Biomedicine: A fertile, challenging and worthy
field for mathematical and engineering research
K. R. Rajagopal
Texas A&M University
College Station, Texas 77845
”Most of what is characteristic of living organisms cannot be
expressed in mathematical terms of the simplistic laws of physics.”
– E. Mayr
”The existence of life must be considered as an elementary fact
that cannot be explained, but must be taken as a starting point in
biology, in a similar way as the quantum of action, which appears
as an irrational element from the point of view of classical
mechanical physics, taken together with the existence of
elementary particles, forms the foundation of atomic physics.”
– N. Bohr
”Surely no biologists would even express such a hope. It would be
difficult to expect the incredible diversity of nature, the complexity
of the process of ontogenetic differentiation and of the nervous
system, or the qualitative uniqueness of each kind of
macromolecule, could be expressed in the form of a few simple
general laws.”
– E. Mayr
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
”Most of what is characteristic of living organisms cannot be
expressed in mathematical terms of the simplistic laws of physics.”
– E. Mayr
”The existence of life must be considered as an elementary fact
that cannot be explained, but must be taken as a starting point in
biology, in a similar way as the quantum of action, which appears
as an irrational element from the point of view of classical
mechanical physics, taken together with the existence of
elementary particles, forms the foundation of atomic physics.”
– N. Bohr
”Surely no biologists would even express such a hope. It would be
difficult to expect the incredible diversity of nature, the complexity
of the process of ontogenetic differentiation and of the nervous
system, or the qualitative uniqueness of each kind of
macromolecule, could be expressed in the form of a few simple
general laws.”
– E. Mayr
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
”Most of what is characteristic of living organisms cannot be
expressed in mathematical terms of the simplistic laws of physics.”
– E. Mayr
”The existence of life must be considered as an elementary fact
that cannot be explained, but must be taken as a starting point in
biology, in a similar way as the quantum of action, which appears
as an irrational element from the point of view of classical
mechanical physics, taken together with the existence of
elementary particles, forms the foundation of atomic physics.”
– N. Bohr
”Surely no biologists would even express such a hope. It would be
difficult to expect the incredible diversity of nature, the complexity
of the process of ontogenetic differentiation and of the nervous
system, or the qualitative uniqueness of each kind of
macromolecule, could be expressed in the form of a few simple
general laws.”
– E. Mayr
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
I will confine my discussion to some issues in
1
Cardiac Mechanics
And if time permits to some issues concerning
2
Growth and Remodeling
1
Cardiac Mechanics
Pathologies of Clot formation
Arterial Mechanics
Aortic Dissection
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Cardiac Mechanics
Pathologies of Clot formation and Lysis:
Causes for thrombosis (Virchow Triad)
1
Local flow stasis/stagnation
2
Blood Vessel Injury or Endothelial Dysfunction
Hypercoagulability (an augmented native tendency of blood to
form clots)
3
Disorders of pathologic thrombus formation and maintenance
Disorders characterized by impaired thrombus formation and
Maintenance
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Cardiac Mechanics
Pathologies of Clot formation and Lysis:
Causes for thrombosis (Virchow Triad)
1
Local flow stasis/stagnation
2
Blood Vessel Injury or Endothelial Dysfunction
Hypercoagulability (an augmented native tendency of blood to
form clots)
3
Disorders of pathologic thrombus formation and maintenance
Disorders characterized by impaired thrombus formation and
Maintenance
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Cardiac Mechanics
Pathologies of Clot formation and Lysis:
Causes for thrombosis (Virchow Triad)
1
Local flow stasis/stagnation
2
Blood Vessel Injury or Endothelial Dysfunction
Hypercoagulability (an augmented native tendency of blood to
form clots)
3
Disorders of pathologic thrombus formation and maintenance
Disorders characterized by impaired thrombus formation and
Maintenance
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Cardiac Mechanics
Pathologies of Clot formation and Lysis:
Causes for thrombosis (Virchow Triad)
1
Local flow stasis/stagnation
2
Blood Vessel Injury or Endothelial Dysfunction
Hypercoagulability (an augmented native tendency of blood to
form clots)
3
Disorders of pathologic thrombus formation and maintenance
Disorders characterized by impaired thrombus formation and
Maintenance
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Cardiac Mechanics
Pathologies of Clot formation and Lysis:
Causes for thrombosis (Virchow Triad)
1
Local flow stasis/stagnation
2
Blood Vessel Injury or Endothelial Dysfunction
Hypercoagulability (an augmented native tendency of blood to
form clots)
3
Disorders of pathologic thrombus formation and maintenance
Disorders characterized by impaired thrombus formation and
Maintenance
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
1
Atrial thrombosis
Caused mainly due to atrial dysrhythmias (atrial fibrillation
and atrial flutter)
There is local flow stagnation leading to atrial thrombus
formation.
2
Ventricular thrombosis
Mainly due to severe systolic ventricular dysfunction and
ventricular aneurysms characterized by regional ventricular wall
dilation and thinning that is associated with paradoxical
expansion during ventricular systole that is associated with a
high rate of intra-cavitary thrombus formation.
3
Mostly a problem with artificial mechanical valves due to
non-endothelialized surfaces.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
1
Atrial thrombosis
Caused mainly due to atrial dysrhythmias (atrial fibrillation
and atrial flutter)
There is local flow stagnation leading to atrial thrombus
formation.
2
Ventricular thrombosis
Mainly due to severe systolic ventricular dysfunction and
ventricular aneurysms characterized by regional ventricular wall
dilation and thinning that is associated with paradoxical
expansion during ventricular systole that is associated with a
high rate of intra-cavitary thrombus formation.
3
Mostly a problem with artificial mechanical valves due to
non-endothelialized surfaces.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
1
Atrial thrombosis
Caused mainly due to atrial dysrhythmias (atrial fibrillation
and atrial flutter)
There is local flow stagnation leading to atrial thrombus
formation.
2
Ventricular thrombosis
Mainly due to severe systolic ventricular dysfunction and
ventricular aneurysms characterized by regional ventricular wall
dilation and thinning that is associated with paradoxical
expansion during ventricular systole that is associated with a
high rate of intra-cavitary thrombus formation.
3
Mostly a problem with artificial mechanical valves due to
non-endothelialized surfaces.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
1
Atrial thrombosis
Caused mainly due to atrial dysrhythmias (atrial fibrillation
and atrial flutter)
There is local flow stagnation leading to atrial thrombus
formation.
2
Ventricular thrombosis
Mainly due to severe systolic ventricular dysfunction and
ventricular aneurysms characterized by regional ventricular wall
dilation and thinning that is associated with paradoxical
expansion during ventricular systole that is associated with a
high rate of intra-cavitary thrombus formation.
3
Mostly a problem with artificial mechanical valves due to
non-endothelialized surfaces.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
1
Atrial thrombosis
Caused mainly due to atrial dysrhythmias (atrial fibrillation
and atrial flutter)
There is local flow stagnation leading to atrial thrombus
formation.
2
Ventricular thrombosis
Mainly due to severe systolic ventricular dysfunction and
ventricular aneurysms characterized by regional ventricular wall
dilation and thinning that is associated with paradoxical
expansion during ventricular systole that is associated with a
high rate of intra-cavitary thrombus formation.
3
Mostly a problem with artificial mechanical valves due to
non-endothelialized surfaces.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
1
Atrial thrombosis
Caused mainly due to atrial dysrhythmias (atrial fibrillation
and atrial flutter)
There is local flow stagnation leading to atrial thrombus
formation.
2
Ventricular thrombosis
Mainly due to severe systolic ventricular dysfunction and
ventricular aneurysms characterized by regional ventricular wall
dilation and thinning that is associated with paradoxical
expansion during ventricular systole that is associated with a
high rate of intra-cavitary thrombus formation.
3
Mostly a problem with artificial mechanical valves due to
non-endothelialized surfaces.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
4
Arterial thrombosis
Arterial insufficiency or impaired local arterial blood flow
(ischemia) and oxygen delivery.
Acute coronary syndromes: Thrombus formation over unstable
plaque, critical stenosis reached by a stable plaque, coronary
vasospasm and acute increase in myocardial oxygen
consumption demand.
Extremity arterial insufficiency: Due to thrombus formation
over unstable plaque in patients with pre-existent
atherosclerotic disease. Formation of thrombo- or athero
embolism.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
5
Capillary thrombosis
Not well understood. Associated with disseminated
intravascular coagulation.
6
Venous thrombosis and Pulmonary thrombo-embolism
Genetic disorders in which coagulation factors are synthesized
in excessive amounts or anti-coagulant or fibrinolytic factors
are synthesized in inadequate amounts.
Factor V Leiden
Mutant prothrombin
Protein C deficiency
Protein S deficiency
AT-III deficiency
Endothelial dysfunction or injury
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Bleeding Disorders
(a) Platelet disorders: Thrombocytopenia (decreased blood
platelet concentration) Platelet count ≤ 20000. Many
iatrogenic processes lead to platelet disorders. Also,
endogenous disease states lead to platelet dysfunction.
(b) Disorders of Coagulation factors and Fibrinolysis: Reduced
level of activity of coagulation factors. Pathologically activated
fibrinolysis. Factor deficiencies.
Hemophilia A
Hemophilia B
Liver failure (Depression of coagulation factor levels)
Disseminated intravascular coagulation
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Bleeding Disorders
(a) Platelet disorders: Thrombocytopenia (decreased blood
platelet concentration) Platelet count ≤ 20000. Many
iatrogenic processes lead to platelet disorders. Also,
endogenous disease states lead to platelet dysfunction.
(b) Disorders of Coagulation factors and Fibrinolysis: Reduced
level of activity of coagulation factors. Pathologically activated
fibrinolysis. Factor deficiencies.
Hemophilia A
Hemophilia B
Liver failure (Depression of coagulation factor levels)
Disseminated intravascular coagulation
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Bleeding Disorders
(a) Platelet disorders: Thrombocytopenia (decreased blood
platelet concentration) Platelet count ≤ 20000. Many
iatrogenic processes lead to platelet disorders. Also,
endogenous disease states lead to platelet dysfunction.
(b) Disorders of Coagulation factors and Fibrinolysis: Reduced
level of activity of coagulation factors. Pathologically activated
fibrinolysis. Factor deficiencies.
Hemophilia A
Hemophilia B
Liver failure (Depression of coagulation factor levels)
Disseminated intravascular coagulation
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Bleeding Disorders
(a) Platelet disorders: Thrombocytopenia (decreased blood
platelet concentration) Platelet count ≤ 20000. Many
iatrogenic processes lead to platelet disorders. Also,
endogenous disease states lead to platelet dysfunction.
(b) Disorders of Coagulation factors and Fibrinolysis: Reduced
level of activity of coagulation factors. Pathologically activated
fibrinolysis. Factor deficiencies.
Hemophilia A
Hemophilia B
Liver failure (Depression of coagulation factor levels)
Disseminated intravascular coagulation
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Bleeding Disorders
(a) Platelet disorders: Thrombocytopenia (decreased blood
platelet concentration) Platelet count ≤ 20000. Many
iatrogenic processes lead to platelet disorders. Also,
endogenous disease states lead to platelet dysfunction.
(b) Disorders of Coagulation factors and Fibrinolysis: Reduced
level of activity of coagulation factors. Pathologically activated
fibrinolysis. Factor deficiencies.
Hemophilia A
Hemophilia B
Liver failure (Depression of coagulation factor levels)
Disseminated intravascular coagulation
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Bleeding Disorders
(a) Platelet disorders: Thrombocytopenia (decreased blood
platelet concentration) Platelet count ≤ 20000. Many
iatrogenic processes lead to platelet disorders. Also,
endogenous disease states lead to platelet dysfunction.
(b) Disorders of Coagulation factors and Fibrinolysis: Reduced
level of activity of coagulation factors. Pathologically activated
fibrinolysis. Factor deficiencies.
Hemophilia A
Hemophilia B
Liver failure (Depression of coagulation factor levels)
Disseminated intravascular coagulation
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Composition of Whole Blood
Cell matter ≈ 46 % by volume of blood
(RBC) erythrocytes: ≈ 98 % of cell matter
(WBC) leukocytes
Platelets
Plasma is primarily water (92 - 93 %) in which various
proteins are dissolved along with various ions.
Proteins: f-I: brinogen, f-II:prothrombin, f-V, f-VIII, f-IX, f-X,
f-XI, f-XII, f-XIII, antithrombin III, Tissue factor pathway
inhibitor, protein C, protein S, Plasminogen, α1 -antitrypsin,
α2 -antiplasmin
Ions: N a+ , K + , Ca2+ , M g 2+ , Cl− , HCO3− , P O43−
Thus blood is a very complex mixture.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Composition of Whole Blood
Cell matter ≈ 46 % by volume of blood
(RBC) erythrocytes: ≈ 98 % of cell matter
(WBC) leukocytes
Platelets
Plasma is primarily water (92 - 93 %) in which various
proteins are dissolved along with various ions.
Proteins: f-I: brinogen, f-II:prothrombin, f-V, f-VIII, f-IX, f-X,
f-XI, f-XII, f-XIII, antithrombin III, Tissue factor pathway
inhibitor, protein C, protein S, Plasminogen, α1 -antitrypsin,
α2 -antiplasmin
Ions: N a+ , K + , Ca2+ , M g 2+ , Cl− , HCO3− , P O43−
Thus blood is a very complex mixture.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Composition of Whole Blood
Cell matter ≈ 46 % by volume of blood
(RBC) erythrocytes: ≈ 98 % of cell matter
(WBC) leukocytes
Platelets
Plasma is primarily water (92 - 93 %) in which various
proteins are dissolved along with various ions.
Proteins: f-I: brinogen, f-II:prothrombin, f-V, f-VIII, f-IX, f-X,
f-XI, f-XII, f-XIII, antithrombin III, Tissue factor pathway
inhibitor, protein C, protein S, Plasminogen, α1 -antitrypsin,
α2 -antiplasmin
Ions: N a+ , K + , Ca2+ , M g 2+ , Cl− , HCO3− , P O43−
Thus blood is a very complex mixture.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Composition of Whole Blood
Cell matter ≈ 46 % by volume of blood
(RBC) erythrocytes: ≈ 98 % of cell matter
(WBC) leukocytes
Platelets
Plasma is primarily water (92 - 93 %) in which various
proteins are dissolved along with various ions.
Proteins: f-I: brinogen, f-II:prothrombin, f-V, f-VIII, f-IX, f-X,
f-XI, f-XII, f-XIII, antithrombin III, Tissue factor pathway
inhibitor, protein C, protein S, Plasminogen, α1 -antitrypsin,
α2 -antiplasmin
Ions: N a+ , K + , Ca2+ , M g 2+ , Cl− , HCO3− , P O43−
Thus blood is a very complex mixture.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Composition of Whole Blood
Cell matter ≈ 46 % by volume of blood
(RBC) erythrocytes: ≈ 98 % of cell matter
(WBC) leukocytes
Platelets
Plasma is primarily water (92 - 93 %) in which various
proteins are dissolved along with various ions.
Proteins: f-I: brinogen, f-II:prothrombin, f-V, f-VIII, f-IX, f-X,
f-XI, f-XII, f-XIII, antithrombin III, Tissue factor pathway
inhibitor, protein C, protein S, Plasminogen, α1 -antitrypsin,
α2 -antiplasmin
Ions: N a+ , K + , Ca2+ , M g 2+ , Cl− , HCO3− , P O43−
Thus blood is a very complex mixture.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Composition of Whole Blood
Cell matter ≈ 46 % by volume of blood
(RBC) erythrocytes: ≈ 98 % of cell matter
(WBC) leukocytes
Platelets
Plasma is primarily water (92 - 93 %) in which various
proteins are dissolved along with various ions.
Proteins: f-I: brinogen, f-II:prothrombin, f-V, f-VIII, f-IX, f-X,
f-XI, f-XII, f-XIII, antithrombin III, Tissue factor pathway
inhibitor, protein C, protein S, Plasminogen, α1 -antitrypsin,
α2 -antiplasmin
Ions: N a+ , K + , Ca2+ , M g 2+ , Cl− , HCO3− , P O43−
Thus blood is a very complex mixture.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Composition of Whole Blood
Cell matter ≈ 46 % by volume of blood
(RBC) erythrocytes: ≈ 98 % of cell matter
(WBC) leukocytes
Platelets
Plasma is primarily water (92 - 93 %) in which various
proteins are dissolved along with various ions.
Proteins: f-I: brinogen, f-II:prothrombin, f-V, f-VIII, f-IX, f-X,
f-XI, f-XII, f-XIII, antithrombin III, Tissue factor pathway
inhibitor, protein C, protein S, Plasminogen, α1 -antitrypsin,
α2 -antiplasmin
Ions: N a+ , K + , Ca2+ , M g 2+ , Cl− , HCO3− , P O43−
Thus blood is a very complex mixture.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Composition of Whole Blood
Cell matter ≈ 46 % by volume of blood
(RBC) erythrocytes: ≈ 98 % of cell matter
(WBC) leukocytes
Platelets
Plasma is primarily water (92 - 93 %) in which various
proteins are dissolved along with various ions.
Proteins: f-I: brinogen, f-II:prothrombin, f-V, f-VIII, f-IX, f-X,
f-XI, f-XII, f-XIII, antithrombin III, Tissue factor pathway
inhibitor, protein C, protein S, Plasminogen, α1 -antitrypsin,
α2 -antiplasmin
Ions: N a+ , K + , Ca2+ , M g 2+ , Cl− , HCO3− , P O43−
Thus blood is a very complex mixture.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Composition of Whole Blood
Cell matter ≈ 46 % by volume of blood
(RBC) erythrocytes: ≈ 98 % of cell matter
(WBC) leukocytes
Platelets
Plasma is primarily water (92 - 93 %) in which various
proteins are dissolved along with various ions.
Proteins: f-I: brinogen, f-II:prothrombin, f-V, f-VIII, f-IX, f-X,
f-XI, f-XII, f-XIII, antithrombin III, Tissue factor pathway
inhibitor, protein C, protein S, Plasminogen, α1 -antitrypsin,
α2 -antiplasmin
Ions: N a+ , K + , Ca2+ , M g 2+ , Cl− , HCO3− , P O43−
Thus blood is a very complex mixture.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Q: Is it reasonable to treat blood as a single component fluid?
A: Depends on the application.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Q: Is it reasonable to treat blood as a single component fluid?
A: Depends on the application.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Plasma - Newtonian fluid ≈ 1.2 cP.
Erythrocytes (RBCs): biconcave deformable discs that lack
nuclei.
Membrane: approximately 3 % by weight of RBC
cytoplasm - solution of hemoglobin in water - viscoelastic
(Evans and Hochmuth, 1976; based on micropipette
experiments).
Leukocytes : approximately 1 % volume of blood.
Granulocytes - viscoelastic (Schmidschonbein and Sung,1981).
Monocytes
Lymphocytes
Platelets: Elastic/Viscoelastic
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Plasma - Newtonian fluid ≈ 1.2 cP.
Erythrocytes (RBCs): biconcave deformable discs that lack
nuclei.
Membrane: approximately 3 % by weight of RBC
cytoplasm - solution of hemoglobin in water - viscoelastic
(Evans and Hochmuth, 1976; based on micropipette
experiments).
Leukocytes : approximately 1 % volume of blood.
Granulocytes - viscoelastic (Schmidschonbein and Sung,1981).
Monocytes
Lymphocytes
Platelets: Elastic/Viscoelastic
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Plasma - Newtonian fluid ≈ 1.2 cP.
Erythrocytes (RBCs): biconcave deformable discs that lack
nuclei.
Membrane: approximately 3 % by weight of RBC
cytoplasm - solution of hemoglobin in water - viscoelastic
(Evans and Hochmuth, 1976; based on micropipette
experiments).
Leukocytes : approximately 1 % volume of blood.
Granulocytes - viscoelastic (Schmidschonbein and Sung,1981).
Monocytes
Lymphocytes
Platelets: Elastic/Viscoelastic
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Plasma - Newtonian fluid ≈ 1.2 cP.
Erythrocytes (RBCs): biconcave deformable discs that lack
nuclei.
Membrane: approximately 3 % by weight of RBC
cytoplasm - solution of hemoglobin in water - viscoelastic
(Evans and Hochmuth, 1976; based on micropipette
experiments).
Leukocytes : approximately 1 % volume of blood.
Granulocytes - viscoelastic (Schmidschonbein and Sung,1981).
Monocytes
Lymphocytes
Platelets: Elastic/Viscoelastic
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Plasma - Newtonian fluid ≈ 1.2 cP.
Erythrocytes (RBCs): biconcave deformable discs that lack
nuclei.
Membrane: approximately 3 % by weight of RBC
cytoplasm - solution of hemoglobin in water - viscoelastic
(Evans and Hochmuth, 1976; based on micropipette
experiments).
Leukocytes : approximately 1 % volume of blood.
Granulocytes - viscoelastic (Schmidschonbein and Sung,1981).
Monocytes
Lymphocytes
Platelets: Elastic/Viscoelastic
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Plasma - Newtonian fluid ≈ 1.2 cP.
Erythrocytes (RBCs): biconcave deformable discs that lack
nuclei.
Membrane: approximately 3 % by weight of RBC
cytoplasm - solution of hemoglobin in water - viscoelastic
(Evans and Hochmuth, 1976; based on micropipette
experiments).
Leukocytes : approximately 1 % volume of blood.
Granulocytes - viscoelastic (Schmidschonbein and Sung,1981).
Monocytes
Lymphocytes
Platelets: Elastic/Viscoelastic
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Plasma - Newtonian fluid ≈ 1.2 cP.
Erythrocytes (RBCs): biconcave deformable discs that lack
nuclei.
Membrane: approximately 3 % by weight of RBC
cytoplasm - solution of hemoglobin in water - viscoelastic
(Evans and Hochmuth, 1976; based on micropipette
experiments).
Leukocytes : approximately 1 % volume of blood.
Granulocytes - viscoelastic (Schmidschonbein and Sung,1981).
Monocytes
Lymphocytes
Platelets: Elastic/Viscoelastic
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Plasma - Newtonian fluid ≈ 1.2 cP.
Erythrocytes (RBCs): biconcave deformable discs that lack
nuclei.
Membrane: approximately 3 % by weight of RBC
cytoplasm - solution of hemoglobin in water - viscoelastic
(Evans and Hochmuth, 1976; based on micropipette
experiments).
Leukocytes : approximately 1 % volume of blood.
Granulocytes - viscoelastic (Schmidschonbein and Sung,1981).
Monocytes
Lymphocytes
Platelets: Elastic/Viscoelastic
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Plasma - Newtonian fluid ≈ 1.2 cP.
Erythrocytes (RBCs): biconcave deformable discs that lack
nuclei.
Membrane: approximately 3 % by weight of RBC
cytoplasm - solution of hemoglobin in water - viscoelastic
(Evans and Hochmuth, 1976; based on micropipette
experiments).
Leukocytes : approximately 1 % volume of blood.
Granulocytes - viscoelastic (Schmidschonbein and Sung,1981).
Monocytes
Lymphocytes
Platelets: Elastic/Viscoelastic
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Models from the perspective of hemodynamics
Homogenized Fluid Models
Newtonian Fluid: May be reasonable in large vessels
Generalized Newtonian Fluid: Necessary in smaller vessels
Viscoelastic fluid capable of shear thinning and relaxation time
depending on the shear rate.
Reviews of One dimensional continuum models - Cho and
Kensey (1991)
Three dimensional continuum models - Yeleswarapu (1996)
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Models from the perspective of hemodynamics
Homogenized Fluid Models
Newtonian Fluid: May be reasonable in large vessels
Generalized Newtonian Fluid: Necessary in smaller vessels
Viscoelastic fluid capable of shear thinning and relaxation time
depending on the shear rate.
Reviews of One dimensional continuum models - Cho and
Kensey (1991)
Three dimensional continuum models - Yeleswarapu (1996)
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Models from the perspective of hemodynamics
Homogenized Fluid Models
Newtonian Fluid: May be reasonable in large vessels
Generalized Newtonian Fluid: Necessary in smaller vessels
Viscoelastic fluid capable of shear thinning and relaxation time
depending on the shear rate.
Reviews of One dimensional continuum models - Cho and
Kensey (1991)
Three dimensional continuum models - Yeleswarapu (1996)
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Models from the perspective of hemodynamics
Homogenized Fluid Models
Newtonian Fluid: May be reasonable in large vessels
Generalized Newtonian Fluid: Necessary in smaller vessels
Viscoelastic fluid capable of shear thinning and relaxation time
depending on the shear rate.
Reviews of One dimensional continuum models - Cho and
Kensey (1991)
Three dimensional continuum models - Yeleswarapu (1996)
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Models from the perspective of hemodynamics
Homogenized Fluid Models
Newtonian Fluid: May be reasonable in large vessels
Generalized Newtonian Fluid: Necessary in smaller vessels
Viscoelastic fluid capable of shear thinning and relaxation time
depending on the shear rate.
Reviews of One dimensional continuum models - Cho and
Kensey (1991)
Three dimensional continuum models - Yeleswarapu (1996)
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Models from the perspective of hemodynamics
Homogenized Fluid Models
Newtonian Fluid: May be reasonable in large vessels
Generalized Newtonian Fluid: Necessary in smaller vessels
Viscoelastic fluid capable of shear thinning and relaxation time
depending on the shear rate.
Reviews of One dimensional continuum models - Cho and
Kensey (1991)
Three dimensional continuum models - Yeleswarapu (1996)
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Models based on mixture theory
Mixture models - Trowbridge (1984); Kline (1972); Chathurani
and Upadhya (1979)
We also need to take into account:
The complex biochemical reactions that take place.
Shear-thinning property of blood Charm and Kurland, 1965;
Chien et al., 1966;
Stress-relaxation Thurston, 1972
Viscoelasticity of RBC membrance Evans and Hochmuth,
1976
Variation of stress-relaxation with shear rate Thurston, 1973
Numerous generalized Newtonian uid models have been
proposed for blood.
Rate-type fluid model for blood: Thurston, 1972; Yeleswarapu
et al.,1998
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Models based on mixture theory
Mixture models - Trowbridge (1984); Kline (1972); Chathurani
and Upadhya (1979)
We also need to take into account:
The complex biochemical reactions that take place.
Shear-thinning property of blood Charm and Kurland, 1965;
Chien et al., 1966;
Stress-relaxation Thurston, 1972
Viscoelasticity of RBC membrance Evans and Hochmuth,
1976
Variation of stress-relaxation with shear rate Thurston, 1973
Numerous generalized Newtonian uid models have been
proposed for blood.
Rate-type fluid model for blood: Thurston, 1972; Yeleswarapu
et al.,1998
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Models based on mixture theory
Mixture models - Trowbridge (1984); Kline (1972); Chathurani
and Upadhya (1979)
We also need to take into account:
The complex biochemical reactions that take place.
Shear-thinning property of blood Charm and Kurland, 1965;
Chien et al., 1966;
Stress-relaxation Thurston, 1972
Viscoelasticity of RBC membrance Evans and Hochmuth,
1976
Variation of stress-relaxation with shear rate Thurston, 1973
Numerous generalized Newtonian uid models have been
proposed for blood.
Rate-type fluid model for blood: Thurston, 1972; Yeleswarapu
et al.,1998
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Models based on mixture theory
Mixture models - Trowbridge (1984); Kline (1972); Chathurani
and Upadhya (1979)
We also need to take into account:
The complex biochemical reactions that take place.
Shear-thinning property of blood Charm and Kurland, 1965;
Chien et al., 1966;
Stress-relaxation Thurston, 1972
Viscoelasticity of RBC membrance Evans and Hochmuth,
1976
Variation of stress-relaxation with shear rate Thurston, 1973
Numerous generalized Newtonian uid models have been
proposed for blood.
Rate-type fluid model for blood: Thurston, 1972; Yeleswarapu
et al.,1998
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Models based on mixture theory
Mixture models - Trowbridge (1984); Kline (1972); Chathurani
and Upadhya (1979)
We also need to take into account:
The complex biochemical reactions that take place.
Shear-thinning property of blood Charm and Kurland, 1965;
Chien et al., 1966;
Stress-relaxation Thurston, 1972
Viscoelasticity of RBC membrance Evans and Hochmuth,
1976
Variation of stress-relaxation with shear rate Thurston, 1973
Numerous generalized Newtonian uid models have been
proposed for blood.
Rate-type fluid model for blood: Thurston, 1972; Yeleswarapu
et al.,1998
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Models based on mixture theory
Mixture models - Trowbridge (1984); Kline (1972); Chathurani
and Upadhya (1979)
We also need to take into account:
The complex biochemical reactions that take place.
Shear-thinning property of blood Charm and Kurland, 1965;
Chien et al., 1966;
Stress-relaxation Thurston, 1972
Viscoelasticity of RBC membrance Evans and Hochmuth,
1976
Variation of stress-relaxation with shear rate Thurston, 1973
Numerous generalized Newtonian uid models have been
proposed for blood.
Rate-type fluid model for blood: Thurston, 1972; Yeleswarapu
et al.,1998
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Models based on mixture theory
Mixture models - Trowbridge (1984); Kline (1972); Chathurani
and Upadhya (1979)
We also need to take into account:
The complex biochemical reactions that take place.
Shear-thinning property of blood Charm and Kurland, 1965;
Chien et al., 1966;
Stress-relaxation Thurston, 1972
Viscoelasticity of RBC membrance Evans and Hochmuth,
1976
Variation of stress-relaxation with shear rate Thurston, 1973
Numerous generalized Newtonian uid models have been
proposed for blood.
Rate-type fluid model for blood: Thurston, 1972; Yeleswarapu
et al.,1998
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Models based on mixture theory
Mixture models - Trowbridge (1984); Kline (1972); Chathurani
and Upadhya (1979)
We also need to take into account:
The complex biochemical reactions that take place.
Shear-thinning property of blood Charm and Kurland, 1965;
Chien et al., 1966;
Stress-relaxation Thurston, 1972
Viscoelasticity of RBC membrance Evans and Hochmuth,
1976
Variation of stress-relaxation with shear rate Thurston, 1973
Numerous generalized Newtonian uid models have been
proposed for blood.
Rate-type fluid model for blood: Thurston, 1972; Yeleswarapu
et al.,1998
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Models based on mixture theory
Mixture models - Trowbridge (1984); Kline (1972); Chathurani
and Upadhya (1979)
We also need to take into account:
The complex biochemical reactions that take place.
Shear-thinning property of blood Charm and Kurland, 1965;
Chien et al., 1966;
Stress-relaxation Thurston, 1972
Viscoelasticity of RBC membrance Evans and Hochmuth,
1976
Variation of stress-relaxation with shear rate Thurston, 1973
Numerous generalized Newtonian uid models have been
proposed for blood.
Rate-type fluid model for blood: Thurston, 1972; Yeleswarapu
et al.,1998
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Platelet Activation and Aggregation
Platelets constitute approximately 3 % of blood - discoid cell
fragment - approximately 6 µm3 .
Platelet activation occurs due to interaction with collagens
and adhesive glycoproteins exposed by damage with thrombin
or adenosine diphosphate (ADP).
Activation - Organelles within the platelet are centralized. Glycoproteins on the platelet membrane undergo a change in
conformation.
Pseudopods are extended so that the platelet is a sticky spiny
sphere.
Platelet activation is followed by interaction with plasma
proteins like Factor IX, Factor V, vWF, fibrinogen and fibrin
so as to adhere to sub-endothelial tissue and leads to
aggregation.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Platelet Activation and Aggregation
Platelets constitute approximately 3 % of blood - discoid cell
fragment - approximately 6 µm3 .
Platelet activation occurs due to interaction with collagens
and adhesive glycoproteins exposed by damage with thrombin
or adenosine diphosphate (ADP).
Activation - Organelles within the platelet are centralized. Glycoproteins on the platelet membrane undergo a change in
conformation.
Pseudopods are extended so that the platelet is a sticky spiny
sphere.
Platelet activation is followed by interaction with plasma
proteins like Factor IX, Factor V, vWF, fibrinogen and fibrin
so as to adhere to sub-endothelial tissue and leads to
aggregation.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Platelet Activation and Aggregation
Platelets constitute approximately 3 % of blood - discoid cell
fragment - approximately 6 µm3 .
Platelet activation occurs due to interaction with collagens
and adhesive glycoproteins exposed by damage with thrombin
or adenosine diphosphate (ADP).
Activation - Organelles within the platelet are centralized. Glycoproteins on the platelet membrane undergo a change in
conformation.
Pseudopods are extended so that the platelet is a sticky spiny
sphere.
Platelet activation is followed by interaction with plasma
proteins like Factor IX, Factor V, vWF, fibrinogen and fibrin
so as to adhere to sub-endothelial tissue and leads to
aggregation.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Platelet Activation and Aggregation
Platelets constitute approximately 3 % of blood - discoid cell
fragment - approximately 6 µm3 .
Platelet activation occurs due to interaction with collagens
and adhesive glycoproteins exposed by damage with thrombin
or adenosine diphosphate (ADP).
Activation - Organelles within the platelet are centralized. Glycoproteins on the platelet membrane undergo a change in
conformation.
Pseudopods are extended so that the platelet is a sticky spiny
sphere.
Platelet activation is followed by interaction with plasma
proteins like Factor IX, Factor V, vWF, fibrinogen and fibrin
so as to adhere to sub-endothelial tissue and leads to
aggregation.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Platelet Activation and Aggregation
Platelets constitute approximately 3 % of blood - discoid cell
fragment - approximately 6 µm3 .
Platelet activation occurs due to interaction with collagens
and adhesive glycoproteins exposed by damage with thrombin
or adenosine diphosphate (ADP).
Activation - Organelles within the platelet are centralized. Glycoproteins on the platelet membrane undergo a change in
conformation.
Pseudopods are extended so that the platelet is a sticky spiny
sphere.
Platelet activation is followed by interaction with plasma
proteins like Factor IX, Factor V, vWF, fibrinogen and fibrin
so as to adhere to sub-endothelial tissue and leads to
aggregation.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Intrinsic Pathway to Coagulation: Factor XII→Fibrin
Extrinsic Pathway to Coagulation: Factor VII→Fibrin
Starts with the exposure of TF (a cell membrane bound
protein) in the subendothelium to blood which leads to a
chain of coagulation reactions.
Formation of TF-VIIa→Enzymes, Factor IXa, Xa→Factor Va
and VIIIa
The enzyme complex IXa-VIIIa bound to the membrane of the
activated platelet catalyzes the formation of Xa from X.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Intrinsic Pathway to Coagulation: Factor XII→Fibrin
Extrinsic Pathway to Coagulation: Factor VII→Fibrin
Starts with the exposure of TF (a cell membrane bound
protein) in the subendothelium to blood which leads to a
chain of coagulation reactions.
Formation of TF-VIIa→Enzymes, Factor IXa, Xa→Factor Va
and VIIIa
The enzyme complex IXa-VIIIa bound to the membrane of the
activated platelet catalyzes the formation of Xa from X.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Intrinsic Pathway to Coagulation: Factor XII→Fibrin
Extrinsic Pathway to Coagulation: Factor VII→Fibrin
Starts with the exposure of TF (a cell membrane bound
protein) in the subendothelium to blood which leads to a
chain of coagulation reactions.
Formation of TF-VIIa→Enzymes, Factor IXa, Xa→Factor Va
and VIIIa
The enzyme complex IXa-VIIIa bound to the membrane of the
activated platelet catalyzes the formation of Xa from X.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Intrinsic Pathway to Coagulation: Factor XII→Fibrin
Extrinsic Pathway to Coagulation: Factor VII→Fibrin
Starts with the exposure of TF (a cell membrane bound
protein) in the subendothelium to blood which leads to a
chain of coagulation reactions.
Formation of TF-VIIa→Enzymes, Factor IXa, Xa→Factor Va
and VIIIa
The enzyme complex IXa-VIIIa bound to the membrane of the
activated platelet catalyzes the formation of Xa from X.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Intrinsic Pathway to Coagulation: Factor XII→Fibrin
Extrinsic Pathway to Coagulation: Factor VII→Fibrin
Starts with the exposure of TF (a cell membrane bound
protein) in the subendothelium to blood which leads to a
chain of coagulation reactions.
Formation of TF-VIIa→Enzymes, Factor IXa, Xa→Factor Va
and VIIIa
The enzyme complex IXa-VIIIa bound to the membrane of the
activated platelet catalyzes the formation of Xa from X.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Next, the enzyme complex Xa-Va formed on the membrane of
the activated platelet catalyzes the production of thrombin
from prothrombin. Thrombin acts on fibrinogen to yield fibrin
monomers that polymerize and are cross-linked to form a
fibrin matrix.
Three inhibitory mechanisms in blood:
AntiThrombin III (AT III), TF Pathway Inhibitor (TFPI), APC
Bauer and Rosenberg, 1995, November, 1995.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Next, the enzyme complex Xa-Va formed on the membrane of
the activated platelet catalyzes the production of thrombin
from prothrombin. Thrombin acts on fibrinogen to yield fibrin
monomers that polymerize and are cross-linked to form a
fibrin matrix.
Three inhibitory mechanisms in blood:
AntiThrombin III (AT III), TF Pathway Inhibitor (TFPI), APC
Bauer and Rosenberg, 1995, November, 1995.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Next, the enzyme complex Xa-Va formed on the membrane of
the activated platelet catalyzes the production of thrombin
from prothrombin. Thrombin acts on fibrinogen to yield fibrin
monomers that polymerize and are cross-linked to form a
fibrin matrix.
Three inhibitory mechanisms in blood:
AntiThrombin III (AT III), TF Pathway Inhibitor (TFPI), APC
Bauer and Rosenberg, 1995, November, 1995.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Next, the enzyme complex Xa-Va formed on the membrane of
the activated platelet catalyzes the production of thrombin
from prothrombin. Thrombin acts on fibrinogen to yield fibrin
monomers that polymerize and are cross-linked to form a
fibrin matrix.
Three inhibitory mechanisms in blood:
AntiThrombin III (AT III), TF Pathway Inhibitor (TFPI), APC
Bauer and Rosenberg, 1995, November, 1995.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Fibrinolysis and Clot Dissolution
Enzymatic reactions initiated when thrombin and fibrin
activate endothelial cells resulting in enhanced production of
tissue PLS activator (tPA) and urokinase-like PLS activator
(uPA).
tPA and uPA catalyze the transformation of PLS into active
enzyme plasmin.
Plasmin degrades the fibrin polymer into smaller units leading
to dissolution of the clot.
Fibrinolysis has its share of regulatory mechanisms. We shall
not discuss these in detail.
Clot dissolution can also occur due to high shear stress which
causes the fibrin to rupture. (See Riha et al., 1999).
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Fibrinolysis and Clot Dissolution
Enzymatic reactions initiated when thrombin and fibrin
activate endothelial cells resulting in enhanced production of
tissue PLS activator (tPA) and urokinase-like PLS activator
(uPA).
tPA and uPA catalyze the transformation of PLS into active
enzyme plasmin.
Plasmin degrades the fibrin polymer into smaller units leading
to dissolution of the clot.
Fibrinolysis has its share of regulatory mechanisms. We shall
not discuss these in detail.
Clot dissolution can also occur due to high shear stress which
causes the fibrin to rupture. (See Riha et al., 1999).
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Fibrinolysis and Clot Dissolution
Enzymatic reactions initiated when thrombin and fibrin
activate endothelial cells resulting in enhanced production of
tissue PLS activator (tPA) and urokinase-like PLS activator
(uPA).
tPA and uPA catalyze the transformation of PLS into active
enzyme plasmin.
Plasmin degrades the fibrin polymer into smaller units leading
to dissolution of the clot.
Fibrinolysis has its share of regulatory mechanisms. We shall
not discuss these in detail.
Clot dissolution can also occur due to high shear stress which
causes the fibrin to rupture. (See Riha et al., 1999).
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Fibrinolysis and Clot Dissolution
Enzymatic reactions initiated when thrombin and fibrin
activate endothelial cells resulting in enhanced production of
tissue PLS activator (tPA) and urokinase-like PLS activator
(uPA).
tPA and uPA catalyze the transformation of PLS into active
enzyme plasmin.
Plasmin degrades the fibrin polymer into smaller units leading
to dissolution of the clot.
Fibrinolysis has its share of regulatory mechanisms. We shall
not discuss these in detail.
Clot dissolution can also occur due to high shear stress which
causes the fibrin to rupture. (See Riha et al., 1999).
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Fibrinolysis and Clot Dissolution
Enzymatic reactions initiated when thrombin and fibrin
activate endothelial cells resulting in enhanced production of
tissue PLS activator (tPA) and urokinase-like PLS activator
(uPA).
tPA and uPA catalyze the transformation of PLS into active
enzyme plasmin.
Plasmin degrades the fibrin polymer into smaller units leading
to dissolution of the clot.
Fibrinolysis has its share of regulatory mechanisms. We shall
not discuss these in detail.
Clot dissolution can also occur due to high shear stress which
causes the fibrin to rupture. (See Riha et al., 1999).
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Model Characteristics
Whole blood is modeled as a shear thinning viscoelastic fluid
that contains the various reactants involved in clot formation.
Development of convection-reaction-diffusion equations that
govern the generation/depletion of the reactants.
Platelet activation either due to action by thrombin and
agonists like ADP or prolonged exposure to shear stresses.
Boundary conditions that represent the level of injury and
endothelial cell activity
Prescription of a threshold concentration of surface bound
TF-VIIa Complex.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Model Characteristics
Whole blood is modeled as a shear thinning viscoelastic fluid
that contains the various reactants involved in clot formation.
Development of convection-reaction-diffusion equations that
govern the generation/depletion of the reactants.
Platelet activation either due to action by thrombin and
agonists like ADP or prolonged exposure to shear stresses.
Boundary conditions that represent the level of injury and
endothelial cell activity
Prescription of a threshold concentration of surface bound
TF-VIIa Complex.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Model Characteristics
Whole blood is modeled as a shear thinning viscoelastic fluid
that contains the various reactants involved in clot formation.
Development of convection-reaction-diffusion equations that
govern the generation/depletion of the reactants.
Platelet activation either due to action by thrombin and
agonists like ADP or prolonged exposure to shear stresses.
Boundary conditions that represent the level of injury and
endothelial cell activity
Prescription of a threshold concentration of surface bound
TF-VIIa Complex.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Model Characteristics
Whole blood is modeled as a shear thinning viscoelastic fluid
that contains the various reactants involved in clot formation.
Development of convection-reaction-diffusion equations that
govern the generation/depletion of the reactants.
Platelet activation either due to action by thrombin and
agonists like ADP or prolonged exposure to shear stresses.
Boundary conditions that represent the level of injury and
endothelial cell activity
Prescription of a threshold concentration of surface bound
TF-VIIa Complex.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Model Characteristics
Whole blood is modeled as a shear thinning viscoelastic fluid
that contains the various reactants involved in clot formation.
Development of convection-reaction-diffusion equations that
govern the generation/depletion of the reactants.
Platelet activation either due to action by thrombin and
agonists like ADP or prolonged exposure to shear stresses.
Boundary conditions that represent the level of injury and
endothelial cell activity
Prescription of a threshold concentration of surface bound
TF-VIIa Complex.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
A model for the clot as a viscoelastic fluid with a much
greater viscosity than pure blood.
Clot growth, the boundary defined by a fibrin concentration.
Clot dissolution due to decrease in fibrin concentration and
shear stresses.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
A model for the clot as a viscoelastic fluid with a much
greater viscosity than pure blood.
Clot growth, the boundary defined by a fibrin concentration.
Clot dissolution due to decrease in fibrin concentration and
shear stresses.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
A model for the clot as a viscoelastic fluid with a much
greater viscosity than pure blood.
Clot growth, the boundary defined by a fibrin concentration.
Clot dissolution due to decrease in fibrin concentration and
shear stresses.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Basic Kinematics
κτ ,κt - Placers
κτ (B),κt (B) - Configurations
Motion is a one-parameter family
of placers.
X ∈ B, X = κR (X) ,x = κt (X)
Identify motion by
x = χκR (X, t).
ξ = χκR (X, τ ) = χκR (χ−1
κR (X, t), τ )
= χκt (x, τ )
...Relative Motion
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Deformation Gradient
F κR :=
∂χκR
,
∂X
(1)
F κR is a linear transformation from the tangent space at X to the
tangent space at x
Relative Deformation Gradient
F κR :=
∂χt
∂X
(2)
φ = φ̂(X, t) = φ̃(x, t)
5φ :=
∂ φ̂
∂X
,
dφ
∂ φ̂
:=
,
dt
∂t
Lagrangean
K. R. Rajagopal
gradφ :=
(3)
∂ φ̃
,
∂x
∂φ
∂ φ̃
:=
∂t
∂t
Eulerian
(4)
(5)
Biomedicine: A fertile, challenging and worthy field for mathemat
CONFIGURATION OF A BODY
Figure: κp (τ ) - Natural configuration corresponding to κτ
κp (t) - Natural configuration corresponding to κt
If one inhomogeneously deforms a body and then removes the
traction, it is possible that the unloaded body will not fit
together compatably and be simultaneously stress free in an
Euclidean space.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
However, it can be unloaded in a non-Euclidean space in
which it fits together and is stress free (Eckart 1940S )
However, a ”sufficiently small” neighborhood of a material
point can be unloaded to a stress free state in Euclidean
space, i.e., if the deformation is reasonably smooth, we can
pick sufficiently small neighborhoods wherein the deformation
is homogeneous. The notion of a configuration really applies
to an appropriately small neighborhood of a point.
Henceforth, for the sake of illustration, let us assume
homogeneous deformations.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
However, it can be unloaded in a non-Euclidean space in
which it fits together and is stress free (Eckart 1940S )
However, a ”sufficiently small” neighborhood of a material
point can be unloaded to a stress free state in Euclidean
space, i.e., if the deformation is reasonably smooth, we can
pick sufficiently small neighborhoods wherein the deformation
is homogeneous. The notion of a configuration really applies
to an appropriately small neighborhood of a point.
Henceforth, for the sake of illustration, let us assume
homogeneous deformations.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
However, it can be unloaded in a non-Euclidean space in
which it fits together and is stress free (Eckart 1940S )
However, a ”sufficiently small” neighborhood of a material
point can be unloaded to a stress free state in Euclidean
space, i.e., if the deformation is reasonably smooth, we can
pick sufficiently small neighborhoods wherein the deformation
is homogeneous. The notion of a configuration really applies
to an appropriately small neighborhood of a point.
Henceforth, for the sake of illustration, let us assume
homogeneous deformations.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Natural Configuration
– Can think of it as a stress-free configuration
– It is a ”local notion”.
– It is really an equivalence class of configurations.
Eg: Classical Plasticity
Figure: Traditional Plasticity
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Classical metal plasticity involves an infinity of natural
configurations, and to determine the stress we require
kinematical information from more than one natural
configuration.
The response is elastic from each of these natural
configurations and the inelasticity is purely due to the change
in the natural configurations.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Classical metal plasticity involves an infinity of natural
configurations, and to determine the stress we require
kinematical information from more than one natural
configuration.
The response is elastic from each of these natural
configurations and the inelasticity is purely due to the change
in the natural configurations.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Eg. 2 Twinning
In twinning there are a finite number. As many as the number
of variants.
Figure: Modulo variants, we have two natural configurations, that
corresponding to O and F, and these two natural configurations have
different material symmetries.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Other Examples:
Viscoelasticity
Superplasticity
Crystallization
Multi-network Polymers
Classical theories are trivial examples:
In classical elasticity the natural configuration does not evolve.
In classical fluids the current configuration is the natural
configuration.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Figure: Configuration as a local
notion
Figure: Spider spinning a web
New material is laid in a stressed state. It can have a different
natural configuration than the material laid down previously.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Restrict ourselves to homogeneous deformation.
Think in terms of Global configurations.
Figure: Non-uniqueness of stress-free state (Modulo rigid motion)
More than one Natural Configuration can be associated with
the current deformed configuration.
Example: Consider a Viscoelastic body capable of
instantaneous elastic response κt
– Natural Configuration reached by instantaneous unloading–An
adiabatic process.
– Natural Configuration reached in an isothermal
stress-relaxation process
Thus, we need to know the process class under consideration.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
SINGLE CONSTITUENT BODY
Balance of Mass:
∂ρ
+ div(ρv) = 0
(6)
∂t
Assumption of the incompressibility implies that the body can
undergo only isochoric motion, i.e.,
div(v) = 0
(7)
Balance of Linear Momentum:
divT + ρb = ρ
K. R. Rajagopal
dv
dt
(8)
Biomedicine: A fertile, challenging and worthy field for mathemat
SINGLE CONSTITUENT BODY
Balance of Angular Momentum:
T = TT
(9)
Balance of Energy:
ρ
d
+ divq − T.L − ρr = 0
dt
(10)
2nd Law:
q ρr
dη
+ div −
:= ρξ >= 0
(11)
dt
θ
θ
T-stress, η - specific entropy, θ-temperature,q-heat flux vector,
r-radiant heating
ρ
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
The evolution of the natural configuration, amongst other
things, is determined by the maximization of the entropy
production.
Ziegler suggested the use of maximization of dissipation, but
not within this context.
The maximization of the entropy production makes choices
amongst possible reponse functions. For instance, it will pick
a rate of dissipation(or entropy production) from amongst a
class of candidates.
For a class of materials, such a choice leads to a Liapunov
function that decreases with time to a minimum value
(Onsager/Prigogine- Minimum entropy production criterion).
Rajagopal and Srinivasa(2003), Proc. Royal Society.
There is no contradiction between these two criteria:
– Maximization of the entropy production to pick constitutive
equations and minimization of entropy production with time
once a choice has been made. (Rajagopal and
Srinivasa(2002)).
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
The evolution of the natural configuration, amongst other
things, is determined by the maximization of the entropy
production.
Ziegler suggested the use of maximization of dissipation, but
not within this context.
The maximization of the entropy production makes choices
amongst possible reponse functions. For instance, it will pick
a rate of dissipation(or entropy production) from amongst a
class of candidates.
For a class of materials, such a choice leads to a Liapunov
function that decreases with time to a minimum value
(Onsager/Prigogine- Minimum entropy production criterion).
Rajagopal and Srinivasa(2003), Proc. Royal Society.
There is no contradiction between these two criteria:
– Maximization of the entropy production to pick constitutive
equations and minimization of entropy production with time
once a choice has been made. (Rajagopal and
Srinivasa(2002)).
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
The evolution of the natural configuration, amongst other
things, is determined by the maximization of the entropy
production.
Ziegler suggested the use of maximization of dissipation, but
not within this context.
The maximization of the entropy production makes choices
amongst possible reponse functions. For instance, it will pick
a rate of dissipation(or entropy production) from amongst a
class of candidates.
For a class of materials, such a choice leads to a Liapunov
function that decreases with time to a minimum value
(Onsager/Prigogine- Minimum entropy production criterion).
Rajagopal and Srinivasa(2003), Proc. Royal Society.
There is no contradiction between these two criteria:
– Maximization of the entropy production to pick constitutive
equations and minimization of entropy production with time
once a choice has been made. (Rajagopal and
Srinivasa(2002)).
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
The evolution of the natural configuration, amongst other
things, is determined by the maximization of the entropy
production.
Ziegler suggested the use of maximization of dissipation, but
not within this context.
The maximization of the entropy production makes choices
amongst possible reponse functions. For instance, it will pick
a rate of dissipation(or entropy production) from amongst a
class of candidates.
For a class of materials, such a choice leads to a Liapunov
function that decreases with time to a minimum value
(Onsager/Prigogine- Minimum entropy production criterion).
Rajagopal and Srinivasa(2003), Proc. Royal Society.
There is no contradiction between these two criteria:
– Maximization of the entropy production to pick constitutive
equations and minimization of entropy production with time
once a choice has been made. (Rajagopal and
Srinivasa(2002)).
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
The evolution of the natural configuration, amongst other
things, is determined by the maximization of the entropy
production.
Ziegler suggested the use of maximization of dissipation, but
not within this context.
The maximization of the entropy production makes choices
amongst possible reponse functions. For instance, it will pick
a rate of dissipation(or entropy production) from amongst a
class of candidates.
For a class of materials, such a choice leads to a Liapunov
function that decreases with time to a minimum value
(Onsager/Prigogine- Minimum entropy production criterion).
Rajagopal and Srinivasa(2003), Proc. Royal Society.
There is no contradiction between these two criteria:
– Maximization of the entropy production to pick constitutive
equations and minimization of entropy production with time
once a choice has been made. (Rajagopal and
Srinivasa(2002)).
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
The evolution of the natural configuration, amongst other
things, is determined by the maximization of the entropy
production.
Ziegler suggested the use of maximization of dissipation, but
not within this context.
The maximization of the entropy production makes choices
amongst possible reponse functions. For instance, it will pick
a rate of dissipation(or entropy production) from amongst a
class of candidates.
For a class of materials, such a choice leads to a Liapunov
function that decreases with time to a minimum value
(Onsager/Prigogine- Minimum entropy production criterion).
Rajagopal and Srinivasa(2003), Proc. Royal Society.
There is no contradiction between these two criteria:
– Maximization of the entropy production to pick constitutive
equations and minimization of entropy production with time
once a choice has been made. (Rajagopal and
Srinivasa(2002)).
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
During the process entropy is produced in a variety of ways:
1
2
3
4
5
Due to conduction
Due to mixing
Due to work being converted to heat (dissipation)
Phase change
Growth
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
During the process entropy is produced in a variety of ways:
1
2
3
4
5
Due to conduction
Due to mixing
Due to work being converted to heat (dissipation)
Phase change
Growth
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
During the process entropy is produced in a variety of ways:
1
2
3
4
5
Due to conduction
Due to mixing
Due to work being converted to heat (dissipation)
Phase change
Growth
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
During the process entropy is produced in a variety of ways:
1
2
3
4
5
Due to conduction
Due to mixing
Due to work being converted to heat (dissipation)
Phase change
Growth
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
During the process entropy is produced in a variety of ways:
1
2
3
4
5
Due to conduction
Due to mixing
Due to work being converted to heat (dissipation)
Phase change
Growth
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
During the process entropy is produced in a variety of ways:
1
2
3
4
5
Due to conduction
Due to mixing
Due to work being converted to heat (dissipation)
Phase change
Growth
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Part of the energy that is supplied to the body is stored in the
body in a variety of ways. The energy supplied
1
2
3
Can change the kinetic energy.
Can change the potential energy.
Is stored as ”strain energy”
(a) that can be recovered in a purely mechanical process
(b) that can only be recovered in a thermal process.
– Part of the energy due to mechanical working is transferred as
energy in its thermal form (Heat).
– Part of the energy changes the ”Latent Energy”.
– Part goes towards ”Latent Heat”
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Part of the energy that is supplied to the body is stored in the
body in a variety of ways. The energy supplied
1
2
3
Can change the kinetic energy.
Can change the potential energy.
Is stored as ”strain energy”
(a) that can be recovered in a purely mechanical process
(b) that can only be recovered in a thermal process.
– Part of the energy due to mechanical working is transferred as
energy in its thermal form (Heat).
– Part of the energy changes the ”Latent Energy”.
– Part goes towards ”Latent Heat”
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Part of the energy that is supplied to the body is stored in the
body in a variety of ways. The energy supplied
1
2
3
Can change the kinetic energy.
Can change the potential energy.
Is stored as ”strain energy”
(a) that can be recovered in a purely mechanical process
(b) that can only be recovered in a thermal process.
– Part of the energy due to mechanical working is transferred as
energy in its thermal form (Heat).
– Part of the energy changes the ”Latent Energy”.
– Part goes towards ”Latent Heat”
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Part of the energy that is supplied to the body is stored in the
body in a variety of ways. The energy supplied
1
2
3
Can change the kinetic energy.
Can change the potential energy.
Is stored as ”strain energy”
(a) that can be recovered in a purely mechanical process
(b) that can only be recovered in a thermal process.
– Part of the energy due to mechanical working is transferred as
energy in its thermal form (Heat).
– Part of the energy changes the ”Latent Energy”.
– Part goes towards ”Latent Heat”
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Part of the energy that is supplied to the body is stored in the
body in a variety of ways. The energy supplied
1
2
3
Can change the kinetic energy.
Can change the potential energy.
Is stored as ”strain energy”
(a) that can be recovered in a purely mechanical process
(b) that can only be recovered in a thermal process.
– Part of the energy due to mechanical working is transferred as
energy in its thermal form (Heat).
– Part of the energy changes the ”Latent Energy”.
– Part goes towards ”Latent Heat”
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Part of the energy that is supplied to the body is stored in the
body in a variety of ways. The energy supplied
1
2
3
Can change the kinetic energy.
Can change the potential energy.
Is stored as ”strain energy”
(a) that can be recovered in a purely mechanical process
(b) that can only be recovered in a thermal process.
– Part of the energy due to mechanical working is transferred as
energy in its thermal form (Heat).
– Part of the energy changes the ”Latent Energy”.
– Part goes towards ”Latent Heat”
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Part of the energy that is supplied to the body is stored in the
body in a variety of ways. The energy supplied
1
2
3
Can change the kinetic energy.
Can change the potential energy.
Is stored as ”strain energy”
(a) that can be recovered in a purely mechanical process
(b) that can only be recovered in a thermal process.
– Part of the energy due to mechanical working is transferred as
energy in its thermal form (Heat).
– Part of the energy changes the ”Latent Energy”.
– Part goes towards ”Latent Heat”
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Part of the energy that is supplied to the body is stored in the
body in a variety of ways. The energy supplied
1
2
3
Can change the kinetic energy.
Can change the potential energy.
Is stored as ”strain energy”
(a) that can be recovered in a purely mechanical process
(b) that can only be recovered in a thermal process.
– Part of the energy due to mechanical working is transferred as
energy in its thermal form (Heat).
– Part of the energy changes the ”Latent Energy”.
– Part goes towards ”Latent Heat”
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Part of the energy that is supplied to the body is stored in the
body in a variety of ways. The energy supplied
1
2
3
Can change the kinetic energy.
Can change the potential energy.
Is stored as ”strain energy”
(a) that can be recovered in a purely mechanical process
(b) that can only be recovered in a thermal process.
– Part of the energy due to mechanical working is transferred as
energy in its thermal form (Heat).
– Part of the energy changes the ”Latent Energy”.
– Part goes towards ”Latent Heat”
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Figure: Biochemical model of selected reactions involved in the extrinsic
and intrinsic coagulation pathways and fibrinolysis. Arrow heads with a
plus sign near them indicate activation or enzymatic stimulation. Arrow
heads with a minus sign near them indicate inactivation or inhibition.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Table: Scheme of enzymatic reactions.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
The constitutive equations derived using the constrained
maximization procedure are
T = −p1 + S,
S = µBκp (t) + η1 D,
5
2µ
Bκp (t) − λ1 ,
η
3
.
trB−1
κp (t)
Bκp (t) = −
λ =
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
The biochemical reactions are given by:
∂[Yi ]
+ div {[Yi ]v} = div {DYi (D)[Yi ]} + GYi
∂t
Here, and elsewhere below, [Yi ] represents the concentration of the
reactant Yi ; GYi represents the production or depletion of Yi due
to the enzymatic reactions, v is the velocity, and DYi represents
the diffusion coefficient of Yi which could be a function of the
shear rate (captured by means of the stretching tensor D).
It is also possible to allow for GYi to depend on the concentration
of the jth reactant.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Model for Clot
Similar form as that for blood. However, the material functions are
different. They reflect the fact that the clot is much more viscous
than whole blood.
Activation Criterion:
1
A(t) = A(0) +
A0
Z
t
exp
k
|Trz |
Tth
H (|Trz | − Tth ) dt.
0
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
The Criterion for the Activation of Resting Platelets [RP]:
If A (t − tact ) > Athr and A(t) < Adam or
A (t − tact ) = Athr , Ȧ (t − tact ) > 0 and A(t) < Adam , then
[AP ](t) = [AP ] (t − tact ) + [RP ] (t − tact ) .
The Criterion for Lysis of Platelets
If A (t − tact ) > Athr and A(t) > Adam then
[AP ](t) = [RP ] (t − tact ) .
Model for Dissolved Clots:
We assume the clot on dissolution reverts to the original model for
whole blood.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Figure: Pressure gradient components in-phase with (P’), and
out-of-phase with (P”) the rms volumetric flow rate (U) during
oscillatory flow in small tubes.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat
Figure: Velocity profiles: Theoretical predictions, for Poiseuille flow, using
the proposed model, the GOB model, and the GM model, are compared
with the data for porcine blood.
K. R. Rajagopal
Biomedicine: A fertile, challenging and worthy field for mathemat