Pierre Larrenie
email: pierre.larrenie@esiee.fr

Jean-François Bercher
email: jean-francois.bercher@esiee.fr

Olivier Venard
email: olivier.venard@esiee.fr

Iyad Lahsen-Cherif
email: lahsencherif@inpt.ac.ma

LOW COMPLEXITY ADAPTIVE MACHINE LEARNING APPROACHES FOR END-TO-END LATENCY PREDICTION *

Keywords: KPI Prediction, Machine Learning, Adaptivity, General Regression, SDN, Networking

Software Defined Networks have opened the door to statistical and AI-based techniques to improve efficiency of networking. Especially to ensure a certain Quality of Service (QoS) for specific applications by routing packets with awareness on content nature (VoIP, video, files, etc.) and its needs (latency, bandwidth, etc.) to use efficiently resources of a network. Monitoring and predicting various Key Performance Indicators (KPIs) at any level may handle such problems while preserving network bandwidth. The question addressed in this work is the design of efficient, low-cost adaptive algorithms for KPI estimation, monitoring and prediction. We focus on end-to-end latency prediction, for which we illustrate our approaches and results on data obtained from a public generator provided after the recent international challenge on GNN [12]. In this paper, we improve our previously proposed low-cost estimators [6] by adding the adaptive dimension, and show that the performances are minimally modified while gaining the ability to track varying networks.

Introduction

Routing while ensuring quality of service (QoS) remains a significant challenge in all networks. Whatever the resources, their use must be optimized to satisfy both throughput and QoS to users. This is true for static wide area networks, but even more so for mobile networks with dynamic topology.

The emergence of software-defined networks (SDNs) [START_REF] Singh | A survey on Software Defined Networking: Architecture for next generation network[END_REF][START_REF] Amin | Hybrid SDN networks: A survey of existing approaches[END_REF] has enabled data to be shared more efficiently across communication layers. Services can provide network requirements to routers; routers acquire data about network performance and allocate resources to meet those requirements as best as possible. However, acquiring overall network performance can result in high network bandwidth consumption for signaling, degrading the available resources, and is particularly limiting for resource-constrained networks such as mobile networks (MANETs).

We consider a network for which we wish to reduce signaling and perform intelligent routing. In order to limit the amount of signaling, the first axis is to estimate some key performance indicators (KPIs) from other KPIs. A second axis would be to perform this prediction locally, at the node level, rather than a global estimation in the network. Finally, if predictions are to be performed locally, the complexity of the algorithms will need to be low while preserving good prediction quality. The last point is to be able to detect and track changes in the state of the network, which implies that the predictors will have to use only a small number of the previous states of the network and be able to readapt continuously.

The question addressed in this work is the design of efficient, low-cost adaptive algorithms for KPI estimation, monitoring and prediction. In the present paper, we improve our previously proposed lowcost estimators [START_REF] Larrenie | Low complexity approaches for end-toend latency prediction[END_REF] by adding the adaptive dimension and show that the performances are minimally modified while gaining the ability to track varying networks. We focus on end-to-end latency prediction, for which we illustrate our approaches and results on data we generated using a public generator made available after the recent international challenge [START_REF] Suárez-Varela | The graph neural networking challenge: a worldwide competition for education in AI/ML for networks[END_REF]. The best performances of the state-of-the-art are obtained with Graph Neural Networks (GNNs) [START_REF] Rusek | Unveiling the potential of graph neural networks for network modeling and optimization in SDN[END_REF][START_REF]Barcelona Neural Networking Center: The graph neural networking challenge[END_REF][START_REF] Suárez-Varela | The graph neural networking challenge: a worldwide competition for education in AI/ML for networks[END_REF]. Although this is a global method while we favor local and adaptive methods, we used these performances as a benchmark.

We present related works in section 2. Then we present in section 3 our main results from [START_REF] Larrenie | Low complexity approaches for end-toend latency prediction[END_REF]. Instead of using high performances global and high-costs methods based on Graph Neural Networks (GNNs) [START_REF] Rusek | Unveiling the potential of graph neural networks for network modeling and optimization in SDN[END_REF][START_REF]Barcelona Neural Networking Center: The graph neural networking challenge[END_REF][START_REF] Suárez-Varela | The graph neural networking challenge: a worldwide competition for education in AI/ML for networks[END_REF], we proposed to use standard machine learning regression methods. We showed that a careful feature engineering and feature selection (based on queue theory and the approach in [START_REF] De Aquino Afonso | GNNet challenge 2021 report (1st place)[END_REF]), as well as the use of a single feature with curve-fitting methods, allows to obtain near state-of-the-art performances with both a very low number of parameters, significantly lower learning and inference times compared to GNNs, and the with the ability to operate at the link level instead of a whole-graph level. In section 4, we show how these block algorithms can be transformed into versions implementable in an iterative way (i.e. by taking into account the data one by one as they become available), with the originality of using a regularization term. Then, time dependent estimations, or the addition of forgetting factor will give them an adaptive character. In section 5 we describe the validation dataset we built from a public generator and then the results of our experimentation. Finally, we conclude, discuss the overall results and draw some perspectives.

2 Related work [START_REF] Chua | Stringer: Balancing latency and resource usage in service function chain provisioning[END_REF] present an heuristic and an Mixed Integer Programming approach to optimize Service Functions Chain provisioning when using Network Functions Virtualization for a service provider. Their approach relies on minimizing a trade-off between the expected latency and infrastructures resources.

Such optimization routing flow in SDN may need additional information to be exchanged between the nodes of a network. This results in an increase of the volume of signalization, by performing some measurements such as in [START_REF] Pasca | AMPS: Application aware multipath flow routing using machine learning in SDN[END_REF]. This is not a consequent problem in unconstrained networks, i.e. static wired networks with near-infinite bandwidth but may decrease performance of wireless network with poor capacity. An interesting solution to save bandwidth would be to predict some of the KPIs from other KPIs and data exchanged globally between nodes.

In [START_REF] Poularakis | SDN-enabled tactical ad hoc networks: Extending programmable control to the edge[END_REF][START_REF] Poularakis | Flexible SDN control in tactical ad hoc networks[END_REF], authors proposed a MANETs application of SDN in the domain of tactical networks. They proposed a multi-level SDN controllers architecture to build both secure and resilient networking. While orchestrating communication efficiently under military constraints such as: high-level of dynamism, frequent network failures, resources-limited devices. The proposed architecture is a trade-off between traditional centralized architecture of SDN and a decentralized architecture to meet dynamic in-network constraints.

[5] proposed a Quality of Experience (QoE) management strategy in a SDN to optimize the loading time of all the tile of a mapping application. They have shown the impact of several KPIs on their application using a Generalized Linear Model (GLM). This mechanism make the application aware of the current network state.

[10] used GNNs for predicting KPIs such as latency, error-rate and jitter. They relied on the Routenet architecture of Figure 1. The idea is to model the problem as a bipartite hypergraph mapping flows to links as depicted on Figure 2. Aggregating messages in such graph may result in predicting KPIs of the network in input. The model needs to know the routing scheme, traffic and links properties. Their result is very promising and has been the subject of two ITU Challenge in 2020 and 2021 [START_REF]Barcelona Neural Networking Center: The graph neural networking challenge[END_REF][START_REF] Suárez-Varela | The graph neural networking challenge: a worldwide competition for education in AI/ML for networks[END_REF]. These ITU challenges have very good results since the top-3 teams are around 2% error in delay prediction in the sense of Mean-Absolute Percentage Error (MAPE).

In [START_REF] De Aquino Afonso | GNNet challenge 2021 report (1st place)[END_REF], very promising results were obtained with a a near 1% GNN model error (in the sense of MAPE) on the test set. The model mix analytical M/M/1/K queueing theory used to create extra-features to feed GNN model. In order to satisfy the constraint of scalability proposed by the challenge, the first part of model operates at the link level.

A B C D E F 3 F 2 F 1 (a) Simple topology F 1 F 2 F 3 L AD L DE L AB L BE L CB (

Simple machine-learning approaches for latency prediction

Our first problem is to define an estimator ŷ of the occupancy y as a function of the different available "features" of the system, with a joint objective of low complexity and performance. To do so, we will look for an approximation function f θ (u) that allows to estimate y from the features u and parameters θ.

ŷ = f θ (u) (1)
Here u and θ are vectors that collect the different features or parameters. Once an estimate of occupancy is obtained, it is possible to get the latency prediction dn for a specific link n by the simple relation

dn = ŷn E(|P n |) c n (2)
where E [|P n |] is the observed average packet size on link n and c n the capacity of this link.

For analytical simplicity, the parameters θ will be sought by minimizing the minimum mean square error

E (y -ŷ) 2 = E (y -f θ (u)) 2 , (3)
although the performances are also often evaluated in the MAPE sense

L (ŷ, y) = 100% N N n=1 ŷn -y n y n (4)
which is preferred to Mean Squared Error (MSE) because of its scale-invariant property.

We will focus here on two very simple models, although other machine learning models have also been considered in [START_REF] Larrenie | Low complexity approaches for end-toend latency prediction[END_REF]. Indeed, these two models lend themselves very easily to an adaptive formulation. In this section, we will first describe these two approaches and their performances, before giving the general adaptive formulation, which we will particularize in both cases.

Feature Engineering and Linear Regression

Based on the assumption that the system may be approximated by a model whose essential features come from M/M/1/K and M/G/1/K queue theory, we took essential parameters characterizing queueing systems, such as: ρ, ρ e , π 0 , π K , etc. and built further features by applying interactions and various non-linearities (powers, log, exponential, square root). Then, we selected features in this set by a forward step-wise selection method; i.e. by adding in turn each feature to potential models and keeping the feature with best performance. Finally, we selected the model with best MAPE error. For a linear regression model, this led us to select and keep a set of 4 simple features, which interestingly enough, have simple interpretations:

         π 0 = 1-ρ 1-ρ K+1 L = ρ + π 0 k kρ k ρ e = λe λ ρ = λe µ S e = k kρ k e ( 5
)
where L is the expected number of packets in the queue according to M/M/1/K, π 0 the probability that the queue is empty according to M/M/1/K theory, ρ e the effective queue utilization, and S e the unnormalized expected value of the effective number of packet in the queue buffer. These features can be thought as a kind of data preprocessing, before applying ML algorithms, and this turns out to be a key to achieving good performances. The 4 previous features have been used as input for several machine learning models like Multi-Layer Perceptron model (MLP), Linear Regression, SVM, Random Forest, Gradient Boosting Regression Tree. We only describe here the case of linear regression, since it is a method for which an adaptive version is readily obtained. In this case, model ( 1) is simply

ŷ = θ 0 + θ 1 π 0 + θ 2 L + θ 3 ρ e + θ 4 S e = θ T u (6)
with θ T = [θ 0 , . . . θ 4 ] and u T = [1, π 0 , L, ρ e , S e ]. For the linear regression model in (( 6), it is well known that the regularized minimum mean squared error

J(θ) = E (y -θ T u) 2 + α||θ|| 2 (7)
is obtained for

θ : (R uu + α1) θ = R yu (8)
where we denoted

R uu = E uu T , the correlation matrix of u R yu = E [yu] ,
the correlation vector of y and u and 1 the identity matrix, α the regularization parameter.

As far as performance is concerned with this approach, it was evaluated using static data from the GNN ITU Challenge 2021 [START_REF] Suárez-Varela | The graph neural networking challenge: a worldwide competition for education in AI/ML for networks[END_REF]. Compared to the state-of-the-art, our linear regression with carefully selected features shows a very slight performance degradation: 1.74% in MAPE while the best state-of-the-art method is at 1.27%. One strong advantage is in term of training and inference time. It has a training time of less than a second when GNN requires more than 8 hours. Moreover, the inference time for the complete network is also much lower, by a factor of almost 1000 (0.296s vs 214s).

Curve Regression by Bernstein polynomials

There is a high interdependence of the features we selected in Equation ( 5), since all these features can be expressed in term of ρ e . Furthermore, it is confirmed by data exploration that ρ e is the prominent feature for occupancy prediction (and in turn latency prediction), as exemplified in Figure 3.

It is then tempting to try to further simplify our features space and estimate the occupancy from a nonlinear transformation of the single feature ρ e , as:

ŷ = g(ρ e ) ( 9
)
where ŷ is the estimate of the occupancy y. The concerns are of course to define simple and efficient functions g, with a low number of parameters, that can model the kind of growth shown in Figure 3, and of course to check that the performance remains interesting. The estimator g is defined as a linear combination of simple functions f n :

ŷ = g(ρ e ) = n θ n • f n (ρ e ) (10)
which is also a linear model in terms of function f n (ρ e ).

Several solutions were considered in [START_REF] Larrenie | Low complexity approaches for end-toend latency prediction[END_REF] to define or choose the functions f n . Since we know that the Bernstein polynomials form a basis in the set of polynomial in the interval [0; 1]; and that the approximation of any continuous function on [0; 1[ by a Bernstein polynomial converges uniformly, we were led to these polynomials:

f K n (x) = K n x n (1 -x) K-n (11)
where K is maximum order of polynomials.

As mentioned, (10) can be rewritten as the linear model

ŷ = g(ρ e ) = n θ n • f n (ρ e ) = θ T u (12)
with θ T = [θ 0 , . . . , θ K ] and u T = [f K 0 (ρ e ), f K 1 (ρ e ), . . . , f K K (ρ e )]. Hence, we have the same form as in (8) for the solution.

In term of performances, we also obtained a minor degradation in MAPE (1.68%) compared to state-ofthe-art (1.29%), while improving by several orders the wall training and inference times (2min/3.14s vs 8hrs/214s); though a bit less than the simple linear regression.

Adaptive versions

We place ourselves in the context where we have regular snapshots of the state of the network, which allows us to both monitor the quality of predictions, and to track changes in the network. For the n-th series of measurements, let us denote y(n) the measured latency and u(n) the features. We can also group several snapshots or several links into a vector of latencies y(n) and matrix U(n). In the following we will derive equations for this block case, which includes immediately the scalar case.

The minimum mean square error [START_REF] Pasca | AMPS: Application aware multipath flow routing using machine learning in SDN[END_REF] which has the explicit solution (8) can also be solved by a gradient algorithm as

θ k+1 = θ k -µ ∇J(θ)| θ=θ k , (13)
= θ k -µ ((R uu + α1) θ k -R yu ) . (14)
In [START_REF] Widrow | Adaptive Signal Processing[END_REF], we can substitute the true values with estimated ones. In order to introduce adaptivity to context changes in the network, these estimates will preserve the temporal dimension. We thus use either a sliding average

Ruu (n) = L l=0 U(n -l)U(n -l) T Ryu (n) = L l=0 U(n -l)y(n -l) (15)
or an exponential mean

Ruu (n) = n l=0 λ l-n U(l)U(l) T = λ Ruu (n -1) + U(n)U(n) T Ryu (n) = λ Ryu (n -1) + U(n)y(n). ( 16
)
where λ ≤ 1 is the forgetting factor.

In the limit case where we take either L = 0 or λ = 0 in the previous formulas, we get the 'instantaneous estimates'

Ruu (n) = U(n)U(n) T Rdu (n) = U(n)y(n). (17)
we obtain

θ(n + 1) = (1 -µα)θ(n) -µU(n) U(n) T θ(n) -y(n) (18
) which reduces to the well known LMS algorithm [START_REF] Widrow | Adaptive Signal Processing[END_REF] in the scalar case and no regularization, α = 0.

Alternatively, one can try to solve the normal equation ( 8), using the time dependent estimates as the exponential mean (16). The difficulty with the solution

θ(n + 1) = Ruu (n + 1) + α1 -1 Ryu (n + 1) (19)
is the inversion, for each n, of the correlation matrix. Let us denote

K(n + 1) = Ruu (n + 1) + α1 -1 . ( 20
)
Using ( 16), we have

K(n + 1) -1 = λ Ruu (n) + U(n + 1)U(n + 1) T + α1 (21) = λ Ruu (n) + α1 + U(n + 1)U(n + 1) T + α(1 -λ)1 (22) = λK(n) -1 + U(n + 1)U(n + 1) T + α(1 -λ)1 (23)
and

K(n + 1) = λK(n) -1 + U(n + 1)U(n + 1) T + α(1 -λ)1 -1 (24) = [Q(n + 1) + δ1] -1
(25) with Q(n + 1) = λK(n) -1 + U(n + 1)U(n + 1) T (26) and δ = α(1 -λ) The matrix inversion lemma enables to reduce the inversion of Q(n + 1) to

Q(n + 1) -1 = 1 λ K(n) - 1 λ 2 K(n)U(n + 1)× 1 + 1 λ U(n + 1) T K(n)U(n + 1) -1 U(n + 1) T K(n), (27)
which simplifies to

Q(n + 1) -1 = 1 λ K(n) - 1 λ 2 K(n)u(n + 1)u(n + 1) T K(n) 1 + 1 λ u(k + 1) T K(n)u(k + 1) , (28)
for scalar observations. Now, we can use the Taylor expansion to get

K(n + 1) = [Q(n + 1) + δ1] -1 = Q(n + 1) -1 -δQ(n + 1) -2 + δ 2 Q(n + 1) -3 + . . . (29
) This gives us a way to compute recursively the inverse of the regularized estimate of the correlation matrix by combining ( 27) and (29) into

K(n + 1) ≈ Q(n + 1) -1 -δQ(n + 1) -2
(30) which, by (27), does not require the inversion of K(n).

In both cases, we have the updating formula

θ(n + 1) = θ(n) + K(n + 1)U(n + 1)[y(n + 1) -θ(n) T U(n + 1)]. ( 31
)
5 Experiments and results

Dataset

We generate a dataset thanks to a public challenge data generator [START_REF] Suárez-Varela | The graph neural networking challenge: a worldwide competition for education in AI/ML for networks[END_REF]. This data generator is based on the well-known OMNET++ discrete event network simulator [START_REF] Varga | An overview of the omnet++ simulation environment[END_REF]. The published simulator is available as a docker image. However, due to the rules of the 2022 edition of the challenge, it is not possible to generate large topologies, i.e. no more than 10 nodes. Since our models are link-based, the use of small topologies does not seem to be a problem. The simulator is parameterized by a traffic matrix and a topological graph that are easy to generate thanks to the provided API.

Our generated dataset, used is this paper, is the result of 11900 simulations of the same topology graph of 10 nodes and 30 links, subject to 100 different traffic matrices. In order to get complex results of simulations but at low cost, we made the choice to model a network with small queue buffers (8000 bits) and possibly subject of high traffic intensities (maximum traffic rate set to 4000 bits/s for each flow). Then for each traffic matrices, we alter the capacity of the network according to a sigmoid, in order to model a network subject to jamming, with 2 stationary states. The proposed jamming may cause a decrease in the capacity of the network links by up to a factor of 5, as depicted on Figure 4a. For simplification purposes, we have considered that each link of the network has the same capacity. This result in a Ushaped distribution of our link data samples according to the link capacity as shown in Figure 4b.

Results

From the generated data, we validate our approach along several axes.

Global performances

First, we establish the benchmark performances based on the global methods presented in the paper [START_REF] Larrenie | Low complexity approaches for end-toend latency prediction[END_REF] and Sections 3.1 and 3.2.

For the linear regression method described in Section 3.1, we obtain an MSE of 5.86e-4 and a MAPE of 9.58% for the queue occupancy prediction and an MSE of 1.10e-3 and a MAPE of 10.26%. for the end-to-end latency prediction.

Concerning the curve regression using Bernstein polynomials (of degree 8) described in in Section 3.2, we obtain an MSE of 4.52e-4 and a MAPE of 8.72% for the queue occupancy prediction and an MSE of 9.35e-4 and a MAPE of 9.95% for the end-to-end latency prediction.

Note that these benchmark performances are below the performances obtained in [START_REF] Larrenie | Low complexity approaches for end-toend latency prediction[END_REF], but the dataset we use here is probably more severe since using ground truth value for occupancy results in an MSE 6.03e-4 of a MAPE 9.34% for the flow delay prediction. That is indeed very close of the obtained results.

Behavior of iterative algorithms

In a second step, we verify that the algorithms presented in section 4 converge and allow us to recover these performances. With a forgetting factor of 1 (use of all data with the same weight) and a block size of 10, we observe, for example in Figure 5, that the model coefficients converge towards a stable value, and that MAPE recovers the value obtained with the global method using all data. The convergence is obtained in less than 10,000 operations. It is thus possible to replace the global method, which is already low-cost, with an approach where the calculations are carried out recursively.

Adaptivity

In a third step, we compare the algorithms to the case of network modifications We consider an abrupt change in the network capacity, which could correspond to a jamming scenario. We then examine how the two adaptive algorithms presented (linear regression with judiciously chosen features; and Bernstein polynomial model) can detect and adapt to these modifications. In this context, we examine the role of the forgetting factor and the regularization parameter. Figure 6 and Figure 7 present the results for the case of a capacity change. We observe that (i) the square of the residual error, smoothed over 100 points, is a remarkable indicator of a change in the network; and (ii) that the model coefficients readjust over the iterations after this change.

Discussion

These experiments show the effectiveness and relevance of our iterative and adaptive versions of endto-end latency estimation procedures. The iterative versions have the same performance as their global counterparts; an even lower cost since they can be implemented iteratively as the data is received or made available. The convergence time for the model coefficients is a few thousand samples while the global model used around 350,000 samples for training, while the GNN models require several million samples. Moreover, we observe that the residual error converges very quickly, in some tens of samples, which means that although the convergence of the models' coefficients does not seem to be complete, they are equivalent from the point of view of performance for occupancy prediction. From an operational point of view, the model can be refreshed regularly, and the predicted KPIs between these updates can be used for intelligent routing. As we have observed, residual error monitoring is an excellent indicator of changes in the network state.

Conclusion

In this paper, we considered the problem of designing efficient and low-cost algorithms for KPI prediction that are locally implementable and adaptive to network changes. Based on a previous work, we have argued and developed adaptive solutions, introducing in addition a regularization term in order to stabilize the results. We used a public domain simulator to simulate networks and generate relevant data.

The experiments demonstrate the effectiveness and relevance of these algorithms. Thus, we now have low-complexity models that can be implemented iteratively at the level of local links. We have the possibility to predict the occupancy of the different links, and the end-to-end latencies (the models predict the occupancy of the queues, then compute analytically the delay for each link and finally aggregate along the path). Moreover, the adaptability of the solution allows to follow changes in the network state, always at a minimal cost, by re-adapting from the current solution and new data. The continuation of the work will focus on the choice criteria of the forgetting factor, on the impact of the regularization factor, in order to find automatic selection methods. Of course, the approaches considered here will have to be considered and adapted for other types of KPI, such as error rate or jitter.

Figure 1 :

 1 Figure 1: Routenet Architecture [10]

Figure 2 :

 2 Figure 2: Routenet [10] paths-links hypergraph transformation applied on a simple topology graph carrying 3 flows. (a) Black circles represents communication node, double headed arrows between them denotes available symmetric communications links and dotted arrows shows flows path. (b) Circle (resp. dotted) represents links (resp. flows) entities defined in the first graph (Lij is the symmetric link between node i and node j.). Unidirectional arrows encode the relation "<flow> is carried by <link>".

Figure 3 :

 3 Figure 3: Data of ITU Challenge 2021 [12], ρ e vs queue occupancy. Color-scale is an indicator of points cloud density.

  links (bits/s) (a) Capacity alteration to model jamming with a decrease of the capacity up to a factor of 5. (b) Link capacity distribution of the generated dataset.

Figure 4 :

 4 Figure 4: Overview of the generated dataset

  Iterative curve-fitting based on Bernstein polynomials of degree 8.

Figure 5 :

 5 Figure 5: Evolution of weights for our iterative methods without forgetting (non-adaptive) while fitting the whole dataset.

  Evolution of weights along the scenario.

  Evolution of the RMSE along the scenario.

Figure 6 :

 6 Figure6: Evolution of weights and √ MSE (RMSE) for our adaptive approach of Bernstein polynomial curve regression of degree 8,λ = 0.9, α = 0.08. Scenario describes a nominal period between 2 periods of jamming. #39 corresponds to a link capacity of 11 Kbits/s, #10 40 Kbits/s and #1 49 Kbts/s. Figureissmoothed over 100 points.

  Evolution of the RMSE along the scenario.

Figure 7 :

 7 Figure 7: Evolution of weights and √ MSE (RMSE) for our adaptive approach of Linear Regression,λ = 0.9, α = 0.08. Scenario describes a nominal period between 2 periods of jamming. #39 corresponds to a link capacity of 11 Kbits/s, #10 40 Kbits/s and #1 49 Kbts/s. Figure is smoothed over 100 points.


Aurèle Buzancais

Vincent Brunot

Romaric Larcher

Jean-Jacques Tudesq

Laura Platon

Noémie Besnard

Matthieu Amalric

Delphine Daubin

Philippe Corne

Valérie Moulaire

Aurele Buzancais1

Vincent Brunot1

Romaric Larcher1

Jean-Jacques Tudesq1

Noemie Lauraplaton1

Besnard

Matthieu Amalric1

Delphine Daubin1

Phiuppe Comet1

Valerie Moulaire1

Bemard Canaud3

Jean-Paul Cristol

Kada K|ouche1

Kada Klouche
email: k-klouche@chu-montpellier.fr

Sodium flux during hemodialysis and hemodiafiltration treatment of acute kidney injury: effects of dialysate and infusate sodium concentration at 140 and 145 mmol/L

Keywords: acute kidney injury, fluid volume balance, hemodiafiltration, hemodialysis, ionic dialysance, sodium flux, sodium mass removal

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

BACKGROUND

Acute kidney injury (AKI) is common in critically ill patients and is associated with a significant morbi-mortality. About 5% to 6% of these patients will develop a severe AKI that requires renal replacement therapy (RRT).I 2 Despite technological progress, RRT remains a critical issue in ICU settings, and its use independently increases the mortality, bringing it to 3o%_5o%.2 Several complications may be associated with RRT including vascular access-associated infections, bleeding and, the most prevalent, hemodynamlc instability (related to RRT: HIRRT). HIRRT complicates both intermittent and continuous modalities and has been reported to occur in more than 20% of patients.3j Nonetheless, continuous RRT has gained wide application in ICUs, often supplanting intermittent modalities because of the belief that it is better tolerated ln hemodynamically unstable patients.6-[° A similar hemodynamic tolerance would be, however, obtained with intermittent compared with continuous techniques lf certain recommendations are followed including cooler dialysate, longer duration of RRT sessions and most importantly high dialysate sodium." In fact, a Na dialysate >145 mmol/L has been associated with a better hemodynamlc tolerance ln ICU patients treated by IHD,12 but it would induce a Na overload to the patient. Given that natremia is mostly less than 145 mmol/L, the osmotic gradient between blood and dialysate induces a Na transfer from dialysate to the patient and a fluid load, In chronic hemodialyzed patients, the elevation of the dialysate Na levels prevented intradialytic hypotension, but it may also promote hypernatremia, thirst, hypertension, and increased interdlalytlc weight gain.]3-t5 Data on AK| patients treated by RRT are scarce. However, fluid overload has been associated with altered lung and function and increased mortality.'6.]7 We hypothesized that RRT contributes to a patient load independent of fluid balance; and evaluated Na transfers according to different Na dialysate and infusate concentrations (140 and 145 mmol/L) with both intermittent HD and HDF in AKI patients.

I METHODS

ln our Department of Intensive Care Medicine (I.apeyronie University Hospital, Montpellier, France). we routinely use online intermittent hemodialysis (HD) and hemodiaffiltratton (HDF) for the treatment of AKI. Usually, dialysate Na concentrations are set at values varying from 140 to 150mmol/L as recommended I i We performed a prospective crossover study to evaluate Na flue according to two Na dlalysate and infusate concentrations -140 and 145mmol/L-during intermittent HD and HDF sessions. This work represents a preliminary study of a prospective randomized study NADIRA -Evaluation of hemodynamic effects of Na dialysate and infusate concentration variations during on-line HDF-which was approved by the institutional review board Comit¢ ds Protection des Persormels N° ID-RCB/AudraCT 2017-A0170449, Our local institutional review board waived the need of informed consent for this study.

I Population

Baseline patient characteristics were recorded, and the severity of illness was determined using the simplified acute physiologic (SAPS) 11 and SOFA scores. During the lCU stay, patients consecutively admitted to ICU with AKI were eligible for the study. They uere included when RRT was initiated,]I and prior RRT natremia above 135 and less than 145 mmol/L. Exclusioncriteriaincludedpregnaney,age<18yearsold,previouschronicrenalfailure( glomerular filtration rate <30ml/min/I.73m2 as estimated by CKD EPI). AKI of obstructive, glomerular, or vascular etiology. treatment limitation or presumed survival <48h. Fourteen patients were included and every patient was submitted to 4-h online intermittent HD or HDF, The need for invasive mechanical ventilation and the use of vasoactive drugs during ICU stay were recorded.

I Renal replacement therapy parameters

Online-HD/F was performed using an RRT generator (Fresenius 5008, Fresenius Medical Care. Bad Homburg, Germany) with a standard 1.8 m2 Polysulfone hemodiafilter HF80. Countercurrent dialysate flow was routinely set at 200 ml/min; online infusate fl(rw in predilution mode at 60ml/min and blood flow at 200ml/min. Dialysate and infusate temperature was adjusted to 36°C. The Na dlalysate and infusate concentrations were set at 145 and 140mmol/L. We distinguished four types of RRT sessions: online HD or HI)F with Na dialysate and infusate at 140 mmol/L, and at 145 mmol/L. The net-ultrafiltration rate was maintained al 0 mt/h. I.IDF, or HD session followed HD or HDF session after a washout period of 12 h. During the washout period, serum Na concentrations were maintained in normal range (135-145 mmol/L) and the order of sessions was changed simultaneously. During RRT treatment, any perfusion or bolus infusion was not permitted unless an intradialytic hypotension occurred.

I Biological parameters

Blood samples were drawn from the arterial port of the venous catheter immediately before, and at the end of each online HD/F session for measurements of Na, urea, creatinine, albumin, and protein. At the end of treatment, blood sarnples were collected by standard stop-flow technique." Plasma Na concentrations were measured using an indirect potentiometry method.

I Renal replacement therapy dose -quantification

Kt/V were determined by using Daugirdas second generation` Single pool urea model equation (spKt/v)20:

where pre-and post-correspond to pro-and post-treatment conditions, respectively, td is time duration session in minutes, C urea is urea concentration in mmol/L, and W is body weight in kg.

Equilibrated Kt/V (eqKt/V) was calculated as21

During each RRT session, instantaneous ionic clearance (K ocM , ml/min) as well as total ionic clearance (K tocM = K ocM X time duration Session in minutes) were calculated from the measurement of ionic dialysance by the OCM module, standard on the dialysis monitor. Dialysis monitors measured effective treatment time (tHD or tllF). K tocMr /v was calculated by the dialysis monitors after entering patient age, gender. and anthropometric characteristics using the Watson formula.22

I Na transfer estimation

We estimated Na mass removal and transfer using blood based simple kinetic modeling assessment. [START_REF] Lim | Precise quantitative assessment of the clinical performances of two-high-flux Polysulfone hemodialyzers in hemodialysis: validation of a blood.based simple kinetic model versus direct dialysis quantification[END_REF] The patient was arbitrarily designatecl as the reference, that is, in case of positive transfer there is a transfer of sodium from the dialysate/replacement liquid to the patient and conversely in case of a negative transfer.

Measurement of Na mass transfer using blood-side-based calculation on pre-and post-session serum sampling values.

Mean ionic dialysance measured by the dialysance machine( K ocM ) was used as asurrogate of sodium dialysance.

The logarithmic mean ot. Na plasma concentration (pNa) over the session was calculated as where Na pre and Na post correspond to natremia before and after the RRT session, respectively. Na diffusive mass transfer (NaDT) was calculated as the Product of (Na dialysate -PNa) by its effective clearance (KocM, ml/min) and duration of the session (tHD, min) as: Na convective mass transfer (NaCT) must take into account Na flux during substitution (NaFluxsub) and ultrafiltration (NaFluxuF).

NacT was calculated as follows:

Given that none of our patients had any ultrafiltration, NaCT included only Na flue during substitution.

Consequently, NaCT was calculated as where Qsub = substitution Volume (L) x 1000//tHF (min) and DifNa = 0.97 x pNa owing that only 97% of Na diffused because of colligative forces induced by Gibbs-Dorman effect.

During the HD session. total sodium mass transfer (NetNaMT) is equal to NaDT and during the HDF session to the sum of NaDT and NaCT.

I Clinical evaluation of renal replacement therapy sessions

Clinical tolerance of RRT was investigated by collecting at baseline and every 30min during each session: pulse, temperature, and mean arterial blood pressure. An intradialytic hypotensive event was defined by a 20% reduction of mean arterial blood pressure or by an increase in vasocinstrictiive agents' dose. Rates of blood, fluid substitution, and dialysate flows were also monitored every hour.

I Statistical analysis

Results are expressed as mean ± SD or median and interquartile range [lQR], de|)ending on data distribution, for continuous variables and as frequency and percentages for categorical variables. Qualitative values were compared using chi-square test. Quantitative variables were compared using the Student I-test or the nonparametric Wllcoxon test or Mann-Whltney test. Comparative statistical analysis between the four groups was performed using the Kruskall-Wallis test. A p value of <0.05 was considered as statistically significant. Statistical analysis was performed using R Software version 3.3.I (R Foundation for Statistical Computing, Vienna, Austria).

RESULTS

I Population

From December 2017 to July 2018, of the 60 KDIGO 3 AKI patients who received RRT. 14 with a median age of 66.5years old entered the study (Figure 1). Al admission` median SAPSII and SOFA score were at 61.5 and 9, respectively. Main clinical characteristics are reported ln Table 1. Most of the patients were men (78.6%). Mechanically ventilated (11/14, 7997o) and treated by vasoocontrictive agents (10/14, 71%). All patients were anuric. AKI was of septic origin in most of the patients (8/14. 57%).

Renal replacement therapy sessions, efficiency.

RRT included 19 HDI and 20 HDF sessions. Each patient underwent at least two IID or HDF sessions with Na at 140 and 145 mmoL/L Treatment time session average was at 4±0.1hours. Na dialysate and infusate were set at 140mmol/L in 9 HD and 9 HDF sessions and at 145mmol/L ln 10 HDI and 11 HDF sessions. eqKt/V and Kt QCM /V varied from 0.86 to 1.01 and from 0.78 to o.92, respectively, for all RRT sessions without any statistical differences in efficiency according to RRT modality and Na dialysate and infusate (Table 2),

FIGURE I Flow chart of the studied population.

AKI, acute kidney injury; HD hemodialysis; H DF. hcmodiafilttration; RRT, renal replacement therapy.

I Na flux according to renal replacement therapy modality and Na dialysate and infusate concentrations

Protidemia remained unchanged after each session and glycemia was maintained below 8mmoL/L before and after each session, thanks to an insulin Intravenous Infusion whenever necessitated. Median baseline natremia varied from 137 to 139 mmol/L and did not differ between groups (Table 2). After RRT, natremia increased to a median value of 140 (139-143) mmol/L. This increase was, however, significantly different only after HD and HI)F at Na 145. A Na transfer from dialysate or infusate to the patient was observed in all modalities except for HD with Na dialysate at 140mmol/L (Table 3). Na flux to the patient was higher with Na 145 compared with Na 140 with both modalities. The comparison between HD and HDF145 showed (hat Na flux was significantly higher with HDF underlying that a Na overload occurred more importantly with convection mode (Table : 4). By contrast, Na flux to the patient was canceled even inversed with HD140, whereas HDF140 still induced Na flux to the patient but significantly less than HD and HDF145 (Figure 2).

I Clinical and hemodynamic tolerance of renal replacement therapy sessions

Body weight, blood pressure, and heart rate remained unchanged before and after each RRT session. Among the 39 RRT sessions, an intradialytic hypotensive event was observed in three patients during three sessions accounting for a mat-tolerated session's rate at 7%. The occurrence of these episodes was similar in all groups. Nine patients were discharged alive from the ICU that accounted for a mortality of 36%.

DISCUSSION

We found that critically ill patients undergoing RRT received a sodium load with both HD and HDF when Na dialysate and infusate were set at 145 mmol/L. When Na dialysate and infusate were set at 140mmol/L, a Na transfer to the patient persisted but decreased significantly during HDF sessions to cancel out or even negated during HD. A fair intradialytic hemodynamic tolerance was observed regardless of modalities and Na dialysate and infusate concentrations. AKI patients treated try RRT have an in-hospital global mortality rate of 40 to 60%.24 Though RRT represents the cornerstone of the supportive management Of severe AKI, its performances may have untoward effects Including intradialytic hypotension that contribute to the morbimortality. 24 Indeed, intradialytic hypotension may occur in 30% to 70% of intermittent RRT sessions ln ICU patients. 3.2S continuous RRT modalities are presumed to induce less HIRRT however, it still occurs in 19% to 43% of patients.5 Accordingly, interventions to limit I]IRRT might ultimately Improve the persistently dismal outcomes of critically ills treated by RRT. Among all, the use of a high sodium dialysate concentration (>145 mmol/L) to Limit osmotic shift and promote hemodynamic stability has been reported as an effective intervention.Ilr2627 However, lt may induce a cumulative fluid balance which is an independent risk factor for increased mortality in septic shock and AKI.28'29 Our aim in this study was to evaluate the amount of fluid load during an RRT session according to Na dialysate and infusate. We found that a Na dialysate and infusate set at 145 mmol that was above patient natremia induced a significant overload with both intermittent isovolumic HD and HDF.

In chronic hemodialyzed patients, a Na dialysate higher than natremia leads to a Na transfer to the patient according to the osmotic gradient.3" Consequently. This Na overload increased progressively weight gain and Na inflation. In addition, jt has been observed that the use of a Na dialysate >140mmol/L, generating a positive osmotic gradient from the dialysate to the patient, induced an interdialytic weight gain and a higher ultrafiltration volume during RRT sessions." By contrast, the use of a Na dlalysate lower that predialytic natremia will affect patient weight and contribute to a better pressure control. 32 A few data exist regarding to Na dialysate and infusate concentrations and potential overload in critically ill. In a study of 60 ICU patients receiving extended daily dialysis or CRRT (CVVIID and CVVHDF), both extended daily dialysis and CRRT contributed to a positive Na flux with a greater positive sodium flux with continuous modalities. The sodium load was affected by multiple factors especially the gradient between dlalysate and serum Na.33. Our results arc in line with these observations. The major factor that causes the transfer of Na to the patient appears to be the positive gradient between the dialysate/replacement fluid sodium and the patient`s serum Na at the start of the treatment leading to an extracellular fluid expansion. In addition, HDF contributed a higher load of Na to the patient compared with HD. Indeed, HDF cumulates the effects of both diffusion and convection an d, therefore, a greater transfer of Na.

In critically ill patients, Na overload is a matter of concern since it may contribute [o the morbi-mortality. During their ICU stay, the vast majority of patients experience an increased extracellular volume that will be reflected in an exacerbation of interstitial edema in various organs including the lung. In addition, hypervolemia would generate an Increased pressure in the renal capsula by venous congestion that induces a decrease in renal plasmatic flow and glomerular filtration flow.3J There is a proof Of evidence of a deleterious effect of Na overload in the critical]y Ill. AKI patients exhibited a greater Na overload than those without AKI.35 A weight gain >10% during ICU stay increased significantly the mortality ln critically ill patients.2"r` Last, among AKI patients requiring RRT, the presence of a Na overload at the initiation of the treatment increases 90-day mortality 36 and would have a negative impact on the recovery of renal functlon. [START_REF] Heung | Fluid overload at initiation of renal replacement therapy is associated with lack of renal recovery in patients with acute kidney injury[END_REF] We must acknowledge some limitations to the study. First, our work shares the limitations of single-center studies. Our unit is indeed familiar with both hemodialysis and hemofiltration therapies which are not the general rule in ICU settings. In addition, citrate anticoagulation which may be currently used in ICU settings was avoided in this study to prevent any Na loading induced by citrate infusion. Second, the number Of patients included may be considered small but a great number of RRT sessions was evaluated that allowed our investigations. The modelized method we used to calculate Na transfers was validated only in chronic hemodialysis settings but not in AKI. Nevertheless. we assume that such will be extrapolated in acute settings. In addition, we did not include the amount of Na due to different perfusions including antibiotics, sedation, parenteral nutrition... Given the short course of the RRT session less than 4 h, this should not significantly impact our estimation.

CONCLUSIONS

Our study showed that both intermittent HD and HDF with a Na dialysate and infusate set at 145 mmol/L will induce a Na overload to the patient. This Na overload will be lesser with HDF and even negative with HD when the Na dialysate and infusate were set at 140mmol/L. Convective modalities induced more Na overload than diffusive ones. At (hose different values of Na dialysate and infusate. We did not observe any differences in hemodynamic tolerance during RRT sessions. These results need further studies for confirmation, but they should prompt us to pay attention to Na transfers during RRT.

FIGURE 2

 2 FIGURE 2 Na mass transfers during HD and HDF according to Na dialysate and infusate. HD, henodialysis; HDF. hemodiafiltration: RRT, renal replacement therapy. *p < 0.05 between HD 140 and HD 145 and between HDF 140 and HDI`` 145. A positive transfer means a transfer of sodium from the dialysate/ replacement liquid to the patient, Conversely, a negative transfer takes place from (he patient's plasma to the dialysate/replacement liquid.

ACKNOWLEDGMENTS

We would like to thank all patients and researchers, service users, carers, and lay people who contributed to the original datasets.

FUNDING INFORMATION

The authors did not receive any funding.

CONFLICT OF INTEREST

Bemard Canaud is part-time employee of Fresenius Medical Care acting as consultant. The remaining authors have nothing to disclose. The results presented in this article have not been published previously ln whole or part.