hallucinating
in both open- and closed-domain contexts by providing informative yet
misleading responses (Maynez et al., 2020; OpenAI, 2023; Lin et al., 2022).
Nonetheless, GPT-3 has shown strong performance in $n$-shot prompting tasks,
making it a strong suitor, and the model of choice, for this project (Brown et
al., 2020). Beyond hallucinations, it is also a uniquely challenging problem
for researchers to design satisfactory human-AI interactions due to two main
factors: uncharted capabilities of AI and its output complexity (Yang et al.,
2020).

For medical literacy, the SMOG test has shown to be superior on consistency,
accuracy, and speed for medical literature in comparison with other standard
metrics such as Flesch-Kincaid, FOG, Flesch Reading Ease, Fry, and Dale-Chall
(Mc Laughlin, 1969; Wang et al., 2013). In the larger planned evaluation, we
also plan to use the ROUGE-L F1 metric to evaluate the quality of the
synthesized summaries (Lin, 2004).

Prompt \- single summary
---
Your task is to extract relevant information from 15 trials delimited in the
triple backticks labeled from 1 to 15 to construct an argument about the
purpose of Fitbit in general physiology trials. Each trial contains a title
and description. Your reader will be clinical research coordinators.

Write a 200 word thesis with references to the trials in the following format:
[1].

Trials: ```[Redacted]```

Table 1. Example prompt for a single summary for the uses of Fitbits in
general physiology.

Prompt - single summary: Your task is to extract relevant information from 15
trials delimited in the triple backticks labeled from 1 to 15 to construct an
argument about the purpose of Fitbit in general physiology trials. Each trial
contains a title and description. Your reader will be clinical research
coordinators. Write a 200 word thesis with references to the trials in the
following format: [1]. Trials: ```[Redacted]```

Prompt \- multi summary
---
Your task is to extract relevant information from the provided text and
references to construct a cumulative argument about the purpose of Fitbit in
general physiology trials. Each paragraph includes references to clinical
trials in the following format: [1]. Weigh each paragraph according to its
word count, weighing longer paragraphs more than shorter ones. Your reader
will be clinical research coordinators.
Summary: ```[Redacted] ```
References: ```[Redacted] ```
Write a 150-250-word thesis with references to the trials in the following
format: [1].

Table 2. Example concatenated prompt for the uses of Fitbits in general
physiology.

Prompt \- multi summary: Your task is to extract relevant information from the
provided text and references to construct a cumulative argument about the
purpose of Fitbit in general physiology trials. Each paragraph includes
references to clinical trials in the following format: [1]. Weigh each
paragraph according to its word count, weighing longer paragraphs more than
shorter ones. Your reader will be clinical research coordinators. Summary:
```[Redacted] ```
References: ```[Redacted] ```
Write a 150-250-word thesis with references to the trials in the following
format: [1].

##  3\. Methodology

In order to get the summaries we had a multi-step pipeline. First, we
extracted all studies related to Fitbit from ClinicalTrials.gov (Marra et al.,
2020). Of these, we reviewed and annotated 457 trials into its primary medical
field, of which we include: somnology, gynecology, obstetrics, cardiology,
general physiology, endocrinology, bariatrics, psychiatry, oncology,
gastroenterology, pulmonology, chronic pain/diseases, nephrology, and other.
Moreover, we separated the studies into completed (within the past 5 years)
and new (within the past two years). We also excluded trials that were
withdrawn and trials with enrollment numbers less than 50. This reduction led
us to between 1-84 clinical studies for 27 medical field combinations.
Nephrology, as the only one, did not have any studies for completed trials.
This fine-grained sorting of clinical trials was then fed into GPT-3.5 to make
200-word summaries.

Since GPT-3.5 Turbo has a maximum token limit of 4,096, even just clinical
trials associated with one wearable in one medical field exceeds this limit.
For example, a hand-picked subset of 25 oncology trials using the Fitbit
physical fitness monitor well exceeds the restrictive 4,096 token limit;
aggregating them into a compact list of just trial names and brief
descriptions, the example subset works out to be nearly 4,900 tokens. Thus,
one of the biggest obstacles, and biggest design challenge we faced, was to
design a way to circumvent this limit through clever prompting that would
yield a similar result without pushing against the limit. To this end, we
split the studies into batches of 15 clinical trials, which were recursively
summarized until it was condensed into one succinct summary. In the case that
a batch had less than 15 clinical trials, we adjusted the prompt’s word count
to be proportional to the number of clinical trials inputted (13 words per
trial). Figure 2 shows an example workflow for a list of trials subdivided
into four batches, which are iterated upon with GPT-3.5 until it reaches one
condensed summary.

For the prompt design, we first prioritized a design that promotes
readability. To do so, we constructed our prompt in a way that instructs
GPT-3.5 Turbo to use language accessible to clinical research coordinators.
The inputted trials, as referred to as Trials: ```[Redacted]``` in table 1,
each consists of the clinical trial’s title and its brief description listed
on ClinicalTrials.gov. As the brief descriptions usually contain predominantly
medical jargon, summarization without this language simplification
specification is not sufficient for our goal of making clinical trial data
accessible to clinical research coordinators.

Second, we wanted to develop a prompt that is able to synthesize broad
concepts relating to all trials. This marked a major challenge because
lackadaisical prompts often caused GPT-3.5 to simply summarize every study
individually. To mitigate this, we experimented with a variety of prompts that
prioritized generalization through keywords such as “summary,” “essay,” and
“argument.” We found that “thesis” and ”argument’ work the best, which can be
seen in table 1.

Another design challenge we needed to address was circumventing a crucial
limitation of such models: hallucinations. As discussed in section 2,
hallucinations are the bane to effective and accurate summarization of
clinical trials, as it would not only spread misinformation, but it would also
be difficult to comprehensively fact-check every source for accuracy. To
address this, we prompted GPT-3.5 to use references whenever possible, as can
be seen in tables 1 and 2. We found that tuning the temperature, which
controls creativity, to 0 and prompt engineering for explainability reduced
hallucinations. While not a perfect solution, it still aims to reduce the
number of hallucinations that occur in these closed-context circumstances
without large modifications to the overall model. Although large language
models are not designed to be factual, by asking for references, GPT is
required to utilize the inputted clinical trials.

##  4\. Results

In table 3, we see an example output from GPT-3.5 that summarizes 39 completed
clinical trials focusing on general physiology. The output includes 11
references in 201 words. An informal analysis shows that this output is
promising, as it specifically references general physiology trials,
generalizes to include multiple trials, and provides useful information for a
clinical research coordinator. Moreover, it includes in-text references with
relatively uniformly spread citations in addition to explaining key
characteristics of specific trials without naming them (which would take up
words in our limited 150-250 output and is unnecessary because of the
inclusion of references).

Result
---
Fitbit has emerged as a valuable tool in General Physiology trials aimed at
promoting physical activity and improving health outcomes. Fitbit is a
wearable activity tracker that allows researchers to monitor physical activity
levels and track progress towards set goals. It has been used in a variety of
trials, including those focused on cancer survivors [1], adolescents [6], and
older adults [10]. Fitbit has also been utilized in trials that aim to promote
physical activity in specific populations, such as Veterans with lower-limb
amputation [5] and South Asian immigrant women and girls [14]. In addition to
promoting physical activity, Fitbit has been used in trials focused on
improving muscle function [8], treating depressive symptoms [2], and managing
chronic pain [15]. Fitbit has been used in conjunction with other
interventions, such as resistance training and virtual reality therapy, to
improve health outcomes. Furthermore, Fitbit has been utilized to support
lifestyle physical activity interventions and improve health outcomes in
trials [31], [32], and [33]. These trials demonstrate the potential of Fitbit
to support physical activity interventions and improve health outcomes.
Overall, Fitbit has become a valuable tool in General Physiology trials aimed
at promoting sustained exercise participation and improving physical and
psychosocial well-being.

Table 3. Final results: this is a summary on Fitbit’s use in general
physiology trials.

Result: Fitbit has emerged as a valuable tool in General Physiology trials
aimed at promoting physical activity and improving health outcomes. Fitbit is
a wearable activity tracker that allows researchers to monitor physical
activity levels and track progress towards set goals. It has been used in a
variety of trials, including those focused on cancer survivors [1],
adolescents [6], and older adults [10]. Fitbit has also been utilized in
trials that aim to promote physical activity in specific populations, such as
Veterans with lower-limb amputation [5] and South Asian immigrant women and
girls [14]. In addition to promoting physical activity, Fitbit has been used
in trials focused on improving muscle function [8], treating depressive
symptoms [2], and managing chronic pain [15]. Fitbit has been used in
conjunction with other interventions, such as resistance training and virtual
reality therapy, to improve health outcomes. Furthermore, Fitbit has been
utilized to support lifestyle physical activity interventions and improve
health outcomes in trials [31], [32], and [33]. These trials demonstrate the
potential of Fitbit to support physical activity interventions and improve
health outcomes. Overall, Fitbit has become a valuable tool in General
Physiology trials aimed at promoting sustained exercise participation and
improving physical and psychosocial well-being.

Across all 14 fields, CliniDigest generates summaries with $\mu=153,\
\sigma=69$ length and each summary utilizes $\mu=54\%,\ \sigma=30\%$ of the
sources. In the 17 summaries that required the second combination step, the
final summaries have $\mu=192,\ \sigma=27$ words with 100% of them following
between the requested 150-250 word range. The distribution can be seen in
figure 3.

Figure 3. Scatter plot of the number of references included in the final
output summaries compared to the number of inputted trial descriptions. There
is a general linear trend up to 50 trials with slope $m=0.5140$ and
correlation coefficient $r^{2}=0.7547$. After the inflection point of around
50 trials, there are diminishing returns to the number of sources that
CliniDigest includes in its summaries due to the inherent difficulty of
compressing a large number of trials within a small word limit.

Plot of Reference Inclusion

###  4.1. Readability

In order to ensure effective communication, health information materials need
to be written in an understandable manner. To evaluate the readability of the
GPT-3.5 outputs, we utilized the SMOG readability formula (Mc Laughlin, 1969;
Wang et al., 2013). As a quantitative test of accessibility, we conducted a
two-sample T-test of means to compare two corpora: the original study titles
and descriptions, and the summaries synthesized by CliniDigest. Using the SMOG
formula, we calculated $\bar{x}_{1}=19.32$ and $s_{1}=1.220$ for raw data from
ClinicalTrials.gov, and $\bar{x}_{2}=18.49$ and $s_{2}=2.148$ for CliniDigest
summaries. In all, $p=0.0929$ so we find that CliniDigest summaries are not
significantly different in readability from the original trial descriptions.

This result still demonstrates some important points about CliniDigest’s
summaries. First, it shows that CliniDigest’s summaries maintain the level of
diction as its constituent raw descriptions: despite the condensation of
upwards of 80 studies into a neat 200-word output, CliniDigest did not
saturate its summaries with polysyllabic medical jargon to help compress
information. Since researchers are expected to be able to read at this level,
CliniDigest, according to the SMOG metrics, should be something they would be
familiar with reading about.

While the SMOG measure provides quantitative evidence of CliniDigest’s
efficacy, readability formulae are not authoritative, but rather act as
guidelines that serve as an addendum to an actual evaluation such as one
planned in section 4.2 (Meade and Smith, 1991). As for SMOG specifically,
there are a variety of factors of CliniDigest that may inadvertently raise its
SMOG score, causing an artificially inflated score. First, the conciseness of
CliniDigest’s summaries increases the density of polysyllabic words per
sentence, the crucial metric that SMOG operates on for readability. Here we
define polysyllabic as words with three or more syllables for concision and
conformity of SMOG’s definitions. This is compounded by the fact that with
regards to clinical trials involving wearables, there are an inescapable
amount of polysyllabic words that should be incorporated. Since many of these
polysyllabic words are at the core of a clinical trial’s purpose and an
integral part of a clinical researcher’s lexicon, it would be unreasonable to
expect that these words be substituted with other monosyllabically-composed
phrases. Thus, these two important factors—compression and the inclusion of
critical polysyllabic words and phrases—combine to contribute to a higher SMOG
score.

###  4.2. Systematic Evaluation Plan

We expect that the majority of Wearipedia’s users are researchers, clinicians,
clinical research coordinators, or other health professionals that seek to use
wearables as a part of studies, trials, or patient monitoring. Thus, our
evaluation will be focused on catering towards this population. First, we plan
to recruit researchers who are conducting or planning to conduct research
involving wearables according to best practices (Caine, 2016). This could be
easily achieved through Wearipedia’s current userbase, but we believe that it
is also important to include in our participants people that are not familiar
with Wearipedia and its facilities. After the recruiting process, we will
survey the participants on their experience as clinical research coordinators.
Among the various questions will include their highest academic degree, time
working as a clinical research coordinator, as well as experience with using
wearables both in daily life and in clinical trials.

Second, each participant will be asked to learn about two distinct wearables
that they are not as familiar with, in two different medical fields that they
are not familiar with. We will label each wearable-field pair as a test for
the clarity of this paper. By working outside their area of expertise, this
will eliminate confounding variables corresponding to prior knowledge as well
as provide a lower bound for the efficacy of the paradigm in concisely
educating the user about the status quo of various wearables in different
areas of study. Among these two tests, we plan to randomly assign one as a
control, and another as an experimental test. The control test will be the
second method of presentation as introduced in section 1: links on the sidebar
of the device page that collect the studies in a list. On the other hand, the
experimental test is solely the generated output through GPT-3.5 Turbo without
any modifications or external links that are provided, except for the list of
references that is provided at the end of each output containing the name of
each trial summarized. The participants will then be asked to read both tests
one at a time in a random permutation. This is done in order to eliminate any
confounding variables regarding the order that the control and experiment are
presented in. After the participants feel like they understand the information
given to them in each test, they will then be asked to write a brief summary
of what they have learned. We hope that through this writing exercise,
participants can reflect and demonstrate their understanding of the material
in both breadth and depth. This portion will be timed, but participants are
encouraged to use as much time as they feel they need to write a tenable
summary. Furthermore, we will use these participant-written summaries as part
of a more quantitative measure by using them as part of computing the standard
ROUGE metric (specifically the ROUGE-L F1 score) that compares human-written
responses with those synthesized by CliniDigest (Lin, 2004; Gulden et al.,
2019). By using participants’ responses, this will eliminate confounding
variables that arise from in-team human-written summaries that conform with
summaries written by CliniDigest that artificially increase the ROUGE score.
After the participants finish both tests, the evaluation concludes. At this
point, we plan to conduct a post-evaluation questionnaire as a debrief that
targets the more qualitative experience of doing both tests. During this
phase, we plan to ask about the accessibility of both methods beyond the
metrics used above in section 4.1 as well as the physical ease of access of
using both, the perceived challenges and advantages of using each method in
terms of breadth and depth, as well as enjoyment of using each. This
qualitative data, along with the time spent on each test and the manual
assessment of the participants’ responses to the evaluation prompt, we hope
will give us a comprehensive picture of the true value of the summaries in
advancing the goals of accessibility and efficiency.

##  5\. Future Work and Conclusion

While this paper has shown the potential for real-time synthesis of clinical
trials in a digestible and accurate medium, it must be corroborated by
empirical findings through a rigorous evaluation. Section 4.2 outlines the
evaluation plan for Wearipedia’s trial summarizer - CliniDigest. Beyond a
comprehensive evaluation, we will continue to monitor GPT-3.5 Turbo’s outputs
for accuracy and concision with all wearables and medical fields as well as
refine the prompts for better summaries.

Another direction we are looking to explore regarding the usage of GPT’s text
completion paradigm rather than that of text summarization. As speculation,
there could be multiple benefits to such an approach. First, a human-dictated
structure could be better at prompting GPT in its summary. Moreover, this more
controlled environment would produce less variability between summaries of
different wearables and fields. This homogeneity would increase the overall
cohesion of the project, as users would be able to develop a clearer mental
model of the organizational structure of the abstracts.

Finally, while it is essential to strike a balance between readability and
preserving the accuracy of the final results, efforts should be directed
towards finding alternative ways to improve the overall readability without
compromising the validity of the content.

In this paper, we have presented a case study in large-scale accurate and
comprehensive summarization of clinical trials through the development of
CliniDigest. From the initial promising results, we will continue to develop
this project from a case study to actual implementation as a crucial
centerpiece of Wearipedia. With the rapid burgeoning usage of wearables in
clinical trials as well as the exponentially increasing trend of the number of
clinical trials being proposed and conducted, these large language models
prove to be an effective and timely tool for presenting the vast amount of
knowledge in a dense and easily-readable format. As a crucial component of the
Wearipedia project, these summaries allow Wearipedia to progress in line with
its goals of accessibility and comprehensiveness, and gives users more context
to work with when using Wearipedia to evaluate and test wearable devices.

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