thursday 15h45 vicent_ribas esann

Introduction Database Description Risk of Death Assessment from Observed Data Conclusions

A Quotient Basis Kernel for the prediction ofmortality in severe sepsis patients

Vicent J. Ribas

LSI - SOCOTechnical University of Catalonia (UPC)

Barcelona

April 19, 2013

Vicent J. Ribas LSI - SOCO Technical University of Catalonia (UPC) Barcelona

A Quotient Basis Kernel for the prediction of mortality in severe sepsis patients

Contents

1 Introduction

2 Database Description

3 Risk of Death Assessment from Observed Data

4 Conclusions

Contents

1 Introduction

4 Conclusions

Introduction

Sepsis is a clinical syndrome defined by the presence ofinfection and Systemic Inflammatory Response Syndrome(SIRS).

This can lead to severe sepsis or to septic shock (severe sepsiswith hypotension refractory to fluid administration) andmulti-organ failure.

In western countries, septic patients account for as much as25% of ICU bed utilization and the pathology occurs in 1% -2% of all hospitalizations.

The mortality rates of sepsis range from 12.8% for sepsis and20.7% for severe sepsis, and up to 45.7% for septic shock.

Introduction

The medical management of sepsis and the study of itsprognosis and outcome is a relevant medical researchchallenge.

Provided that such methods are to be used in a clinicalenvironment (ICU), it requires prediction methods that arerobust, accurate and readily interpretable.

Contents

1 Introduction

4 Conclusions

Dataset

A prospective observational cohort study of adult patients withsevere sepsis was conducted at the Critical Care Departmentof the Vall d’ Hebron University Hospital (Barcelona, Spain).

Data from 354 patients with severe sepsis was collected inthis ICU between June, 2007 and December, 2010.

55% of cases correspond to ‘medical’ sepsis.

The mean age of the patients in the database was 57.08 (withstandard deviation ±16.65) years.

40% of patients were female.

Dataset

Available Attributes

The collected data show the worst values for all variablesduring the first 24 hours of evolution of Severe Sepsis.

Organ dysfunction was evaluated by means of the SOFA scoresystem, which objectively measures organ dysfunction for 6organs/systems.

Cardiovascular (CV) 2.86 (1.62) Haematologic (HAEMATO) 0.78 (1.14)Respiratory (RESP) 2.31 (1.15) Global SOFA score 7.94 (3.86)

Central Nerv. Sys. (CNS) 0.48 (1.00) Dysf. Organs 1.68 (1.09)Hepatic (HEPA) 0.48 (0.92) Failure Organs 1.51 (1.02)Renal (REN) 1.06 (1.20) Total Dysf. Organs 3.18 (1.32)

The collected data show the worst values for all variablesduring the first 24 hours of evolution of Severe Sepsis.

Organ dysfunction was evaluated by means of the SOFA scoresystem, which objectively measures organ dysfunction for 6organs/systems.

Cardiovascular (CV) 2.86 (1.62) Haematologic (HAEMATO) 0.78 (1.14)Respiratory (RESP) 2.31 (1.15) Global SOFA score 7.94 (3.86)

Central Nerv. Sys. (CNS) 0.48 (1.00) Dysf. Organs 1.68 (1.09)Hepatic (HEPA) 0.48 (0.92) Failure Organs 1.51 (1.02)Renal (REN) 1.06 (1.20) Total Dysf. Organs 3.18 (1.32)

Severity was evaluated by means of the APACHE II score,which was 23.03± 9.62 for the population under study.

Mechanical ventilation was also assessed (66.71%).

Compliance with the SSC resuscitation bundles was 31.41%.

The mortality rate intra-ICU for our study population was26.32%.

Why generative kernels?

Selection of kernel function for a given problem is not trivial.

One normally must have good insight about the problem athand.

Mapping over higher dimensions simplifies the problem butcomputational cost grows with ∼ d3.

Solution: exploit the statistical structure of the data to buildthe kernel.

Requirement: pdf must be a regular exponential family. Thisrequirement is fulfilled by the dataset analysed in this work.

Quotient Basis Kernel

Let A be a set of n unique points A = {a1, . . . , an} and τ a termordering. A Grobner basis of A, G = g1, . . . , gt , is a Grobner basisof I (A). Therefore, the points in A can be presented as the set ofsolutions of

g1(a) = 0g2(a) = 0· · ·

gt(a) = 0

Let A, be a set of n × s unique support points A = {a1, . . . , an}and τ a term ordering. A monomial basis of the set of polynomialfunctions over A is

ESTτ = {xα : xα /∈ 〈LT(g) : g ∈ I (A)〉}

This means that ESTτ comprises the elements xα that are notdivisible by any of the leading terms of the elements of the Grobnerbasis of I (A).

Let τ be a term ordering and let us consider an ordering over thesupport points A = {a1, . . . , an}. We call design matrix (i.e. ESTτevaluated in A) the following n × c matrix

Z = [ESTτ ]∣∣A

where c is the cardinality of ESTτ and n is the number of supportpoints. The covariance of the design matrix of ESTτ , which is akernel, is the QBK.

Regular Exponential Family

Let P = (P|η ∈ N) be a regular exponential family with canonicalsufficient statistic T . Then the log likelihood function takes theform

l(η|T ) = n(ηtT − G (η))

Kernels based on the Jensen Shannon Metric

Let P = (P|η ∈ N) be a regular exponential family with canonicalsufficient statistic T . Then the log likelihood function takes theform

l(η|T ) = n(ηtT − G (η))

This function accepts a convex - conjugate (Legendre Dual) of theform

l(γ|T ) = n(γtT − F (γ))

In our case, the dual F is the negative log-entropy function.

JS(γ1, γ2) =F (γ1) + F (γ2)

2− F

(γ1 + γ2

Centred kernel:φ(x , y) = JS(x , x0) + JS(y , x0)− JS(x , y)− JS(x0, x0).

Exponentiated kernel: φ(x , y) = exp(−tJS(x , y)) ∀t > 0.

Inverse kernel: φ(x , y) = 1t+JS(x ,y) ∀t > 0.

JS(γ1, γ2) =F (γ1) + F (γ2)

2− F

(γ1 + γ2

JS(γ1, γ2) =F (γ1) + F (γ2)

2− F

(γ1 + γ2

Contents

1 Introduction

4 Conclusions

RoD with Generative Kernels - QBK -

x1 is the Number of Dysfunctional Organs as measured by theSOFA Score.

x2 corresponds to Mechanical Ventilation (yes/no).

x3 corresponds to Severity as Measured by the APACHE IIScore.

x4 corresponds to the SSC Resuscitation Bundles (i.e.administration of antibiotics, performance of haemoculturesand so on). This is also a binary variable.

RoD with Generative Kernels - QBK -

ESTτ =

1, x4, x3, x3x4,x2, x2x4, x2x3, x2x3x4,x22 , x

22x4, x

22x3, x

22x3x4,

x32 , x32x4, x

32x3, x

32x3x4,

x42 , x42x4, x

42x3, x

42x3x4,

x52 , x52x4, x

62 , x1,

x1x4, x1x3, x1x3x4, x1x2,x1x2x4, x1x2x3, x1x2x3x4, x1x

x1x22x4, x1x

22x3, x1x

22x3x4, x1x

x1x32x4, x1x

32x3, x1x

32x3x4, x1x

x1x42x4, x1x

42x3, x1x

RoD with Generative Kernels

We have used Matlab’s Support Vector Machine QP solverimplemented in the BioInformatics and Optimization Toolboxes.We have also used 10-fold cross validation to evaluate theclassification performance for the different kernels. A grid searchyielded the appropriate values for C parameters for each Kernel.More particularly,

Quotient Basis and Fisher C = 1.

Generative Kernels C = 10. Also the parameter t for theExponential and Inverse Kernels was set to 2.

Gaussian, Linear and Polynomial Kernels C = 10.

The Risk-of-Death (ROD) formula based on the APACHE II scoreis a standard method in use in the critical care field.

1− ROD

)= −3.517 + 0.146 · A + ε, (1)

where A is the APACHE II score and ε is a correction factor thatdepends on clinical traits at admission in the ICU. For instance, ifthe patient has undergone post-emergency surgery, ε is set to0.613.

Results

Kernel AUC Error Rate Sens. Spec. CPU time [s]

Quotient 0.89 0.18 0.70 0.86 1.45

Exponential 0.75 0.21 0.70 0.82 1.64

Inverse 0.62 0.22 0.70 0.82 1.68

Centred 0.75 0.21 0.70 0.82 1.99

Gaussian 0.83 0.24 0.65 0.81 1.56

Poly (order 2) 0.69 0.28 0.71 0.76 1.59

Linear 0.62 0.26 0.62 0.78 1.35

Apache II 0.70 0.28 0.55 0.82 n/a

Contents

1 Introduction

4 Conclusions

Conclusions

The investigated kernels provided accurate and medicallyactionable results, whilst keeping an acceptable balancebetween the different parameters of interest (accuracy rate,sensitivity and specificity).

The QBK is defined through the Grobner basis of an algebraicideal.

It outperforms all kernels presented in this work as well as theclinical standard method based on the APACHE II score.

All kernels presented outperform the standard APACHE IIROD formula in terms of accuracy.

THANK YOU!!

thursday 15h45 vicent_ribas esann

Technology

severe sepsis patientsvicent

prediction of mortality

sepsis and20

prediction methods

prediction ofmortality

septic patients account

presence ofinfection

clinical syndrome dened