Intestine and routes¶

A route applies a dose of a substance: intravenous (bolus and infusion), oral tablet (with dissolution) and oral solution. A dose is a dose state in mg which an experiment sets or adds to (PKPDSimulationExperiment.add_doses, "PODOSE_api + 10.0"), applied by a first order process, the convention of the hand-written models; the dose in mg becomes mmole with the molecular weight Mr_<s>. The oral routes need an intestine with a lumen: the segmented intestine (the default) or the single lumen, which replace small and large intestine with one submodel GU, own the lumen, apply the oral doses and take the bile.

Routes¶

The dose states and the parameters of the routes are parameters of the top model; the intestine reads the oral ones through its ports and owns their reactions and rate rules. Subcutaneous, intramuscular, inhalation, sublingual and buccal, rectal, transdermal and vaginal routes and the glucose and insulin clamps are designed for (a route is a component with dose parameters) and implemented when the models which need them are migrated.

J_iv_<s>: -> Cve_<s> = Kiv_<s> * IVDOSE_<s> / Mr_<s>
Kiv_<s> = ln(2) / ti_<s> * 60 s_per_min
d(IVDOSE_<s>)/dt = -J_iv_<s> * Mr_<s> + Ri_<s>

J_sol_<s>: -> Clumen_st_<s> = Ka_sol_<s> / 60 min_per_hr * SOLDOSE_<s> / Mr_<s>
d(SOLDOSE_<s>)/dt = -J_sol_<s> * Mr_<s>
J_po_<s>: -> Clumen_st_<s> = Ka_dis_<s> / 60 min_per_hr * PODOSE_<s> / Mr_<s>
J_trs_st_<s>: -> Asolid_du_<s> = PODOSE_<s> / Mr_<s> / ((T_st + FED * (T_st_fed - T_st)) * fimp_t_stomach)
d(PODOSE_<s>)/dt = -(J_po_<s> + J_trs_st_<s>) * Mr_<s>

PODOSE_<s> is the undissolved tablet in the stomach: it dissolves into the stomach lumen and empties undissolved with the stomach into the duodenum. In the next segments the undissolved drug is a species of its own, Asolid_<seg>_<s>, which dissolves there and moves on undissolved into the feces.

Dissolution¶

The dissolution rate kdis (1/min) acts on the undissolved amount of every segment; the formulation of the tablet decides it:

• FirstOrderDissolution(k, fed, release): kdis = k (Ka_dis_<s>, 1/hr), with two optional factors (see below); the default (2/hr, the default of the hand-written models, a choice of the library) when the route states no formulation;
• WeibullDissolution(t50, shape, lag): the fraction dissolved F = 1 - exp(-ln2*((t - lag)/t50)^shape) after the lag time, applied as its hazard on the undissolved amount, with the time since the dose the mean age of the undissolved drug (age_dis_<s>); exact for one dose, doses which overlap share one clock (the amount-weighted mean age);
• DoseDependentDissolution(k, dose50): kdis = k*D50/(D50 + D) with the undissolved amount D (mg) of all segments, a dissolution which saturates with the dose (solubility-limited).

The first order dissolution has two optional factors, each creating a dimensionless parameter of the body only if it is set (without them the model has neither parameter and the rate is Ka_dis_<s>): fed creates f_dis_fed_<s>, the factor of the rate in the fed state (FED = 1; food slows the dissolution), and release creates f_dis_release_<s>, the factor of a modified-release formulation (the given value, 1 for the immediate-release tablet; the experiments of a prolonged-release tablet set it, e.g. 0.2). The single lumen has no fed state, a fed factor there raises.

J_po_<s>: -> Clumen_st_<s> = Ka_dis_<s> / 60 min_per_hr * f_dis_release_<s> * (1 + FED * (f_dis_fed_<s> - 1)) * PODOSE_<s> / Mr_<s>

kdis_<s> = piecewise(0, age_dis_<s> <= lag_dis_<s>, 0.693147180559945 * shape_dis_<s> / t50_dis_<s> * ((age_dis_<s> - lag_dis_<s>) / t50_dis_<s> + 1e-9)^(shape_dis_<s> - 1))
J_dis_du_<s>: Asolid_du_<s> -> Clumen_du_<s> = kdis_<s> * Asolid_du_<s>

Segmented intestine¶

The lumen segments follow the anatomical segmentation of Thelen 2011, as PK-Sim: stomach st, duodenum du, upper and lower jejunum jej1, jej2, upper and lower ileum ile1, ile2, caecum cae, colon ascendens asc, transversum tra, descendens des, sigmoid sig and rectum rec, then the feces; the tokens in the order of the transit are ids.GUT_SEGMENTS, which a model reads instead of a list of its own. A substance which reaches the lumen (oral route, biliary excretion, transport into the lumen) has the dissolved concentration Clumen_<seg>_<s> in every segment, which moves on with the transit V*C/T of the segment and from the rectum into the feces Afeces_<s>. The fed state FED = 1 slows the gastric emptying to the stomach transit time of the fed state.

The absorption rate constant of the small intestine is ka_<s> (1/hr, 1 by default for a substance with an oral route, a choice of the library; 0 for any other); a segment absorbs with ka_<s>*f_area_<seg>, the mucosal surface per lumen volume relative to the small intestine (the stomach does not absorb); optional factors change it (see Fraction absorbed and colon absorption and Regional absorption). Without chemistry of the intestine the substance is absorbed into the plasma of the intestinal wall:

J_tr_st_<s>: Clumen_st_<s> -> Clumen_du_<s> = Vlumen_st * Clumen_st_<s> / ((T_st + FED * (T_st_fed - T_st)) * fimp_t_stomach)
J_tr_du_<s>: Clumen_du_<s> -> Clumen_jej1_<s> = Vlumen_du * Clumen_du_<s> / (T_du * fimp_t_si)
J_tr_rec_<s>: Clumen_rec_<s> -> Afeces_<s> = Vlumen_rec * Clumen_rec_<s> / T_rec
J_abs_du_<s>: Clumen_du_<s> -> Cpl_<s> = ka_<s> / 60 min_per_hr * f_area_du * Vlumen_du * Clumen_du_<s>
J_trs_du_<s>: Asolid_du_<s> -> Asolid_jej1_<s> = Asolid_du_<s> / (T_du * fimp_t_si)
J_trs_rec_<s>: Asolid_rec_<s> -> Afeces_<s> = Asolid_rec_<s> / T_rec

The factors fimp_t_stomach and fimp_t_si are the factors of chronic kidney disease on the gastric emptying time and the small intestinal transit time: they multiply the transit time, a factor above 1 slows the transit. They are 1 until their source is read (see Impairment). The transit time of the stomach of the condition, (T_st + FED*(T_st_fed - T_st))*fimp_t_stomach, is a constant of the top model as well, T_st_eff (ids.T_STOMACH, an output of the intestine evaluated after the changes before the simulation): a formula of the top model reads it, e.g. the rate of a conversion in the stomach which converts a given fraction of a solution, kconv = 60*f/((1 - f)*T_st_eff), fasted and fed, instead of a copy of the fasted 22.79 min. The single lumen has no stomach and no T_st_eff. The data of the segments:

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

Sources of the columns: volume: Mudie2010, Schiller2005; transit: Metcalf1987, Mudie2010; transit fed: Mudie2010; pH: Evans1988, Mudie2010; area factor: Helander2014, Mudie2010, Yu1999; CYP3A4: ModelChoice, Paine1997.

The small intestinal transit time is split onto duodenum, jejunum and ileum by their lengths (Mudie 2010), the colonic transit onto its segments by the segmental transits of Metcalf 1987, and the fluid volume of a region onto its segments in proportion to their transit times. The pH is data only, no process reads it yet.

Fraction absorbed and colon absorption¶

Two optional attributes of SegmentedIntestine change the absorption of the segments, both None by default. Without them the model has neither parameter nor reaction, and the absorption is the one above.

• fabs: the fraction of the absorbed flux of every segment which enters the body, the parameter fabs_<s> of the top model (as in the single lumen). The rest, 1 - fabs_<s>, goes from the lumen of the segment into the feces (J_feces_<seg>_<s>). The lumen empties as fast as without it, so the onset of the absorption stays and less of the drug enters the body: for a substance with incomplete absorption and a fast onset, which a smaller ka_<s> cannot give (it delays the onset and moves the absorption down the intestine).
• f_ka_colon: the factor f_ka_colon_<s> of the absorption of the segments of the large intestine (caecum to rectum, the segments of the region Organ.LARGE_INTESTINE; the shorthand of the group colon of a regional profile), relative to ka_<s>*f_area_<seg>; the area factors stay. For a substance which the colon absorbs less than its area factor gives, or not at all (0): its slow transit otherwise adds a late absorption of what the small intestine left.

A set attribute is one value for every substance with an oral route (the substances whose ka_<s> has the default of the data), or a mapping from the substance id to a value (ka={"cr": 2.0, "crtn": 0.5}, fabs={"cr": 0.6}) for a model with several oral substances. In a mapping ka of a substance which is missing is the default of the data, fabs_<s> and f_ka_colon_<s> exist only for the substances of the mapping, and a key which is no substance of the definition (or, for fabs and f_ka_colon, a substance without an oral route) raises a ValueError; a ka for a substance without an oral route is allowed (its reabsorption after biliary excretion). The same holds for ka and fabs of the single lumen, where a substance missing from fabs has fabs_<s> = 1. A model fits the parameter of its substance by id (fabs_api, f_ka_colon_api). A substance without an oral route (a metabolite which reaches the lumen by the bile, ka_<s> = 0) gets neither parameter and no J_feces_<seg>_<s>, its absorption stays ka_<s>*f_area_<seg>. A value of 1 gives the model without the attribute numerically. SingleLumen.fabs differs: its None is fabs_<s> = 1, a parameter of every substance.

J_abs_du_<s>: Clumen_du_<s> -> Cpl_<s> = fabs_<s> * ka_<s> / 60 min_per_hr * f_area_du * Vlumen_du * Clumen_du_<s>
J_feces_du_<s>: Clumen_du_<s> -> Afeces_<s> = (1 - fabs_<s>) * ka_<s> / 60 min_per_hr * f_area_du * Vlumen_du * Clumen_du_<s>
J_abs_asc_<s>: Clumen_asc_<s> -> Cpl_<s> = fabs_<s> * ka_<s> / 60 min_per_hr * f_area_asc * f_ka_colon_<s> * Vlumen_asc * Clumen_asc_<s>
J_feces_asc_<s>: Clumen_asc_<s> -> Afeces_<s> = (1 - fabs_<s>) * ka_<s> / 60 min_per_hr * f_area_asc * f_ka_colon_<s> * Vlumen_asc * Clumen_asc_<s>

Regional absorption¶

SegmentedIntestine(regional=...) gives a substance with an oral route a regional absorption profile, RegionalAbsorption({"ile2": 0.5, "asc": 0.1}): a factor f_ka_<seg>_<s> per segment (du to rec, the stomach does not absorb) which multiplies the absorption of the segment, ka_<s>*f_area_<seg>*f_ka_<seg>_<s>. The area factor holds the mucosal surface per lumen volume, the profile the permeability of the segment relative to the small intestine; a segment missing from the profile has no factor (1). One profile is that of every substance with an oral route, a mapping from the substance id gives each substance its own (regional={"api": RegionalAbsorption(...)}). A factor is a number or a Value (dimensionless, with its source or fitted). The group colon gives the segments of the large intestine (caecum to rectum) one shared factor, the parameter f_ka_colon_<s>: f_ka_colon=x is the shorthand of RegionalAbsorption({"colon": x}), both make the same model. A segment of the group with a factor of its own as well, or a substance with f_ka_colon and a factor of the large intestine in its profile, is refused (two factors of one absorption). A measured colonic absorption goes into the group (RegionalAbsorption({"ile2": ..., "colon": Value(..., source="Byon2018")})), a segment factor where the study resolves the segment. The notes of f_ka_colon_<s> say where its value comes from: of a Value with a source the regional absorption profile, the value of that source, not fitted (and its citation); of a number or a fitted value that a model fits it.

There is no generic default profile, the default is no factor (1, the absorption above): the absorption in the colon relative to the small intestine is a property of the substance. Of 42 drugs released in the colon in humans the highly permeable ones (BCS class I) had a relative colonic bioavailability above 70 % and those of low permeability (BCS III and IV) below 50 % (Tannergren 2009); a single profile would be wrong for one of the two groups. With the area factors of the large intestine alone a solution released in the caecum is absorbed to 70 % (in the colon ascendens 64 %) of the fraction absorbed of an oral solution at the default ka (1/hr), the range of a highly permeable drug (simulated with the data of this version, test_the_colon_absorption_of_the_docs). A substance whose regional absorption is measured takes its profile from the study, e.g. apixaban released by a capsule in the distal small intestine and the ascending colon (Byon 2018: the AUC 60 % and 84 % lower than after the oral solution).

A release study gives the exposure of a solution released in a segment relative to an oral solution, not the factors: they depend on the rest of the absorption of the model (ka_<s>, fabs_<s>, an efflux, the first pass), so they are derived with the model. release_auc_ratios(r, substance, segments, dose) simulates the oral solution (SOLDOSE_<s>) and the same dose dissolved in the lumen of each release segment (GU__Clumen_<seg>_<s>) and gives the AUC of the arm vein relative to the oral solution (14 days at one point per minute by default); derive_regional_absorption(r, substance, targets, dose) finds the factors which give the reported ratios, one factor per release segment (regional_key: the segment in the small intestine, the group colon in the large one), each the root of its ratio with the others fixed, in turns until no factor moves. A study in another condition than the reference (fed, a body weight, an impairment) passes it as changes, initial values by id, which the initial assignments follow; afterwards the condition is taken back and every value of the roadrunner is reset to its initial value (the factors of the model, not the derived ones). The model needs an oral solution of the substance and a first value of each factor in its profile. Because the factors depend on the fitted absorption, a model derives them on its adopted fit, writes them into its profile (a Value with the source of the study) and fits again from the adopted fit, until the factors derived on the fit are those of the model (apixaban: Byon 2018, the lower ileum 0.40 and the colon ascendens 0.16 give f_ka_ile2_api 0.149 and f_ka_colon_api 0.0567).

# not executed
from pkdb_models.models.apixaban import MODEL
from pkpdsim.pbpk import derive_regional_absorption

r = MODEL.load_roadrunner()
r.integrator.absolute_tolerance = 1e-14
r.integrator.relative_tolerance = 1e-10
factors = derive_regional_absorption(r, "api", {"ile2": 0.40, "asc": 0.16}, dose=2.5)

J_abs_ile2_<s>: Clumen_ile2_<s> -> Cpl_<s> = fabs_<s> * ka_<s> / 60 min_per_hr * f_area_ile2 * f_ka_ile2_<s> * Vlumen_ile2 * Clumen_ile2_<s>
J_abs_asc_<s>: Clumen_asc_<s> -> Cpl_<s> = fabs_<s> * ka_<s> / 60 min_per_hr * f_area_asc * f_ka_colon_<s> * Vlumen_asc * Clumen_asc_<s>

Conversions in the lumen¶

SegmentedIntestine(conversions=[LumenConversion(substrate, product, k, segments)]) converts a dissolved substance into another in the chosen lumen segments, first order with the rate constant kconv_<s>_<m> (1/hr, the unit of ka_<s>, a parameter of the top model), one mole of the product per mole of the substrate: a chemical conversion (creatine to creatinine in the acid of the stomach), the hydrolysis of a prodrug, a bacterial conversion in the colon (the deconjugation of a glucuronide). The undissolved drug of a tablet does not convert. The substrate reaches the lumen (an oral route, biliary excretion, a transport of the intestine or another conversion); the product gets the lumen species, the transit and the absorption of the intestine (its ka_<m> is 0 without an oral route unless ka gives it one). Without conversions the model has no rate constant and no reaction.

J_conv_st_<s>_<m>: Clumen_st_<s> -> Clumen_st_<m> = kconv_<s>_<m> / 60 min_per_hr * Vlumen_st * Clumen_st_<s>

The stomach does not absorb, so the converted fraction f of an oral solution by a conversion in the stomach follows from the gastric transit time T (T_st, 22.79 min fasted, T_st_fed 63.48 min fed) and k = kconv/60 (1/min): the dissolved drug leaves the stomach with 1/T and converts with k,

f = k*T / (1 + k*T)            kconv = 60 * f / ((1 - f) * T)

An immediate-release tablet with the first order dissolution Ka_dis_<s> (1/hr, without the factors f_dis_fed_<s> and f_dis_release_<s>) converts only what dissolves in the stomach, the fraction g = kd*T/(1 + kd*T) with kd = Ka_dis/60 (1/min; not the rate kdis_<s> of a Weibull or dose-dependent dissolution, which is in 1/min already), so f = g*k*T/(1 + k*T). Over several segments without absorption the drug survives each with 1/(1 + k*T_seg). A converted fraction of 0.0069 of a solution at the fasted transit is kconv = 0.0183 1/hr.

Intestinal wall¶

The wall is the standard organ of the volume Vgu = Vsi + Vla, with the volume-weighted vascular and interstitial fractions and the distribution of the small intestine (its Kp; a Kp override of the large intestine is refused). It receives the arterial inflow of both organs, Qgu = Qsi + Qla, and drains into the portal vein.

Vgu = Vsi + Vla
f_vas_gu = (Vsi * f_vas_si + Vla * f_vas_la) / Vgu
Qgu = Qsi + Qla
J_in_ar_pl_<s>: Car_<s> -> Cpl_<s> = Qgu * (1 - HCT) * Car_<s>
J_out_pl_<s>: Cpl_<s> -> Cpo_<s> = Qgu * (1 - HCT) * Cout_<s>

Mucosa¶

A substance with chemistry of the intestine (substrate, product, transported substance, inhibitor or inducer of a protein of a process of Organ.SMALL_INTESTINE) has a mucosa (enterocyte) species per segment, Cmu_<seg>_<s>: flow-limited with the mucosa to plasma coefficient K_mu_<s> (the Kp of the small intestine, 1 without), perfused by arterial blood with the mucosa share of the blood flow of its region times the transit share of the segment in the region, and draining into the portal vein; the absorbed drug enters it. Its volume is carved out of the tissue of the wall, so wall and mucosa together have the capacity of the wall of a substance without mucosa. The mucosa is a tenth of the tissue of the intestine and receives 70 % of its blood flow, choices of the library (data/intestine_human.toml). The chemistry of the small intestine is placed into the mucosa segments by the relative expression of its protein (CYP3A4 of Paine 1997) times the mucosa volume, the sum kept: the Vmax stays per litre of small intestine. A protein without expression data, and a process without protein, is placed by mucosa volume over the segments of the small intestine only, the organ of the process; the colon mucosa carries none of it, so the gut first pass of such a process equals that of the single lumen. An apical efflux moves the drug from the mucosa into the lumen of its segment, an apical influx back.

Vmu_du = f_mucosa_volume * Vsi * (1 - f_vas_si) * 0.0744792
Qmu_du = f_mucosa_flow * Qsi * 0.0744792
Qgu_wall = Qgu - Qmu_total
K_mu_<s> = Kp_<s>
J_mu_in_du_<s>: Car_<s> -> Cmu_du_<s> = Qmu_du * (1 - HCT) * Car_<s>
J_mu_out_du_<s>: Cmu_du_<s> -> Cpo_<s> = Qmu_du * (1 - HCT) * Cmu_du_<s> / K_mu_<s>
J_abs_du_<s>: Clumen_du_<s> -> Cmu_du_<s> = ka_<s> / 60 min_per_hr * f_area_du * Vlumen_du * Clumen_du_<s>
w_cyp3a4_du = 31 * Vmu_du / (31 * Vmu_du + 31 * Vmu_jej1 + 24 * Vmu_jej2 + 17 * Vmu_ile1 + 17 * Vmu_ile2)
DRGGU_du: Cmu_du_drg -> Cmu_du_drg_m = f_cyp3a4 * w_cyp3a4_du * DRGGU_Vmax * Vsi * (fu_drg / K_mu_drg * Cmu_du_drg) / (DRGGU_Km + fu_drg / K_mu_drg * Cmu_du_drg)

The number 0.0744792 is the transit share of the duodenum in the small intestine. A protein which a substance inactivates or induces has one turnover state for the intestine, driven by the unbound mucosa concentration weighted by the placement of the protein. ABCB1, ABCG2 and every protein but CYP3A4 are placed by mucosa volume over the small intestine until their abundances along the intestine (Drozdzik 2014) are transcribed and verified.

Single lumen¶

Variants(organs={Organ.SMALL_INTESTINE: "single_lumen"}) replaces the segments with one lumen, the fluid of the small intestine (105 ml, transit time 192 min): the first order absorption of most hand-written models, from the same interface. The lumen absorbs with ka_<s>, of which the fraction fabs_<s> enters the body and the rest the feces, and it keeps a physiological transit into the feces, so the fraction absorbed is Fa = fabs*ka/(ka + 1/T): the fitted absorption constants of the hand-written models (which had no transit) do not carry over unchanged, a refit is expected anyway.

Readouts¶

The lumen and the fluxes of the intestine are spread over its segments; a model which reads their sums names them in PBPKDefinition.readouts, and the intestine adds each as an output, as the kidney adds Rurine_<s>: the dissolved amount in the lumen Alumen_<s>, the summed fluxes of the absorption Rabs_<s>, of the efflux into the lumen Reff_<s> and of the loss to the feces Rfeces_<s>, the amount absorbed Aabs_<s> and the gross absorbed fraction of the oral dose Fa_<s> (absorbed from the lumen over the oral dose given, before the first pass, above the bioavailability), the same ids for the segmented intestine and the single lumen (see Observers). A model without readouts has none of them.

Bile and enterohepatic circulation¶

The bile Cbile_<s> (compartment Vbile) collects the apical efflux of the liver and drains with the first order k_bile (0.01 1/min, a choice of the library) into the duodenum, of a substance without lumen species into the feces:

J_bile_<s>: Cbile_<s> -> Clumen_du_<s> = k_bile * Vbile * Cbile_<s>

With an absorption of the substance from the lumen (ka_<s>, which is 0 for a substance without an oral route unless the model sets or fits it) the excreted drug is reabsorbed: the enterohepatic circulation.

Sinks¶

Every substance has the amounts in urine Aurine_<s> and feces Afeces_<s> (compartments of amounts, Vurine and Vfeces) and the bile Cbile_<s>. With the standard organs as intestine, which have no lumen, the bile drains into the feces. The breath is a sink of a later project.

Limitations¶

• A conversion in the lumen is first order (LumenConversion): a saturable or enzyme-limited conversion in the lumen is not available, a custom reaction of the intestine is refused.
• The single lumen has no segments: SingleLumen(regional=...) and SingleLumen(conversions=...) are refused with a message, a regional profile and the conversions in the lumen need the segmented intestine.
• A regional profile is given to a substance with an oral route (as fabs and f_ka_colon): the product of a conversion without an oral route absorbs with ka_<m>*f_area_<seg> only.
• Only CYP3A4 has an expression profile along the intestine; every other protein is placed by mucosa volume over the small intestine, so a process expressed in the colon needs expression data (or a hook for the large intestine). The pH of the segments is data no process reads.
• The gastric emptying and small intestinal transit factors of chronic kidney disease are 1 (see Impairment).