Introduction
Recent statistics of the American Society of Nephrology confirm
that over 17 million Americans suffer from Chronic Kidney Disease
and approximately 458.000 Americans suffer from End-Stage renal
Disease (ESRD), 400.000 of whom are either on dialysis or require
kidney transplant. This situation corresponds to a total cost
of 5.8% of the medical care system budget. This results in an
approximate cost of $13.82 billion annually to support and treat
ESRD patients. Moreover, these figures will increase of 165% by
2050. Since two decades, peritoneal dialysis (PD) under its continuous
ambulatory form (CAPD) occupies a well established place among
the therapeutic options for patients with ESRD (1, 2). Blood purification
is obtained by exchanges of chemicals between blood and a solution
injected in the peritoneal cavity. The solution in the peritoneal
cavity is periodically replaced by injections or extractions from
the patient, through an external pump. The exchange of chemicals
takes place across the net of capillaries permeating the peritoneum
(see figure 1).
 |
Fig. 1
• Descrizione sinottica di dialisi peritoneale automatizzata
(APD).
• A synoptic description of Automated Peritoneal
Dialysis. |
However, more recently a stabilisation or even a decline in the
use of this therapy has been observed in several countries (3).
Among the different factors that may explain this trend, a high
drop-out rate is persistently observed. Nowadays most technique
failures are no longer due to recurrent peritonitis episodes but
to alterations in the peritoneal membrane transport characteristics
leading to inefficient small solute and/or water removal (3).
Automated peritoneal dialysis (APD) represents an alternative
that appears not only more convenient to most patients but also
allows to customize the therapy by making it more efficient and/or
more biocompatible. So far, due to the complexity of its prescriptions
and set-up, APD has not gained wide acceptance among the dialysis
personnel. To improve the acceptance of APD and PD in general,
we have developed a mathematical model that allows the optimisation
of PD therapies and facilitate the use of more elaborated therapeutic
options. We report here the different mathematical steps that
have led to this development. Our goal is to develop a procedure
able to find for each patient the dynamics of injections and extractions
patterns that ensure the best blood purification and water removal.
To pursue this aim, the following tasks were accomplished: the
set up of a mathematical model for the exchanges of chemicals
during PD, the validation of this model with respect to measurements
collected from patients and finally the study of a methodology
to improve the efficie of dialysis. Several publications describe
the considered exchanges with mathematical equations, (4, 5).
All the models that have been proposed are basically similar,
indeed they have been derived from equations describing exchanges
of many chemical species across a membrane separating two solutions
with different concentrations. In this work we start from a similar
approach, however we move forward by proposing equations capable
of accounting for a more accurate and realistic description of
the reality. This new mathematical framework makes our model adaptable
and accurate to all the patient’s categories, from low to
high transporters. Moreover, we propose a general algorithm that
is capable of providing an accurate numerical approximation irrespectevely
of how complex the above mathematical model is. The development
of this simulation environment (discussed in detail in (9) and
summarized in figure 2) is the result of a fructful multi-disciplinary
collaboration among a med-tech commercial partner (Debiotech s.a.,
Lausanne) and clinical partners (Inselspital, Bern, Gent University
Hospital and Ospedale le Molinette, Torino) aiming to set up the
concept, the hardware and the software for a new and more efficient
treatment in peritoneal dialysis.
 |
Fig. 2
• L’impostazione dell’ambiente di simulazione
della dialisi peritoneale.
• The set up of the PD simulation environment. |
2.Methodology
2.1 A mathematical model for PD.
During PD therapy, the exchange of chemicals takes place through
the net of capillaries within the folded peritoneal membrane.
For this reason the geometrical modeling of the domain to account
for spatial variations would be extremely difficult. Moreover
the exchanges are very rapid in time, due to a high concentration
gradient between the two sides of the membrane. A space lumped
model, in which the variations in space are neglected, looks therefore
more suitable to study the kinetics of chemicals during the therapy.
Our model considers one compartment accounting for the body (b),
and one for the peritoneal cavity of the patient denoted by the
index (d) that are separated by a semi-permeable membrane that
represents the peritoneal membrane. The latter compartment is
filled by a solution of N chemicals, denoted by the indices i=1,2,…,N.
Assuming that the concentrations are uniform in space, the physical
quantities of interest are the volume of the solution and the
total amount of each solute in the two compartments, namely Vb,
Vd, Vb*cb,i, Vd*cb,i, where cb,i, cd,i are the concentrations
(mass of solute per volume of solution). The interaction between
the two compartments is governed by the equations prescribing
the flux of solvent Jv and of solute Js,i across the membrane.
A well accepted mathematical model for the fluxes of solvent and
solute is due to Kedem and Katchalsky (6). In this model the membrane
is characterized by a set of pores that allow the exchange of
the solvent and the solutes between the two compartments. The
pores can be subdivided in different classes that we denote by
the index j=1,…,M, depending on their size. We introduce
Lp and Pi, the hydraulic conductivity and permeability of the
membrane relative to the ith molecule. Then we consider the Starling
law of filtration which states that the solvent flux across each
class of pores of the membrane is proportional to the pressure
difference between the two compartments. The total flux of solvent
is the sum of the contributions of each class of pores. The pressure
is, on the other hand, split in two parts, the static pressure
and the osmotic pressure. The latter depends on the solute concentration
on the two sides of the membrane, according to the Van't
Hoff's law. On the other hand, the solute flux, Js,i, according
to(7,8) can be interpreted as the sum of a diffusive term (depending
on the jump of concentration across the membrane) and a transport
term (defined as the product of effective solvent flux and the
average concentration within the membrane). By applying these
physical laws, we determine Jv and Js,i and we end up with a system
of 2N+2 equations that describe the rate of change of the unknowns
Vb, Vd, Vb*cb,i, Vd*cb,i I=1,…,N. For further details on
the derivation of the model the reader is referred to the theory
of mass transport through membranes (6,7,8). The general mathematical
setting that we have introduced here, allows the application of
our model to a large number of chemical species, with very weak
limitations. In particular, our model takes into account the basic
chemicals considered in dialysis, as urea, glucose and creatinine.
Furthermore, it can be also applied to sodium, in order to study
the sodium removal, or to large polymers, which is nowadays becoming
an alternative for glucose in dialysate exchanges of long duration.
Unfortunately, due to its complexity it is not possible to solve
the equations of the model exactly (with an explicit formula for
the unknown variables) without resorting to strong simplifications.
With this aim, Pyle Vonesh (4) have found an analytic solution
of the so called two compartment model under suitable simplificative
assumptions. An alternative approach is to approximate directly
the solution of the model by means of numerical techniques since
the fast progress in the development of both computers and algorithms
have made possible a more detailed modelling of physics. Along
this line we apply an efficient, stable and accurate method that
results to be extremely effective. Indeed, the simulation of a
complete PD therapy requires 51 ms on a 2GHz Pentium IV processor.
3. Results
3.1 Prescription in PD.
The aim of the mathematical model that we advocate is to provide
clinicians with information that help to prescribe a PD therapy
suitable to each patient. A key issue is to keep the mathematical
model very flexible with respect to the established clinical experience
and to the clinical needs. In the following sections we describe
some characteristic features of our model that make it easily
adaptable to the user’s needs.
3.2 The isopore and Three-pore models.
Our mathematical model encompasses the two main categories of
models for PD developed so far: the isopore model (by Vonesh et
al. (4) and the three-pore model. In the isopore model the exchange
across the peritoneum occurs through one single pathway, namely
a single class of pores. The isopore model can be recovered from
our equations by restricting the index j to a single value j=1
associated to the only class of pores considered. The three pore
model, proposed in Rippe(5), is another special instance of our
general two compartment model for the specific case of three sets
of pores, j=1,2,3. In the latter case, we take into account the
exchanges across medium-sized pores, large pores (accounting for
the exchange of large macromolecules like, for example, proteins)
and ultra-small pores (accounting for the exchange of water).
The three pore model can predict the amount of fluid removal from
the patient more accurately than the isopore model(5). Since the
number of pores is not restricted, either
the three-pore or the isopore model can be considered in our computer
simulations in a straightforward automatic way.
3.3 General therapy profile.
A second advantage of the numerical solution technique that we
have developed is that it can easily account for very general
injection-extraction profiles to define a specific therapy through
a suitable specification of the input of the cycler that executes
the therapy. Many therapy profiles like the CAPD (continuous ambulatory
peritoneal dialysis), CCPD (continuous cycling peritoneal dialysis),
TPD (tidal peritoneal dialysis), NPD (nocturnal peritoneal dialysis),
APD (automated peritoneal dialysis) have been considered. Thus
a modern tool for the simulation of PD must be able to determine
the efficiency of all the therapies mentioned above. Moreover
our model can easily account for several constraints imposed by
clinicians, such as a prescribed total therapy time and the total
dialysate volume. As a matter of fact one can define a general
injection-extraction pattern that is able to represent all the
aforementioned therapies, through the setting of the input of
the cycler that executes the therapy. This new framework to prescribe
the PD is called DPD (dynamic peritoneal dialys...
3.4 Quantification of the efficacy of the therapy.
After running a numerical simulation of PD the quantities Vb,
Vd, cb,i, cb,i are known at every time and thus the efficacy of
the therapy can be computed. In order to quantify PD efficacy
clinicians mostly focus on two molecules, urea and creatinine.
Moreover the ultrafiltration or net amount of fluid extracted
during a therapy is to be considered too. Consequently, an effective
therapy is characterized by a suitable balance of the following
indicators: the normalized extracted urea over a week, called
KT/Vurea, the creatinine clearance, called Clcreat, the net amount
of fluid extracted during a therapy, called ultrafiltration (UF).
From the quantitative point of view, we define the efficacy parameter
Eff as a weighted combination of the previous indicators, precisely
Eff=w1*KT/Vurea+ w2*Clcreat + w3*UF, where w1, w2, w3 are suitable
weighting coefficients satisfying w1+w2+w3=1, which can be chosen
from the practician. Furthermore, we observe that despite KT/Vurea,
Clcreat and the ultrafiltration are the most common indicators
applied to identify the efficacy of a therapy, we are not restricted
to these choices and thanks to the high flexibility provided by
the numerical simulation approach we can introduce new indicators.
In this perspective, it is possible to consider the glucose exposure
or the sodium balance, precisely the amount of glucose and sodium
that is absorbed by the patient form the dialysate during a therapy,
as additional factors to evaluate the adequacy of PD. By means
of the numerical simulation software, the efficacy of a therapy
can be computed for several values of the inputs, for instance
for several points in the range: 4
liters < Vtot < 16 liters and 4 hours < Ttot < 10
hours. The resulting data are summarized in charts, represented
in figure 3, which describe the trend of KT/Vurea ultrafiltration,
or other indicators, for each specific patient. This global point
of view on the therapy performance on a patient specific basis
can actually help to set up a prescription of the PD treatment
to each specific patient.
|
Fig. 3
• Grafici che sintetizzano la performance della dialisi
peritoneale per un paziente specifico. Si possono considerare
vari indicatori di efficacia. La KT/Vurea è riportata
in alto mentre il drenaggio di sodio è riportato in
basso per varie combinazioni di tempo totale e di volume totale.
• Charts summarizing the peritoneal dialysis performance
for a specific patient. Several efficacy indicators can be
considered, KT/Vurea is reported at the top while sodium removal
is reported at the bottom for several combinations of total
time and total volume. |
4. New Applications
4.1 Computer Optimization of PD.
The Dynamic Peritoneal Dialysis (DPD) is a powerful tool to improve
the peritoneal dialysis treatment on a individual patient. However,
since this therapy enjoys a larger number of degrees of freedom
with respect to more classical CCPD or APD the prescription of
such treatment to a specific patient may be a challenging task.
The prescription of a DPD profile is neither intuitive, nor can
be provided by hand-made computations. For these reasons, mathematical
algorithms and in particular mathematical optimization are the
key tools to allow clinicians to exploit the advantages of DPD
on patients. To achieve our goal we consider a multi-objective
optimization strategy that allows us to take into account several
factors with different weights and seek the maximum of some of
them and the minimum of the others. In fact, we have introduced
a very general definition of the dialysis efficacy that encompasses
urea clearance, creatinine clearance and ultrafiltration. Moreover,
in the framework of multi-objective optimization this definition
might be made even more general and flexible by taking into account
the negative effect of sodium re-intake and glucose exposure.
Finally, all these aspects should be casted in a rigorous mathematical
framework. To this aim, let u be the vector denoting the control
parameters that define the injection-dwell-extraction pattern
and let Eff(u) be the relationship between the efficacy of the
therapy and the vector u of the control parameters. Then the control
problem can be formulated as follows: find u such that uopt=maxu
Eff(u) with the following constraints on the injection-dwell-extraction
pattern: the total duration of the therapy must not be exceeded;
the total amount of dialysate must be fully exploited; the peritoneal
cavity should be emptied at the end of the therapy. The optimization
algorithm consists of an iterative process that starts at an initial
guess u0 (corresponding to a standard therapy, for instance an
APD) and generates a sequence of iterates un n=1,2… (corresponding
to several instances of DPD therapy) that terminates when either
no more progress can be made or when point of maximal efficacy
has been approximated with sufficient accuracy. These algorithms
require, in our case, to run a numerical simulation of PD at each
iteration, for different values of the vector u and seek the maximum
of Eff(u) following a well defined sequence of instructions. We
provide a graphical representation of the optimization algorithm
in figure 4. In this case, we have applied the algorithm to the
optimization of KT/Vurea with respect to a general family of injection-dwell-extraction
patterns described by the frequency of exchanges (identified by
the parameter a) and the frction of volume exchanged (identified
by the parameter ß). Each point on the plots reported in
figure 4 represents a different combination of the parameters
a and ß corresponding to an injection-dwell-extraction pattern
with a particular shape. Given a range of variation of a and ß
we represent on each plot the contour levels of Eff=KT/Vurea,
in order to put into evidence the region where KT/Vurea is maximum
for a given patient. For n=10,30 the steps of the optimization
algorithm applied to approximate a local maximum of Eff are put
into evidence. We observe that after 30 iterations the maximum
is approximated with a satisfactory accuracy.
|
Fig. 4
• Rappresentazione grafica dell'algoritmo di
ottimizzazione applicato alla massimizzazione della KT/Vurea
rispetto ai parametri a e ß che determinano la forma
dello schema di infusione-sosta-drenaggio. Le linee colorate
rappresentano i livelli di efficacia (KT/Vurea) calcolati
per un paziente specifico. I colori caldi rappresentano i
picchi e i colori freddi invece le zone di bassa efficacia.
La linea rossa dimostra i passaggi dell'algoritmo di ottimizzazione
iterativa che ricava i picchi di efficacia da una più
ampia gamma di terapie DPD caratterizzate da diversi valori
di a e ß.
• graphical representation of the optimization algorithm
applied to the maximization of KT/Vurea with respect to the
parameters a and ß that govern the shape of the injection-dwell-extraction
pattern. The color lines represent the levels of efficacy
(KT/Vurea) computed for a specific patient. Hor colors represent
peaks while cold colors represent regions of low efficacy.
The red line shows the steps of the iterative optimization
algorithm that seeks the peaks of efficacy among a wide range
of DPD therapies, characterized by different values of a and
ß. |
4.2 Potential Improvement
We finally observe that the example of the optimization of KT/Vurea
can be made more general in many ways. First of all, a more general
description of the injection-dwell-extraction pattern can be considered,
by enlarging the set of control parameters. Moreover, the optimization
algorithm can deal with the maximization of some targets and the
minimization of some others, as for example the glucose exposure
or the sodium re-intake. This optimization procedure results in
providing the injection-dwell-extraction pattern that ensures
a suitable balance between a high efficacy of small solute removal
and a low glucose exposure and sodium re-intake. This information
could be of remarkable help in putting into evidence which therapy
is more adequate for a given patient.
Conclusions
The new simulation environment we have discussed here should become
a very attractive way to reconsider PD treatment at home, in particular
by using the optimization strategy it can offer. By virtue of
this system, PD patients could enjoy an increased time flexibility,
comfort and life quality. On the other hand, from the point of
view of the nephrologist, this system provides quantitative information
on the performance of a specific PD therapy.
Alfio Quarteroni
MOX, Dipartimento di Matematica,
Politecnico di Milano CMCS
(Chair of Modeling and
Scientific Computing), EPFL, Lausanne.
