pbpk physiology¶

The standard human physiology and its scaling to an individual.

The physiology is a set of SBML parameters of the top model. The covariates of the individual (BW, HEIGHT, AGE) and the factors of organ function are constant parameters which an experiment changes before the simulation, as it changes BW today; everything else follows from them by initial assignments on constant parameters, which sbmlsim evaluates again after the changes of an experiment (a value which no rule recomputes during the integration keeps the compiled model small and fast to load):

• the reference values of the sex of the build (ICRP 89, data/physiology_human.toml); the sex is chosen when the model is built, a study of both sexes uses one model per sex;
• the default individual (BW, HEIGHT, AGE) is the reference individual of the sex of the population of the build (pkpdsim.pbpk.populations, ICRP 89 by default); the reference anatomy stays ICRP 89 for every population, the organs of another population's individual follow from the scaling;
• body mass index, body surface area (Du Bois 1916) and fat-free mass (Janmahasatian 2005);
• the tissues scale as in Huisinga 2012 (eqs. 9, 10, 12): brain is constant, skin scales with the body surface area, the other lean organs and the blood with the fat-free mass less brain and skin (SV_tis); adipose is the remainder of the body weight;
• the cardiac output scales as the tissues, SV_tis (Huisinga 2012 eq. 13), and the organ flows are fractions of it, renormalised over the organs so that they sum to the cardiac output;
• the GFR scales with the body surface area.

An organ volume is the volume of its tissue and its blood, M<o>/density + blood*Vblood; the body weight is the mass of the tissues without blood and of the blood, sum(M<o>) + Vblood*rho_blood. The vascular fraction of an organ volume is its blood, blood*Vblood/V<o>, so that the vascular spaces of the organs and the pools are the blood volume for every individual; the interstitial fraction is the interstitial fraction of the tissue without blood (Schmitt 2008) times the tissue share 1 - f_vas.

A covariate or a factor of organ function is an initial value: changed before the simulation (a change of an experiment, init(...) in roadrunner), every quantity which depends on it follows; changed during a simulation, nothing follows (unless the definition names it dynamic, see pkpdsim.pbpk.builder).

The impaired organ function (pkpdsim.pbpk.impairment) multiplies cardiac output, the blood flows, the GFR, the hematocrit and the kidney mass by its factors (factors of physiology_objects); without them the physiology is the healthy one.

LEAN_ORGANS = tuple(
    o for o in Organ if o is not Organ.ADIPOSE
)

DEFAULT_POPULATION = 'european_icrp'

AGE_DEFAULT = 40.0

MODEL_DEFAULT = 'Model default without a source.'

LIVER_DEFAULTS = {ids.F_SHUNT: 0.0, ids.F_FUNCTIONAL: 1.0}

ReferenceIndividual(
    sex,
    bw,
    height,
    gfr,
    volumes,
    tissue_volumes,
    intracellular_volumes,
    lumen_volumes,
)

physiology_objects(
    sex=MALE, factors=None, population=DEFAULT_POPULATION
)

reference_fractions(organ)

normal_gfr(sex)

reference_tissue_volumes(sex)

reference_hematocrit(sex)

reference_individual(sex)

The populations: the reference individual and the distributions of the covariates.

A population is a set of data of the library (data/populations/<key>.toml): the reference covariates of each sex (body weight, height and, where the data give it, the age), which become the default individual of a model, and the distributions of the covariates for a later sampler of virtual individuals. The reference anatomy stays ICRP 89 (data/physiology_human.toml): the organ volumes and flows of a population's reference individual follow from the covariate scaling of pkpdsim.pbpk.physiology (Huisinga 2012) at its body weight and height. The population is chosen when the model is built (PBPKDefinition.population), the European reference individuals of ICRP 89 are the default.

The distributions are data of the library and never SBML distrib uncertainties: run_simulation --uncertainty draws every parameter with a distrib uncertainty and must only draw the uncertainty of a fit.

The populations:

• european_icrp: the ICRP 89 reference individuals (the default); ICRP 89 gives reference values, not distributions, so the population has none;
• white_american_nhanes, black_american_nhanes, mexican_american_nhanes: adults of 20 to 49 years of the NHANES examinations 2013-2018 as preprocessed by httk (HTTK-Pop, Ring 2017), survey-weighted means and standard deviations;
• east_asian_tanaka: the Asian Reference Man of Tanaka 1996 (Tables 9 and 10, 20 to 49 years) with the distributions of the Japanese of the same tables.

PK-Sim also offers a Japanese population (based on Japanese national statistics of 2015). Its primary data were not read, so the library has no japanese population; the Japanese data of Tanaka 1996 (1976 to 1988) are the distributions of east_asian_tanaka.

DEFAULT_POPULATION = 'european_icrp'

COVARIATES = (ids.BW, ids.HEIGHT, ids.AGE)

POPULATIONS = _populations()

Distribution(kind, mean, sd, unit, source, notes=None)

Population(key, name, source, reference, covariates)

population(key)

The impaired organ function: parameter vectors on the physiology.

Impairment is never a structurally different model: three severities of the top model set factors on the quantities of the healthy physiology, the liver and the binding to plasma proteins (data/impairment.toml, every number with its source; a value which no publication read gives is a model choice):

• CP_severity: liver cirrhosis, 0 healthy, 1, 2, 3 the Child-Pugh classes A, B, C; a severity between the classes interpolates every factor linearly (scans and fits use intermediate values). It sets the functional fraction f_functional and the portosystemic shunt f_shunt of the liver, the hepatic arterial and the renal blood flow, the cardiac output (cardiac index; the organs without a factor of their own close the cardiac output to it), the GFR, albumin and alpha-1-acid glycoprotein (and so fu) and the activity of the enzymes and transporters of the liver.
• f_renal_function: chronic kidney disease by the GFR relative to healthy (the factor of the physiology); its GFR per 1.73 m2 (eGFR) gives the stage (normal, mild, moderate, severe, end stage), which sets the binding capacity of the plasma proteins (and so fu). The factors are piecewise linear in the eGFR through anchor points at the stage midpoints ([ckd] stage_anchor of the data, a model choice: continuous covariate for scans and fits; stage midpoints): 1 from 90 ml/min/1.73 m2, the mild value at 75, the moderate value at 45, the severe value at 22.5 and the end stage value at 7.5 and below, linear in between (from 90 to 75 from 1 to the mild value), so a scan or fit of f_renal_function sees no step. The binding to albumin and alpha-1-acid glycoprotein follows the CKD profile of its protein ([ckd.binding] of the data): albumin falls in concentration and in the binding function of its molecule, AAG rises in concentration; the binding to an unknown protein follows the binding capacity of the plasma proteins (plasma_protein). The switch CKD_hepatic (0 off, 1 on) adds the reduced activity of hepatic CYP2D6 and OATP1B. The tubular capacity of the kidney (every metabolism and transport of the kidney) follows f_renal_function continuously, 1 - f_tubular + f_tubular*f_renal_function with the share f_tubular 1 of the intact nephron hypothesis ([ckd] tubular_capacity of the data, a model choice; the kidney owns the factor, pkpdsim.pbpk.organ).
• HF_severity: heart failure, 0 healthy, 1 to 4 the NYHA classes I to IV, interpolated: the blood flows of liver and portal organs, kidney and skin, adipose and muscle are reduced, the cardiac output is the sum of the flows, the GFR follows the renal blood flow (which is not monotonic in the class, NYHA IV above NYHA III, as measured by Leithe 1984).

The hematocrit (anaemia of cirrhosis and CKD, combined multiplicatively on the hematocrit of the sex of the build), the kidney volume and blood flow and the gastric emptying and small intestinal transit of CKD have their factors, which are 1 until a publication with the healthy reference is read (the data say why).

Each condition has its factors fcp_<key>, fckd_<key>, fhf_<key>; the factor of all conditions on a quantity is fimp_<id> (fimp_GFR, fimp_Qki, fimp_HCT), their product. Blood flows: every organ flow is its fraction of the cardiac output without impairment (CO_base) times fimp_Q<o>, and the cardiac output is CO_base times fimp_CO = sum fQ<o>*fimp_Q<o>, so the flows sum to it. Plasma binding (McNamara 2002): fu = 1/(1 + f*(1 - fu_ref)/fu_ref) with the binding factor f, the factor of the binding protein in cirrhosis (fimp_albumin, fimp_agp, 1 for an unknown protein) times the CKD factor of the binding: the CKD profile of the protein (fimp_albumin_ckd, fimp_agp_ckd, the product of its rows of [ckd.binding]), for an unknown protein one over the factor of the binding capacity fimp_plasma_protein (fu of a strongly bound substance rises by it). fu stays below 1. A binding with ckd=False (measured unchanged in renal impairment) leaves out the CKD factor, so f is the factor of the binding protein alone. The parameters of the CKD profile of a protein are created only when a substance binds to it with the CKD factor. Enzymes and transporters: a rate of a protein in the liver runs with f_<protein>*fimp_<protein>, the organ-local base activity (genotype, interactions of an experiment, LI__f_cyp3a4) times the input fimp_<protein> of the impairment (fimp_li_<protein> of the body), so both act multiplicatively.

All are initial assignments (pkpdsim.pbpk.physiology): set the severities before the simulation, as a covariate; a severity which changes during a simulation is named in PBPKDefinition.dynamic. f_shunt and f_functional stay parameters an experiment may set directly. All severities 0 give the healthy model: every factor is 1.

DATA = 'impairment'

GASTRIC_EMPTYING = ids.impairment_factor('t_stomach')

SI_TRANSIT = ids.impairment_factor('t_si')

organ_activity(organ, protein)

physiology_factors()

unbound_fraction_notes(s)

unbound_fraction(s)

ckd_unbound_fractions(s)

impairment_objects(definition)

The observers of a PBPK model: assignment rules of the top model, no states.

An observer reads the pool species and the outflow concentrations of the organs and changes nothing, so it conserves mass. Per substance <s>:

• C<pool>_free_<s> = fu*C<pool>_<s>, the unbound plasma concentration of a pool (ar, ve, po);
• C<pool>_blood_<s>, the blood concentration of a pool, (1-HCT)*C + HCT*C_bc: BP*C unless the blood cells are kinetic;
• Chv_<s>, the hepatic vein: the outflow of the liver mixed with the portosystemic shunt by their flows, ((1-f_shunt)*Qh*Cout_li + f_shunt*(Qha*Car + Qpo*Cpo))/Qh, with Chv_free_<s> and Chv_blood_<s>;
• Carm_<s> = sum w*C, the arm vein (peripheral venous) sampling site: the venous plasma outflows of organs and the arterial plasma mixed with the weights w_arm_<site> of a preset of data/arm_vein.toml (PBPKDefinition.arm_vein, by default the mix of skin and muscle of the PK-Sim documentation); Carm_free_<s> and Carm_blood_<s> as for the pools.

The tissue concentrations are the species of the organs themselves, their unbound concentrations Cu_<part>_<s> observers of the organs, the amounts in urine and feces the sink species, the urinary excretion rate Rurine_<s> an output of the kidney.

A definition adds observers of its own (Observer, PBPKDefinition.observers, e.g. the sum of two metabolites): parameters of the top model over the ids of the top model, appended after the observers of the library. An observer names a species or an unbound concentration of an organ by its flat id (MU__Cti_<s>, LI__Cu_ic_<s>, an amount MU__A<x>), which the organ exposes as a PD input is exposed (pkpdsim.pbpk.builder). An observer which reads a state, directly or through other observers (a species, an observer of the library, an output of an organ or a PD component, a dynamic parameter, time), is an assignment rule; an observer of constants only is a constant with an initial assignment, evaluated after the changes before the simulation (an assignment rule of constants would slow the compile of the model).

The intestine adds readouts on request (PBPKDefinition.readouts, intestine_readouts): the dissolved amount in its lumen Alumen_<s> and the summed fluxes of the absorption Rabs_<s>, the efflux into the lumen Reff_<s> and the loss to the feces Rfeces_<s>, outputs of the intestine as Rurine_<s> is one of the kidney.

TIME = 'time'

Observer(sid, formula, unit, name, notes=None)

OrganOutput(unit, name, kind, text, constant=False)

formula_symbols(formula)

formula_functions(formula)

definition_observer_objects(observers, constant=())

readout_ids(substances)

intestine_readouts(md, readouts, substances)

arm_vein_weights(preset)

observer_objects(definition)

The sinks of the body: urine, feces and bile.

Every substance has the amounts in urine Aurine_<s> and feces Afeces_<s> (compartments of amounts, Vurine and Vfeces) and the bile Cbile_<s> in Vbile, which drains with first order k_bile into the duodenum of the intestine (pkpdsim.pbpk.variants.intestine, which owns the lumen and that transit); with the standard organs as intestine, which have no lumen, the bile drains into the feces.

sink_objects(definition, lumen=True)