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Kazumasa Nomura* and Paul Terwilliger
e-Polymers 2019; 19: 385–410
Review Article
Tejas V. Shah and Dilip V. Vasava*
Self-dual Leonard pairs
https://doi.org/10.1515/spma-2019-0001
Received May 8, 2018; accepted September 22, 2018
Open Access
A glimpse of biodegradable
polymers
and
a pair A, A of diagonalizable
F-linear maps
on V, their
each of which acts on an eigenba
irreducible tridiagonal fashion. Such a pair is called a Leonard pair. We consider th
biomedical applications
there exists an automorphism of the endomorphism algebra of V that swaps A and A
Abstract: Let F denote a field and let V denote a vector space over F with finite pos

is unique, and called the duality A ↔ A∗ . In the present paper we give a compreh
https://doi.org/10.1515/epoly-2019-0041
NIPAMwe
– display
N-isopropylacrylamide
duality. In particular,
an invertible F-linear map T on V such that the map
Received December 04, 2018; accepted March 29, 2019.

DEAM
N,
N-diethyl
acrylamide
A ↔ A . We express T as a polynomial
in A and A∗ . We describe how T acts on 4
– N-vinylcaprolactam
and 24 bases forNVC
V.
Abstract: Over the past two decades, biodegradable
MVE – Methyl vinyl ether
polymers (BPs) have been widely used in biomedical
Keywords: Leonard
tridiagonal matrix,
PEO pair,
– poly(ethylene
oxide) self-dual
applications such as drug carrier, gene delivery, tissue
PPO
– poly(propylene oxide)
engineering, diagnosis, medical devices, Classification:
and antibac- 17B37, 15A21
AAc
– Acrylic acid
terial/antifouling biomaterials. This can be attributed
AAm – Acrylamide
to numerous factors such as chemical, mechanical and
physiochemical properties of BPs, their improved processibility, functionality and sensitivity towards stimuli.
1 Introduction
The present review intended to highlight main results
of research on advances and improvements
of a field and let V denote a vector space over F with finite positive d
LetinFterms
denote
Petroleum-based polymers (commonly known as

synthesis, physical properties, stimuli response,
pair A,and/or
A of diagonalizable
maps
on V,
eachtime
of which
acts
Plastics) haveF-linear
been used
since
a long
due to
its on an eigenba
applicability of biodegradable plastics (BPs) during last
irreducible tridiagonal
fashion.
Such
a
pair
is
called
a
Leonard
pair
(see [13, Definit
handiness, durability, flexibility and less reactivity
two decades, and its biomedical applications.
∗ Recent
A, A is said to towards
be self-dual
exists
an automorphism
the whenever
water andthere
other
chemicals,
also these of the endom
literature relevant to this study has been cited and their ∗
are such
easy an
and
cheaper to is
process.
swaps A and A products
. In this case
automorphism
unique,These
and called the dua
developing trends and challenges of BPs have also been
properties
allow
synthetic
plastics
to
be
broadly
used
in For instance (
The literature contains many examples of self-dual Leonard pairs.
discussed.
many fields
(1). The
for plastic
is boosting
day
ated with an irreducible
module
for demand
the Terwilliger
algebra
of the hypercube
(see [4,
by day, the production of plastic was between 280-290
Leonard
pair of Krawtchouk type (see [10, Definition 6.1]); (iii) the Leonard pair asso
Keywords: biodegradable polymers; polylactic
acid; drug
million tons in 2012, exceeded 300 million tons in 2014.
module for the Terwilliger algebra of a distance-regular graph that has a spin mode
delivery; orthopedic application; tissue engineering
It is alleged that only half of this plastic, has been
bra (see [1, Theorem], [3, Theorems 4.1, 5.5]); (iv) an appropriately normalized tota
collected for recycling or reuse, while others have been
Abbreviations
(see [11, Lemmaremained
14.8]); (v)littered
the Leonard
pair away
consisting
any two
of a modular Leo
or thrown
in theofoceans
(2-4).
Definition
1.4]);
(vi)
the
Leonard
pair
consisting
of
a
pair
of
opposite
PGA
– poly(glycolic acid)
Most of the petroleum-derived polymers are resistant generators fo
bra, acting on an
module
(see of
[5,them
Proposition
example (i) is a s
PCL
– poly(ε-caprolactone)
toevaluation
degradation
and many
release 9.2]).
toxic The
gases,
examples
(iii),
(iv)
are
special
cases
of
(v).
PBS
– polybutylene succinate
upon incineration, they trigger the pollution and global
PBSA – poly(butylene succinate-co-adipate) Let A, A ∗ denote
warming.
Whereas,
bury
waste are
also A, A ∗ is selfa Leonard
pairsites
on V.toWe
canthis
determine
whether
PTMAT – poly (methylene adipate-co- terephthalate)
limited.
upon being
they put aone.
serious
By [13, Lemma 1.3]
eachThus,
eigenspace
of A,discarded,
A∗ has dimension
Let {θ i }di=0 denote
PVA
– polyvinyl alcohol
impact
on
the
environment
and
surroundings
(5-9).
values of A. For 0 ≤ i ≤ d let v i denote a θ i -eigenvector for A. The ordering {θ i }
PLA
– poly(lactic acid)
However, to conquer
drawbacks of the synthetic
whenever A∗ acts
on the basis {v i }di=0the
in an
irreducible tridiagonal fashion. If the or
PDLA – poly (D- lactic acid)
plastics, dresearchers are intensifying their efforts in
then the ordering {θ d−i }i=0 is also standard, and no further ordering is standard. S
PHB
– poly(hydroxybutyrate)
the development of biodegradable plastics (BPs).
A∗ . Let {θ i }di=0 denote a standard ordering of the eigenvalues of A. Then A, A∗ is self
PBS
– polybutylene succinate
These polymers are vulnerable ∗by variation in pH and
is
a
standard
ordering
of the eigenvalues
of A non-enzymatic
(see [7, Proposition
8.7]).
PBAT – poly (butylene adipate-co-terephthalate)
temperature,
enzymatic and
actions
of micro-organisms, oxidation, reduction, light which
scissions them in smaller portions which are further
degraded into simpler and non-harmful products such
* Corresponding author: Dilip V. Vasava, Department*Corresponding
of Chemistry,
Author:
Nomura:
Tokyo Medical
and Dental
University,
Ichikawa, 2
as CO2, Kazumasa
CH4, water,
and hummus.
BPs can
be used
for
School of Sciences, Gujarat University, Ahmedabad,E-mail: knomura@pop11.odn.ne.jp
packaging materials, disposable non-woven, hygiene
Gujarat- 380009, India, email: dilipvasava20@gmail.com.
Paul Terwilliger: Department of Mathematics, University of Wisconsin, Madison, WI53706, USA,
products, consumer goods, agricultural tools, and much
Tejas V. Shah, Department of Chemistry, School of Sciences, Gujarat
terwilli@math.wisc.edu
more (10-13).
University, Ahmedabad, Gujarat- 380009, India.
1 Introduction
Open Access. © 2019 Shah and Vasava, published byOpen
De Gruyter.
This
work
is licensed
under
the
Creative
Commons
Access.
©
2019
Kazumasa
Nomura
and
Paul
Terwilliger,
published by De Gruyter. This work i
Attribution alone 4.0 License.
Attribution alone 4.0 License.
386
T.V. Shah and D.V. Vasava: A glimpse of biodegradable polymers and their biomedical applications
by authors over the globe with their claimed biomedical
uses have been communicated in detail. Drug delivery,
orthopedics and couple of other applications and trends
of degradable polymers are also highlighted.
2 Common polymers used for
biomedical applications
2.1 Polyhydroxyalkanoates (PHA)
Figure 1: Classification of biodegradable plastics.
BPs can be classified into three major classes (3,10,14)
(Figure 1):
(I)
Natural polymers: These are the polymers which are
derived directly from the natural sources
(II) Semi-synthetic polymers: These are the polymers
in which raw material is obtained from nature, but
polymerization takes place after some chemical
modification.
(III) Synthetic polymers: These are purely synthetic
polymers derived from chemical synthesis.
Owing to their outstanding biocompatibility, low
toxicity and chemically tunable properties, biodegradable
polymers have become excellent materials for biomedical
applications and by observing the recent literature, it is
not exaggerating to say that, BPs have rife applications in
therapeutics (15-17). These applications embrace skillful
release of vaccine, drugs, and proteins (18-20), tissue
engineering and scaffolds (21-23), cardiology (24), urology,
and orthopedics (25-27). However, all BPs does not possess
the characteristics to be used directly; but intense research
has been done and is still going on for the development of
properties of BPs by applying synthetic tactics to amend
the mechanical properties, degradation rates, surface
properties, glass transition temperatures (21,28-33).
The primary intention of this study is to provide
the insights into the properties of the biodegradable
polymers used for biomedical applications along with
the chemical and synthetic developments accomplished
so far to mitigate the inadequacies possessed by the
virgin polymer. Selected chemical, as well as physical
vital parameters of BPs which are decisive for BPs to
be used for specific biomedical applications are also
summarized and discussed. The various BPs, their
blends, composites and nanoscale materials reported
Polyhydroxyalkanoates are the versatile class of biodegradable and biocompatible polyesters produced by microorganisms as an energy storage material via fermentation
process under stressed conditions. A number of bacterial
species from both gram-positive and gram-negative species
are capable of producing PHA ranging from 30-80% of their
cellular dry weight. The general formula of PHA is given in
Figure 2, and till date more than 150 types of different PHA
monomers have been identified and classified, thus allowing
a possibility of preparing an extensive range of polymers and
copolymers with different properties (34-37).
Few common monomers of this family are presented
in Table 1.
Monomer units in PHA possess D (-) configuration and
the molecular mass of PHA depending upon the source,
ranges from 50,000 to 1,000,000 Da. In general, PHAs
contain 1 to 14 carbon atoms at C-3 position, and thus
can also be classified into three subcategories depending
upon number of carbons at C-3 position as:
a) short chain length PHAs (scl-PHA) (up to 5 carbon
atoms),
b) medium chain length PHAs (mcl-PHA) (6 to 14 carbon
atoms), and
c) long chain length PHA (lcl-PHA) (>14 carbon atoms).
The scl-PHAs are brittle in nature and resemble
conventional plastics in terms of properties and are
Figure 2: General formula of PHA.
T.V. Shah and D.V. Vasava: A glimpse of biodegradable polymers and their biomedical applications
387
Table 1: Examples of members of the PHA family (34). (© Reprinted with the permission from Royal Society of Chemistry)
Name
Abbreviation
Side group (R)
Polymer Structure
Poly (3-hydroxybutyrate)
Poly (4-hydroxybutyrate)
Poly (3-hydroxyvalerate)
Poly (3-hydroxyhexanoate)
Poly (3-hydroxyheptanoate)
Poly (3-hydroxyoctanoate)
Poly (3-hydroxynonanoate)
(3-hydroxydecanoate)
Poly (3-hydroxybutyrate-co-3-hydroxyvalerate)
Poly (3-hydroxybutyrate-co-4-hydroxybutyrate)
Poly (3-hydroxyoctanoate-co-hydroyhexanoate)
Poly (3- hydroxybutyrate-co-3-hydroxyoctanoate)
3-hydroxybutyrate-co-3-hydroxydecanoate
P3HB
P4HB
3HV
P3HHx
P3HH
3HO
P3HN
3HD
PHBV
P3/4HB
PHBHHx
PHBO
PHBD
Methyl (-CH3)
Hydrogen (-H)
Ethyl (-C2H5)
Propyl (-C3H7)
Butyl (-C4H9)
Pentyl (-C5H11)
Hexyl (-C6H13)
Heptyl (-C7H15)
-CH3 and -C2H5
-CH3 and -H
-CH3 and -C3H7
-CH3 and -C7H15
-CH3 and -C5H11
Homopolymer
Homopolymer
Homopolymer
Homopolymer
Homopolymer
Homopolymer
Homopolymer
Homopolymer
Copolymer
Copolymer
Copolymer
Copolymer
Copolymer
Table 2: Potential biomedical applications of PHAs (44). (© Reprinted with the permission from Elsevier)
Type of application
Products
Wound Management
Sutures, skin substitutes, nerve cuffs, surgical, meshes, staples, swabs
Vascular System
Heart valves, Cardiovascular fabrics, pericardial patches, vascular grafts
Cartilage
Orthopaedy
cartilage tissue engineering, spinal cages, bone graft substitutes, internal
fixation devices
Drug delivery
Micro- and Nano-spheres for anticancer therapy
Urological
Urological stents
Dental
Barrier material for guided tissue regeneration in periodontitis
Computer-assisted tomography and ultrasound imaging
Contrast Agents
crystalline, stiff and brittle in nature, whereas mcl-PHAs
are amorphous or semi-crystalline, elastic and rubbery in
nature (38,39). Degradation of PHA proceeds via abiotic
pathway where ester bonds are hydrolyzed without any
catalytic enzymes; however, the residual products are
degraded by enzymes during biodegradation (40). PHAs
had been approved by the United States Foods and Drugs
Administration (USFDA) for clinical applications and
used in orthopedics, repair patches, tissue engineering
applications, sustained and controlled drug delivery,
biomedical devices, and artificial organs as shown in
Table 2 (41-43).
2.2 Poly (lactic acid) (PLA) and
poly (glycolic acid) PGA and
poly(lactic acid-co-glycolic acid) PLGA
PGA, PLA belongs to PHA family and they are one of the
widely used BPs for bio-applications along with their
copolymer PLGA which is made up of both PLA and PGA
units (45,46). PGA is greatly hydrophilic and is made
up of glycolic acid (GA) monomer viz lightest weighed
constituent of PHA family having carbon atoms at carboxyl
group and hydroxyl group respectively. Ring-opening
polymerization (ROP) of glycolide (a cyclic diester of GA)
or polycondensation of GA is employed for the preparation
of PGA (47,48). While the degradation of PGA occurs
via citric acid cycle, inflammatory response has been
related to the high concentration of PGA. The precipitous
degradation rate abates the mechanical strength of PGA
and its insolubility in common solvents circumvent the
application of PGA. Because of this, it is either employed
as a filler with other biodegradable polymers or utilized in
short-term tissue scaffold application (49).
PLA is naturally occurring aliphatic polyester obtained
from natural resources like wheat, rice bran, potato starch,
corn and biomass. It is widely utilized in the commodity
as well as biomedical applications owing to its excellent
renewability, biocompatibility and biodegradability. PLA
is one of the widely explored materials to be applied
for various biomedical applications; in fact, it is USFDA
388
T.V. Shah and D.V. Vasava: A glimpse of biodegradable polymers and their biomedical applications
approved material used extensively for therapeutics
applications including surgical implants, tissue culture,
resorbable surgical sutures, wound closure, controlled
release systems, and prosthetic devices (50-52).
The constituent monomer of PLA is 2-hydroxypropionic acid, it is a chiral molecule existing in D- and
L-enantiomeric form and properties of the polymer depend
on stereoregularity of monomers in polymeric backbone;
thus, polymer with required properties can be achieved by
varying amount of enantiomeric stereocenters (D- or L-) in
resulting PLA (53-56). The general properties of lactic acid
polymers are depicted in Table 3.
PLA can be synthesized from lactic acid monomers by
various processes like polycondensation, ROP of lactone
ring (dimer of lactic acid known as lactide) or direct
methods like azeotropic dehydration and enzymatic
polymerization (Figure 3). Among this, polycondensation
is the cheapest method. However, it does not produce
high molecular weight PLA. To achieve high molecular
weight PLA, azeotropic dehydrative condensation or
ROP is generally being employed (57,58). Azeotropic
polycondensation method includes removal of water
molecules using appropriate azeotropic solvents, which
manipulates the equilibrium of polymerization process
to produce relatively high molecular weight polymers at a
temperature lower than melting point temperature of the
polymer (59). Kim and Woo (60) obtained high molecular
weight PLA using Dean-Stark trap and molecular sieve
(3 Å) to remove the water molecules formed during the
polymerization process.
The low to intermediate molecular weight PLA
obtained by polycondensation can be converted to high
molecular weight PLA (>30,000 gmol-1) either using chain
extending agents such as diols or dicarboxylic acids or
esterification adjuvants such as diisocyanates (61-63).
The polar oxygen linkages in PLA are accountable for its
hydrophilic nature, however, the presence of methyl group
side chain imparts hydrophobicity to this polymer and
Table 3: Properties of lactic acid polymer (57). (© Reprinted with the permission from Elsevier)
Lactic acid polymers
Glass transition temperature
Tg (°C)
Melting temperature
Tm (°C)
Density
(g/cm3)
55-80
43-53
40-50
173-178
120-170
120-150
1.290
1.25
1.248
PLLA
PDLLA
PDLA
Figure 3: Synthesis methods of PLA.
Solubility
CHCl3, furan, dioxane, and dioxolane
PLLA solvents and acetone
Ethyl lactate, THF, ethyl acetate,
DMSO, N, N-xylene, and DMF
T.V. Shah and D.V. Vasava: A glimpse of biodegradable polymers and their biomedical applications
thus PLA mildly decomposes under humid conditions.
The degradation of PLA embarks with reduction of
molecular weight (

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