RICHMOND, Va.  — As abortion opponents cheer the passage of fetal heartbeat laws and other bans on the procedure, abortion-rights groups have been waging a quieter battle in courthouses around the country to overturn less direct restrictions passed in recent years.

At least a dozen lawsuits have been filed over the last two years challenging what abortion-rights groups call TRAP laws, Targeted Restrictions on Abortion Providers.

Anti-abortion groups say the laws are needed to protect women’s health. Abortion-rights groups say the laws are medically unnecessary regulations designed to drive abortion clinics out of business and make it more difficult for women to end pregnancies.

The first lawsuit to make it to trial is set to begin Monday in U.S. District Court in Richmond.

Women’s health groups suing Virginia in the case won a brief victory this month when the presiding judge struck down a law that said only physicians can perform first-trimester abortions. A week later, the judge rescinded his ruling and said he needed more evidence before deciding if the law places an “undue burden” on women seeking abortions.

“We are excited to put this case before the judge,” said Jenny Ma, a senior staff attorney at the Center for Reproductive Rights, a legal advocacy organization.

TRAP laws “have been instrumental in making it very difficult for our clients to run their facilities, and it makes it very difficult for women to access abortion care, both in Virginia and in other states,” she said.

The center has filed similar lawsuits challenging abortion restrictions in Arizona, Louisiana and Mississippi.

After the Supreme Court’s landmark 1973 decision in Roe v. Wade, many states began imposing restrictions on abortion clinics.

Laws focusing on clinic regulation made a comeback in the 1990s and had another resurgence starting in 2011. About one-third of all abortion restrictions passed since Roe v. Wade have been enacted since January 2011, according to the Guttmacher Institute, a research group that supports abortion rights.

On Wednesday, Alabama’s Republican governor signed the nation’s strictest abortion ban into law, making performing an abortion a felony in nearly all cases, punishable by up to life in prison, and with no exceptions for rape and incest.

Hours later, Missouri’s Republican-led Senate passed a wide-ranging bill to ban abortions at eight weeks of pregnancy. The bill needs another vote of approval in the GOP-led House before it can go to Republican Gov. Mike Parson, who voiced support for an earlier version.

It includes exceptions for medical emergencies, but not for pregnancies caused by rape or incest. Doctors would face five to 15 years in prison for violating the eight-week cutoff. Women who receive abortions would not be prosecuted.

Four states — Mississippi, Georgia, Ohio and Kentucky — have approved bans on abortion once a fetal heartbeat is detected, which can occur in about the sixth week of pregnancy, before many women realize they are pregnant.