A mathematical method for constraint-based cluster analysis towards optimized constrictive diameter smoothing of saphenous vein grafts

  • Thomas FranzEmail author
  • B. Daya Reddy
  • Paul Human
  • Peter Zilla
Original Paper


This study was concerned with the cluster analysis of saphenous vein graft data to determine a minimum number of diameters, and their values, for the constrictive smoothing of diameter irregularities of a cohort of veins. Mathematical algorithms were developed for data selection, transformation and clustering. Constrictive diameter values were identified with interactive pattern evaluation and subsequently facilitated in decision-tree algorithms for the data clustering. The novel method proved feasible for the analysis of data of 118 veins grafts, identifying the minimum of two diameter classes. The results were compared to outcome of a statistical recursive partitioning analysis of the data set. The method can easily be implemented in computer-based intelligent systems for the analysis of larger data sets using the diameter classes identified as initial cluster structure.


Saphenous vein Diameter irregularity Constrictive smoothing Mathematical model Pattern recognition 



Constriction degree required for complete smoothing of vein i, by reducing the vein’s maximum diameter D max,i, to the minimum diameter D min,i , where i = 1 to n


Constriction degree applied to vein i, to reduce the maximum diameter D max,i to the constricted diameter d (k), where i = 1 to n and k = 1 to l


Constriction degree applied to vein i, to reduce the diameter D i,j at each measurement position x i,j along the length of the vein to the constriction diameter \( d_{\text{P}}^{s}\) proposed with the recursive partitioning, where i = 1 to n and s = 1 to l P

\( C_{\text{A}}^{\max } \)

Maximum permissible constriction degree applied to a vein to reduce the maximum diameter D max,i to the constricted diameter d (k), where i = 1 to n and k = 1 to l


Mean of applied constriction degree C A,i for all veins i, with i = 1 to n across all constriction diameters d (k), with k = 1 to l

\( C_{\text{A}}^{(k)} \)

Mean of applied constriction degree C A,i for all veins z with z = 1 to n k for one constriction diameters d (k), with k = 1 to l


Outer diameter of vein i measured at luminal pressure associated with post-harvest leak test syringe inflation at measurement position x i,j , where i = 1 to n and j = 1 to m i (mm)


Constriction diameter for vein i, i.e. outer diameter of vein i after constriction, where i = 1 to n (mm)


Maximum value of the outer diameter D i,j of vein i at luminal pressure associated with post-harvest leak test syringe inflation, where i = 1 to n (mm)


Minimum value of the outer diameter, D i,j , of vein i at luminal pressure associated with post-harvest leak test syringe inflation, where i = 1 to n (mm)

\( D_{\text{max,z}}^{(k)} \)

Maximum outer diameter of vein z of subset k, where z = 1 to n k and k = 1 to l (mm)

\( D_{\text{min,z}}^{(k)} \)

Minimum outer diameter of vein z of subset k, where z = 1 to n k and k = 1 to l (mm)


Constrictive smoothing diameter for all veins i, where i = 1 to n (mm)


Constrictive smoothing diameter for a subset k of veins z, where k = 1 to l and z = 1 to n k (mm)


Partitions of D min,i determined with the recursive partitioning method, that represent constrictive smoothing diameters with i = 1 to n and s = 1 to l P (mm)


Identifier of a vein in full set (i = 1 to n)


Identifier of the measurement position along a vein (j = 1 to m i )


Identifier of a vein set (k = 1 to l)


Harvested length of vein i, where i = 1 to n (mm)


Number of vein sets: l = 1 for the entire set of veins i with i = 1 to n and l > 1 if it is required to divide the set of n veins into subsets k of veins


Number of partitions \(d_{\text{P}}^{s}\) determined with the recursive partitioning method


Number of measurement points for the outer diameter D i,j along vein i with the length L i , where m i  = int (L i /20)


Number of veins in analysis


Number of veins excluded from analysis


Number of veins in subset k where k = 1 to l

\( \hat{n}_{k} \)

Normalized number of veins in subset k: \( \hat{n}_{k} = {{n_{k} } \mathord{\left/ {\vphantom {{n_{k} } {\sum\nolimits_{k = 1}^{l} {n_{k} } }}} \right. \kern-\nulldelimiterspace} {\sum\nolimits_{k = 1}^{l} {n_{k} } }}, \) where k = 1 to l (%)


Identifier of partitions of D min,i , \(d_{\text{P}}^{s} ,\) determined with the recursive partitioning method, where s = 1 to l P


Position of D i,j measurement along vein i, where i = 1 to n and j = 1 to m i


Identifier of a vein in subset k, where z = 1 to n k and k = 1 to l



The authors wish to acknowledge Dr. L. Moodley, Dr. W. Lichtenberg, Dr. J. Scherman, Dr. H. Legodi, Dr. G. Mphahlele and Dr. I. Taunyane of the Chris Barnard Division of Cardiothoracic Surgery, Groote Schuur Hospital, Cape Town, for performing the saphenous vein dimensional measurements. This research was mainly funded through a research collaboration grant by Medtronic Inc. (Minneapolis, MN, USA) to the University of Cape Town and a THRIP grant of the National Research Foundation (South Africa). The salary of TF was fully funded and that of PH was partially funded from the Medtronic research grant.


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Copyright information

© International Federation for Medical and Biological Engineering 2010

Authors and Affiliations

  • Thomas Franz
    • 1
    • 3
    Email author
  • B. Daya Reddy
    • 2
  • Paul Human
    • 1
  • Peter Zilla
    • 1
  1. 1.Chris Barnard Division of Cardiothoracic SurgeryUniversity of Cape TownCape TownSouth Africa
  2. 2.Department of Mathematics and Applied MathematicsUniversity of Cape TownCape TownSouth Africa
  3. 3.Faculty of Health Sciences, Cardiovascular Research UnitUniversity of Cape TownObservatory, Cape TownSouth Africa

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