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9th IFAC Symposium on Biological and Medical Systems 9th on Biological and Medical Systems 9th IFAC Symposium on Biological and Aug.IFAC 31 - Symposium Sept. 2, 2015. Germany 9th IFAC onBerlin, Biological and Medical Medical Systems Systems Aug. 31 -- Symposium Sept. 2, 2015. Berlin, Germany Available online at www.sciencedirect.com Aug. 31 Sept. 2, 2015. Berlin, Germany Aug. 31 - Sept. 2, 2015. Berlin, Germany

ScienceDirect IFAC-PapersOnLine 48-20 (2015) 249–254

Novel data-driven stochastic model Novel data-driven stochastic model Novelantibody data-driven stochastic model dynamics in kidney antibody dynamics in kidney antibody dynamics in kidney transplantation transplantation transplantation

Lowe ∗∗ David Briggs ∗∗ Robert Higgins ∗∗∗ ∗∗ ∗∗ David Briggs∗∗∗ ∗∗ Robert Higgins ∗∗∗ ∗∗∗ Lowe ∗∗ David ∗∗ Robert Higgins ∗∗∗ Lowe Briggs Natasha Khovanova Lowe David Briggs ∗ ∗ Natasha Khovanova Khovanova ∗ Robert Higgins Natasha Natasha Khovanova ∗ School of Engineering, University of Warwick, UK (corresponding ∗ ∗ School of Engineering, University of Warwick, UK (corresponding ∗ School of author Engineering, of e-mails:University [email protected] and School of author Engineering, University of Warwick, Warwick, UK UK (corresponding (corresponding e-mails: [email protected] and author e-mails: [email protected] and [email protected]). author e-mails: [email protected] and [email protected]). ∗∗ [email protected]). NHS Blood and Transplant, Birmingham [email protected]). ∗∗ ∗∗ Blood and Transplant, Birmingham ∗∗∗ ∗∗ NHS NHS Blood and Birmingham University Hospitals and Warwickshire NHS Blood Coventry and Transplant, Transplant, BirminghamNHS Trust, ∗∗∗ ∗∗∗ University Hospitals Coventry and Warwickshire NHS Trust, ∗∗∗ University Hospitals Coventry and Warwickshire NHS Coventry University Hospitals Coventry and Warwickshire NHS Trust, Trust, Coventry Coventry Coventry

Yan Yan Yan Yan

Zhang ∗∗ ∗ Zhang Zhang Zhang ∗

for for for

David David David David

Abstract: Falls in the serum levels of donor specific HLA antibodies (DSA) after kidney Abstract: Falls in serum levels of donor specific HLA antibodies (DSA) after kidney Abstract: Falls in the serum levels of specific HLA (DSA) kidney transplantation are ofthe great clinical interest, as they are associated with resolution of rejection Abstract: Fallsare in of the serum levels of donor donor specific HLA antibodies antibodies (DSA) after after kidney transplantation great clinical interest, as they are associated with resolution of rejection transplantation are of great clinical interest, as they are associated with resolution of rejection and good long term outcomes in patients at high riskare of associated graft loss. A data-driven model in the transplantation are of great clinical interest, as they with resolution of rejection and long term outcomes in patients high risk of graft loss. A data-driven model in the and good long term outcomes in patients at high risk of graft loss. A data-driven model in the formgood of third order differential equation hasat been developed to describe the dynamics of the falls and good long term outcomes in patients at high risk of graft loss. A data-driven model in the form of third order differential equation has been developed to describe the dynamics of the falls form of third order differential equation has been developed to describe the dynamics of the falls in DSA after renal transplantation. The model characterises the post transplant DSA behaviour form of third order differential equation has been developed to describe the dynamics of the falls in renal transplantation. The model those characterises the post transplant DSA behaviour in DSA after renal transplantation. The characterises the DSA forDSA two after groups of renal transplant recipients: who experienced acute antibody mediated in DSA after renal transplantation. The model model those characterises the post post transplant transplant DSA behaviour behaviour for two groups of renal transplant recipients: who experienced acute antibody mediated for two groups of renal transplant recipients: those who experienced acute antibody mediated rejection (AMR)ofinrenal the first days after operation andwho those who did not. A variational Bayesian for two groups transplant recipients: those experienced acute antibody mediated rejection the first days after operation and who did not. A variational Bayesian rejection (AMR) in the first days after operation and those who did not. A variational Bayesian inference (AMR) methodin was employed to find the form ofthose the model, infer the system parameters rejection (AMR) in the first days after operation and those who did not. A variational Bayesian inference method was employed find the form of theand model, infer thefeatures system parameters inference method was to find of model, infer system parameters and extract the information of theto recognisable patterns the common DSA post inference method was employed employed torecognisable find the the form form of the theand model, infer the thefeatures systemin parameters and extract the information of the patterns the common in DSA post and extract the information of the recognisable patterns and the common features in post transplant dynamics. Three models of different order have been investigated, and the third and extractdynamics. the information ofmodels the recognisable patterns and the common featuresand in DSA DSA post transplant Three of different order have been investigated, the third transplant dynamics. Three models of different order have been investigated, and the third order lineardynamics. model withThree four parameters outperformed the models ofinvestigated, lower orders.and Thethe inferred transplant models of different order have been third order linear model with four parameters outperformed the models of lower orders. The inferred order linear with parameters the models lower The inferred deterministic parameters were found to outperformed be significantly different between the two groups of order linear model model with four four parameters outperformed thedifferent models of ofbetween lower orders. orders. Thegroups inferred deterministic parameters were found to be significantly the two of deterministic parameters were found to be significantly different between the two groups of people with and without AMR. The eigenvalues for each DSA time series have been calculated deterministic parameters were found to be significantly different between the two groups of people with without The for DSA series have calculated people with and and without AMR. The eigenvalues for each each DSA time time and series have been been calculated and compared between theAMR. groups. A eigenvalues higher frequency of oscillation a faster dissipation rate people with and without AMR. The eigenvalues for each DSA time series have been calculated and compared between the groups. A higher frequency of oscillation and a faster dissipation rate and compared between the groups. A higher frequency of oscillation and a faster dissipation rate of antibodies have beenthe found in the AMR group, which demonstrate a apotential for intelligent and compared between groups. A higher frequency of oscillation and faster dissipation rate of antibodies have been found the AMR group, which demonstrate aa potential intelligent of antibodies have in the which for intelligent laboratory interrogation of thein underlying immunological mechanisms, which atfor present are of antibodies have been been found found in underlying the AMR AMR group, group, which demonstrate demonstrate a potential potential forpresent intelligent laboratory interrogation of the immunological mechanisms, which at are laboratory interrogation of the underlying immunological mechanisms, which at present are entirely opaque. laboratory interrogation of the underlying immunological mechanisms, which at present are entirely entirely opaque. opaque. entirely opaque. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Kidney transplantation, Antibody dynamics, Ordinary differential equations, Data Keywords: Kidney transplantation, Antibody dynamics, Keywords: Kidney transplantation, Antibody dynamics, Ordinary differential differential equations, equations, Data Data driven model, Eigenvalue, Variational Bayesian inferenceOrdinary Keywords: Kidney transplantation, Antibody dynamics, Ordinary differential equations, Data driven model, Eigenvalue, Variational Bayesian inference driven model, model, Eigenvalue, Eigenvalue, Variational Variational Bayesian Bayesian inference inference driven 1. INTRODUCTION 1. INTRODUCTION INTRODUCTION 1. 1. INTRODUCTION Kidney transplantation is commonly performed despite Kidney transplantation is commonly performed between despite Kidney transplantation commonly performed despite human leukocyte antigen is (HLA) incompatibilities Kidney transplantation is commonly performed between despite human leukocyte antigen (HLA) incompatibilities human leukocyte antigen (HLA) incompatibilities between the donor and theantigen recipient. HLA defines the immunologhuman leukocyte (HLA) incompatibilities between the donor and the recipient. HLA defines the immunologthe and the recipient. HLA defines immunological donor character of each of us, and there arethe many different the donor and of theeach recipient. HLAthere defines immunological character of us, and arethe many different ical character of each of us, and there are many different HLA types that allow for inter-individual variation in the ical character of each of us, and there are many different HLA types that allow allow for to inter-individual variation in the the HLA types that for inter-individual variation in immune response, likely confer a survival benefit to HLA types that allow for inter-individual variation in the immune response, likely to confer a survival benefit to immune response, likely to confer a survival benefit to the species. Differences between donor and recipient HLA immune response, likely to confer a survival benefit to the species. Differences between donor and recipient HLA the species. Differences between donor and recipient HLA drivespecies. an immune response (called rejection) againstHLA the the Differences between donor and recipient drive an immune response (called rejection) rejection) against the drive (called the graft. an Theimmune portionresponse of this response driven by against T lymphodrive an immune response (called rejection) against the graft. The portion of this response driven by T lymphograft. The portion of this response by T lymphocytes has effectively been managed driven by improvements in graft. The portion of this response driven by T lymphocytes has effectively effectively been managed by improvements in cytes has improvements in drug therapy over thebeen last managed 55 years. by However, some indicytes has effectively been managed by improvements in drug therapy over the the last 55 years. years. However, some indidrug therapy over 55 However, individuals also develop anlast antibody response. Thissome antibody drug therapy over the last 55 years. However, some individuals also develop an antibody response. This antibody viduals develop an antibody antibody responsealso to non-self HLA can be response. produced This by pregnancy viduals also develop an antibody response. This antibody response totransfusion non-self HLA HLA canasbe be produced by pregnancy pregnancy response non-self can produced by and bloodto as well organ transplantation; can response totransfusion non-self HLA canasbe produced by pregnancy and blood as well organ transplantation; can and blood transfusion as well as organ transplantation; can persist for transfusion life; can cause severe rejection and graft loss. and blood as well as organ transplantation; can persist for life; life; can cause cause severe rejection andproduction graft loss. loss. persist for can severe rejection and graft Mechanisms underlying the control of antibody persist for life; can cause severe rejection and graft loss. Mechanisms underlying the the control of antibody antibody production Mechanisms underlying of production are poorly understood, andcontrol treatments given to patients Mechanisms underlying the control of antibody production are poorly understood, and treatments given to patients are poorly understood, and treatments given to patients withpoorly HLA antibody mediated rejection (AMR) are often are understood, and treatments given to patients with HLA antibody antibody mediated rejection (AMR) (AMR) are often often with HLA mediated rejection are ineffective. In recent years, the development of a microbead with HLA antibody mediated rejection (AMR) are often ineffective. In In recent years, years, the development development of aa microbead microbead ineffective. ineffective. In recent recent years, the the development of of a microbead

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assay using purified HLA protein has made the identificaassay using purified HLA protein has made the identificaassay using purified protein made the tion and quantification of different types of DSA in patient assay using purified HLA HLA protein has has made the identificaidentification and quantification of different types of DSA in tion and and quantification of different different typesetof ofal. DSA in patient patient sera possible, as explained in Thaunat (2009). High tion quantification of types DSA in patient sera possible, as explained in Thaunat et al. (2009). High sera possible, as explained in Thaunat et al. (2009). High titrespossible, of preformed DSA have been recognised as the cause sera as explained in Thaunat et al. (2009). High titres of preformed DSA have been recognised as the cause titres of preformed DSA have been recognised as the cause of acute AMR since strong evidence was provided by Patel titres of preformed DSA have been recognised as the cause of acute AMR since strong evidence was provided Patel of acute AMR since evidence was by Patel and Terasaki Therefore, removal of the by existing of acute AMR (1969). since strong strong evidence was provided provided by Patel and Terasaki (1969). Therefore, removal of the existing and Terasaki Terasaki (1969). Therefore, Therefore, removal of the the existing existing DSA before transplantation through plasmapheresis has and (1969). removal of DSA through plasmapheresis has DSA before transplantation through plasmapheresis has been before widely transplantation applied in practice for desensitisation and DSA before transplantation through plasmapheresis has been widely applied in practice for desensitisation and been widely applied in practice for desensitisation and prevent acute AMR, as explained in Krishnan et al. (2013) been widely applied in practice for desensitisation and prevent acute AMR, explained et al.between (2013) prevent acute AMR, as explained in Krishnan et (2013) and Terasaki Caias (2008). But in theKrishnan relationship prevent acuteand AMR, as explained in Krishnan et al. al.between (2013) and Terasaki and Cai (2008). But the relationship and Terasaki and Cai (2008). But the relationship between DSATerasaki and chronic rejection is still not clear, as between various and and Cai (2008). But the relationship DSA and chronic rejection is still not clear, as various DSA is not various factorsand arechronic involvedrejection (see review Nankivell and as Chapman DSA and chronic rejection is still still not clear, clear, as various factors are involved (see review Nankivell and Chapman factors are involved (see review Nankivell and Chapman (2006)). In recent years, a number of publications such as factors are involved (see review Nankivell and Chapman (2006)). In recent years, aa number of publications such as (2006)). In recent years, number of publications such Sellar´ e s et al. (2012), Terasaki and Cai (2008), Zachary (2006)). In recent years,Terasaki a number of Cai publications such as as Sellar´ e s et al. (2012), and (2008), Zachary Sellar´ e s et al. (2012), Terasaki and Cai (2008), Zachary and Leffell (2008), Feucht and and Opelz (1996), Sellar´ es et al. (2012), Terasaki Cai (2008),Hourmant Zachary and Leffell (2008), Feucht and Opelz (1996), Hourmant and Feucht Opelz Hourmant et al.Leffell (2005)(2008), have confirmed that HLA(1996), antibodies, espeand Leffell (2008), Feucht and and Opelz (1996), Hourmant et al. (2005) have confirmed that HLA antibodies, espeet al. (2005) have confirmed that HLA antibodies, especially DSA, are the major cause of acute AMR and chronic et al. (2005) have confirmed that HLA antibodies, especially DSA, are the major cause of acute AMR and chronic cially DSA, are the major cause of acute AMR and chronic rejection, the latter accounting for most graft failure. Even cially DSA, are the major cause of acute AMR and chronic rejection, therisk latter accounting for is most graft failure. failure. Even rejection, the latter for most graft Even though the of accounting graft failure positively correlated rejection, the latter accounting for most graft failure. Even though the risk of graft failure is positively correlated though the risk of graft failure is positively correlated with DSA level, the association canisvary betweencorrelated patients. though the risk of graft failure positively with DSA level, the association can vary across between patients. with DSA level, the can between patients. In the acute transplantation high DSA with DSA level,setting, the association association can vary vary across between patients. In the acute setting, transplantation high DSA In the themay acute setting, transplantation across high DSA levels result in 50% graft loss, but datahigh based on In acute setting, transplantation across DSA levels may result in 50% graft loss, but data based on levels may result in 50% graft loss, but data based levels may result in 50% graft loss, but data based on on

Copyright 249 Hosting by Elsevier Ltd. All rights reserved. 2405-8963©©2015 2015,IFAC IFAC (International Federation of Automatic Control) Copyright © 2015 249 Copyright © IFAC 249 Peer review underIFAC responsibility of International Federation of Automatic Copyright © 2015 2015 IFAC 249Control. 10.1016/j.ifacol.2015.10.147

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the currently available assays cannot discriminate between successes and failures (see reference Puttarajappa et al. (2012)). Likewise, in the chronic setting there is not a clear relationship between the occurrence of AMR and the DSA as detected in the blood. Preformed DSA, compared with de novo DSA, usually show dramatic rises and falls over a period of a few days after operation. However the dynamic behaviour of post transplant DSA varies from case to case; even different DSA in the same patient (targeting different HLA) can reveal diverse patterns. We have obtained a unique dataset with detailed antibody levels spanning three to six months starting around 10 days before transplantation. A previous study in Higgins et al. (2009) carried out for this data has suggested that repetitive patterns occur in patients with or without acute AMR episodes: some DSA time series show a rapid rise during the first week followed by a rapid fall to almost undetectable levels, which then remain low. This finding was striking; in many of these patients, the DSA had been persistent for many years before transplantation, and therapies used experimentally have been unable to stop antibody production before transplantation. Therefore an understanding of this remarkable phenomenon could lead to ‘desensitisation’ of a patient before transplantation, potentially even removing the antibody barrier to engraftment completely. However this phenomenon has only recently been described, and it is not at all clear what the immunological mechanisms might be. There are several hypotheses that might explain the disappearance of DSA, each of which is potentially difficult and time consuming to test. The focus of our work is on the development of a dynamic model to describe the dynamics of DSA after transplantation, hypothesising that these results would enable intelligent application of laboratory testing to patient samples, ultimately leading to therapies which might be able selectively to control this antibody response and improve clinical transplant outcomes. 2. DATA DESCRIPTION Twenty-one post transplant DSA time series from twelve patients who underwent antibody incompatible renal transplantation between 2003 and 2007 were investigated in this study. The group characteristics and details of therapy have previously been described in Higgins et al. (2009). Note that some of our patients had multiple DSA. Twelve DSA time series belong to six patients that experienced acute AMR in the first 30 days after transplantation (AMR group), and nine DSA time series belong to the other six patients who did not have an acute AMR (no-AMR group). We selected these DSA time series based on the common feature of a rapid rise and fall in DSA levels after kidney transplantation. Rejection episodes were diagnosed by renal biopsy or clinically if there was rapid onset of oliguria with a rise in both serum creatinine and in DSA levels. Serum samples for DSA analysis were taken daily in the first three to four weeks, as most dynamic behaviour appears during that period, and sampling becomes more sparse later when antibodies tend to be more stable. Peak DSA was defined as the highest level of DSA within the first six weeks post transplant. The antibody level was measured using microbead assay manufactured by One 250

Lambda Inc (Canoga Park, CA, USA), analysed on the Luminex platform (XMap 200, Austin, TX, USA). The assay measures the Mean Fluorescence Intensity (MFI) which corresponds to antibody level although their relationship is linear only over a limited range. As described in Higgins et al. (2009), when the MFI value is higher than 10,000 AU (arbitrary units) and below about 1,000 AU, the linear correlation breaks. An example of the MFI time series for a patient from a no-AMR group is shown in Fig. 1, where day 0 is the day of transplantation.

Fig. 1. Mean fluorescence intensity measurement time series of DSA 62, DSA 60 from a patient from noAMR group. Each measurement is indicated by a cross. The initial drop in DSA levels before day 0 was caused by double filtration plasmapheresis that started 10 days before transplantation. On average, two to five alternate day sessions were performed before operation to remove the existing DSA. The rise in DSA over the first few days after transplantation was partly caused by plasmapheresis stopping, but also by an increased rate of synthesis of DSA, the expected immunological memory response. The different patterns of falls can be easily distinguished from B62(DSA) and B60(DSA) in Fig. 1. A previous study done by Higgins et al. (2009) suggested that the fall is greater in the AMR group compared with the no-AMR group, and the rate of fall of DSA exceeds the rate of fall of other nonDSA antibodies. In this work we concentrate our attention on the falling dynamics from the peak, which is typically reached within two weeks after operation, down to a steady state. 3. MODELS AND METHODS 3.1 Model formulation DSA falls in the antibody response to a transplanted kidney during the acute stage after operation for people with and without AMR can be described by the following model: dn dn−1 d xt + θn n−1 xt + ...θ2 xt + θ1 xt − θ0 = 0 n dt dt dt y t = xt + ε t

(1) (2)

Eqn. (1) is an evolution equation of nth order, where xt is a function of t that describes the MFI dynamics,

9th IFAC BMS Aug. 31 - Sept. 2, 2015. Berlin, Germany Yan Zhang et al. / IFAC-PapersOnLine 48-20 (2015) 249–254

yt is the measured MFI time series and εt is measurement noise. The order of the system equation n is to be decided together with unknown system’s parameters θi (i = 0, 1, 2, ..., n − 1). n initial conditions are required to obtain a closed form solution. 3.2 Model selection and parameter inference Model (1)-(2) covers a variety of dynamic patterns depending on the order of the system equation. As described in Occam’s razor (refer to Mackay (2003)), a more complex model is able to explain a wider range of system behaviour in the data at the risk of overfitting. Our aim was to select a parsimonious model for all the DSA time series in both AMR and no-AMR groups. Starting from the first order (M1 ), the model order n was increased until the model Mn explained the data sufficiently well. A variational Bayesian learning algorithm (refer to Daunizeau et al. (2009)) was employed to decide the order of the model for each DSA time series. It calculates the probabilities p(yt |M ) (where M is M1 , M2 ,..., Mn ) of observing the time series yt given different models M , so that the model with the highest value of p(yt |M ) can be selected for that specific DSA time series. An attention has to be paid to the features in the dataset that can be explained by a model with a higher order but cannot be explained by the model with a lower order, and to decide if the features are general enough to make the final decision of the order for all DSA time series under investigation. In particular, a variational Bayesian toolbox (see SPM9 reference) was used with the corresponding modifications to our specific data. For each model candidate, the value of probability p(yt |M ), which is also referred as the model evidence, can be approached by iteratively optimising the model parameters one by one until a local maximum value of p(yt |M ) is reached. Following Bayes’ rule (refer to Mackay (2003)), the model evidence p(yt |M ) was found by integrating the product of the prior distributions of the parameters p(ϑ|M ) and the likelihood of observing time series yt given the parameters ϑ and the model M over the distribution of all the parameters ϑ: p(yt |M ) = p(ϑ|M )p(yt |ϑ, M )dϑ (3) In Eqn. (3), ϑ represents the space of all parameters of Eqn. (1)-(2). The integral in Eqn. (3) is analytically intractable; therefore Laplace and mean-field approximations were applied. The distribution of each parameter was approximated by the first two moments (mean and variance), known as Laplace approximation (see details in Daunizeau et al. (2009)). The combined distribution of all the parameters was approximated by the product of the individual parameter distributions, known as mean-field approximation. The intractable p(ϑ|M ) was then replaced by the approximated distribution q(ϑ), making the integral in Eqn. (3) solvable. Friston et al. (2010) has pointed out that the estimated model evidence is smaller than the true value due to decreased entropy introduced by these two approximations. The logarithm of p(yt |M ) is known as ‘free energy’ F(q(ϑ), yt ), a term borrowed from statistical physics. This value is maximised, and, among other criteria (RMSE value and the stability of the immune response,

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both of which are discussed in later), defines the goodness of fit. In the Bayesian approach, a careful parameter prior selection is vital, since it may carry an important weight on the posterior distributions of the inferred parameters. As there was no preliminary information on possible parameter values, the mean values of the parameter priors were set to zero. To allow the algorithm searching in a relatively wide region for the optimal parameters, all variances were set to be large (104 ), i.e. priors with flat distributions were considered. The noise precision, which is inversely proportional to noise intensity, was modelled by a Gamma distribution with two hyperparameters (shape σa and rate σb ). A weakly informative Jeffrey’s prior, as described in Gelman (2006), were chosen for the precision of the noise, with both shape and rate parameters set to 1. The initial conditions were all modelled as Gaussian distributions. The prior means of the initial conditions were defined from the measurement time series, and the prior variances were set as 104 . 3.3 Data fitting Three models (first order M1 , second order M2 and third order M3 ) were used to fit the DSA time series and compared to select the best model. For each DSA time series, free energy F has been maximised by tuning system’s parameters in an iterative manner for each model. The comparison of the free energy between models could be problematic due to a heavy penalisation of the model complexity embedded in variational Bayesian method as explained in Beal (2003). Increasing the order of the system by one would increase the degree of freedom dramatically because the degree of the covariance matrix is n2 . Therefore, alongside the free energy use for model selection, the following three criteria were also utilised. First of all, as the aim was to find a model capable of capturing the common patterns revealed in all time series, a model that can only describe part of the DSA time series was disregarded. Second, the inferred parameters θi were applied back to the system equation to generate deterministic time series. Then the differences between the measurement data and the deterministic time series were calculated via the root mean squared error (RMSE): n 2 t=1 (yˆt − yt ) (4) RM SE = n Here yt is the MFI time series and yˆt is the inferred deterministic time series. The model with the lowest value of RMSE is favourable. Third, the model has to have a unique stable steady state, which implies that system’s response decays with time. This has been checked via calculations of real parts of eigenvalues which have to be negative for stability. Note, even though the steady state of the immune homoeostasis was disturbed by transplantation, the antibody levels settled rapidly to a new steady state except for the extreme cases such as hyperacute rejection of the transplanted kidney, which is not our concern in this paper.


Table 1. Summary for the free energy and RMSE values for three models in Fig. 2 Free Energy

RMSE [AU]

M1

M2

M3

M1

M2

M3

(a)

-99

-164

-233

1573

305

81

(b)

-182

-346

-507

329

338

52

3.4 Statistical analysis Statistical analysis of the models’ parameters was performed by using the Wilcoxon rank sum test. The null hypothesis of no difference between the groups of interest was tested at the 5% level of significance, and this is presented by p-values. 4. RESULTS AND DISCUSSIONS 4.1 The order of the model Fig. 2 shows typical fittings for two DSA time series, one from each group (AMR and no-AMR), by the three suggested models. The fitting results for models M1 -M3 in Fig. 2 (a) indicate a winning model candidate M3 . M1 failed to describe the dynamics of the time series, which corresponds to a large RMSE value of 1573 shown in Table 1. M2 successfully captured the initial fall, but failed to follow the trend of DSA after day 50, which can also be indicated by a large RMSE value of 305 shown in Table. 1. M3 , on the other hand, captured the falling part and the later trend with a small RMSE value of 81. Fig. 2 (b) represents different dynamics, where a cluster of data gathered around day 20 during the course of the fall. This was a common feature observed from the majority of the time series under investigation, and therefore required attention. Both the first and second order system could not capture the temporary stall of falling; however, the third order system was able to show the change in speed presented in the magnified box 1 in Fig. 2 (b). Also, the fitting by M1 and M2 was almost not distinguishable after day 70, but both models underestimated the settling level of DSAs; the fitting from M3 otherwise correctly estimated the settled value and gave better description at another clustered region near day 30 (see magnified box 2 in Fig. 2 (b)). For both cases, M1 and M2 were ruled out based on their incapability of describing important features. As M1 and M2 are the particular cases of the high order model M3 which successfully captured more detailed information, M3 was chosen to be the successful model. It is worth noticing that the free energy of M1 was the largest and the free energy of M3 was the smallest due to heavy penalisation of complex structure by the inference method; however, RMSD comparison from Table 1 confirmed a dramatic improvement in fitting when the order of the model was increased to three. 4.2 Parameter comparison between no-AMR and AMR groups Organ transplantation leads to major external disturbance of the immune system. A successful transplantation re252

Fig. 2. Typical fitting results compared among the three models for (a) B39 DSA for a patient from no-AMR group; (b) DSA DRB301 for a patient from AMR group. The measured values are indicated by circles.

Fig. 3. Boxplot for the settling values compared between no-AMR and AMR groups quires a new homoeostasis to be established in the immune system such that the measured amount of DSA settles at a low and relatively stable level once the rates of change in the production and the removal of DSA reduce. The new settling level is of interest since the DSA level is highly correlated with graft loss. From the Eqn. (1), θ0 /θ1 defines the settling level assuming the system reaches a stable state. A comparison of the settling value between two groups is shown in Fig. 3. Since the time series are selected based on their common dynamics of rising and falling, the majority of the settling values in both groups are less than 1000 AU, showing no significant differences between AMR and no-AMR groups. However, a comparison of the parameters (θ0 , θ1 , θ2 and θ3 ) separately between the two groups shows significant differences. From the boxplots in Fig. 4 (a) - (d), the median values of all four parameters show significant differences between the AMR and no-AMR groups. It is also worth noticing that the range of the parameter values

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Fig. 5. Boxplot of the square root of the inferred intensity of the noise compared between the no-AMR group and AMR group Fig. 4. Boxplot for the inferred parameters θ0 , θ1 , θ2 , θ3 in the AMR group is much wider than in the no-AMR group, which indicates more diverse dynamic behaviour in DSA in the AMR group. Even though, DSA levels settle at low levels for both groups, the contrast between Fig. 3 and Fig. 4 implies that the dynamic behaviour in AMR group might be controlled by more complex and diverse underlying mechanisms. Noise accounts for measurement error due to the inaccuracy of the MFI readings, and the imperfection of the model. The intensities of noise were compared between no-AMR and AMR groups as shown in Fig. 5. The pvalue (0.08) between the two groups is higher than 0.05; however, a smaller noise intensity and more compact range of value across the no-AMR group compared with AMR group is noticeable. Since we used the same method to measure the DSA level in both groups, it is sensible to believe that the difference between the distributions of the noise in the two groups is mainly caused by a different level of model imperfection. A smaller noise indicates a better model description for no-AMR group; the higher level of noise in AMR group could be caused by some nonlinear dynamic features which cannot be captured by the linear model. A further development of a nonlinear model is under investigation. It is also worth noticing that the choice of Jeffery’s priors for the noise intensity at the beginning of the iterative optimisation procedure is not ideal for the parameter estimation. A more informative prior may limit the flexibility of the model, but a carefully chosen informative prior could improve the estimation of deterministic parameters and parameters related to noise description. The choice of the priors is not straightforward, and an appropriate methodology is also under development. 4.3 Eigenvalues As known (Arnold (1973)), the solution of the third order linear state space model: x˙ t xt 0 0 x ¨ x ˙ = A + (5) t t ... xt θ0 x ¨t where 0 1 0 0 0 1 (6) A= −θ1 −θ2 −θ3 253

is defined by eigenvalues λ1 , λ2 , λ3 of matrix A, eigenvectors, and corresponding initial conditions. Sum of the eigenvalues defines the divergence of vector field (phase volume V (t)) in the state space (Arnold (1973)): V (t) = V0 e(λ1 +λ2 +λ3 )t = V0 eRt , (7) where R can be interpreted the dissipation rate of the antibodies. The eigenvalues for every DSA time series were calculated using the inferred parameters θ1 , θ2 and θ3 . Each DSA time series in the cohort are characterised by 3 eigenvalues, one of which is real, λ1 , and two are complex conjugate, λ2,3 = λr ± iλi . All eigenvalues λ1 and real parts of λ2 and λ3 were negative confirming that the system generates stable solutions for each DSA type. The system’s dynamics for each DSA demonstrates two characteristic times (decay rates), defined by λ1 and λr , with some oscillations, the frequency of which is determined by λi . Characteristic times, frequencies of oscillations and antibody dissipation rates were compared for AMR and noAMR groups: there were 42 distinctive real parts, λr , and 21 imaginary parts, λi , for the statistical analysis. There was no significant difference (p > 0.05) in either characteristic times, associated with the largest or smallest real parts of the eigenvalues, for AMR and no-AMR groups. However, the sum of eigenvalues R was greater in AMR group (Fig. 6 (a)), which indicated that dissipation rate of DSA in AMR group is faster than in no-AMR group. There was also a significant difference in the values of imaginary parts between AMR and no-AMR groups. Fig. 6 (b) shows the comparison of the imaginary parts of the eigenvalues between the two groups. The larger values of the imaginary parts in AMR group represent a higher frequency of oscillation, which indicates a stronger regulation during the transient process for the patients with AMR. One hypothesis of this regulation is the possible production of a secondary antibody (such as anti-idiotype) which targets at the dramatically increased DSA, resulting in a battling force between the DSA production and secondary antibody production (see reference Reed et al. (1987)). 5. CONCLUSIONS A data driven mathematical model in the form of differential equations has been developed for the first time to describe the post transplant dynamics of ‘falls’ in DSA


Fig. 6. Boxplots for (a) the dissipation rate R and (b) the imaginary parts λi of the eigenvalues for no-AMR and AMR groups for patients with and without acute AMR episodes. A variational Bayesian method was successfully applied to select the order of the model and infer the parameters of the model. A third order linear model outperformed the models with lower orders and is capable of capturing the common features in DSA dynamics during the early stage after transplantation. The inferred deterministic parameters were found to be significantly different for AMR and no-AMR groups. The intensities of noise obtained by a hierarchical inference procedure were also found to be different in these two groups. The results showed a higher frequency of oscillations and a faster antibody decay rate for the AMR group. This study comprises a pilot research on data-driven model development for early post transplant antibody dynamics. The properties that we have observed in system’s dynamics require further investigations, which are in progress. REFERENCES Arnold, L. (1973). Stochastic Differential Equations : Theory and Applications. Beal, M.J. (2003). Variational algorithms for approximate Bayesian inference. Ph.D. thesis. Daunizeau, J., Friston, K.J., and Kiebel, S.J. (2009). Variational Bayesian identification and prediction of stochastic nonlinear dynamic causal models. Physica D. Nonlinear phenomena, 238(21), 2089–2118. Feucht, H.E. and Opelz, G. (1996). The humoral immune response towards HLA class II determinants in renal transplantation. 50, 1464–1475. Friston, K., Stephan, K., Li, B., and Daunizeau, J. (2010). Generalised Filtering. Mathematical Problems in Engineering, 1–34. Gelman, A. (2006). Prior distributions for variance parameters in hierarchical models. Bayesian analysis, 1(3), 515–533. Higgins, R., Lowe, D., Hathaway, M., Kashi, H., Tan, L.C., Imray, C., Fletcher, S., Chen, K., Krishnan, N., Hamer, R., and Others (2009). Rises and falls in donor-specific and third-party HLA antibody levels after antibody incompatible transplantation. Transplantation, 87(6), 882–888. 254

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