Organ variants¶

An organ variant is an alternative component for an organ with the boundary of the standard organ: blood in and blood out (plasma and blood cells, per substance), the flow, the organ volume, the hematocrit and the parameters of the substances and proteins it reads. The circulation sees nothing else, so swapping a variant changes nothing outside the organ: not the other organs, not the observers, not the experiments. A variant places the chemistry of its organ into its parts, so the chemistry is written once per organ whatever variant the model chooses.

Catalogue¶

A definition selects a variant by name or gives a component with its options:

from pkpdsim.pbpk import Organ, Variants, ZonatedLiver

variants = Variants(
    organs={
        Organ.LIVER: ZonatedLiver(n_zones=4, profiles={"CYP3A4": (1.0, 1.0, 2.0, 4.0)}),
        Organ.KIDNEY: "nephron",
        Organ.SMALL_INTESTINE: "single_lumen",
    }
)
print({organ.name: getattr(v, "name", v) for organ, v in variants.organs.items()})

{'LIVER': 'zonated', 'KIDNEY': 'nephron', 'SMALL_INTESTINE': 'single_lumen'}

The intestine variants are chosen on Organ.SMALL_INTESTINE; the large intestine has no variant of its own while an intestine replaces it. Coupled definitions (PBPKDefinition.combine) take the variant which one of them chooses for an organ the others leave at the default, and refuse two different choices.

The equivalence contract¶

Every variant ships an equivalence test: in its limiting case it reproduces the standard organ, the outflow concentrations and the extraction at steady state.

Every variant also has the same boundary as the standard organ (a test compares the inputs and outputs) and conserves mass in a closed system.

Zonated liver¶

ZonatedLiver builds the liver as n well-stirred zones in series along the sinusoid, from the periportal zone z1 to the pericentral zone z<n>, the pattern of the sinusoidal unit of the zonation models of pkdb_models (misc/zonation). Every zone has the parts of the standard liver which a substance needs, with the share f_zone = 1/n of their volumes and passive exchanges and the coefficients of the liver. The blood flows from the hepatic artery and the portal vein into the first zone, from zone to zone and from the last one into the venous pool; the shunt f_shunt bypasses all zones, the functional fraction f_functional scales the tissue and the capacity of every zone. The species and processes of a zone carry the zone in their id (Cpl_z1_<s>, DRG2M_z1), the outflow Cout_<s> is that of the last zone, so the hepatic and the arm vein observers are those of the standard liver.

Every metabolism and transport of the liver is placed into every zone with the share w of the capacity of the liver: by volume (f_zone), or by the profile of its protein, the relative abundance p_<protein>_<zone> per zone (a pericentral CYP3A4), w = p*V_zone/sum(p*V_zone) = p/sum(p), so that the total capacity Vmax*V is that of the standard liver. A metabolism with a profile takes the profile of that name, any other process the profile of its protein. The equations of a liver of three zones with a CYP3A4 profile (1, 2, 4):

J_in_ar_pl_<s>: Car_<s> -> Cpl_z1_<s> = (1 - f_shunt) * Qha * (1 - HCT) * Car_<s>
J_sin_z1_pl_<s>: Cpl_z1_<s> -> Cpl_z2_<s> = (1 - f_shunt) * Qh * (1 - HCT) * Cpl_z1_<s>
J_sin_z2_pl_<s>: Cpl_z2_<s> -> Cpl_z3_<s> = (1 - f_shunt) * Qh * (1 - HCT) * Cpl_z2_<s>
J_out_pl_<s>: Cpl_z3_<s> -> Cve_<s> = (1 - f_shunt) * Qh * (1 - HCT) * Cout_<s>
Cout_<s> = Cpl_z3_<s>
Vti_z1 = Vli * (1 - f_vas_li) * f_functional * f_zone
J_ti_z1_<s>: Cpl_z1_<s> -> Cti_z1_<s> = PS_<s> * f_zone * fu_<s> * (Cpl_z1_<s> - Cti_z1_<s> / Kp_<s>)
w_cyp3a4_z1 = p_cyp3a4_z1 / (p_cyp3a4_z1 + p_cyp3a4_z2 + p_cyp3a4_z3)
DRG2M_z1: Cti_z1_drg -> Cpl_z1_drg_m = f_cyp3a4 * fimp_cyp3a4 * f_functional * w_cyp3a4_z1 * DRG2M_Vmax * Vli * (fu_drg / Kp_drg * Cti_z1_drg) / (DRG2M_Km + fu_drg / Kp_drg * Cti_z1_drg)

With one zone the zonated liver is the standard liver. With n zones and linear kinetics the availability of a flow-limited liver at steady state is (1 + x/n)^-n with x = fu*CLint/(Q*BP) instead of the well-stirred 1/(1 + x): with the same intrinsic clearance the zonated liver extracts more, and for large n it approaches the parallel tube, exp(-x). A protein which a substance inactivates or induces has one turnover state for the liver, driven by the unbound concentration in the cells of the zones weighted by the placement of the protein (exact for the same concentration in every zone, the mean of the zones along a gradient).

Nephron kidney¶

NephronKidney builds the kidney with the tubule of the nephron. The unbound afferent (arterial) plasma concentration is filtered into the lumen of the proximal tubule, as in the standard kidney, and the tubular fluid flows through the lumens of the segments in series, proximal tubule pt, loop of Henle lh, distal tubule dt and collecting duct cd, into the urine. Every segment reabsorbs water: the flow leaving a segment is the fraction f_tub_<seg> of the GFR (Scotcher 2016), so the tubular flows follow the GFR of the individual and the renal function, and the drug concentrates as the water is reabsorbed. The lumens are well-stirred compartments of constant volume (Huang 2018, both kidneys), lumen and not tissue: the blood and tissue of the kidney keep the volumes of the standard kidney.

J_gfr_<s>: Cpl_<s> -> Ctub_pt_<s> = fu_<s> * GFR * Car_<s>
J_tub_pt_<s>: Ctub_pt_<s> -> Ctub_lh_<s> = GFR * f_tub_pt * Ctub_pt_<s>
J_tub_lh_<s>: Ctub_lh_<s> -> Ctub_dt_<s> = GFR * f_tub_lh * Ctub_lh_<s>
J_tub_dt_<s>: Ctub_dt_<s> -> Ctub_cd_<s> = GFR * f_tub_dt * Ctub_dt_<s>
J_tub_cd_<s>: Ctub_cd_<s> -> Aurine_<s> = GFR * f_tub_cd * Ctub_cd_<s> * (1 - f_reabsorbed_<s>)
J_reab_<s>: Ctub_cd_<s> -> Cpl_<s> = GFR * f_tub_cd * Ctub_cd_<s> * f_reabsorbed_<s>
J_out_pl_<s>: Cpl_<s> -> Cve_<s> = Qki * (1 - HCT) * Cout_<s> - fu_<s> * GFR * Cout_<s> * (1 - f_reabsorbed_<s>)
Rurine_<s> = J_tub_cd_<s>

Without secretion and reabsorption every filtered molecule reaches the urine: the urinary excretion at steady state is the filtration of the standard kidney, whatever the volumes and flows of the segments; the tubule delays the excretion by minutes. The plasma leaving the kidney lacks the net filtrate of the standard kidney: at steady state the reabsorption returns the fraction reabsorbed of the filtrate into the plasma of the kidney, so the kidney plasma and the clearance of the arterial plasma are those of the standard kidney. The fraction reabsorbed (Excretion.fraction_reabsorbed) is an empirical passive reabsorption at the end of the collecting duct, which acts on secreted drug as well, the lumped (fu*GFR + CL_sec)*(1 - F_reabs) of the literature. A threshold reabsorption (Excretion.reabsorption=ThresholdReabsorption(...)) returns the fraction of the standard kidney at the afferent plasma, f(Car) = 1/(1 + (fu*|Car|/reab_threshold)^reab_n)^(1/reab_n), of the flow out of the collecting duct up to the filtered load fu*GFR*Car (Car_<s> a modifier of both reactions out of the collecting duct): it never exceeds the filtration, also with a secretion into the tubule, whose drug leaves into the urine, and at steady state without a secretion the circulation sees the standard kidney.

J_tub_cd_<s>: Ctub_cd_<s> -> Aurine_<s> = GFR * f_tub_cd * Ctub_cd_<s> - min(GFR * f_tub_cd * Ctub_cd_<s>, fu_<s> * GFR * Car_<s>) * 1 / (1 + (fu_<s> * abs(Car_<s>) / reab_threshold_<s>)^reab_n_<s>)^(1 / reab_n_<s>)
J_reab_<s>: Ctub_cd_<s> -> Cpl_<s> = min(GFR * f_tub_cd * Ctub_cd_<s>, fu_<s> * GFR * Car_<s>) * 1 / (1 + (fu_<s> * abs(Car_<s>) / reab_threshold_<s>)^reab_n_<s>)^(1 / reab_n_<s>)

Generated from data/kidney_human.toml by scripts/pbpk_tables.py; the notes of every number are in the data file.

Sources of the columns: flow: Scotcher2016; volume: Huang2018, ModelChoice.

Intestine variants¶

The segmented intestine and the single lumen replace small and large intestine with one submodel GU, with the boundary of the two standard organs on the blood side; they own the intestinal lumen, apply the oral doses and take the bile. They are described with the routes on Intestine and routes. Selecting the standard organs for the intestine (Variants(organs={Organ.SMALL_INTESTINE: "standard"})) builds no lumen: a model with an oral route or an apical transport of the intestine then fails with an error.

Writing a variant¶

A variant is an OrganComponent: an object with a name and a method build(ctx: OrganContext) -> tuple[ModelDefinition, Boundary]. The context gives the organ, the substances with their parts in the organ (parts_for), the chemistry, the inhibitions and inductions and the routes; the boundary names, for every input of the model definition, the element of the body which replaces it, and the outputs which replace parameters of the body (Cout_<s> -> Cout_<o>_<s>). The builder generates the ports and replacements from the boundary; a variant writes no ReplacedElement.

An intestine which replaces several organs with one submodel is a GutComponent: in addition it names the organs it replaces (replaces, the small intestine first) and gives the parameters of the top model it reads (body_objects(definition), e.g. ka_<s> and FED), as the segmented intestine and the single lumen do.

The base classes do the bookkeeping: pkpdsim.pbpk.organ_builder.OrganBuilder collects the inputs with their ports, the compartments, species, rules, initial assignments, constants with their provenance and the activities of the proteins, and pkpdsim.pbpk.organ.StandardBuilder builds the parts, flows and exchanges of the standard organ (with zones, as the zonated liver uses it); pkpdsim.pbpk.placement places the chemistry. A new variant derives from them, keeps the boundary of the standard organ, places the chemistry by volume fraction and ships its equivalence test.

A model package can write its own variant against the same interface first and pass the instance, Variants(organs={Organ.LIVER: MyLiver()}); a variant which several models need moves into the library and is registered by name (pkpdsim.pbpk.variants.register(organ, name, component)). The registry stores component instances, which are immutable dataclasses.

Limitations¶

• The variants refuse a CustomReaction of their organ (with an error naming the process): an arbitrary formula cannot be split over zones or segments. A model with custom liver chemistry uses the standard liver until a hook for it exists.
• A zonal profile of zeros gives no placement weights: the build refuses an all-zero profile, but a profile changed to zeros during a simulation gives NaN.
• The volume of the collecting duct of the nephron is a choice of the library (0.005 l, no source could be verified), and the tubular lumens do not scale with the individual.