Studijní materiály
Chapter 2. Mathematical Modeling
Chapter 3. Static analysis of Physiological Systems
Proprietární kód klasických programovacích jazyků (např. Ada, Fortran, C,...)
Proprietární kód
Definice systému
Dekompozice systému
Modelování subsystémů Odvození kauzality výpočtu
(manuální odvození
vstupně/výstupních vztahů) Implementace Simulace
Modelování
Blokové modelovací jazyky
Blokové modelovací jazyky (např. Simulink, ...)
……
…
Vstupy do bloku jsou vnější vstupy nebo výstupy z jiných bloků
Výstupy z bloku jsou vnější výstupy nebo vstupy do dalších bloků
Modelování a simulace
Formalizace (modelování)
Počítačová simulace
Softwarové nástroje pro tvorbu modelů
VECml
DistNaFlow
200
Venous conductance[ml/min/torr]
7
Vena renalis pressure [torr]
-C-
Totasl body water content [ml]
Scope1 Scope
0
Renal artery pressure clamp
drop [torr]
0.07
Plasma protein cnoncentration
[g/ml]
6.25
Normal proximal tubule conductance
[ml/min/torr]
1
Normal macula densafeedback signal
16
Normal glomerular fi ltration coeffitient
[ml/min/torr]
0.5
Normal distalfractional reabsorbtion for Na
25
Normal conductanceof Efferent artery
[ml/min/torr]
30
Normal conductanceof Afferent artery
[ml/min/torr]
0.93
Normal collecting duct fractional reabsorbtion for Na
0.8
Normal Na proximal fractional reabsorbtion
0.125
NaDiet [mmol/min]
MDNaFlow
LogA2
MDNorm
MDSig
M A C U L A D E N S A
INPUTS :MDNaFlow - Macula densa sodium flow [mmol/min]
logA2 - Logarithm of plasma angiotensin concentration [pG/ml]MDNorm - Normal macula densa feedback signal [ x Normal]
OUTPUT :MDSig - Macula densa feedback signal [ x Normal]
Macula densa feedback signal calculation based on macula densa sodium flow and angiotensin concentration
0.44
Hematocrit
NaDiet
NaUrine
VECml
ZNAE
PNa
S I M P L E S O D I U M B A L A N C E
INPUTS :NaDIet - Dietary sodium intake [mmol/min]
NaUrine - Sodium urine outflow [mmol/min]VECml - Extracellular fluid volume [ml]
OUTPUT :ZNAE - ECF sodium content [mmol]
PNa - Plasma sodium concentration [mmol/ml]
Extracellular sodium quantity is the integral over time dietary sodium intake minus urinary sodium loss
15000
Extracellular fluid volume[mmol/min]1
0.5086
1.855
18.84
1.275
6.846
0.144
2159
125.1
3.648
14.37
0.7975
0.1248
0.9999
99.65
1182
662.1
29.83
25.1
99.651.006
0.9418
1.792
0.9304
1.668
1
Converting Enzyme Activity
ZNAE
LogA2
APNorm
AP
A R T E R I A L P R E S S U R E
INPUTS :ZNAE - ECF sodium content [mmol]
logA2 - Logarithm of plasma angiotensin concentration [pG/ml]APNorm - Normal value of arterial pressurel [torr]
OUTPUT :AP Arterial pressure [torr]
Control of arterial pressure by angiotensin and extracellular sodium content
RAP Af f My o1
M Y O G E N I C R E S P O N S E
INPUT :RAP - Renal artery pressure [torr]
OUTPUT :AffC - Myogenic effect [ x Normal]
Calculation of the myogenic response to changes in renal perfusion pressure(afferent conductance responds to changes in perfusion pressure,
with pressure increases causing vasoconstriction)
Af f C
Ef f C
RenVenC
AP
VP
Hct
Clamp
RAP
RBF
RPF
R E N A L P E R F U S I O N
INPUTS :AffC - Afferent artery conductance [ml/min/torr]EffC Efferent artery conductance [ml/min/torr]
RenVenC - Renal venous conductance [ml/min/torr]AP - Arterial pressure [torr]
VP - Vena renalis pressure [torr]Hct - Hematocrit [relative number]
Clamp - Renal artery pressure drop caused by renal artery clamp [torr]
OUTPUTS :RAP - Renal artery pressure [torr]
RBF - Renal blood flow rate[ml/min]RPF - Renal plasma flow rate[ml/min]
Calculation of renal artery pressure and renal blood flow rate
PNa
GFR
LogA2
PrxFNaNorm
MDNaFlow
PdxNaReab
PrxFNa
N A T R I U M - P R O X I M A L T U B U L E
INPUTS :PNa - Plasma sodium concentration [mmol/ml]
GFR - GLomerulal fi ltration rate [ml/min]LogA2 - Logarithm of plasma angiotensin concentration
[pg/ml]PrxFNaNorm - Normal value of sodium proximal
fractional reabsorbtion [relative number]
OUTPUTS :MDNaFlow - Sodium outflow [mmol/min]
PdxNaReab - Proximal sodium reabsorbrtion [mmol/l]PrxFNa - Proximal fractional sodium reabsorbtion
[relative number]
Calculation of proximal tubule sodium reabsorbtion
MDSig
VECml
PRA
R E N I N
INPUTS :MDSig - Macula densa feedback signal [ x Normal]
VECml - Ectracellular fluid volume
OUTPUT :PRA - Plasma renin activity [ Units/ml]
Calculation of plasma renin activity
PRA
CEAct
A2Inf
A2
logA2
A N G I O T E N S I N
INPUTS :PRA - Plasma renin activity [ Units/ml]
CEAct - Converting enzyme activity [x Normal]A2Inf Angiotensin 2 infusion rate [nG/min]
OUTPUTS :A2 - Plasma angiotensin 2 concentration [ pG/ml]
LogA2 - logarithm of plasma angiotensin concentration [ pG/ml])
Calculation of plasma angiotensin concentration and logarithm of plasma angiotensin concentration (most of the action of angiotensin are logarithmic in nature: concentration changes at higher concentrations produce less of an
effect than changes of the same size at lower concentrations)
logA2
AldoInf
VTWml
Aldo
A L D O S T E R O N E
INPUTS :logA2 - Logarithm of plasma angiotensin concentration [pG/ml]
AldoInf - Aldosterone infusion rate [nG/min]VTW - Total body water content [ml]
OUTPUT :Aldo - Plasma aldosterone concentration [ nG/dl]
Calculation of plasma aldosterone concentration
RAP
Af f C
TubC
RBF
RPF
APr
GKf
GFR
G L O M E R U L A R F I L T R A T I O N
INPUTS :RAP - Renal artery pressure[torr]
Affc - Afferent artery conductance [mll/min/torr]TubC - Proximal tubule conductaqnce [ml/min/torr]
RBF - Renal blood flow [ml/min]RPF - Renal plasma flow
APr - Plasma protein concentration (in afferent artery) [g/ml]GKf - Glomerular fi ltration coeffitient [ml/min/torr]
OUTPUT :GFR - Glomerular fi ltration rate [ml/min]
Calculation of glomerular fi ltration rate1
MDNaFlow
Aldo
DisFNaNorm
DisNaFlow
DisNaReab
DisFNa
N A T R I U M - D I S T A L T U B U L E
INPUTS :MDNaFlow - Sodium inflow [mmol/min]Aldo - Plasma aldosterone level [pg/ml]
DisFNaNorm - Normal value of sodium distal fractional reabsorbtion [relative number]
OUTPUTS :DisNaFlow - Sodium outflow [mmol/min]
DisNaReab - Distal sodium reabsorbrtion [mmol/min]DisFNa - Distal fractional sodium reabsorbtion
[relative number]
Calculation of distal tubule sodium reabsorbtion
DisNaFlow
CDFNaNorm
NaUrine
CDNaReab
CDFNa
N A T R I U M - C O L L E C T I N G D U C T
INPUTS :DisNaFlow - Sodium inflow [mmol/min]
CDFNaNorm - Normal value of sodium distal fractional reabsorbtion [relative number]
OUTPUTS :NaUrine - Sodium urine outflow [mmol/min]
CDNaReab - Collecting duct sodium reabsorbrtion [mmol/min]
CDFNa - Collecting duct fractional sodium reabsorbtion [relative number]
Calculation of collecing duct sodium reabsorbtion
logA2
MDSig
Ef f Norm
Ef f C
E F F E R E N T A R T E R Y
INPUTS :logA2 - logarithm of angiotensin concentration
MDSig - Macula densa feedback signal [ x Normal]EffNorm - Normal conductance in afferent artery [ml/min/torr]
OUTPUT :EffC - Vascular conductance [ml/min/torr]
Calculates conductance of efferent artery
Af f My o
MDSig
Af f Norm
Af f C
A F F E R E N T A R T E R Y
INPUTS :AffMyo - Myogenic effect [ x Nomal]
MDSig - Macula densa feedback signal [ x Normal]AffNorm - Normal conductance in afferent artery [ml/min/torr]
OUTPUT :AffC - Vascular conductance [ml/min/torr]
Calculates conductance of afferent artery
0
Angiotensin Infusion Rate
0
Aldosteron Infusion Rate
100
APNorm [torr]
Ef f C
AP
AP
Aldo
LogA2
LogA2
LogA2
Matlab/Simulink
Softwarové nástroje pro tvorbu modelů
VECml
DistNaFlow
200
Venous conductance[ml/min/torr]
7
Vena renalis pressure [torr]
-C-
Totasl body water content [ml]
Scope1 Scope
0
Renal artery pressure clamp
drop [torr]
0.07
Plasma protein cnoncentration
[g/ml]
6.25
Normal proximal tubule conductance
[ml/min/torr]
1
Normal macula densafeedback signal
16
Normal glomerular filtration coeffitient
[ml/min/torr]
0.5
Normal distalfractional reabsorbtion for Na
25
Normal conductanceof Efferent artery
[ml/min/torr]
30
Normal conductanceof Afferent artery
[ml/min/torr]
0.93
Normal collecting duct fractional reabsorbtion for Na
0.8
Normal Na proximal fractional reabsorbtion
0.125
NaDiet [mmol/min]
MDNaFlow
LogA2
MDNorm
MDSig
M A C U L A D E N S A
INPUTS :MDNaFlow - Macula densa sodium flow [mmol/min]
logA2 - Logarithm of plasma angiotensin concentration [pG/ml]MDNorm - Normal macula densa feedback signal [ x Normal]
OUTPUT :MDSig - Macula densa feedback signal [ x Normal]
Macula densa feedback signal calculation based on macula densa sodium flow and angiotensin concentration
0.44
Hematocrit
NaDiet
NaUrine
VECml
ZNAE
PNa
S I M P L E S O D I U M B A L A N C E
INPUTS :NaDIet - Dietary sodium intake [mmol/min]
NaUrine - Sodium urine outflow [mmol/min]VECml - Extracellular fluid volume [ml]
OUTPUT :ZNAE - ECF sodium content [mmol]
PNa - Plasma sodium concentration [mmol/ml]
Extracellular sodium quantity is the integral over time dietary sodium intake minus urinary sodium loss
15000
Extracellular fluid volume[mmol/min]1
0.5086
1.855
18.84
1.275
6.846
0.144
2159
125.1
3.648
14.37
0.7975
0.1248
0.9999
99.65
1182
662.1
29.83
25.1
99.651.006
0.9418
1.792
0.9304
1.668
1
Converting Enzyme Activity
ZNAE
LogA2
APNorm
AP
A R T E R I A L P R E S S U R E
INPUTS :ZNAE - ECF sodium content [mmol]
logA2 - Logarithm of plasma angiotensin concentration [pG/ml]APNorm - Normal value of arterial pressurel [torr]
OUTPUT :AP Arterial pressure [torr]
Control of arterial pressure by angiotensin and extracellular sodium content
RAP Af f My o1
M Y O G E N I C R E S P O N S E
INPUT :RAP - Renal artery pressure [torr]
OUTPUT :AffC - Myogenic effect [ x Normal]
Calculation of the myogenic response to changes in renal perfusion pressure(afferent conductance responds to changes in perfusion pressure,
with pressure increases causing vasoconstriction)
Af f C
Ef f C
RenVenC
AP
VP
Hct
Clamp
RAP
RBF
RPF
R E N A L P E R F U S I O N
INPUTS :AffC - Afferent artery conductance [ml/min/torr]EffC Efferent artery conductance [ml/min/torr]
RenVenC - Renal venous conductance [ml/min/torr]AP - Arterial pressure [torr]
VP - Vena renalis pressure [torr]Hct - Hematocrit [relative number]
Clamp - Renal artery pressure drop caused by renal artery clamp [torr]
OUTPUTS :RAP - Renal artery pressure [torr]
RBF - Renal blood flow rate[ml/min]RPF - Renal plasma flow rate[ml/min]
Calculation of renal artery pressure and renal blood flow rate
PNa
GFR
LogA2
PrxFNaNorm
MDNaFlow
PdxNaReab
PrxFNa
N A T R I U M - P R O X I M A L T U B U L E
INPUTS :PNa - Plasma sodium concentration [mmol/ml]
GFR - GLomerulal filtration rate [ml/min]LogA2 - Logarithm of plasma angiotensin concentration
[pg/ml]PrxFNaNorm - Normal value of sodium proximal
fractional reabsorbtion [relative number]
OUTPUTS :MDNaFlow - Sodium outflow [mmol/min]
PdxNaReab - Proximal sodium reabsorbrtion [mmol/l]PrxFNa - Proximal fractional sodium reabsorbtion
[relative number]
Calculation of proximal tubule sodium reabsorbtion
MDSig
VECml
PRA
R E N I N
INPUTS :MDSig - Macula densa feedback signal [ x Normal]
VECml - Ectracellular fluid volume
OUTPUT :PRA - Plasma renin activity [ Units/ml]
Calculation of plasma renin activity
PRA
CEAct
A2Inf
A2
logA2
A N G I O T E N S I N
INPUTS :PRA - Plasma renin activity [ Units/ml]
CEAct - Converting enzyme activity [x Normal]A2Inf Angiotensin 2 infusion rate [nG/min]
OUTPUTS :A2 - Plasma angiotensin 2 concentration [ pG/ml]
LogA2 - logarithm of plasma angiotensin concentration [ pG/ml])
Calculation of plasma angiotensin concentration and logarithm of plasma angiotensin concentration (most of the action of angiotensin are logarithmic in nature: concentration changes at higher concentrations produce less of an
effect than changes of the same size at lower concentrations)
logA2
AldoInf
VTWml
Aldo
A L D O S T E R O N E
INPUTS :logA2 - Logarithm of plasma angiotensin concentration [pG/ml]
AldoInf - Aldosterone infusion rate [nG/min]VTW - Total body water content [ml]
OUTPUT :Aldo - Plasma aldosterone concentration [ nG/dl]
Calculation of plasma aldosterone concentration
RAP
Af f C
TubC
RBF
RPF
APr
GKf
GFR
G L O M E R U L A R F I L T R A T I O N
INPUTS :RAP - Renal artery pressure[torr]
Affc - Afferent artery conductance [mll/min/torr]TubC - Proximal tubule conductaqnce [ml/min/torr]
RBF - Renal blood flow [ml/min]RPF - Renal plasma flow
APr - Plasma protein concentration (in afferent artery) [g/ml]GKf - Glomerular filtration coeffitient [ml/min/torr]
OUTPUT :GFR - Glomerular filtration rate [ml/min]
Calculation of glomerular filtration rate1
MDNaFlow
Aldo
DisFNaNorm
DisNaFlow
DisNaReab
DisFNa
N A T R I U M - D I S T A L T U B U L E
INPUTS :MDNaFlow - Sodium inflow [mmol/min]Aldo - Plasma aldosterone level [pg/ml]
DisFNaNorm - Normal value of sodium distal fractional reabsorbtion [relative number]
OUTPUTS :DisNaFlow - Sodium outflow [mmol/min]
DisNaReab - Distal sodium reabsorbrtion [mmol/min]DisFNa - Distal fractional sodium reabsorbtion
[relative number]
Calculation of distal tubule sodium reabsorbtion
DisNaFlow
CDFNaNorm
NaUrine
CDNaReab
CDFNa
N A T R I U M - C O L L E C T I N G D U C T
INPUTS :DisNaFlow - Sodium inflow [mmol/min]
CDFNaNorm - Normal value of sodium distal fractional reabsorbtion [relative number]
OUTPUTS :NaUrine - Sodium urine outflow [mmol/min]
CDNaReab - Collecting duct sodium reabsorbrtion [mmol/min]
CDFNa - Collecting duct fractional sodium reabsorbtion [relative number]
Calculation of collecing duct sodium reabsorbtion
logA2
MDSig
Ef f Norm
Ef f C
E F F E R E N T A R T E R Y
INPUTS :logA2 - logarithm of angiotensin concentration
MDSig - Macula densa feedback signal [ x Normal]EffNorm - Normal conductance in afferent artery [ml/min/torr]
OUTPUT :EffC - Vascular conductance [ml/min/torr]
Calculates conductance of efferent artery
Af f My o
MDSig
Af f Norm
Af f C
A F F E R E N T A R T E R Y
INPUTS :AffMyo - Myogenic effect [ x Nomal]
MDSig - Macula densa feedback signal [ x Normal]AffNorm - Normal conductance in afferent artery [ml/min/torr]
OUTPUT :AffC - Vascular conductance [ml/min/torr]
Calculates conductance of afferent artery
0
Angiotensin Infusion Rate
0
Aldosteron Infusion Rate
100
APNorm [torr]
Ef f C
AP
AP
Aldo
LogA2
LogA2
LogA2
Matlab/Simulink
Softwarové nástroje pro tvorbu modelů
VECml
DistNaFlow
200
Venous conductance[ml/min/torr]
7
Vena renalis pressure [torr]
-C-
Totasl body water content [ml]
Scope1 Scope
0
Renal artery pressure clamp
drop [torr]
0.07
Plasma protein cnoncentration
[g/ml]
6.25
Normal proximal tubule conductance
[ml/min/torr]
1
Normal macula densafeedback signal
16
Normal glomerular filtration coeffitient
[ml/min/torr]
0.5
Normal distalfractional reabsorbtion for Na
25
Normal conductanceof Efferent artery
[ml/min/torr]
30
Normal conductanceof Afferent artery
[ml/min/torr]
0.93
Normal collecting duct fractional reabsorbtion for Na
0.8
Normal Na proximal fractional reabsorbtion
0.125
NaDiet [mmol/min]
MDNaFlow
LogA2
MDNorm
MDSig
M A C U L A D E N S A
INPUTS :MDNaFlow - Macula densa sodium flow [mmol/min]
logA2 - Logarithm of plasma angiotensin concentration [pG/ml]MDNorm - Normal macula densa feedback signal [ x Normal]
OUTPUT :MDSig - Macula densa feedback signal [ x Normal]
Macula densa feedback signal calculation based on macula densa sodium flow and angiotensin concentration
0.44
Hematocrit
NaDiet
NaUrine
VECml
ZNAE
PNa
S I M P L E S O D I U M B A L A N C E
INPUTS :NaDIet - Dietary sodium intake [mmol/min]
NaUrine - Sodium urine outflow [mmol/min]VECml - Extracellular fluid volume [ml]
OUTPUT :ZNAE - ECF sodium content [mmol]
PNa - Plasma sodium concentration [mmol/ml]
Extracellular sodium quantity is the integral over time dietary sodium intake minus urinary sodium loss
15000
Extracellular fluid volume[mmol/min]1
0.5086
1.855
18.84
1.275
6.846
0.144
2159
125.1
3.648
14.37
0.7975
0.1248
0.9999
99.65
1182
662.1
29.83
25.1
99.651.006
0.9418
1.792
0.9304
1.668
1
Converting Enzyme Activity
ZNAE
LogA2
APNorm
AP
A R T E R I A L P R E S S U R E
INPUTS :ZNAE - ECF sodium content [mmol]
logA2 - Logarithm of plasma angiotensin concentration [pG/ml]APNorm - Normal value of arterial pressurel [torr]
OUTPUT :AP Arterial pressure [torr]
Control of arterial pressure by angiotensin and extracellular sodium content
RAP Af f My o1
M Y O G E N I C R E S P O N S E
INPUT :RAP - Renal artery pressure [torr]
OUTPUT :AffC - Myogenic effect [ x Normal]
Calculation of the myogenic response to changes in renal perfusion pressure(afferent conductance responds to changes in perfusion pressure,
with pressure increases causing vasoconstriction)
Af f C
Ef f C
RenVenC
AP
VP
Hct
Clamp
RAP
RBF
RPF
R E N A L P E R F U S I O N
INPUTS :AffC - Afferent artery conductance [ml/min/torr]EffC Efferent artery conductance [ml/min/torr]
RenVenC - Renal venous conductance [ml/min/torr]AP - Arterial pressure [torr]
VP - Vena renalis pressure [torr]Hct - Hematocrit [relative number]
Clamp - Renal artery pressure drop caused by renal artery clamp [torr]
OUTPUTS :RAP - Renal artery pressure [torr]
RBF - Renal blood flow rate[ml/min]RPF - Renal plasma flow rate[ml/min]
Calculation of renal artery pressure and renal blood flow rate
PNa
GFR
LogA2
PrxFNaNorm
MDNaFlow
PdxNaReab
PrxFNa
N A T R I U M - P R O X I M A L T U B U L E
INPUTS :PNa - Plasma sodium concentration [mmol/ml]
GFR - GLomerulal fi ltration rate [ml/min]LogA2 - Logarithm of plasma angiotensin concentration
[pg/ml]PrxFNaNorm - Normal value of sodium proximal
fractional reabsorbtion [relative number]
OUTPUTS :MDNaFlow - Sodium outflow [mmol/min]
PdxNaReab - Proximal sodium reabsorbrtion [mmol/l]PrxFNa - Proximal fractional sodium reabsorbtion
[relative number]
Calculation of proximal tubule sodium reabsorbtion
MDSig
VECml
PRA
R E N I N
INPUTS :MDSig - Macula densa feedback signal [ x Normal]
VECml - Ectracellular fluid volume
OUTPUT :PRA - Plasma renin activity [ Units/ml]
Calculation of plasma renin activity
PRA
CEAct
A2Inf
A2
logA2
A N G I O T E N S I N
INPUTS :PRA - Plasma renin activity [ Units/ml]
CEAct - Converting enzyme activity [x Normal]A2Inf Angiotensin 2 infusion rate [nG/min]
OUTPUTS :A2 - Plasma angiotensin 2 concentration [ pG/ml]
LogA2 - logarithm of plasma angiotensin concentration [ pG/ml])
Calculation of plasma angiotensin concentration and logarithm of plasma angiotensin concentration (most of the action of angiotensin are logarithmic in nature: concentration changes at higher concentrations produce less of an
effect than changes of the same size at lower concentrations)
logA2
AldoInf
VTWml
Aldo
A L D O S T E R O N E
INPUTS :logA2 - Logarithm of plasma angiotensin concentration [pG/ml]
AldoInf - Aldosterone infusion rate [nG/min]VTW - Total body water content [ml]
OUTPUT :Aldo - Plasma aldosterone concentration [ nG/dl]
Calculation of plasma aldosterone concentration
RAP
Af f C
TubC
RBF
RPF
APr
GKf
GFR
G L O M E R U L A R F I L T R A T I O N
INPUTS :RAP - Renal artery pressure[torr]
Affc - Afferent artery conductance [mll/min/torr]TubC - Proximal tubule conductaqnce [ml/min/torr]
RBF - Renal blood flow [ml/min]RPF - Renal plasma flow
APr - Plasma protein concentration (in afferent artery) [g/ml]GKf - Glomerular filtration coeffitient [ml/min/torr]
OUTPUT :GFR - Glomerular filtration rate [ml/min]
Calculation of glomerular filtration rate1
MDNaFlow
Aldo
DisFNaNorm
DisNaFlow
DisNaReab
DisFNa
N A T R I U M - D I S T A L T U B U L E
INPUTS :MDNaFlow - Sodium inflow [mmol/min]Aldo - Plasma aldosterone level [pg/ml]
DisFNaNorm - Normal value of sodium distal fractional reabsorbtion [relative number]
OUTPUTS :DisNaFlow - Sodium outflow [mmol/min]
DisNaReab - Distal sodium reabsorbrtion [mmol/min]DisFNa - Distal fractional sodium reabsorbtion
[relative number]
Calculation of distal tubule sodium reabsorbtion
DisNaFlow
CDFNaNorm
NaUrine
CDNaReab
CDFNa
N A T R I U M - C O L L E C T I N G D U C T
INPUTS :DisNaFlow - Sodium inflow [mmol/min]
CDFNaNorm - Normal value of sodium distal fractional reabsorbtion [relative number]
OUTPUTS :NaUrine - Sodium urine outflow [mmol/min]
CDNaReab - Collecting duct sodium reabsorbrtion [mmol/min]
CDFNa - Collecting duct fractional sodium reabsorbtion [relative number]
Calculation of collecing duct sodium reabsorbtion
logA2
MDSig
Ef f Norm
Ef f C
E F F E R E N T A R T E R Y
INPUTS :logA2 - logarithm of angiotensin concentration
MDSig - Macula densa feedback signal [ x Normal]EffNorm - Normal conductance in afferent artery [ml/min/torr]
OUTPUT :EffC - Vascular conductance [ml/min/torr]
Calculates conductance of efferent artery
Af f My o
MDSig
Af f Norm
Af f C
A F F E R E N T A R T E R Y
INPUTS :AffMyo - Myogenic effect [ x Nomal]
MDSig - Macula densa feedback signal [ x Normal]AffNorm - Normal conductance in afferent artery [ml/min/torr]
OUTPUT :AffC - Vascular conductance [ml/min/torr]
Calculates conductance of afferent artery
0
Angiotensin Infusion Rate
0
Aldosteron Infusion Rate
100
APNorm [torr]
Ef f C
AP
AP
Aldo
LogA2
LogA2
LogA2
Matlab/Simulink
RPF - Renal plasma flow [ml/min]
GKf - Glomerular fi ltration coeffitient [ml/min/torr]
GFR - Glomerular fi ltration rate [ml/min]
AVeCOP - Average colloid osmotic pressure [torr]
ECOP - Efferent colloid osmotic pressure [torr]
APr - Afferent protein concentration [g/ml]
EPr - Efferent protein concentration [g/ml]
GFR - Glomerular fi ltration rate [ml/min]
PTP - Proximal tubule pressure [torr]TubC - Tubule conductance [ml/min/torr]
FF - Filtration fraction [relative number]
AffC - Afferent artery conductance [ml/min/torr]
RBF - Renal blood flow [ml/min]
ACOP - Afferent colloid osmotic pressure [torr]
GFR - Glomerular fi ltration rate [ml/min]
RAP - Renal artery pressure [torr]
GKf - Glomerular fi ltration coeffitient [ml/min/torr]
NETP - Net pressure gradient in glomerulus [torr]
PAff - Afferent artery pressure [torr]
GP - Glomerulal pressure [torr]
simulation chip: G L O M E R U L A R F I L T R A T I O N
1
GFR
PTP = GFR/TubC
PAff=RBF*AffC
NetP = GP - PTP - AVECOP
GP = RAP - PAff
GFR = NETP*GKf
FF = GFR/RPF
EPr = APr / (1-FF)
ECOP = A*Epr + B*(EPr)^2
1160
B - Landis-Pappenheimer coeffitient [torr/g/ml]1
B * (Epr)^2
B * (Apr)^2
AVeCOP = (ACOP+ECOP)/2
ACOP+ECOP
ACOP = A*Apr + B*(APr)^2
A*Epr
A*Apr
320
A - Landis-Pappenheimer coeffitient [torr/g/ml]
1 - FF
u2
(EPr)^2
u2
(APr)^2
2
1
7
GKf
6
APr
5
RPF
4
RBF
3
TubC
2
AffC
1
RAP
GP
PAf f
RAP
GKf
Matlab/Simulink
Grafické zobrazení matematických vztahů?
RPF - Renal plasma flow [ml/min]
GKf - Glomerular fi ltration coeffitient [ml/min/torr]
GFR - Glomerular fi ltration rate [ml/min]
AVeCOP - Average colloid osmotic pressure [torr]
ECOP - Efferent colloid osmotic pressure [torr]
APr - Afferent protein concentration [g/ml]
EPr - Efferent protein concentration [g/ml]
GFR - Glomerular fi ltration rate [ml/min]
PTP - Proximal tubule pressure [torr]TubC - Tubule conductance [ml/min/torr]
FF - Filtration fraction [relative number]
AffC - Afferent artery conductance [ml/min/torr]
RBF - Renal blood flow [ml/min]
ACOP - Afferent colloid osmotic pressure [torr]
GFR - Glomerular fi ltration rate [ml/min]
RAP - Renal artery pressure [torr]
GKf - Glomerular fi ltration coeffitient [ml/min/torr]
NETP - Net pressure gradient in glomerulus [torr]
PAff - Afferent artery pressure [torr]
GP - Glomerulal pressure [torr]
simulation chip: G L O M E R U L A R F I L T R A T I O N
1
GFR
PTP = GFR/TubC
PAff=RBF*AffC
NetP = GP - PTP - AVECOP
GP = RAP - PAff
GFR = NETP*GKf
FF = GFR/RPF
EPr = APr / (1-FF)
ECOP = A*Epr + B*(EPr)^2
1160
B - Landis-Pappenheimer coeffitient [torr/g/ml]1
B * (Epr)^2
B * (Apr)^2
AVeCOP = (ACOP+ECOP)/2
ACOP+ECOP
ACOP = A*Apr + B*(APr)^2
A*Epr
A*Apr
320
A - Landis-Pappenheimer coeffitient [torr/g/ml]
1 - FF
u2
(EPr)^2
u2
(APr)^2
2
1
7
GKf
6
APr
5
RPF
4
RBF
3
TubC
2
AffC
1
RAP
GP
PAf f
RAP
GKf
FF=GFR/RPF
GFR
RPFRPF
APrAPr
BA
RBF
AffC
TubC
RAP
GFR=NETP*GKf
NETP=GP-PTP-AVeCOP
PTP=GFR/TubC
PAff=RBF/AffCGP=RAP-PAff
ACOP=A*Apr+B*(APr)^2A*Apr
B*(APr)^2(APr)^2
EPr=APr/(1-FF)
(1-FF)
EPr^2
A*EPr
B*EPr^2
ECOP=A*EPr+B*EPr^2
ACOP+ECOP
AVeCOP=(ACOP+ECOP)/2
AVeCOPPTP
GP
GFR
TubC
GFRGKf
GKf
Matlab/Simulink
Grafické zobrazení matematických vztahů?
RPF - Renal plasma flow [ml/min]
GKf - Glomerular fi ltration coeffitient [ml/min/torr]
GFR - Glomerular fi ltration rate [ml/min]
AVeCOP - Average colloid osmotic pressure [torr]
ECOP - Efferent colloid osmotic pressure [torr]
APr - Afferent protein concentration [g/ml]
EPr - Efferent protein concentration [g/ml]
GFR - Glomerular fi ltration rate [ml/min]
PTP - Proximal tubule pressure [torr]TubC - Tubule conductance [ml/min/torr]
FF - Filtration fraction [relative number]
AffC - Afferent artery conductance [ml/min/torr]
RBF - Renal blood flow [ml/min]
ACOP - Afferent colloid osmotic pressure [torr]
GFR - Glomerular fi ltration rate [ml/min]
RAP - Renal artery pressure [torr]
GKf - Glomerular fi ltration coeffitient [ml/min/torr]
NETP - Net pressure gradient in glomerulus [torr]
PAff - Afferent artery pressure [torr]
GP - Glomerulal pressure [torr]
simulation chip: G L O M E R U L A R F I L T R A T I O N
1
GFR
PTP = GFR/TubC
PAff=RBF*AffC
NetP = GP - PTP - AVECOP
GP = RAP - PAff
GFR = NETP*GKf
FF = GFR/RPF
EPr = APr / (1-FF)
ECOP = A*Epr + B*(EPr)^2
1160
B - Landis-Pappenheimer coeffitient [torr/g/ml]1
B * (Epr)^2
B * (Apr)^2
AVeCOP = (ACOP+ECOP)/2
ACOP+ECOP
ACOP = A*Apr + B*(APr)^2
A*Epr
A*Apr
320
A - Landis-Pappenheimer coeffitient [torr/g/ml]
1 - FF
u2
(EPr)^2
u2
(APr)^2
2
1
7
GKf
6
APr
5
RPF
4
RBF
3
TubC
2
AffC
1
RAP
GP
PAf f
RAP
GKf
FF=GFR/RPF
GFR
RPFRPF
APrAPr
BA
RBF
AffC
TubC
RAP
GFR=NETP*GKf
NETP=GP-PTP-AVeCOP
PTP=GFR/TubC
PAff=RBF/AffCGP=RAP-PAff
ACOP=A*Apr+B*(APr)^2A*Apr
B*(APr)^2(APr)^2
EPr=APr/(1-FF)
(1-FF)
EPr^2
A*EPr
B*EPr^2
ECOP=A*EPr+B*EPr^2
ACOP+ECOP
AVeCOP=(ACOP+ECOP)/2
AVeCOPPTP
GP
GFR
TubC
GFRGKf
GKf
Matlab/Simulink
Grafické zobrazení matematických vztahů?
RPF - Renal plasma flow [ml/min]
GKf - Glomerular fi ltration coeffitient [ml/min/torr]
GFR - Glomerular fi ltration rate [ml/min]
AVeCOP - Average colloid osmotic pressure [torr]
ECOP - Efferent colloid osmotic pressure [torr]
APr - Afferent protein concentration [g/ml]
EPr - Efferent protein concentration [g/ml]
GFR - Glomerular fi ltration rate [ml/min]
PTP - Proximal tubule pressure [torr]TubC - Tubule conductance [ml/min/torr]
FF - Filtration fraction [relative number]
AffC - Afferent artery conductance [ml/min/torr]
RBF - Renal blood flow [ml/min]
ACOP - Afferent colloid osmotic pressure [torr]
GFR - Glomerular fi ltration rate [ml/min]
RAP - Renal artery pressure [torr]
GKf - Glomerular fi ltration coeffitient [ml/min/torr]
NETP - Net pressure gradient in glomerulus [torr]
PAff - Afferent artery pressure [torr]
GP - Glomerulal pressure [torr]
simulation chip: G L O M E R U L A R F I L T R A T I O N
1
GFR
PTP = GFR/TubC
PAff=RBF*AffC
NetP = GP - PTP - AVECOP
GP = RAP - PAff
GFRold - GFRnew
GFR = NETP*GKf
FF = GFR/RPF
EPr = APr / (1-FF)
ECOP = A*Epr + B*(EPr)^2
1160
B - Landis-Pappenheimer coeffitient [torr/g/ml]1
B * (Epr)^2
B * (Apr)^2
f (z) zSolve
f(z) = 0
Algebraic Constraint
AVeCOP = (ACOP+ECOP)/2
ACOP+ECOP
ACOP = A*Apr + B*(APr)^2
A*Epr
A*Apr
320
A - Landis-Pappenheimer coeffitient [torr/g/ml]
1 - FF
u2
(EPr)^2
u2
(APr)^2
2
1
7
GKf
6
APr
5
RPF
4
RBF
3
TubC
2
AffC
1
RAP
GP
PAf f
RAP
GKf
FF=GFR/RPF
GFR
RPFRPF
APrAPr
BA
RBF
AffC
TubC
RAP
GFR=NETP*GKf
NETP=GP-PTP-AVeCOP
PTP=GFR/TubC
PAff=RBF/AffCGP=RAP-PAff
ACOP=A*Apr+B*(APr)^2A*Apr
B*(APr)^2(APr)^2
EPr=APr/(1-FF)
(1-FF)
EPr^2
A*EPr
B*EPr^2
ECOP=A*EPr+B*EPr^2
ACOP+ECOP
AVeCOP=(ACOP+ECOP)/2
AVeCOPPTP
GP
GFR
TubC
GFRGKf
GKf
GFRold-GFRnew
Algebraic Constraint
Matlab/Simulink
Grafické zobrazení matematických vztahů?
RPF - Renal plasma flow [ml/min]
GKf - Glomerular fi ltration coeffitient [ml/min/torr]
GFR - Glomerular fi ltration rate [ml/min]
AVeCOP - Average colloid osmotic pressure [torr]
ECOP - Efferent colloid osmotic pressure [torr]
APr - Afferent protein concentration [g/ml]
EPr - Efferent protein concentration [g/ml]
GFR - Glomerular fi ltration rate [ml/min]
PTP - Proximal tubule pressure [torr]TubC - Tubule conductance [ml/min/torr]
FF - Filtration fraction [relative number]
AffC - Afferent artery conductance [ml/min/torr]
RBF - Renal blood flow [ml/min]
ACOP - Afferent colloid osmotic pressure [torr]
GFR - Glomerular fi ltration rate [ml/min]
RAP - Renal artery pressure [torr]
GKf - Glomerular fi ltration coeffitient [ml/min/torr]
NETP - Net pressure gradient in glomerulus [torr]
PAff - Afferent artery pressure [torr]
GP - Glomerulal pressure [torr]
simulation chip: G L O M E R U L A R F I L T R A T I O N
1
GFR
PTP = GFR/TubC
PAff=RBF*AffC
NetP = GP - PTP - AVECOP
GP = RAP - PAff
GFRold - GFRnew
GFR = NETP*GKf
FF = GFR/RPF
EPr = APr / (1-FF)
ECOP = A*Epr + B*(EPr)^2
1160
B - Landis-Pappenheimer coeffitient [torr/g/ml]1
B * (Epr)^2
B * (Apr)^2
f (z) zSolve
f(z) = 0
Algebraic Constraint
AVeCOP = (ACOP+ECOP)/2
ACOP+ECOP
ACOP = A*Apr + B*(APr)^2
A*Epr
A*Apr
320
A - Landis-Pappenheimer coeffitient [torr/g/ml]
1 - FF
u2
(EPr)^2
u2
(APr)^2
2
1
7
GKf
6
APr
5
RPF
4
RBF
3
TubC
2
AffC
1
RAP
GP
PAf f
RAP
GKf
Matlab/Simulink
Grafické zobrazení matematických vztahů?
Grafické zobrazení transformace
vstupních hodnot na výstupní
Ne!
Softwarové nástroje pro tvorbu modelů
VECml
DistNaFlow
200
Venous conductance[ml/min/torr]
7
Vena renalis pressure [torr]
-C-
Totasl body water content [ml]
Scope1 Scope
0
Renal artery pressure clamp
drop [torr]
0.07
Plasma protein cnoncentration
[g/ml]
6.25
Normal proximal tubule conductance
[ml/min/torr]
1
Normal macula densafeedback signal
16
Normal glomerular filtration coeffitient
[ml/min/torr]
0.5
Normal distalfractional reabsorbtion for Na
25
Normal conductanceof Efferent artery
[ml/min/torr]
30
Normal conductanceof Afferent artery
[ml/min/torr]
0.93
Normal collecting duct fractional reabsorbtion for Na
0.8
Normal Na proximal fractional reabsorbtion
0.125
NaDiet [mmol/min]
MDNaFlow
LogA2
MDNorm
MDSig
M A C U L A D E N S A
INPUTS :MDNaFlow - Macula densa sodium flow [mmol/min]
logA2 - Logarithm of plasma angiotensin concentration [pG/ml]MDNorm - Normal macula densa feedback signal [ x Normal]
OUTPUT :MDSig - Macula densa feedback signal [ x Normal]
Macula densa feedback signal calculation based on macula densa sodium flow and angiotensin concentration
0.44
Hematocrit
NaDiet
NaUrine
VECml
ZNAE
PNa
S I M P L E S O D I U M B A L A N C E
INPUTS :NaDIet - Dietary sodium intake [mmol/min]
NaUrine - Sodium urine outflow [mmol/min]VECml - Extracellular fluid volume [ml]
OUTPUT :ZNAE - ECF sodium content [mmol]
PNa - Plasma sodium concentration [mmol/ml]
Extracellular sodium quantity is the integral over time dietary sodium intake minus urinary sodium loss
15000
Extracellular fluid volume[mmol/min]1
0.5086
1.855
18.84
1.275
6.846
0.144
2159
125.1
3.648
14.37
0.7975
0.1248
0.9999
99.65
1182
662.1
29.83
25.1
99.651.006
0.9418
1.792
0.9304
1.668
1
Converting Enzyme Activity
ZNAE
LogA2
APNorm
AP
A R T E R I A L P R E S S U R E
INPUTS :ZNAE - ECF sodium content [mmol]
logA2 - Logarithm of plasma angiotensin concentration [pG/ml]APNorm - Normal value of arterial pressurel [torr]
OUTPUT :AP Arterial pressure [torr]
Control of arterial pressure by angiotensin and extracellular sodium content
RAP Af f My o1
M Y O G E N I C R E S P O N S E
INPUT :RAP - Renal artery pressure [torr]
OUTPUT :AffC - Myogenic effect [ x Normal]
Calculation of the myogenic response to changes in renal perfusion pressure(afferent conductance responds to changes in perfusion pressure,
with pressure increases causing vasoconstriction)
Af f C
Ef f C
RenVenC
AP
VP
Hct
Clamp
RAP
RBF
RPF
R E N A L P E R F U S I O N
INPUTS :AffC - Afferent artery conductance [ml/min/torr]EffC Efferent artery conductance [ml/min/torr]
RenVenC - Renal venous conductance [ml/min/torr]AP - Arterial pressure [torr]
VP - Vena renalis pressure [torr]Hct - Hematocrit [relative number]
Clamp - Renal artery pressure drop caused by renal artery clamp [torr]
OUTPUTS :RAP - Renal artery pressure [torr]
RBF - Renal blood flow rate[ml/min]RPF - Renal plasma flow rate[ml/min]
Calculation of renal artery pressure and renal blood flow rate
PNa
GFR
LogA2
PrxFNaNorm
MDNaFlow
PdxNaReab
PrxFNa
N A T R I U M - P R O X I M A L T U B U L E
INPUTS :PNa - Plasma sodium concentration [mmol/ml]
GFR - GLomerulal filtration rate [ml/min]LogA2 - Logarithm of plasma angiotensin concentration
[pg/ml]PrxFNaNorm - Normal value of sodium proximal
fractional reabsorbtion [relative number]
OUTPUTS :MDNaFlow - Sodium outflow [mmol/min]
PdxNaReab - Proximal sodium reabsorbrtion [mmol/l]PrxFNa - Proximal fractional sodium reabsorbtion
[relative number]
Calculation of proximal tubule sodium reabsorbtion
MDSig
VECml
PRA
R E N I N
INPUTS :MDSig - Macula densa feedback signal [ x Normal]
VECml - Ectracellular fluid volume
OUTPUT :PRA - Plasma renin activity [ Units/ml]
Calculation of plasma renin activity
PRA
CEAct
A2Inf
A2
logA2
A N G I O T E N S I N
INPUTS :PRA - Plasma renin activity [ Units/ml]
CEAct - Converting enzyme activity [x Normal]A2Inf Angiotensin 2 infusion rate [nG/min]
OUTPUTS :A2 - Plasma angiotensin 2 concentration [ pG/ml]
LogA2 - logarithm of plasma angiotensin concentration [ pG/ml])
Calculation of plasma angiotensin concentration and logarithm of plasma angiotensin concentration (most of the action of angiotensin are logarithmic in nature: concentration changes at higher concentrations produce less of an
effect than changes of the same size at lower concentrations)
logA2
AldoInf
VTWml
Aldo
A L D O S T E R O N E
INPUTS :logA2 - Logarithm of plasma angiotensin concentration [pG/ml]
AldoInf - Aldosterone infusion rate [nG/min]VTW - Total body water content [ml]
OUTPUT :Aldo - Plasma aldosterone concentration [ nG/dl]
Calculation of plasma aldosterone concentration
RAP
Af f C
TubC
RBF
RPF
APr
GKf
GFR
G L O M E R U L A R F I L T R A T I O N
INPUTS :RAP - Renal artery pressure[torr]
Affc - Afferent artery conductance [mll/min/torr]TubC - Proximal tubule conductaqnce [ml/min/torr]
RBF - Renal blood flow [ml/min]RPF - Renal plasma flow
APr - Plasma protein concentration (in afferent artery) [g/ml]GKf - Glomerular filtration coeffitient [ml/min/torr]
OUTPUT :GFR - Glomerular filtration rate [ml/min]
Calculation of glomerular filtration rate1
MDNaFlow
Aldo
DisFNaNorm
DisNaFlow
DisNaReab
DisFNa
N A T R I U M - D I S T A L T U B U L E
INPUTS :MDNaFlow - Sodium inflow [mmol/min]Aldo - Plasma aldosterone level [pg/ml]
DisFNaNorm - Normal value of sodium distal fractional reabsorbtion [relative number]
OUTPUTS :DisNaFlow - Sodium outflow [mmol/min]
DisNaReab - Distal sodium reabsorbrtion [mmol/min]DisFNa - Distal fractional sodium reabsorbtion
[relative number]
Calculation of distal tubule sodium reabsorbtion
DisNaFlow
CDFNaNorm
NaUrine
CDNaReab
CDFNa
N A T R I U M - C O L L E C T I N G D U C T
INPUTS :DisNaFlow - Sodium inflow [mmol/min]
CDFNaNorm - Normal value of sodium distal fractional reabsorbtion [relative number]
OUTPUTS :NaUrine - Sodium urine outflow [mmol/min]
CDNaReab - Collecting duct sodium reabsorbrtion [mmol/min]
CDFNa - Collecting duct fractional sodium reabsorbtion [relative number]
Calculation of collecing duct sodium reabsorbtion
logA2
MDSig
Ef f Norm
Ef f C
E F F E R E N T A R T E R Y
INPUTS :logA2 - logarithm of angiotensin concentration
MDSig - Macula densa feedback signal [ x Normal]EffNorm - Normal conductance in afferent artery [ml/min/torr]
OUTPUT :EffC - Vascular conductance [ml/min/torr]
Calculates conductance of efferent artery
Af f My o
MDSig
Af f Norm
Af f C
A F F E R E N T A R T E R Y
INPUTS :AffMyo - Myogenic effect [ x Nomal]
MDSig - Macula densa feedback signal [ x Normal]AffNorm - Normal conductance in afferent artery [ml/min/torr]
OUTPUT :AffC - Vascular conductance [ml/min/torr]
Calculates conductance of afferent artery
0
Angiotensin Infusion Rate
0
Aldosteron Infusion Rate
100
APNorm [torr]
Ef f C
AP
AP
Aldo
LogA2
LogA2
LogA2
Matlab/Simulink
Blokově orientované modelovací nástroje
VECml
DistNaFlow
200
Venous conductance[ml/min/torr]
7
Vena renalis pressure [torr]
-C-
Totasl body water content [ml]
Scope1 Scope
0
Renal artery pressure clamp
drop [torr]
0.07
Plasma protein cnoncentration
[g/ml]
6.25
Normal proximal tubule conductance
[ml/min/torr]
1
Normal macula densafeedback signal
16
Normal glomerular fi ltration coeffitient
[ml/min/torr]
0.5
Normal distalfractional reabsorbtion for Na
25
Normal conductanceof Efferent artery
[ml/min/torr]
30
Normal conductanceof Afferent artery
[ml/min/torr]
0.93
Normal collecting duct fractional reabsorbtion for Na
0.8
Normal Na proximal fractional reabsorbtion
0.125
NaDiet [mmol/min]
MDNaFlow
LogA2
MDNorm
MDSig
M A C U L A D E N S A
INPUTS :MDNaFlow - Macula densa sodium flow [mmol/min]
logA2 - Logarithm of plasma angiotensin concentration [pG/ml]MDNorm - Normal macula densa feedback signal [ x Normal]
OUTPUT :MDSig - Macula densa feedback signal [ x Normal]
Macula densa feedback signal calculation based on macula densa sodium flow and angiotensin concentration
0.44
Hematocrit
NaDiet
NaUrine
VECml
ZNAE
PNa
S I M P L E S O D I U M B A L A N C E
INPUTS :NaDIet - Dietary sodium intake [mmol/min]
NaUrine - Sodium urine outflow [mmol/min]VECml - Extracellular fluid volume [ml]
OUTPUT :ZNAE - ECF sodium content [mmol]
PNa - Plasma sodium concentration [mmol/ml]
Extracellular sodium quantity is the integral over time dietary sodium intake minus urinary sodium loss
15000
Extracellular fluid volume[mmol/min]1
0.5086
1.855
18.84
1.275
6.846
0.144
2159
125.1
3.648
14.37
0.7975
0.1248
0.9999
99.65
1182
662.1
29.83
25.1
99.651.006
0.9418
1.792
0.9304
1.668
1
Converting Enzyme Activity
ZNAE
LogA2
APNorm
AP
A R T E R I A L P R E S S U R E
INPUTS :ZNAE - ECF sodium content [mmol]
logA2 - Logarithm of plasma angiotensin concentration [pG/ml]APNorm - Normal value of arterial pressurel [torr]
OUTPUT :AP Arterial pressure [torr]
Control of arterial pressure by angiotensin and extracellular sodium content
RAP Af f My o1
M Y O G E N I C R E S P O N S E
INPUT :RAP - Renal artery pressure [torr]
OUTPUT :AffC - Myogenic effect [ x Normal]
Calculation of the myogenic response to changes in renal perfusion pressure(afferent conductance responds to changes in perfusion pressure,
with pressure increases causing vasoconstriction)
Af f C
Ef f C
RenVenC
AP
VP
Hct
Clamp
RAP
RBF
RPF
R E N A L P E R F U S I O N
INPUTS :AffC - Afferent artery conductance [ml/min/torr]EffC Efferent artery conductance [ml/min/torr]
RenVenC - Renal venous conductance [ml/min/torr]AP - Arterial pressure [torr]
VP - Vena renalis pressure [torr]Hct - Hematocrit [relative number]
Clamp - Renal artery pressure drop caused by renal artery clamp [torr]
OUTPUTS :RAP - Renal artery pressure [torr]
RBF - Renal blood flow rate[ml/min]RPF - Renal plasma flow rate[ml/min]
Calculation of renal artery pressure and renal blood flow rate
PNa
GFR
LogA2
PrxFNaNorm
MDNaFlow
PdxNaReab
PrxFNa
N A T R I U M - P R O X I M A L T U B U L E
INPUTS :PNa - Plasma sodium concentration [mmol/ml]
GFR - GLomerulal fi ltration rate [ml/min]LogA2 - Logarithm of plasma angiotensin concentration
[pg/ml]PrxFNaNorm - Normal value of sodium proximal
fractional reabsorbtion [relative number]
OUTPUTS :MDNaFlow - Sodium outflow [mmol/min]
PdxNaReab - Proximal sodium reabsorbrtion [mmol/l]PrxFNa - Proximal fractional sodium reabsorbtion
[relative number]
Calculation of proximal tubule sodium reabsorbtion
MDSig
VECml
PRA
R E N I N
INPUTS :MDSig - Macula densa feedback signal [ x Normal]
VECml - Ectracellular fluid volume
OUTPUT :PRA - Plasma renin activity [ Units/ml]
Calculation of plasma renin activity
PRA
CEAct
A2Inf
A2
logA2
A N G I O T E N S I N
INPUTS :PRA - Plasma renin activity [ Units/ml]
CEAct - Converting enzyme activity [x Normal]A2Inf Angiotensin 2 infusion rate [nG/min]
OUTPUTS :A2 - Plasma angiotensin 2 concentration [ pG/ml]
LogA2 - logarithm of plasma angiotensin concentration [ pG/ml])
Calculation of plasma angiotensin concentration and logarithm of plasma angiotensin concentration (most of the action of angiotensin are logarithmic in nature: concentration changes at higher concentrations produce less of an
effect than changes of the same size at lower concentrations)
logA2
AldoInf
VTWml
Aldo
A L D O S T E R O N E
INPUTS :logA2 - Logarithm of plasma angiotensin concentration [pG/ml]
AldoInf - Aldosterone infusion rate [nG/min]VTW - Total body water content [ml]
OUTPUT :Aldo - Plasma aldosterone concentration [ nG/dl]
Calculation of plasma aldosterone concentration
RAP
Af f C
TubC
RBF
RPF
APr
GKf
GFR
G L O M E R U L A R F I L T R A T I O N
INPUTS :RAP - Renal artery pressure[torr]
Affc - Afferent artery conductance [mll/min/torr]TubC - Proximal tubule conductaqnce [ml/min/torr]
RBF - Renal blood flow [ml/min]RPF - Renal plasma flow
APr - Plasma protein concentration (in afferent artery) [g/ml]GKf - Glomerular fi ltration coeffitient [ml/min/torr]
OUTPUT :GFR - Glomerular fi ltration rate [ml/min]
Calculation of glomerular fi ltration rate1
MDNaFlow
Aldo
DisFNaNorm
DisNaFlow
DisNaReab
DisFNa
N A T R I U M - D I S T A L T U B U L E
INPUTS :MDNaFlow - Sodium inflow [mmol/min]Aldo - Plasma aldosterone level [pg/ml]
DisFNaNorm - Normal value of sodium distal fractional reabsorbtion [relative number]
OUTPUTS :DisNaFlow - Sodium outflow [mmol/min]
DisNaReab - Distal sodium reabsorbrtion [mmol/min]DisFNa - Distal fractional sodium reabsorbtion
[relative number]
Calculation of distal tubule sodium reabsorbtion
DisNaFlow
CDFNaNorm
NaUrine
CDNaReab
CDFNa
N A T R I U M - C O L L E C T I N G D U C T
INPUTS :DisNaFlow - Sodium inflow [mmol/min]
CDFNaNorm - Normal value of sodium distal fractional reabsorbtion [relative number]
OUTPUTS :NaUrine - Sodium urine outflow [mmol/min]
CDNaReab - Collecting duct sodium reabsorbrtion [mmol/min]
CDFNa - Collecting duct fractional sodium reabsorbtion [relative number]
Calculation of collecing duct sodium reabsorbtion
logA2
MDSig
Ef f Norm
Ef f C
E F F E R E N T A R T E R Y
INPUTS :logA2 - logarithm of angiotensin concentration
MDSig - Macula densa feedback signal [ x Normal]EffNorm - Normal conductance in afferent artery [ml/min/torr]
OUTPUT :EffC - Vascular conductance [ml/min/torr]
Calculates conductance of efferent artery
Af f My o
MDSig
Af f Norm
Af f C
A F F E R E N T A R T E R Y
INPUTS :AffMyo - Myogenic effect [ x Nomal]
MDSig - Macula densa feedback signal [ x Normal]AffNorm - Normal conductance in afferent artery [ml/min/torr]
OUTPUT :AffC - Vascular conductance [ml/min/torr]
Calculates conductance of afferent artery
0
Angiotensin Infusion Rate
0
Aldosteron Infusion Rate
100
APNorm [torr]
Ef f C
AP
AP
Aldo
LogA2
LogA2
LogA2
Matlab/Simulink
Model v Simulinku vyjadřuje spíše
způsob výpočtu než strukturu
modelované reality
Musíme definovat
postup výpočtu
Proprietární kód klasických programovacích jazyků (např. Ada, Fortran, C,...)
Proprietární kód
Definice systému
Dekompozice systému
Modelování subsystémů Odvození kauzality výpočtu
(manuální odvození
vstupně/výstupních vztahů) Implementace Simulace
Modelování
Blokové modelovací jazyky
Blokové modelovací jazyky (např. Simulink, ...)
……
…
Vstupy do bloku jsou vnější vstupy nebo výstupy z jiných bloků
Výstupy z bloku jsou vnější výstupy nebo vstupy do dalších bloků
Modelování a simulace
d
Zobecněné systémové vlastnosti
formalizace
Zobecněné systémové vlastnosti
F = vRm
dP = QR1
dT = QR1
dc = QRc
Mechanická doménaF
v
Q
dP = P1-P2
P1 P2
Termodynamická doména
dT= t°1-t°2
Q
Chemická doménaQ
dc = c1-c2 c1 c2
Hydraulická doména
Elektrická doména R uR = iRRur = u1-u2
u1 u2 iR
Zobecněné úsilí „e“Zobecněný tok „f“
e=rf
Zobecněné systémové vlastnostiElektrická doména
Q=C *uC
1uC = Q
C = iC dt C
1
Mechanická doména
pružinaF
x
x=C *F 1
F = x C = vC dt
C
1
v - rychlost)
Hydraulická doména1
P = V C = fC dt
C
1V=C *P
přítok fc P
V
Termodynamická doména
q=C *dT dT= t°1-t°2
Q - skladované teplo
1 dT = q
C = fq dt C
1fq - tepelný tok
q
fq
t°1
t°2
Zobecněné úsilí „e“Zobecněný tok „f“
e=1/c * f dt
e
f
p q
Zobecnělé úsilí (effort)
Zobecnělý tok (flow)
R
e=Rf
Zobecnělá akumulace (quantity)
ò
Cq=Ce
Zobecnělá hybnost
ò
Lp=Lf
Zobecněné systémové vlastnosti
Zobecněné systémové vlastnosti
e
f
p q
Zobecnělé úsilí (effort)
Zobecnělý tok (flow)
R
e=Rf
Zobecnělá akumulace (quantity)
ò
Cq=Ce
Zobecnělá hybnost
ò
Lp=Lf
úsilí hybnost tok akumulace
e fp
q
úsilí hybnost tok akumulace
napětí proud nábojindukční tok
síla rychlost polohaimpuls síly
moment úhlová rychlost úhelimpuls momentu síly
tlak objemový průtok objemprůtočná hybnost
koncentrace molární průtok množství
teplota tepelný tok teplo
teplota entropický průtokentropie
òò
Obecné systémové vlastnosti
e fp
q
úsilí hybnost tok akumulace
napětí proud nábojindukční tok
síla rychlost polohaimpuls síly
moment úhlová rychlost úhelimpuls momentu síly
tlak objemový průtok objemprůtočná hybnost
koncentrace molární průtok množství
teplota tepelný tok teplo
teplota entropický průtokentropie
energie
òò
Obecné systémové vlastnosti
e
f
p q
Zobecnělé úsilí (effort)
Zobecnělý tok (flow)
R
e=Rf
Zobecnělá akumulace (quantity)
ò
Cq=Ce
Zobecnělá hybnost
ò
Lp=Lf
energie
Obecné systémové vlastnosti
Bond Graphs - Výkonové grafy
prvek prvekf (flow)e (effort)
Přenos energie
prvekprvekf (flow)
e (flow)
Přenos energie
bránabrána
bránabrána
Bond Graphs - Výkonové grafy
1
R:R
l:LSr:us
C:C
i
i
i
i
us
ur
ul
ucStejný tok =uzeltypu „1“
R
us
L
C
Bond Graphs - Výkonové grafy
0
R:R
l:LSr:us
C:C
i
i
i
i
us
ur
ul
ucStejné úsilí = uzel typu „0“,
suma toků nulová
R
us
L
C
Bond Graphs - Výkonové grafy
f (flow)e (effort)
Přenos energie
e - směr výpočtu
f - směr výpočtu
prvek prvek
f (flow)e (effort)
e - směr výpočtu
f - směr výpočtu
prvek prvek
Bond Graphs - Výkonové grafy
Přenos energie
f e
f e
Přenos energie
f e
f e
Přenos energie
Přenos energie
e - směr výpočtu
e - směr výpočtu
e - směr výpočtu
e - směr výpočtu
Bond Graphs - Výkonové grafy
f e
SE
Ideální zdroj
Zdroj úsilířízení
FSEřízení e
f e
SF
Zdroj tokuřízení
F-1SFřízení f
Bond Graphs - Výkonové grafy
Jednobran typu akumulátor (kapacitor)
f
f e C
FCe
dq/dt
q
f
f e C
F-1Ce
q
ò
Bond Graphs - Výkonové grafy
Jednobran typu induktor
f
f e I
e
F-1I
pF-1IF-1IF-1I
f
f
e I
dq/dte
p
FI
Bond Graphs - Výkonové grafy
Jednobran typu induktor
f
f e I
e
F-1I
pF-1IF-1IF-1I
f
f
e I
dq/dte
p
FI
• Podrobnosti v článku „Introduction to Physical System Modeling with Bond Graphs
https://cw.felk.cvut.cz/lib/exe/fetch.php/courses/a6m33mos/intro_bondgraphs.pdf
Využití výkonových grafů k určení kauzálního směru výpočtu
Proprietární kód klasických programovacích jazyků (např. Ada, Fortran, C,...)
Proprietární kód
Definice systému
Dekompozice systému
Modelování subsystémů Odvození kauzality výpočtu
(manuální odvození
vstupně/výstupních vztahů) Implementace Simulace
Modelování
Blokové modelovací jazyky
Blokové modelovací jazyky (např. Simulink, ...)
……
…
Vstupy do bloku jsou vnější vstupy nebo výstupy z jiných bloků
Výstupy z bloku jsou vnější výstupy nebo vstupy do dalších bloků
Modelování a simulace
d
Zobecněné systémové vlastnosti
formalizace
Proprietární kód klasických programovacích jazyků (např. Ada, Fortran, C,...)
Proprietární kód
Definice systému
Dekompozice systému
Modelování subsystémů Odvození kauzality výpočtu
(manuální odvození
vstupně/výstupních vztahů) Implementace Simulace
Modelování
Blokové modelovací jazyky
Blokové modelovací jazyky (např. Simulink, ...)
……
…
Vstupy do bloku jsou vnější vstupy nebo výstupy z jiných bloků
Výstupy z bloku jsou vnější výstupy nebo vstupy do dalších bloků
Modelování a simulace
Zobecněné systémové vlastnosti
formalizace
Modelica
Blokové schéma systému
S xs
x1
x2
.
.
.
xm
u
u1
u2
un
.
.
.
vstupy
y1
y2
y
yr
výstupy
.
.
.
Stavové proměnné xs
Separabilita systému
Podmínka separability systému - systém je separabilní, jestliže jeho výstupy zpětně vlivem prostředí neovlivňují podstatně vstupy.
Příklad:
· termoregulační systém živého organismu - systém můžeme považovat za separabilní, pokud organismus svou tepelnou energií významně neovlivňuje teplotu prostředí, ve kterém se nachází;
·
S Xu y
u1u2
un
x1
x2
.
.
.
xm
y1y2
yr
.
.
.
!!! Výstupy nesmí ovlivňovat vstupy pře okolí systému !!!!
Základní atributy systému
Stav systému - souhrn přesně definovaných podmínek nebo vlastností daného systému, které lze v daném časovém okamžiku rozpoznat. Stavu systému lze v libovolném časovém okamžiku t (z nějakého zvoleného časového intervalu) přiřadit vektor hodnot x(t) , který nazýváme stavovým vektorem, složky xi vektoru x nazýváme
stavovými veličinami (proměnnými) a prostor všech možných hodnot stavových veličin nazýváme stavovým prostorem.
S Xu
y
u1
u2
un
x1
x2
.
.
.
xm
y1
y2
yr
.
.
.
.
.
.
Blokové schéma systému
S xs
x1
x2
.
.
.
xm
u
u1
u2
un
.
.
.
vstupy
y1
y2
y
yr
výstupy
.
.
.
Stavové proměnné xs
Blokové schéma systému
S xs
x1
x2
.
.
.
xm
u
u1
u2
un
.
.
.
vstupy
y1
y2
y
yr
výstupy
.
.
.
Stavové proměnné xs
(vnitřní) nestavové proměnné xs
xn
Stavové proměnnéAlgebrodiferenciální rovnice v Modelice
g(x(t), x‘(t), y(t), u(t))=0
h1(x(t), y(t), u(t))=0
x‘(t) = f (x(t), u(t))
y(t)=h2(x(t), u(t))=0
x(t) - dynamické proměnné
u(t) - vstupní proměnné
y(t) - výstupní proměnné
ODE
Algebr. Eq.
xs(t)
xn(t)x(t)
stavové proměnné (vzájemně nezávislé)
nestavové proměnné (vypočítatelné ze stavových a vstupních proměnných)
xs(t)
xn(t)x(t)
stavové proměnné (vzájemně nezávislé)
nestavové proměnné (vypočítatelné ze stavových a vstupních proměnných)
StateSelect.default - nech to na Modelice,
StateSelect.never – nikdy to nebude stavová, vypočítej vždy z ostatních
StateSelect.avoid – použij ji jako stavovou, jen když to už jinak nejde
StateSelect.prefer - preferuj ji jako stavovou před default
StateSelect.alwais – vždy ji ber jako srtavovou
Real x (min=0, max = 100, … StateSelect.prefer)
Stavové proměnnéAlgebrodiferenciální rovnice v Modelice
Regulace srdečního výdejeNa konci diastoly:
Na konci systoly:
Systolický objem:
Minutový objem:
Qc>=0
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