# Model Mathematics

### Component: PCA

$ddtimeP_p=2⁢Q_Lp⁢R_a-2⁢P_p+2⁢P_aR_a⁢C_pddtimeQ_Lp=P_ai-P_p-Q_Lp⁢R_pL_p$

### Component: PCA_Components

$C_p=3⁢π⁢r_p2⁢r_ph_p+12E_p⁢2⁢r_ph_p+1⁢l_pL_p=Rho_bπ⁢r_p2⁢l_p⁢0.0000075R_p=8⁢Eta_bπ⁢r_p4⁢l_p$

### Component: Arteriolar_Circulation

$ddtimer_a=r_a0⁢P_p⁢R_a+R_v⁢r_a+P_v⁢R_a⁢r_a-T_ae+T_am+P_ic⁢r_a+h_a⁢2⁢R_a+R_vr_a0⁢r_a2⁢K_Va⁢R_a⁢R_a+R_v+Eta_a⁢h_a⁢2⁢R_a+R_vP_a=T_a+P_ic⁢r_a+h_ar_a$

### Component: AC_Components

$R_a=K_Rar_a4C_a=1.0312R_a$

### Component: AC_Flow

$Q_a=2⁢P_a-P_vR_a+R_v$

### Component: Venous_Circulation

$ddtimeP_v=2C_v⁢P_a-P_vR_a+R_v-P_v-P_voR_v$

### Component: AC_Tension

$T_a=T_ae+T_am+T_avT_ae=h_a⁢sigma_ae0⁢ⅇK_a_sigma⁢r_a-r_a0r_a0-1-sigma_acT_am=T_a_max⁢ⅇ-|r_a-r_amr_at-r_am|n_amT_av=Eta_ar_a0⁢ddtimer_a⁢h_a$

### Component: AC_Tension_Variables

$h_a=r_a2+2⁢r_a0⁢h_a0+h_a02-r_aT_a_max=T_a_max0⁢1+M_s$

### Component: CBF_Regulation

$M_s=ⅇ2⁢M_s1-1ⅇ2⁢M_s1+1M_s1=G_myo⁢x_myo+G_shear⁢x_shear+G_neuro⁢x_neuro+G_meta⁢x_meta+x_ini$

### Component: Myo_Regulation

$A_myo=T_a-T_myo0T_myo_sddtimex_myo=A_myo-x_myoTau_myo$

### Component: Shear_Regulation

$A_shear=K_shear⁢Q_ar_a3-1ddtimex_shear=A_shear-x_shearTau_shear$

### Component: Neuro_Regulation

$ddtimex_neuro=A_neuro-x_neuroTau_neuroA_neuro=1iftime≥10otherwise$

### Component: Meta_Regulation

$A_Meta=G_a_Meta⁢conc_CO2_t-conc_CO2_t_spddtimex_meta=A_Meta-x_metaTau_Meta$

### Component: Meta_CO2_Kinetics

$ddtimeconc_CO2_t=1vol_PVC⁢M_CO2-Q_a⁢conc_CO2_v-conc_CO2_aconc_CO2_v=alpha_tv⁢conc_CO2_t+beta_tv$

### Component: VC_Meta_Stim

$M_CO2=M_CO2_0⁢unity+A_neuro⁢f_Q-unityn_QM$
Source
Derived from workspace Spronck, 2012 at changeset 02012ee8bbdf.
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